id	sid	tid	token	lemma	pos
ejde-741	1	1	electronic	electronic	ADJ
ejde-741	1	2	journal	journal	NOUN
ejde-741	1	3	of	of	ADP
ejde-741	1	4	differential	differential	ADJ
ejde-741	1	5	equations	equation	NOUN
ejde-741	1	6	,	,	PUNCT
ejde-741	1	7	vol	vol	NOUN
ejde-741	1	8	.	.	PUNCT
ejde-741	1	9	2025	2025	NUM
ejde-741	1	10	(	(	PUNCT
ejde-741	1	11	2025	2025	NUM
ejde-741	1	12	)	)	PUNCT
ejde-741	1	13	,	,	PUNCT
ejde-741	1	14	no	no	INTJ
ejde-741	1	15	.	.	NOUN
ejde-741	1	16	15	15	NUM
ejde-741	1	17	,	,	PUNCT
ejde-741	1	18	pp	pp	ADJ
ejde-741	1	19	.	.	PUNCT
ejde-741	2	1	1–22	1–22	PROPN
ejde-741	2	2	.	.	PUNCT
ejde-741	3	1	issn	issn	PROPN
ejde-741	3	2	:	:	PUNCT
ejde-741	3	3	1072	1072	NUM
ejde-741	3	4	-	-	SYM
ejde-741	3	5	6691	6691	NUM
ejde-741	3	6	.	.	PUNCT
ejde-741	4	1	url	url	PROPN
ejde-741	4	2	:	:	PUNCT
ejde-741	4	3	https://ejde.math.txstate.edu	https://ejde.math.txstate.edu	PROPN
ejde-741	4	4	,	,	PUNCT
ejde-741	4	5	https://ejde.math.unt.edu	https://ejde.math.unt.edu	PROPN
ejde-741	4	6	doi	doi	PROPN
ejde-741	4	7	:	:	PUNCT
ejde-741	4	8	10.58997	10.58997	NUM
ejde-741	4	9	/	/	SYM
ejde-741	4	10	ejde.2025.15	ejde.2025.15	NOUN
ejde-741	4	11	null	null	NOUN
ejde-741	4	12	-	-	PUNCT
ejde-741	4	13	controllability	controllability	NOUN
ejde-741	4	14	for	for	ADP
ejde-741	4	15	1	1	NUM
ejde-741	4	16	-	-	PUNCT
ejde-741	4	17	d	d	NOUN
ejde-741	4	18	degenerate	degenerate	ADJ
ejde-741	4	19	quasilinear	quasilinear	PROPN
ejde-741	4	20	parabolic	parabolic	NOUN
ejde-741	4	21	equations	equation	NOUN
ejde-741	4	22	pitágoras	pitágoras	PROPN
ejde-741	4	23	p.	p.	PROPN
ejde-741	4	24	de	de	PROPN
ejde-741	4	25	carvalho	carvalho	PROPN
ejde-741	4	26	,	,	PUNCT
ejde-741	4	27	reginaldo	reginaldo	NOUN
ejde-741	4	28	demarque	demarque	NOUN
ejde-741	4	29	,	,	PUNCT
ejde-741	4	30	juan	juan	PROPN
ejde-741	4	31	límaco	límaco	PROPN
ejde-741	4	32	,	,	PUNCT
ejde-741	4	33	luiz	luiz	PROPN
ejde-741	4	34	viana	viana	PROPN
ejde-741	4	35	abstract	abstract	PROPN
ejde-741	4	36	.	.	PUNCT
ejde-741	5	1	in	in	ADP
ejde-741	5	2	this	this	DET
ejde-741	5	3	article	article	NOUN
ejde-741	5	4	,	,	PUNCT
ejde-741	5	5	we	we	PRON
ejde-741	5	6	prove	prove	VERB
ejde-741	5	7	local	local	ADJ
ejde-741	5	8	null	null	NOUN
ejde-741	5	9	-	-	PUNCT
ejde-741	5	10	controllability	controllability	NOUN
ejde-741	5	11	for	for	ADP
ejde-741	5	12	one	one	NUM
ejde-741	5	13	-	-	PUNCT
ejde-741	5	14	dimensional	dimensional	ADJ
ejde-741	5	15	degenerate	degenerate	ADJ
ejde-741	5	16	quasilinear	quasilinear	NOUN
ejde-741	5	17	parabolic	parabolic	PROPN
ejde-741	5	18	equations	equation	NOUN
ejde-741	5	19	.	.	PUNCT
ejde-741	6	1	we	we	PRON
ejde-741	6	2	apply	apply	VERB
ejde-741	6	3	a	a	DET
ejde-741	6	4	well	well	ADV
ejde-741	6	5	-	-	PUNCT
ejde-741	6	6	known	know	VERB
ejde-741	6	7	local	local	ADJ
ejde-741	6	8	inversion	inversion	NOUN
ejde-741	6	9	argument	argument	NOUN
ejde-741	6	10	used	use	VERB
ejde-741	6	11	by	by	ADP
ejde-741	6	12	fursikov	fursikov	NOUN
ejde-741	6	13	and	and	CCONJ
ejde-741	6	14	imanuvilov	imanuvilov	NOUN
ejde-741	6	15	.	.	PUNCT
ejde-741	7	1	the	the	DET
ejde-741	7	2	strategy	strategy	NOUN
ejde-741	7	3	is	be	AUX
ejde-741	7	4	to	to	PART
ejde-741	7	5	use	use	VERB
ejde-741	7	6	carleman	carleman	ADJ
ejde-741	7	7	estimates	estimate	NOUN
ejde-741	7	8	,	,	PUNCT
ejde-741	7	9	previously	previously	ADV
ejde-741	7	10	obtained	obtain	VERB
ejde-741	7	11	for	for	ADP
ejde-741	7	12	weak	weak	ADJ
ejde-741	7	13	and	and	CCONJ
ejde-741	7	14	strong	strong	ADJ
ejde-741	7	15	degenerate	degenerate	ADJ
ejde-741	7	16	parabolic	parabolic	NOUN
ejde-741	7	17	problems	problem	NOUN
ejde-741	7	18	.	.	PUNCT
ejde-741	8	1	1	1	X
ejde-741	8	2	.	.	X
ejde-741	8	3	introduction	introduction	NOUN
ejde-741	8	4	in	in	ADP
ejde-741	8	5	this	this	DET
ejde-741	8	6	article	article	NOUN
ejde-741	8	7	,	,	PUNCT
ejde-741	8	8	we	we	PRON
ejde-741	8	9	investigate	investigate	VERB
ejde-741	8	10	the	the	DET
ejde-741	8	11	controllability	controllability	NOUN
ejde-741	8	12	of	of	ADP
ejde-741	8	13	the	the	DET
ejde-741	8	14	quasilinear	quasilinear	NOUN
ejde-741	8	15	degenerate	degenerate	ADJ
ejde-741	8	16	parabolic	parabolic	NOUN
ejde-741	8	17	system	system	NOUN
ejde-741	8	18	ut	ut	PROPN
ejde-741	8	19	−	−	PROPN
ejde-741	8	20	ℓ(au	ℓ(au	NUM
ejde-741	8	21	)	)	PUNCT
ejde-741	8	22	(	(	PUNCT
ejde-741	8	23	a(x)ux	a(x)ux	ADP
ejde-741	8	24	)	)	PUNCT
ejde-741	8	25	x	x	PUNCT
ejde-741	9	1	+	+	CCONJ
ejde-741	9	2	f(t	f(t	NOUN
ejde-741	9	3	,	,	PUNCT
ejde-741	9	4	x	x	NOUN
ejde-741	9	5	,	,	PUNCT
ejde-741	9	6	u	u	NOUN
ejde-741	9	7	)	)	PUNCT
ejde-741	9	8	=	=	SYM
ejde-741	9	9	hχω	hχω	PROPN
ejde-741	9	10	,	,	PUNCT
ejde-741	9	11	(	(	PUNCT
ejde-741	9	12	t	t	PROPN
ejde-741	9	13	,	,	PUNCT
ejde-741	9	14	x	x	X
ejde-741	9	15	)	)	PUNCT
ejde-741	9	16	∈	∈	PROPN
ejde-741	9	17	q	q	NOUN
ejde-741	9	18	,	,	PUNCT
ejde-741	9	19	u(t	u(t	NOUN
ejde-741	9	20	,	,	PUNCT
ejde-741	9	21	1	1	NUM
ejde-741	9	22	)	)	PUNCT
ejde-741	9	23	=	=	SYM
ejde-741	9	24	0	0	NUM
ejde-741	9	25	,	,	PUNCT
ejde-741	9	26	in	in	ADP
ejde-741	9	27	(	(	PUNCT
ejde-741	9	28	0	0	NUM
ejde-741	9	29	,	,	PUNCT
ejde-741	9	30	t	t	NOUN
ejde-741	9	31	)	)	PUNCT
ejde-741	9	32	,	,	PUNCT
ejde-741	9	33			PROPN
ejde-741	9	34	u(t	u(t	NOUN
ejde-741	9	35	,	,	PUNCT
ejde-741	9	36	0	0	NUM
ejde-741	9	37	)	)	PUNCT
ejde-741	9	38	=	=	SYM
ejde-741	9	39	0	0	NUM
ejde-741	9	40	,	,	PUNCT
ejde-741	9	41	(	(	PUNCT
ejde-741	9	42	weak	weak	ADJ
ejde-741	9	43	)	)	PUNCT
ejde-741	9	44	,	,	PUNCT
ejde-741	9	45	t	t	PROPN
ejde-741	9	46	∈	∈	PROPN
ejde-741	9	47	(	(	PUNCT
ejde-741	9	48	0	0	NUM
ejde-741	9	49	,	,	PUNCT
ejde-741	9	50	t	t	NOUN
ejde-741	9	51	)	)	PUNCT
ejde-741	9	52	,	,	PUNCT
ejde-741	9	53	or	or	CCONJ
ejde-741	9	54	(	(	PUNCT
ejde-741	9	55	aux)(t	aux)(t	NOUN
ejde-741	9	56	,	,	PUNCT
ejde-741	9	57	0	0	NUM
ejde-741	9	58	)	)	PUNCT
ejde-741	9	59	=	=	SYM
ejde-741	9	60	0	0	PUNCT
ejde-741	9	61	(	(	PUNCT
ejde-741	9	62	strong	strong	ADJ
ejde-741	9	63	)	)	PUNCT
ejde-741	9	64	,	,	PUNCT
ejde-741	9	65	t	t	PROPN
ejde-741	9	66	∈	∈	PROPN
ejde-741	9	67	(	(	PUNCT
ejde-741	9	68	0	0	NUM
ejde-741	9	69	,	,	PUNCT
ejde-741	9	70	t	t	NOUN
ejde-741	9	71	)	)	PUNCT
ejde-741	9	72	,	,	PUNCT
ejde-741	9	73	u(0	u(0	PROPN
ejde-741	9	74	,	,	PUNCT
ejde-741	9	75	x	x	NOUN
ejde-741	9	76	)	)	PUNCT
ejde-741	9	77	=	=	SYM
ejde-741	9	78	u0(x	u0(x	NUM
ejde-741	9	79	)	)	PUNCT
ejde-741	9	80	,	,	PUNCT
ejde-741	9	81	x	x	PUNCT
ejde-741	9	82	∈	∈	PROPN
ejde-741	9	83	(	(	PUNCT
ejde-741	9	84	0	0	NUM
ejde-741	9	85	,	,	PUNCT
ejde-741	9	86	1	1	NUM
ejde-741	9	87	)	)	PUNCT
ejde-741	9	88	,	,	PUNCT
ejde-741	9	89	(	(	PUNCT
ejde-741	9	90	1.1	1.1	NUM
ejde-741	9	91	)	)	PUNCT
ejde-741	9	92	where	where	SCONJ
ejde-741	9	93	t	t	PROPN
ejde-741	9	94	>	>	X
ejde-741	9	95	0	0	NUM
ejde-741	9	96	is	be	AUX
ejde-741	9	97	given	give	VERB
ejde-741	9	98	,	,	PUNCT
ejde-741	9	99	q	q	NOUN
ejde-741	9	100	:	:	PUNCT
ejde-741	9	101	=	=	SYM
ejde-741	9	102	(	(	PUNCT
ejde-741	9	103	0	0	NUM
ejde-741	9	104	,	,	PUNCT
ejde-741	9	105	t	t	NOUN
ejde-741	9	106	)	)	PUNCT
ejde-741	9	107	×	×	NOUN
ejde-741	9	108	(	(	PUNCT
ejde-741	9	109	0	0	NUM
ejde-741	9	110	,	,	PUNCT
ejde-741	9	111	1	1	NUM
ejde-741	9	112	)	)	PUNCT
ejde-741	9	113	,	,	PUNCT
ejde-741	9	114	ω	ω	X
ejde-741	9	115	=	=	SYM
ejde-741	9	116	(	(	PUNCT
ejde-741	9	117	α	α	X
ejde-741	9	118	,	,	PUNCT
ejde-741	9	119	β	β	NOUN
ejde-741	9	120	)	)	PUNCT
ejde-741	9	121	⊂	⊂	PROPN
ejde-741	9	122	(	(	PUNCT
ejde-741	9	123	0	0	NUM
ejde-741	9	124	,	,	PUNCT
ejde-741	9	125	1	1	NUM
ejde-741	9	126	)	)	PUNCT
ejde-741	9	127	,	,	PUNCT
ejde-741	9	128	u0	u0	PROPN
ejde-741	9	129	∈	∈	PROPN
ejde-741	9	130	l2(0	l2(0	NOUN
ejde-741	9	131	,	,	PUNCT
ejde-741	9	132	1	1	NUM
ejde-741	9	133	)	)	PUNCT
ejde-741	9	134	and	and	CCONJ
ejde-741	9	135	h	h	NOUN
ejde-741	9	136	∈	∈	PROPN
ejde-741	9	137	l2(qω	l2(qω	PROPN
ejde-741	9	138	)	)	PUNCT
ejde-741	9	139	is	be	AUX
ejde-741	9	140	a	a	DET
ejde-741	9	141	control	control	NOUN
ejde-741	9	142	that	that	PRON
ejde-741	9	143	acts	act	VERB
ejde-741	9	144	on	on	ADP
ejde-741	9	145	the	the	DET
ejde-741	9	146	system	system	NOUN
ejde-741	9	147	through	through	ADP
ejde-741	9	148	qω	qω	NOUN
ejde-741	9	149	:	:	PUNCT
ejde-741	9	150	=	=	SYM
ejde-741	9	151	(	(	PUNCT
ejde-741	9	152	0	0	NUM
ejde-741	9	153	,	,	PUNCT
ejde-741	9	154	t	t	PROPN
ejde-741	9	155	)	)	PUNCT
ejde-741	9	156	×	×	PROPN
ejde-741	9	157	ω	ω	PROPN
ejde-741	9	158	.	.	PUNCT
ejde-741	10	1	during	during	ADP
ejde-741	10	2	this	this	DET
ejde-741	10	3	section	section	NOUN
ejde-741	10	4	,	,	PUNCT
ejde-741	10	5	we	we	PRON
ejde-741	10	6	will	will	AUX
ejde-741	10	7	specify	specify	VERB
ejde-741	10	8	some	some	DET
ejde-741	10	9	conditions	condition	NOUN
ejde-741	10	10	on	on	ADP
ejde-741	10	11	the	the	DET
ejde-741	10	12	functions	function	NOUN
ejde-741	10	13	a	a	DET
ejde-741	10	14	:	:	PUNCT
ejde-741	10	15	[	[	X
ejde-741	10	16	0	0	NUM
ejde-741	10	17	,	,	PUNCT
ejde-741	10	18	1	1	NUM
ejde-741	10	19	]	]	PUNCT
ejde-741	10	20	→	→	SYM
ejde-741	10	21	r	r	X
ejde-741	10	22	,	,	PUNCT
ejde-741	10	23	ℓ	ℓ	INTJ
ejde-741	10	24	:	:	PUNCT
ejde-741	10	25	r	r	NOUN
ejde-741	10	26	→	→	SYM
ejde-741	10	27	r	r	NOUN
ejde-741	10	28	and	and	CCONJ
ejde-741	10	29	f	f	NOUN
ejde-741	10	30	:	:	PUNCT
ejde-741	11	1	[	[	X
ejde-741	11	2	0	0	NUM
ejde-741	11	3	,	,	PUNCT
ejde-741	11	4	t	t	X
ejde-741	11	5	]	]	PUNCT
ejde-741	11	6	×	×	NOUN
ejde-741	12	1	[	[	X
ejde-741	12	2	0	0	NUM
ejde-741	12	3	,	,	PUNCT
ejde-741	12	4	1	1	NUM
ejde-741	12	5	]	]	SYM
ejde-741	12	6	×	×	NOUN
ejde-741	12	7	r	r	NOUN
ejde-741	12	8	→	→	SYM
ejde-741	12	9	r	r	NOUN
ejde-741	12	10	,	,	PUNCT
ejde-741	12	11	under	under	ADP
ejde-741	12	12	which	which	PRON
ejde-741	12	13	the	the	DET
ejde-741	12	14	discussion	discussion	NOUN
ejde-741	12	15	will	will	AUX
ejde-741	12	16	be	be	AUX
ejde-741	12	17	developed	develop	VERB
ejde-741	12	18	.	.	PUNCT
ejde-741	13	1	assumption	assumption	NOUN
ejde-741	13	2	1.1	1.1	NUM
ejde-741	13	3	.	.	PUNCT
ejde-741	14	1	let	let	VERB
ejde-741	14	2	a	a	DET
ejde-741	14	3	∈	∈	PROPN
ejde-741	14	4	c([0	c([0	NOUN
ejde-741	14	5	,	,	PUNCT
ejde-741	14	6	1])∩c1((0	1])∩c1((0	NUM
ejde-741	14	7	,	,	PUNCT
ejde-741	14	8	1	1	NUM
ejde-741	14	9	]	]	PUNCT
ejde-741	14	10	)	)	PUNCT
ejde-741	14	11	be	be	AUX
ejde-741	14	12	a	a	DET
ejde-741	14	13	nondecreasing	nondecrease	VERB
ejde-741	14	14	function	function	NOUN
ejde-741	14	15	satisfying	satisfy	VERB
ejde-741	14	16	a(0	a(0	PROPN
ejde-741	14	17	)	)	PUNCT
ejde-741	15	1	=	=	SYM
ejde-741	15	2	0	0	NUM
ejde-741	15	3	and	and	CCONJ
ejde-741	15	4	a	a	DET
ejde-741	15	5	>	>	X
ejde-741	15	6	0	0	PUNCT
ejde-741	16	1	on	on	ADP
ejde-741	16	2	(	(	PUNCT
ejde-741	16	3	0	0	NUM
ejde-741	16	4	,	,	PUNCT
ejde-741	16	5	1	1	NUM
ejde-741	16	6	]	]	PUNCT
ejde-741	16	7	.	.	PUNCT
ejde-741	17	1	additionally	additionally	ADV
ejde-741	17	2	,	,	PUNCT
ejde-741	17	3	suppose	suppose	VERB
ejde-741	17	4	that	that	SCONJ
ejde-741	17	5	there	there	PRON
ejde-741	17	6	exists	exist	VERB
ejde-741	17	7	k	k	PROPN
ejde-741	17	8	∈	∈	PROPN
ejde-741	17	9	r	r	NOUN
ejde-741	17	10	such	such	ADJ
ejde-741	17	11	that	that	DET
ejde-741	17	12	xa′(x	xa′(x	NOUN
ejde-741	17	13	)	)	PUNCT
ejde-741	17	14	≤	≤	NOUN
ejde-741	17	15	ka(x	ka(x	NOUN
ejde-741	17	16	)	)	PUNCT
ejde-741	17	17	,	,	PUNCT
ejde-741	17	18	∀x	∀x	VERB
ejde-741	17	19	∈	∈	PROPN
ejde-741	18	1	[	[	X
ejde-741	18	2	0	0	NUM
ejde-741	18	3	,	,	PUNCT
ejde-741	18	4	1	1	NUM
ejde-741	18	5	]	]	PUNCT
ejde-741	18	6	,	,	PUNCT
ejde-741	18	7	(	(	PUNCT
ejde-741	18	8	1.2	1.2	NUM
ejde-741	18	9	)	)	PUNCT
ejde-741	18	10	2020	2020	NUM
ejde-741	19	1	mathematics	mathematic	NOUN
ejde-741	19	2	subject	subject	ADJ
ejde-741	19	3	classification	classification	NOUN
ejde-741	19	4	.	.	PUNCT
ejde-741	20	1	35k65	35k65	NUM
ejde-741	20	2	,	,	PUNCT
ejde-741	20	3	35k59	35k59	NUM
ejde-741	20	4	,	,	PUNCT
ejde-741	20	5	93b05	93b05	NUM
ejde-741	20	6	,	,	PUNCT
ejde-741	20	7	35k55	35k55	NUM
ejde-741	20	8	.	.	PUNCT
ejde-741	21	1	key	key	ADJ
ejde-741	21	2	words	word	NOUN
ejde-741	21	3	and	and	CCONJ
ejde-741	21	4	phrases	phrase	NOUN
ejde-741	21	5	.	.	PUNCT
ejde-741	22	1	degenerate	degenerate	ADJ
ejde-741	22	2	parabolic	parabolic	ADJ
ejde-741	22	3	equations	equation	NOUN
ejde-741	22	4	;	;	PUNCT
ejde-741	22	5	quasilinear	quasilinear	PROPN
ejde-741	22	6	parabolic	parabolic	PROPN
ejde-741	22	7	equations	equation	NOUN
ejde-741	22	8	;	;	PUNCT
ejde-741	22	9	controllability	controllability	NOUN
ejde-741	22	10	;	;	PUNCT
ejde-741	22	11	nonlinear	nonlinear	ADJ
ejde-741	22	12	parabolic	parabolic	ADJ
ejde-741	22	13	equations	equation	NOUN
ejde-741	22	14	.	.	PUNCT
ejde-741	23	1	©	©	PROPN
ejde-741	23	2	2025	2025	NUM
ejde-741	23	3	.	.	PUNCT
ejde-741	24	1	this	this	DET
ejde-741	24	2	work	work	NOUN
ejde-741	24	3	is	be	AUX
ejde-741	24	4	licensed	license	VERB
ejde-741	24	5	under	under	ADP
ejde-741	24	6	a	a	DET
ejde-741	24	7	cc	cc	NOUN
ejde-741	24	8	by	by	ADP
ejde-741	24	9	4.0	4.0	NUM
ejde-741	24	10	license	license	NOUN
ejde-741	24	11	.	.	PUNCT
ejde-741	25	1	submitted	submit	VERB
ejde-741	25	2	june	june	PROPN
ejde-741	25	3	2	2	NUM
ejde-741	25	4	,	,	PUNCT
ejde-741	25	5	2024	2024	NUM
ejde-741	25	6	.	.	PUNCT
ejde-741	26	1	published	publish	VERB
ejde-741	26	2	february	february	PROPN
ejde-741	26	3	19	19	NUM
ejde-741	26	4	,	,	PUNCT
ejde-741	26	5	2025	2025	NUM
ejde-741	26	6	.	.	PUNCT
ejde-741	27	1	1	1	NUM
ejde-741	27	2	2	2	NUM
ejde-741	27	3	p.	p.	NOUN
ejde-741	27	4	p.	p.	NOUN
ejde-741	27	5	de	de	PROPN
ejde-741	27	6	carvalho	carvalho	PROPN
ejde-741	27	7	,	,	PUNCT
ejde-741	27	8	r.	r.	PROPN
ejde-741	27	9	demarque	demarque	PROPN
ejde-741	27	10	,	,	PUNCT
ejde-741	27	11	j.	j.	PROPN
ejde-741	27	12	límaco	límaco	PROPN
ejde-741	27	13	,	,	PUNCT
ejde-741	27	14	l.	l.	PROPN
ejde-741	27	15	viana	viana	PROPN
ejde-741	27	16	ejde-2025/15	ejde-2025/15	PROPN
ejde-741	27	17	where	where	SCONJ
ejde-741	27	18	k	k	PROPN
ejde-741	27	19	∈	∈	PROPN
ejde-741	28	1	[	[	X
ejde-741	28	2	0	0	NUM
ejde-741	28	3	,	,	PUNCT
ejde-741	28	4	1	1	NUM
ejde-741	28	5	)	)	PUNCT
ejde-741	28	6	,	,	PUNCT
ejde-741	28	7	for	for	ADP
ejde-741	28	8	the	the	DET
ejde-741	28	9	weakly	weakly	ADJ
ejde-741	28	10	degenerate	degenerate	ADJ
ejde-741	28	11	case	case	NOUN
ejde-741	28	12	(	(	PUNCT
ejde-741	28	13	wdc	wdc	PROPN
ejde-741	28	14	)	)	PUNCT
ejde-741	28	15	,	,	PUNCT
ejde-741	28	16	and	and	CCONJ
ejde-741	28	17	k	k	PROPN
ejde-741	28	18	∈	∈	PROPN
ejde-741	29	1	[	[	X
ejde-741	29	2	1	1	NUM
ejde-741	29	3	,	,	PUNCT
ejde-741	29	4	2	2	NUM
ejde-741	29	5	)	)	PUNCT
ejde-741	29	6	,	,	PUNCT
ejde-741	29	7	for	for	ADP
ejde-741	29	8	the	the	DET
ejde-741	29	9	strongly	strongly	ADV
ejde-741	29	10	degenerate	degenerate	ADJ
ejde-741	29	11	case	case	NOUN
ejde-741	29	12	(	(	PUNCT
ejde-741	29	13	sdc	sdc	NOUN
ejde-741	29	14	)	)	PUNCT
ejde-741	29	15	.	.	PUNCT
ejde-741	30	1	only	only	ADV
ejde-741	30	2	for	for	ADP
ejde-741	30	3	the	the	DET
ejde-741	30	4	(	(	PUNCT
ejde-741	30	5	sdc	sdc	NOUN
ejde-741	30	6	)	)	PUNCT
ejde-741	30	7	,	,	PUNCT
ejde-741	30	8	we	we	PRON
ejde-741	30	9	also	also	ADV
ejde-741	30	10	assume	assume	VERB
ejde-741	30	11	that	that	SCONJ
ejde-741	30	12	∃θ	∃θ	PROPN
ejde-741	30	13	∈	∈	PROPN
ejde-741	30	14	(	(	PUNCT
ejde-741	30	15	1,k	1,k	PROPN
ejde-741	30	16	]	]	PUNCT
ejde-741	30	17	such	such	ADJ
ejde-741	30	18	that	that	SCONJ
ejde-741	30	19	θa	θa	NUM
ejde-741	30	20	≤	≤	NUM
ejde-741	30	21	xa′	xa′	NOUN
ejde-741	30	22	near	near	ADP
ejde-741	30	23	zero	zero	NUM
ejde-741	30	24	,	,	PUNCT
ejde-741	30	25	if	if	SCONJ
ejde-741	30	26	k	k	PROPN
ejde-741	30	27	>	>	X
ejde-741	30	28	1	1	NUM
ejde-741	30	29	;	;	PUNCT
ejde-741	30	30	∃θ	∃θ	PROPN
ejde-741	30	31	∈	∈	PROPN
ejde-741	30	32	(	(	PUNCT
ejde-741	30	33	0	0	NUM
ejde-741	30	34	,	,	PUNCT
ejde-741	30	35	1	1	NUM
ejde-741	30	36	)	)	PUNCT
ejde-741	30	37	such	such	ADJ
ejde-741	30	38	that	that	PRON
ejde-741	30	39	θa	θa	NUM
ejde-741	30	40	≤	≤	NUM
ejde-741	30	41	xa′	xa′	NOUN
ejde-741	30	42	near	near	ADP
ejde-741	30	43	zero	zero	NUM
ejde-741	30	44	,	,	PUNCT
ejde-741	30	45	if	if	SCONJ
ejde-741	30	46	k	k	PROPN
ejde-741	30	47	=	=	NOUN
ejde-741	30	48	1	1	X
ejde-741	30	49	.	.	PUNCT
ejde-741	30	50	(	(	PUNCT
ejde-741	30	51	1.3	1.3	NUM
ejde-741	30	52	)	)	PUNCT
ejde-741	30	53	next	next	ADV
ejde-741	30	54	,	,	PUNCT
ejde-741	30	55	we	we	PRON
ejde-741	30	56	provide	provide	VERB
ejde-741	30	57	some	some	DET
ejde-741	30	58	examples	example	NOUN
ejde-741	30	59	and	and	CCONJ
ejde-741	30	60	comments	comment	NOUN
ejde-741	30	61	about	about	ADP
ejde-741	30	62	assumption	assumption	NOUN
ejde-741	30	63	1.1	1.1	NUM
ejde-741	30	64	.	.	PUNCT
ejde-741	31	1	(	(	PUNCT
ejde-741	31	2	a	a	X
ejde-741	31	3	)	)	PUNCT
ejde-741	31	4	for	for	ADP
ejde-741	31	5	γ	γ	X
ejde-741	31	6	∈	∈	PROPN
ejde-741	31	7	(	(	PUNCT
ejde-741	31	8	0	0	NUM
ejde-741	31	9	,	,	PUNCT
ejde-741	31	10	1	1	NUM
ejde-741	31	11	)	)	PUNCT
ejde-741	31	12	and	and	CCONJ
ejde-741	31	13	α	α	DET
ejde-741	31	14	≥	≥	NOUN
ejde-741	31	15	0	0	NUM
ejde-741	31	16	,	,	PUNCT
ejde-741	31	17	putting	put	VERB
ejde-741	31	18	β	β	X
ejde-741	31	19	=	=	SYM
ejde-741	31	20	arctan(α	arctan(α	NOUN
ejde-741	31	21	)	)	PUNCT
ejde-741	31	22	,	,	PUNCT
ejde-741	31	23	the	the	DET
ejde-741	31	24	function	function	NOUN
ejde-741	31	25	a1(x	a1(x	NOUN
ejde-741	31	26	)	)	PUNCT
ejde-741	31	27	=	=	SYM
ejde-741	31	28	xγ	xγ	PROPN
ejde-741	31	29	cos(βx	cos(βx	NOUN
ejde-741	31	30	)	)	PUNCT
ejde-741	31	31	satisfies	satisfie	NOUN
ejde-741	31	32	(	(	PUNCT
ejde-741	31	33	1.2	1.2	NUM
ejde-741	31	34	)	)	PUNCT
ejde-741	31	35	for	for	ADP
ejde-741	31	36	the	the	DET
ejde-741	31	37	(	(	PUNCT
ejde-741	31	38	wdc	wdc	PROPN
ejde-741	31	39	)	)	PUNCT
ejde-741	31	40	.	.	PUNCT
ejde-741	32	1	on	on	ADP
ejde-741	32	2	the	the	DET
ejde-741	32	3	other	other	ADJ
ejde-741	32	4	hand	hand	NOUN
ejde-741	32	5	,	,	PUNCT
ejde-741	32	6	if	if	SCONJ
ejde-741	32	7	γ	γ	X
ejde-741	32	8	∈	∈	PROPN
ejde-741	32	9	(	(	PUNCT
ejde-741	32	10	1	1	NUM
ejde-741	32	11	,	,	PUNCT
ejde-741	32	12	2	2	NUM
ejde-741	32	13	)	)	PUNCT
ejde-741	32	14	,	,	PUNCT
ejde-741	32	15	then	then	ADV
ejde-741	32	16	a1	a1	NOUN
ejde-741	32	17	becomes	become	VERB
ejde-741	32	18	an	an	DET
ejde-741	32	19	example	example	NOUN
ejde-741	32	20	for	for	ADP
ejde-741	32	21	the	the	DET
ejde-741	32	22	(	(	PUNCT
ejde-741	32	23	sdc	sdc	NOUN
ejde-741	32	24	)	)	PUNCT
ejde-741	32	25	;	;	PUNCT
ejde-741	32	26	(	(	PUNCT
ejde-741	32	27	b	b	X
ejde-741	32	28	)	)	PUNCT
ejde-741	32	29	for	for	ADP
ejde-741	32	30	each	each	DET
ejde-741	32	31	p	p	PROPN
ejde-741	32	32	∈	∈	PROPN
ejde-741	32	33	(	(	PUNCT
ejde-741	32	34	0	0	NUM
ejde-741	32	35	,	,	PUNCT
ejde-741	32	36	1	1	NUM
ejde-741	32	37	)	)	PUNCT
ejde-741	32	38	,	,	PUNCT
ejde-741	32	39	the	the	DET
ejde-741	32	40	function	function	NOUN
ejde-741	32	41	a2(x	a2(x	PROPN
ejde-741	32	42	)	)	PUNCT
ejde-741	32	43	=	=	SYM
ejde-741	33	1	xp	xp	NOUN
ejde-741	34	1	+	+	CCONJ
ejde-741	34	2	x	x	SYM
ejde-741	34	3	satisfies	satisfie	NOUN
ejde-741	34	4	(	(	PUNCT
ejde-741	34	5	1.2	1.2	NUM
ejde-741	34	6	)	)	PUNCT
ejde-741	34	7	for	for	ADP
ejde-741	34	8	the	the	DET
ejde-741	34	9	(	(	PUNCT
ejde-741	34	10	wdc	wdc	PROPN
ejde-741	34	11	)	)	PUNCT
ejde-741	34	12	.	.	PUNCT
ejde-741	35	1	analogously	analogously	ADV
ejde-741	35	2	,	,	PUNCT
ejde-741	35	3	if	if	SCONJ
ejde-741	35	4	p	p	X
ejde-741	35	5	∈	∈	PROPN
ejde-741	35	6	(	(	PUNCT
ejde-741	35	7	1	1	NUM
ejde-741	35	8	,	,	PUNCT
ejde-741	35	9	2	2	NUM
ejde-741	35	10	)	)	PUNCT
ejde-741	35	11	,	,	PUNCT
ejde-741	35	12	then	then	ADV
ejde-741	35	13	a3(x	a3(x	PRON
ejde-741	35	14	)	)	PUNCT
ejde-741	35	15	=	=	SYM
ejde-741	35	16	xp	xp	NOUN
ejde-741	36	1	+	+	CCONJ
ejde-741	36	2	x	x	SYM
ejde-741	36	3	satisfies	satisfie	NOUN
ejde-741	36	4	(	(	PUNCT
ejde-741	36	5	1.2	1.2	NUM
ejde-741	36	6	)	)	PUNCT
ejde-741	36	7	for	for	ADP
ejde-741	36	8	the	the	DET
ejde-741	36	9	(	(	PUNCT
ejde-741	36	10	sdc	sdc	NOUN
ejde-741	36	11	)	)	PUNCT
ejde-741	36	12	.	.	PUNCT
ejde-741	37	1	since	since	SCONJ
ejde-741	37	2	our	our	PRON
ejde-741	37	3	main	main	ADJ
ejde-741	37	4	results	result	NOUN
ejde-741	37	5	are	be	AUX
ejde-741	37	6	associated	associate	VERB
ejde-741	37	7	with	with	ADP
ejde-741	37	8	(	(	PUNCT
ejde-741	37	9	1.1	1.1	NUM
ejde-741	37	10	)	)	PUNCT
ejde-741	37	11	,	,	PUNCT
ejde-741	37	12	we	we	PRON
ejde-741	37	13	should	should	AUX
ejde-741	37	14	make	make	VERB
ejde-741	37	15	some	some	DET
ejde-741	37	16	comments	comment	NOUN
ejde-741	37	17	about	about	ADP
ejde-741	37	18	the	the	DET
ejde-741	37	19	controllability	controllability	NOUN
ejde-741	37	20	of	of	ADP
ejde-741	37	21	one	one	NUM
ejde-741	37	22	-	-	PUNCT
ejde-741	37	23	dimensional	dimensional	ADJ
ejde-741	37	24	degenerate	degenerate	ADJ
ejde-741	37	25	or	or	CCONJ
ejde-741	37	26	quasilinear	quasilinear	NOUN
ejde-741	37	27	problems	problem	NOUN
ejde-741	37	28	.	.	PUNCT
ejde-741	38	1	many	many	ADJ
ejde-741	38	2	applied	apply	VERB
ejde-741	38	3	phenomena	phenomenon	NOUN
ejde-741	38	4	are	be	AUX
ejde-741	38	5	closely	closely	ADV
ejde-741	38	6	related	relate	VERB
ejde-741	38	7	to	to	ADP
ejde-741	38	8	degenerate	degenerate	ADJ
ejde-741	38	9	parabolic	parabolic	ADJ
ejde-741	38	10	equations	equation	NOUN
ejde-741	38	11	,	,	PUNCT
ejde-741	38	12	calling	call	VERB
ejde-741	38	13	a	a	DET
ejde-741	38	14	notorious	notorious	ADJ
ejde-741	38	15	attention	attention	NOUN
ejde-741	38	16	to	to	ADP
ejde-741	38	17	their	their	PRON
ejde-741	38	18	mathematical	mathematical	ADJ
ejde-741	38	19	point	point	NOUN
ejde-741	38	20	of	of	ADP
ejde-741	38	21	view	view	NOUN
ejde-741	38	22	.	.	PUNCT
ejde-741	39	1	motivated	motivate	VERB
ejde-741	39	2	by	by	ADP
ejde-741	39	3	the	the	DET
ejde-741	39	4	properties	property	NOUN
ejde-741	39	5	already	already	ADV
ejde-741	39	6	known	know	VERB
ejde-741	39	7	for	for	ADP
ejde-741	39	8	the	the	DET
ejde-741	39	9	uniformly	uniformly	ADV
ejde-741	39	10	parabolic	parabolic	ADJ
ejde-741	39	11	case	case	NOUN
ejde-741	39	12	,	,	PUNCT
ejde-741	39	13	a	a	DET
ejde-741	39	14	complete	complete	ADJ
ejde-741	39	15	qualitative	qualitative	ADJ
ejde-741	39	16	investigation	investigation	NOUN
ejde-741	39	17	for	for	ADP
ejde-741	39	18	degenerate	degenerate	ADJ
ejde-741	39	19	operators	operator	NOUN
ejde-741	39	20	is	be	AUX
ejde-741	39	21	also	also	ADV
ejde-741	39	22	expected	expect	VERB
ejde-741	39	23	(	(	PUNCT
ejde-741	39	24	see	see	VERB
ejde-741	39	25	a	a	DET
ejde-741	39	26	well	well	ADJ
ejde-741	39	27	-	-	PUNCT
ejde-741	39	28	posedness	posedness	NOUN
ejde-741	39	29	result	result	NOUN
ejde-741	39	30	in	in	ADP
ejde-741	39	31	[	[	X
ejde-741	39	32	7	7	NUM
ejde-741	39	33	]	]	PUNCT
ejde-741	39	34	,	,	PUNCT
ejde-741	39	35	for	for	ADP
ejde-741	39	36	instance	instance	NOUN
ejde-741	39	37	)	)	PUNCT
ejde-741	39	38	.	.	PUNCT
ejde-741	40	1	this	this	DET
ejde-741	40	2	brief	brief	ADJ
ejde-741	40	3	comment	comment	NOUN
ejde-741	40	4	certainly	certainly	ADV
ejde-741	40	5	includes	include	VERB
ejde-741	40	6	control	control	NOUN
ejde-741	40	7	theory	theory	NOUN
ejde-741	40	8	,	,	PUNCT
ejde-741	40	9	where	where	SCONJ
ejde-741	40	10	much	much	ADV
ejde-741	40	11	more	more	ADJ
ejde-741	40	12	development	development	NOUN
ejde-741	40	13	is	be	AUX
ejde-741	40	14	still	still	ADV
ejde-741	40	15	desired	desire	VERB
ejde-741	40	16	.	.	PUNCT
ejde-741	41	1	in	in	ADP
ejde-741	41	2	one	one	NUM
ejde-741	41	3	dimension	dimension	NOUN
ejde-741	41	4	,	,	PUNCT
ejde-741	41	5	it	it	PRON
ejde-741	41	6	seems	seem	VERB
ejde-741	41	7	to	to	ADP
ejde-741	41	8	us	we	PRON
ejde-741	41	9	that	that	SCONJ
ejde-741	42	1	[	[	X
ejde-741	42	2	12	12	NUM
ejde-741	42	3	]	]	PUNCT
ejde-741	42	4	and	and	CCONJ
ejde-741	42	5	[	[	X
ejde-741	42	6	13	13	NUM
ejde-741	42	7	]	]	PUNCT
ejde-741	42	8	are	be	AUX
ejde-741	42	9	the	the	DET
ejde-741	42	10	two	two	NUM
ejde-741	42	11	first	first	ADJ
ejde-741	42	12	articles	article	NOUN
ejde-741	42	13	dealing	deal	VERB
ejde-741	42	14	with	with	ADP
ejde-741	42	15	the	the	DET
ejde-741	42	16	controllability	controllability	NOUN
ejde-741	42	17	of	of	ADP
ejde-741	42	18	degenerate	degenerate	ADJ
ejde-741	42	19	parabolic	parabolic	NOUN
ejde-741	42	20	equations	equation	NOUN
ejde-741	42	21	,	,	PUNCT
ejde-741	42	22	which	which	PRON
ejde-741	42	23	clearly	clearly	ADV
ejde-741	42	24	inspired	inspire	VERB
ejde-741	42	25	much	much	ADJ
ejde-741	42	26	relevant	relevant	ADJ
ejde-741	42	27	work	work	NOUN
ejde-741	42	28	since	since	SCONJ
ejde-741	42	29	then	then	ADV
ejde-741	42	30	(	(	PUNCT
ejde-741	42	31	see	see	VERB
ejde-741	42	32	[	[	X
ejde-741	42	33	3	3	NUM
ejde-741	42	34	,	,	PUNCT
ejde-741	42	35	6	6	NUM
ejde-741	42	36	,	,	PUNCT
ejde-741	42	37	9	9	NUM
ejde-741	42	38	,	,	PUNCT
ejde-741	42	39	14	14	NUM
ejde-741	42	40	,	,	PUNCT
ejde-741	42	41	15	15	NUM
ejde-741	42	42	,	,	PUNCT
ejde-741	42	43	22	22	NUM
ejde-741	42	44	,	,	PUNCT
ejde-741	42	45	24	24	NUM
ejde-741	42	46	,	,	PUNCT
ejde-741	42	47	29	29	NUM
ejde-741	42	48	,	,	PUNCT
ejde-741	42	49	33	33	NUM
ejde-741	42	50	]	]	PUNCT
ejde-741	42	51	and	and	CCONJ
ejde-741	42	52	the	the	DET
ejde-741	42	53	references	reference	NOUN
ejde-741	42	54	therein	therein	ADV
ejde-741	42	55	)	)	PUNCT
ejde-741	42	56	.	.	PUNCT
ejde-741	43	1	on	on	ADP
ejde-741	43	2	the	the	DET
ejde-741	43	3	other	other	ADJ
ejde-741	43	4	hand	hand	NOUN
ejde-741	43	5	,	,	PUNCT
ejde-741	43	6	to	to	ADP
ejde-741	43	7	our	our	PRON
ejde-741	43	8	best	good	ADJ
ejde-741	43	9	knowledge	knowledge	NOUN
ejde-741	43	10	,	,	PUNCT
ejde-741	43	11	there	there	PRON
ejde-741	43	12	are	be	VERB
ejde-741	43	13	not	not	PART
ejde-741	43	14	many	many	ADJ
ejde-741	43	15	controllability	controllability	NOUN
ejde-741	43	16	results	result	NOUN
ejde-741	43	17	involving	involve	VERB
ejde-741	43	18	quasilinear	quasilinear	NOUN
ejde-741	43	19	equations	equation	NOUN
ejde-741	43	20	,	,	PUNCT
ejde-741	43	21	where	where	SCONJ
ejde-741	43	22	the	the	DET
ejde-741	43	23	second	second	ADJ
ejde-741	43	24	-	-	PUNCT
ejde-741	43	25	order	order	NOUN
ejde-741	43	26	differential	differential	NOUN
ejde-741	43	27	operator	operator	NOUN
ejde-741	43	28	is	be	AUX
ejde-741	43	29	associated	associate	VERB
ejde-741	43	30	with	with	ADP
ejde-741	43	31	a	a	DET
ejde-741	43	32	nonlinearity	nonlinearity	NOUN
ejde-741	43	33	which	which	PRON
ejde-741	43	34	depends	depend	VERB
ejde-741	43	35	on	on	ADP
ejde-741	43	36	the	the	DET
ejde-741	43	37	state	state	NOUN
ejde-741	43	38	(	(	PUNCT
ejde-741	43	39	see	see	VERB
ejde-741	43	40	[	[	X
ejde-741	43	41	28	28	NUM
ejde-741	43	42	]	]	PUNCT
ejde-741	43	43	,	,	PUNCT
ejde-741	43	44	for	for	ADP
ejde-741	43	45	instance	instance	NOUN
ejde-741	43	46	)	)	PUNCT
ejde-741	43	47	.	.	PUNCT
ejde-741	44	1	so	so	ADV
ejde-741	44	2	that	that	SCONJ
ejde-741	44	3	,	,	PUNCT
ejde-741	44	4	in	in	ADP
ejde-741	44	5	this	this	DET
ejde-741	44	6	paper	paper	NOUN
ejde-741	44	7	,	,	PUNCT
ejde-741	44	8	the	the	DET
ejde-741	44	9	main	main	ADJ
ejde-741	44	10	intention	intention	NOUN
ejde-741	44	11	is	be	AUX
ejde-741	44	12	a	a	DET
ejde-741	44	13	investigation	investigation	NOUN
ejde-741	44	14	about	about	ADP
ejde-741	44	15	the	the	DET
ejde-741	44	16	controllability	controllability	NOUN
ejde-741	44	17	of	of	ADP
ejde-741	44	18	onedimensional	onedimensional	ADJ
ejde-741	44	19	degenerate	degenerate	ADJ
ejde-741	44	20	quasilinear	quasilinear	NOUN
ejde-741	44	21	equations	equation	NOUN
ejde-741	44	22	.	.	PUNCT
ejde-741	45	1	to	to	PART
ejde-741	45	2	be	be	AUX
ejde-741	45	3	more	more	ADV
ejde-741	45	4	precise	precise	ADJ
ejde-741	45	5	,	,	PUNCT
ejde-741	45	6	we	we	PRON
ejde-741	45	7	will	will	AUX
ejde-741	45	8	prove	prove	VERB
ejde-741	45	9	a	a	DET
ejde-741	45	10	local	local	ADJ
ejde-741	45	11	null	null	ADJ
ejde-741	45	12	-	-	PUNCT
ejde-741	45	13	controllability	controllability	NOUN
ejde-741	45	14	result	result	NOUN
ejde-741	45	15	for	for	ADP
ejde-741	45	16	(	(	PUNCT
ejde-741	45	17	1.1	1.1	NUM
ejde-741	45	18	)	)	PUNCT
ejde-741	45	19	,	,	PUNCT
ejde-741	45	20	at	at	ADP
ejde-741	45	21	any	any	DET
ejde-741	45	22	time	time	NOUN
ejde-741	45	23	,	,	PUNCT
ejde-741	45	24	with	with	ADP
ejde-741	45	25	controls	control	NOUN
ejde-741	45	26	acting	act	VERB
ejde-741	45	27	on	on	ADP
ejde-741	45	28	a	a	DET
ejde-741	45	29	small	small	ADJ
ejde-741	45	30	subinterval	subinterval	NOUN
ejde-741	45	31	ω	ω	PROPN
ejde-741	45	32	⊂	⊂	PROPN
ejde-741	45	33	(	(	PUNCT
ejde-741	45	34	0	0	NUM
ejde-741	45	35	,	,	PUNCT
ejde-741	45	36	1	1	NUM
ejde-741	45	37	)	)	PUNCT
ejde-741	45	38	.	.	PUNCT
ejde-741	46	1	in	in	ADP
ejde-741	46	2	other	other	ADJ
ejde-741	46	3	words	word	NOUN
ejde-741	46	4	,	,	PUNCT
ejde-741	46	5	given	give	VERB
ejde-741	46	6	any	any	DET
ejde-741	46	7	time	time	NOUN
ejde-741	46	8	t	t	X
ejde-741	46	9	>	>	X
ejde-741	46	10	0	0	PUNCT
ejde-741	46	11	and	and	CCONJ
ejde-741	46	12	a	a	DET
ejde-741	46	13	sufficiently	sufficiently	ADV
ejde-741	46	14	small	small	ADJ
ejde-741	46	15	initial	initial	ADJ
ejde-741	46	16	data	datum	NOUN
ejde-741	46	17	u0	u0	ADJ
ejde-741	46	18	,	,	PUNCT
ejde-741	46	19	there	there	PRON
ejde-741	46	20	exists	exist	VERB
ejde-741	46	21	a	a	DET
ejde-741	46	22	state	state	NOUN
ejde-741	46	23	-	-	PUNCT
ejde-741	46	24	control	control	NOUN
ejde-741	46	25	pair	pair	NOUN
ejde-741	46	26	(	(	PUNCT
ejde-741	46	27	uh	uh	INTJ
ejde-741	46	28	,	,	PUNCT
ejde-741	46	29	h	h	NOUN
ejde-741	46	30	)	)	PUNCT
ejde-741	46	31	for	for	ADP
ejde-741	46	32	(	(	PUNCT
ejde-741	46	33	1.1	1.1	NUM
ejde-741	46	34	)	)	PUNCT
ejde-741	46	35	,	,	PUNCT
ejde-741	46	36	such	such	ADJ
ejde-741	46	37	that	that	DET
ejde-741	46	38	uh(t	uh(t	ADV
ejde-741	46	39	,	,	PUNCT
ejde-741	46	40	·	·	PUNCT
ejde-741	46	41	)	)	PUNCT
ejde-741	47	1	=	=	SYM
ejde-741	47	2	0	0	NUM
ejde-741	47	3	in	in	ADP
ejde-741	47	4	[	[	X
ejde-741	47	5	0	0	NUM
ejde-741	47	6	,	,	PUNCT
ejde-741	47	7	1	1	NUM
ejde-741	47	8	]	]	PUNCT
ejde-741	47	9	.	.	PUNCT
ejde-741	48	1	the	the	DET
ejde-741	48	2	proof	proof	NOUN
ejde-741	48	3	will	will	AUX
ejde-741	48	4	be	be	AUX
ejde-741	48	5	based	base	VERB
ejde-741	48	6	on	on	ADP
ejde-741	48	7	[	[	X
ejde-741	48	8	30	30	NUM
ejde-741	48	9	]	]	PUNCT
ejde-741	48	10	,	,	PUNCT
ejde-741	48	11	where	where	SCONJ
ejde-741	48	12	a	a	DET
ejde-741	48	13	meticulous	meticulous	ADJ
ejde-741	48	14	local	local	ADJ
ejde-741	48	15	inversion	inversion	NOUN
ejde-741	48	16	argument	argument	NOUN
ejde-741	48	17	is	be	AUX
ejde-741	48	18	developed	develop	VERB
ejde-741	48	19	,	,	PUNCT
ejde-741	48	20	using	use	VERB
ejde-741	48	21	lyusternik	lyusternik	PROPN
ejde-741	48	22	’s	’s	PART
ejde-741	48	23	theorem	theorem	ADJ
ejde-741	48	24	.	.	PUNCT
ejde-741	49	1	this	this	DET
ejde-741	49	2	goal	goal	NOUN
ejde-741	49	3	passes	pass	VERB
ejde-741	49	4	by	by	ADP
ejde-741	49	5	a	a	DET
ejde-741	49	6	certain	certain	ADJ
ejde-741	49	7	linearization	linearization	NOUN
ejde-741	49	8	of	of	ADP
ejde-741	49	9	(	(	PUNCT
ejde-741	49	10	1.1	1.1	NUM
ejde-741	49	11	)	)	PUNCT
ejde-741	49	12	,	,	PUNCT
ejde-741	49	13	for	for	ADP
ejde-741	49	14	which	which	PRON
ejde-741	49	15	a	a	DET
ejde-741	49	16	global	global	ADJ
ejde-741	49	17	null	null	ADJ
ejde-741	49	18	-	-	PUNCT
ejde-741	49	19	controllability	controllability	NOUN
ejde-741	49	20	result	result	NOUN
ejde-741	49	21	and	and	CCONJ
ejde-741	49	22	some	some	DET
ejde-741	49	23	additional	additional	ADJ
ejde-741	49	24	estimates	estimate	NOUN
ejde-741	49	25	will	will	AUX
ejde-741	49	26	also	also	ADV
ejde-741	49	27	be	be	AUX
ejde-741	49	28	obtained	obtain	VERB
ejde-741	49	29	.	.	PUNCT
ejde-741	50	1	in	in	ADP
ejde-741	50	2	the	the	DET
ejde-741	50	3	current	current	ADJ
ejde-741	50	4	literature	literature	NOUN
ejde-741	50	5	,	,	PUNCT
ejde-741	50	6	it	it	PRON
ejde-741	50	7	is	be	AUX
ejde-741	50	8	undeniable	undeniable	ADJ
ejde-741	50	9	the	the	DET
ejde-741	50	10	strength	strength	NOUN
ejde-741	50	11	of	of	ADP
ejde-741	50	12	the	the	DET
ejde-741	50	13	carleman	carleman	ADJ
ejde-741	50	14	estimates	estimate	NOUN
ejde-741	50	15	method	method	VERB
ejde-741	50	16	,	,	PUNCT
ejde-741	50	17	because	because	SCONJ
ejde-741	50	18	it	it	PRON
ejde-741	50	19	provides	provide	VERB
ejde-741	50	20	a	a	DET
ejde-741	50	21	refined	refined	ADJ
ejde-741	50	22	technique	technique	NOUN
ejde-741	50	23	that	that	PRON
ejde-741	50	24	makes	make	VERB
ejde-741	50	25	the	the	DET
ejde-741	50	26	one	one	NUM
ejde-741	50	27	-	-	PUNCT
ejde-741	50	28	dimensional	dimensional	ADJ
ejde-741	50	29	degenerate	degenerate	ADJ
ejde-741	50	30	controllability	controllability	NOUN
ejde-741	50	31	field	field	NOUN
ejde-741	50	32	well	well	ADV
ejde-741	50	33	-	-	PUNCT
ejde-741	50	34	understood	understand	VERB
ejde-741	50	35	(	(	PUNCT
ejde-741	50	36	see	see	VERB
ejde-741	50	37	[	[	X
ejde-741	50	38	1	1	NUM
ejde-741	50	39	,	,	PUNCT
ejde-741	50	40	8	8	NUM
ejde-741	50	41	,	,	PUNCT
ejde-741	50	42	10	10	NUM
ejde-741	50	43	,	,	PUNCT
ejde-741	50	44	11	11	NUM
ejde-741	50	45	,	,	PUNCT
ejde-741	50	46	32	32	NUM
ejde-741	50	47	]	]	PUNCT
ejde-741	50	48	and	and	CCONJ
ejde-741	50	49	the	the	DET
ejde-741	50	50	references	reference	NOUN
ejde-741	50	51	aforementioned	aforementione	VERB
ejde-741	50	52	)	)	PUNCT
ejde-741	50	53	.	.	PUNCT
ejde-741	51	1	in	in	ADP
ejde-741	51	2	[	[	X
ejde-741	51	3	28	28	NUM
ejde-741	51	4	]	]	PUNCT
ejde-741	51	5	,	,	PUNCT
ejde-741	51	6	the	the	DET
ejde-741	51	7	local	local	ADJ
ejde-741	51	8	null	null	ADJ
ejde-741	51	9	-	-	PUNCT
ejde-741	51	10	controllability	controllability	NOUN
ejde-741	51	11	result	result	NOUN
ejde-741	51	12	,	,	PUNCT
ejde-741	51	13	proved	prove	VERB
ejde-741	51	14	for	for	ADP
ejde-741	51	15	nondegenerate	nondegenerate	ADJ
ejde-741	51	16	quasilinear	quasilinear	NOUN
ejde-741	51	17	equations	equation	NOUN
ejde-741	51	18	,	,	PUNCT
ejde-741	51	19	also	also	ADV
ejde-741	51	20	follows	follow	VERB
ejde-741	51	21	carleman	carleman	NOUN
ejde-741	51	22	’s	’s	PART
ejde-741	51	23	approach	approach	NOUN
ejde-741	51	24	.	.	PUNCT
ejde-741	52	1	to	to	PART
ejde-741	52	2	summarize	summarize	VERB
ejde-741	52	3	,	,	PUNCT
ejde-741	52	4	up	up	ADP
ejde-741	52	5	to	to	ADP
ejde-741	52	6	this	this	DET
ejde-741	52	7	moment	moment	NOUN
ejde-741	52	8	,	,	PUNCT
ejde-741	52	9	the	the	DET
ejde-741	52	10	controllability	controllability	NOUN
ejde-741	52	11	of	of	ADP
ejde-741	52	12	quasilinear	quasilinear	NOUN
ejde-741	52	13	equations	equation	NOUN
ejde-741	52	14	,	,	PUNCT
ejde-741	52	15	where	where	SCONJ
ejde-741	52	16	the	the	DET
ejde-741	52	17	diffusion	diffusion	NOUN
ejde-741	52	18	term	term	NOUN
ejde-741	52	19	depends	depend	VERB
ejde-741	52	20	nonlinearly	nonlinearly	ADV
ejde-741	52	21	on	on	ADP
ejde-741	52	22	the	the	DET
ejde-741	52	23	state	state	NOUN
ejde-741	52	24	,	,	PUNCT
ejde-741	52	25	has	have	AUX
ejde-741	52	26	not	not	PART
ejde-741	52	27	been	be	AUX
ejde-741	52	28	widely	widely	ADV
ejde-741	52	29	investigated	investigate	VERB
ejde-741	52	30	,	,	PUNCT
ejde-741	52	31	even	even	ADV
ejde-741	52	32	for	for	ADP
ejde-741	52	33	the	the	DET
ejde-741	52	34	nondegenerate	nondegenerate	ADJ
ejde-741	52	35	case	case	NOUN
ejde-741	52	36	.	.	PUNCT
ejde-741	53	1	it	it	PRON
ejde-741	53	2	is	be	AUX
ejde-741	53	3	exactly	exactly	ADV
ejde-741	53	4	the	the	DET
ejde-741	53	5	motivation	motivation	NOUN
ejde-741	53	6	for	for	ADP
ejde-741	53	7	the	the	DET
ejde-741	53	8	current	current	ADJ
ejde-741	53	9	research	research	NOUN
ejde-741	53	10	,	,	PUNCT
ejde-741	53	11	where	where	SCONJ
ejde-741	53	12	we	we	PRON
ejde-741	53	13	would	would	AUX
ejde-741	53	14	like	like	VERB
ejde-741	53	15	to	to	PART
ejde-741	53	16	contribute	contribute	VERB
ejde-741	53	17	providing	provide	VERB
ejde-741	53	18	a	a	DET
ejde-741	53	19	controllability	controllability	NOUN
ejde-741	53	20	study	study	NOUN
ejde-741	53	21	for	for	ADP
ejde-741	53	22	the	the	DET
ejde-741	53	23	degenerate	degenerate	ADJ
ejde-741	53	24	quasilinear	quasilinear	NOUN
ejde-741	53	25	problem	problem	NOUN
ejde-741	53	26	(	(	PUNCT
ejde-741	53	27	1.1	1.1	NUM
ejde-741	53	28	)	)	PUNCT
ejde-741	53	29	.	.	PUNCT
ejde-741	54	1	to	to	PART
ejde-741	54	2	complement	complement	VERB
ejde-741	54	3	the	the	DET
ejde-741	54	4	state	state	NOUN
ejde-741	54	5	of	of	ADP
ejde-741	54	6	the	the	DET
ejde-741	54	7	art	art	NOUN
ejde-741	54	8	associated	associate	VERB
ejde-741	54	9	with	with	ADP
ejde-741	54	10	degenerate	degenerate	ADJ
ejde-741	54	11	problems	problem	NOUN
ejde-741	54	12	,	,	PUNCT
ejde-741	54	13	we	we	PRON
ejde-741	54	14	also	also	ADV
ejde-741	54	15	mention	mention	VERB
ejde-741	54	16	[	[	X
ejde-741	54	17	4	4	NUM
ejde-741	54	18	]	]	PUNCT
ejde-741	54	19	,	,	PUNCT
ejde-741	54	20	where	where	SCONJ
ejde-741	54	21	the	the	DET
ejde-741	54	22	boundary	boundary	ADJ
ejde-741	54	23	null	null	ADJ
ejde-741	54	24	controllability	controllability	NOUN
ejde-741	54	25	of	of	ADP
ejde-741	54	26	the	the	DET
ejde-741	54	27	degenerate	degenerate	ADJ
ejde-741	54	28	heat	heat	NOUN
ejde-741	54	29	equation	equation	NOUN
ejde-741	54	30	was	be	AUX
ejde-741	54	31	obtained	obtain	VERB
ejde-741	54	32	as	as	ADP
ejde-741	54	33	the	the	DET
ejde-741	54	34	limit	limit	NOUN
ejde-741	54	35	of	of	ADP
ejde-741	54	36	internal	internal	ADJ
ejde-741	54	37	controllability	controllability	NOUN
ejde-741	54	38	.	.	PUNCT
ejde-741	55	1	to	to	PART
ejde-741	55	2	be	be	AUX
ejde-741	55	3	more	more	ADV
ejde-741	55	4	precise	precise	ADJ
ejde-741	55	5	,	,	PUNCT
ejde-741	55	6	ejde-2025/15	ejde-2025/15	PROPN
ejde-741	55	7	null	null	ADJ
ejde-741	55	8	-	-	PUNCT
ejde-741	55	9	controllability	controllability	NOUN
ejde-741	55	10	degenerate	degenerate	ADJ
ejde-741	55	11	quasilinear	quasilinear	NOUN
ejde-741	55	12	equations	equation	NOUN
ejde-741	55	13	3	3	NUM
ejde-741	55	14	taking	take	VERB
ejde-741	55	15	ω	ω	NOUN
ejde-741	55	16	=	=	SYM
ejde-741	55	17	ωε	ωε	NOUN
ejde-741	55	18	:	:	PUNCT
ejde-741	55	19	=	=	SYM
ejde-741	55	20	(	(	PUNCT
ejde-741	55	21	1	1	NUM
ejde-741	55	22	−	−	PROPN
ejde-741	55	23	ε	ε	PROPN
ejde-741	55	24	,	,	PUNCT
ejde-741	55	25	1	1	NUM
ejde-741	55	26	)	)	PUNCT
ejde-741	55	27	,	,	PUNCT
ejde-741	55	28	for	for	ADP
ejde-741	55	29	each	each	DET
ejde-741	55	30	ε	ε	PROPN
ejde-741	55	31	∈	∈	PROPN
ejde-741	55	32	(	(	PUNCT
ejde-741	55	33	0	0	NUM
ejde-741	55	34	,	,	PUNCT
ejde-741	55	35	1	1	NUM
ejde-741	55	36	)	)	PUNCT
ejde-741	55	37	,	,	PUNCT
ejde-741	55	38	is	be	AUX
ejde-741	55	39	built	build	VERB
ejde-741	55	40	a	a	DET
ejde-741	55	41	family	family	NOUN
ejde-741	55	42	of	of	ADP
ejde-741	55	43	state	state	NOUN
ejde-741	55	44	-	-	PUNCT
ejde-741	55	45	control	control	NOUN
ejde-741	55	46	pairs	pair	NOUN
ejde-741	55	47	{	{	PUNCT
ejde-741	55	48	(	(	PUNCT
ejde-741	55	49	uε	uε	NOUN
ejde-741	55	50	,	,	PUNCT
ejde-741	55	51	hε	hε	PROPN
ejde-741	55	52	)	)	PUNCT
ejde-741	55	53	;	;	PUNCT
ejde-741	55	54	ε	ε	PROPN
ejde-741	55	55	∈	∈	PROPN
ejde-741	55	56	(	(	PUNCT
ejde-741	55	57	0	0	NUM
ejde-741	55	58	,	,	PUNCT
ejde-741	55	59	1	1	NUM
ejde-741	55	60	)	)	PUNCT
ejde-741	55	61	}	}	PUNCT
ejde-741	55	62	solving	solve	VERB
ejde-741	55	63	(	(	PUNCT
ejde-741	55	64	2.2	2.2	NUM
ejde-741	55	65	)	)	PUNCT
ejde-741	55	66	,	,	PUNCT
ejde-741	55	67	with	with	ADP
ejde-741	55	68	c	c	PROPN
ejde-741	55	69	≡	≡	PROPN
ejde-741	55	70	0	0	PUNCT
ejde-741	55	71	and	and	CCONJ
ejde-741	55	72	g	g	PROPN
ejde-741	55	73	≡	≡	PROPN
ejde-741	55	74	0	0	NUM
ejde-741	55	75	,	,	PUNCT
ejde-741	55	76	with	with	ADP
ejde-741	55	77	the	the	DET
ejde-741	55	78	following	follow	VERB
ejde-741	55	79	property	property	NOUN
ejde-741	55	80	:	:	PUNCT
ejde-741	55	81	(	(	PUNCT
ejde-741	55	82	uε	uε	INTJ
ejde-741	55	83	,	,	PUNCT
ejde-741	55	84	hε	hε	PROPN
ejde-741	55	85	)	)	PUNCT
ejde-741	55	86	→	→	SYM
ejde-741	55	87	(	(	PUNCT
ejde-741	55	88	u	u	NOUN
ejde-741	55	89	,	,	PUNCT
ejde-741	55	90	g	g	NOUN
ejde-741	55	91	)	)	PUNCT
ejde-741	55	92	in	in	ADP
ejde-741	55	93	a	a	DET
ejde-741	55	94	suitable	suitable	ADJ
ejde-741	55	95	functional	functional	ADJ
ejde-741	55	96	space	space	NOUN
ejde-741	55	97	,	,	PUNCT
ejde-741	55	98	as	as	ADP
ejde-741	55	99	ε	ε	PROPN
ejde-741	55	100	→	→	SYM
ejde-741	55	101	0	0	NUM
ejde-741	55	102	,	,	PUNCT
ejde-741	55	103	where	where	SCONJ
ejde-741	55	104	(	(	PUNCT
ejde-741	55	105	u	u	NOUN
ejde-741	55	106	,	,	PUNCT
ejde-741	55	107	g	g	NOUN
ejde-741	55	108	)	)	PUNCT
ejde-741	55	109	solves	solve	VERB
ejde-741	55	110	the	the	DET
ejde-741	55	111	boundary	boundary	ADJ
ejde-741	55	112	null	null	ADJ
ejde-741	55	113	controllability	controllability	NOUN
ejde-741	55	114	problem	problem	NOUN
ejde-741	55	115	for	for	ADP
ejde-741	55	116	the	the	DET
ejde-741	55	117	degenerate	degenerate	ADJ
ejde-741	55	118	heat	heat	NOUN
ejde-741	55	119	equation	equation	NOUN
ejde-741	55	120	.	.	PUNCT
ejde-741	56	1	however	however	ADV
ejde-741	56	2	,	,	PUNCT
ejde-741	56	3	this	this	DET
ejde-741	56	4	kind	kind	NOUN
ejde-741	56	5	of	of	ADP
ejde-741	56	6	question	question	NOUN
ejde-741	56	7	keeps	keeps	AUX
ejde-741	56	8	not	not	PART
ejde-741	56	9	understood	understand	VERB
ejde-741	56	10	if	if	SCONJ
ejde-741	56	11	ω	ω	PROPN
ejde-741	56	12	=	=	SYM
ejde-741	56	13	(	(	PUNCT
ejde-741	56	14	0	0	NUM
ejde-741	56	15	,	,	PUNCT
ejde-741	56	16	ε	ε	PROPN
ejde-741	56	17	)	)	PUNCT
ejde-741	56	18	,	,	PUNCT
ejde-741	56	19	as	as	SCONJ
ejde-741	56	20	explained	explain	VERB
ejde-741	56	21	in	in	ADP
ejde-741	56	22	[	[	X
ejde-741	56	23	4	4	NUM
ejde-741	56	24	,	,	PUNCT
ejde-741	56	25	section	section	NOUN
ejde-741	56	26	5	5	NUM
ejde-741	56	27	]	]	PUNCT
ejde-741	56	28	.	.	PUNCT
ejde-741	57	1	on	on	ADP
ejde-741	57	2	the	the	DET
ejde-741	57	3	other	other	ADJ
ejde-741	57	4	hand	hand	NOUN
ejde-741	57	5	,	,	PUNCT
ejde-741	57	6	following	follow	VERB
ejde-741	57	7	this	this	DET
ejde-741	57	8	direction	direction	NOUN
ejde-741	57	9	,	,	PUNCT
ejde-741	57	10	some	some	DET
ejde-741	57	11	effort	effort	NOUN
ejde-741	57	12	has	have	AUX
ejde-741	57	13	been	be	AUX
ejde-741	57	14	made	make	VERB
ejde-741	57	15	to	to	PART
ejde-741	57	16	prove	prove	VERB
ejde-741	57	17	an	an	DET
ejde-741	57	18	analogous	analogous	ADJ
ejde-741	57	19	fact	fact	NOUN
ejde-741	57	20	for	for	ADP
ejde-741	57	21	the	the	DET
ejde-741	57	22	degenerate	degenerate	ADJ
ejde-741	57	23	wave	wave	NOUN
ejde-741	57	24	equation	equation	NOUN
ejde-741	57	25	(	(	PUNCT
ejde-741	57	26	see	see	VERB
ejde-741	57	27	[	[	X
ejde-741	57	28	5	5	NUM
ejde-741	57	29	]	]	NUM
ejde-741	57	30	)	)	PUNCT
ejde-741	57	31	.	.	PUNCT
ejde-741	58	1	naturally	naturally	ADV
ejde-741	58	2	,	,	PUNCT
ejde-741	58	3	it	it	PRON
ejde-741	58	4	would	would	AUX
ejde-741	58	5	also	also	ADV
ejde-741	58	6	be	be	AUX
ejde-741	58	7	very	very	ADV
ejde-741	58	8	interesting	interesting	ADJ
ejde-741	58	9	to	to	PART
ejde-741	58	10	analyze	analyze	VERB
ejde-741	58	11	the	the	DET
ejde-741	58	12	asymptotic	asymptotic	ADJ
ejde-741	58	13	behavior	behavior	NOUN
ejde-741	58	14	of	of	ADP
ejde-741	58	15	families	family	NOUN
ejde-741	58	16	of	of	ADP
ejde-741	58	17	state	state	NOUN
ejde-741	58	18	-	-	PUNCT
ejde-741	58	19	control	control	NOUN
ejde-741	58	20	pairs	pair	NOUN
ejde-741	58	21	corresponding	correspond	VERB
ejde-741	58	22	to	to	ADP
ejde-741	58	23	nonlinear	nonlinear	ADJ
ejde-741	58	24	evolution	evolution	NOUN
ejde-741	58	25	pdes	pde	NOUN
ejde-741	58	26	(	(	PUNCT
ejde-741	58	27	degenerate	degenerate	ADJ
ejde-741	58	28	and	and	CCONJ
ejde-741	58	29	nondegenerate	nondegenerate	ADJ
ejde-741	58	30	cases	case	NOUN
ejde-741	58	31	.	.	PUNCT
ejde-741	59	1	the	the	DET
ejde-741	59	2	discussion	discussion	NOUN
ejde-741	59	3	above	above	ADV
ejde-741	59	4	is	be	AUX
ejde-741	59	5	completely	completely	ADV
ejde-741	59	6	related	relate	VERB
ejde-741	59	7	to	to	ADP
ejde-741	59	8	the	the	DET
ejde-741	59	9	relevance	relevance	NOUN
ejde-741	59	10	of	of	ADP
ejde-741	59	11	degenerate	degenerate	ADJ
ejde-741	59	12	operators	operator	NOUN
ejde-741	59	13	in	in	ADP
ejde-741	59	14	partial	partial	ADJ
ejde-741	59	15	differential	differential	NOUN
ejde-741	59	16	equations	equation	NOUN
ejde-741	59	17	,	,	PUNCT
ejde-741	59	18	including	include	VERB
ejde-741	59	19	the	the	DET
ejde-741	59	20	control	control	NOUN
ejde-741	59	21	theory	theory	NOUN
ejde-741	59	22	setting	set	VERB
ejde-741	59	23	.	.	PUNCT
ejde-741	60	1	specifically	specifically	ADV
ejde-741	60	2	talking	talk	VERB
ejde-741	60	3	about	about	ADP
ejde-741	60	4	this	this	DET
ejde-741	60	5	theme	theme	NOUN
ejde-741	60	6	,	,	PUNCT
ejde-741	60	7	in	in	ADP
ejde-741	60	8	the	the	DET
ejde-741	60	9	presence	presence	NOUN
ejde-741	60	10	of	of	ADP
ejde-741	60	11	nonlinearities	nonlinearitie	NOUN
ejde-741	60	12	,	,	PUNCT
ejde-741	60	13	we	we	PRON
ejde-741	60	14	should	should	AUX
ejde-741	60	15	mention	mention	VERB
ejde-741	60	16	[	[	X
ejde-741	60	17	20	20	NUM
ejde-741	60	18	,	,	PUNCT
ejde-741	60	19	21	21	NUM
ejde-741	60	20	,	,	PUNCT
ejde-741	60	21	19	19	NUM
ejde-741	60	22	]	]	PUNCT
ejde-741	60	23	,	,	PUNCT
ejde-741	60	24	where	where	SCONJ
ejde-741	60	25	the	the	DET
ejde-741	60	26	authors	author	NOUN
ejde-741	60	27	have	have	AUX
ejde-741	60	28	obtained	obtain	VERB
ejde-741	60	29	the	the	DET
ejde-741	60	30	local	local	ADJ
ejde-741	60	31	nullcontrollability	nullcontrollability	NOUN
ejde-741	60	32	for	for	ADP
ejde-741	60	33	a	a	DET
ejde-741	60	34	class	class	NOUN
ejde-741	60	35	of	of	ADP
ejde-741	60	36	degenerate	degenerate	ADJ
ejde-741	60	37	parabolic	parabolic	ADJ
ejde-741	60	38	problems	problem	NOUN
ejde-741	60	39	with	with	ADP
ejde-741	60	40	nonlocal	nonlocal	ADJ
ejde-741	60	41	terms	term	NOUN
ejde-741	60	42	,	,	PUNCT
ejde-741	60	43	dealing	deal	VERB
ejde-741	60	44	with	with	ADP
ejde-741	60	45	theoretical	theoretical	ADJ
ejde-741	60	46	and	and	CCONJ
ejde-741	60	47	numerical	numerical	ADJ
ejde-741	60	48	aspects	aspect	NOUN
ejde-741	60	49	.	.	PUNCT
ejde-741	61	1	we	we	PRON
ejde-741	61	2	emphasize	emphasize	VERB
ejde-741	61	3	that	that	SCONJ
ejde-741	61	4	this	this	DET
ejde-741	61	5	work	work	NOUN
ejde-741	61	6	relies	rely	VERB
ejde-741	61	7	on	on	ADP
ejde-741	61	8	those	those	DET
ejde-741	61	9	carleman	carleman	ADJ
ejde-741	61	10	estimates	estimate	NOUN
ejde-741	61	11	achieved	achieve	VERB
ejde-741	61	12	in	in	ADP
ejde-741	61	13	[	[	X
ejde-741	61	14	20	20	NUM
ejde-741	61	15	]	]	PUNCT
ejde-741	61	16	and	and	CCONJ
ejde-741	61	17	[	[	X
ejde-741	61	18	19	19	NUM
ejde-741	61	19	]	]	PUNCT
ejde-741	61	20	for	for	ADP
ejde-741	61	21	the	the	DET
ejde-741	61	22	(	(	PUNCT
ejde-741	61	23	wdc	wdc	PROPN
ejde-741	61	24	)	)	PUNCT
ejde-741	61	25	and	and	CCONJ
ejde-741	61	26	the	the	DET
ejde-741	61	27	(	(	PUNCT
ejde-741	61	28	sdc	sdc	NOUN
ejde-741	61	29	)	)	PUNCT
ejde-741	61	30	,	,	PUNCT
ejde-741	61	31	respectively	respectively	ADV
ejde-741	61	32	.	.	PUNCT
ejde-741	62	1	next	next	ADV
ejde-741	62	2	,	,	PUNCT
ejde-741	62	3	we	we	PRON
ejde-741	62	4	present	present	VERB
ejde-741	62	5	some	some	DET
ejde-741	62	6	important	important	ADJ
ejde-741	62	7	functional	functional	ADJ
ejde-741	62	8	spaces	space	NOUN
ejde-741	62	9	,	,	PUNCT
ejde-741	62	10	introduced	introduce	VERB
ejde-741	62	11	in	in	ADP
ejde-741	62	12	[	[	X
ejde-741	62	13	1	1	NUM
ejde-741	62	14	]	]	PUNCT
ejde-741	62	15	,	,	PUNCT
ejde-741	62	16	which	which	PRON
ejde-741	62	17	are	be	AUX
ejde-741	62	18	closely	closely	ADV
ejde-741	62	19	related	relate	VERB
ejde-741	62	20	to	to	ADP
ejde-741	62	21	the	the	DET
ejde-741	62	22	initial	initial	ADJ
ejde-741	62	23	data	datum	NOUN
ejde-741	62	24	of	of	ADP
ejde-741	62	25	(	(	PUNCT
ejde-741	62	26	1.1	1.1	NUM
ejde-741	62	27	)	)	PUNCT
ejde-741	62	28	and	and	CCONJ
ejde-741	62	29	its	its	PRON
ejde-741	62	30	linearization	linearization	NOUN
ejde-741	62	31	.	.	PUNCT
ejde-741	63	1	other	other	ADJ
ejde-741	63	2	than	than	ADP
ejde-741	63	3	that	that	PRON
ejde-741	63	4	,	,	PUNCT
ejde-741	63	5	it	it	PRON
ejde-741	63	6	also	also	ADV
ejde-741	63	7	has	have	VERB
ejde-741	63	8	to	to	PART
ejde-741	63	9	do	do	VERB
ejde-741	63	10	with	with	ADP
ejde-741	63	11	the	the	DET
ejde-741	63	12	statement	statement	NOUN
ejde-741	63	13	of	of	ADP
ejde-741	63	14	our	our	PRON
ejde-741	63	15	main	main	ADJ
ejde-741	63	16	result	result	NOUN
ejde-741	63	17	.	.	PUNCT
ejde-741	64	1	definition	definition	NOUN
ejde-741	64	2	1.2	1.2	NUM
ejde-741	64	3	(	(	PUNCT
ejde-741	64	4	weighted	weight	VERB
ejde-741	64	5	sobolev	sobolev	NOUN
ejde-741	64	6	spaces	space	NOUN
ejde-741	64	7	)	)	PUNCT
ejde-741	64	8	.	.	PUNCT
ejde-741	65	1	let	let	VERB
ejde-741	65	2	us	we	PRON
ejde-741	65	3	consider	consider	VERB
ejde-741	65	4	a	a	DET
ejde-741	65	5	real	real	ADJ
ejde-741	65	6	function	function	NOUN
ejde-741	65	7	a	a	DET
ejde-741	65	8	=	=	SYM
ejde-741	65	9	a(x	a(x	NOUN
ejde-741	65	10	)	)	PUNCT
ejde-741	65	11	,	,	PUNCT
ejde-741	65	12	as	as	ADP
ejde-741	65	13	in	in	ADP
ejde-741	65	14	(	(	PUNCT
ejde-741	65	15	1.1	1.1	NUM
ejde-741	65	16	)	)	PUNCT
ejde-741	65	17	.	.	PUNCT
ejde-741	66	1	(	(	PUNCT
ejde-741	66	2	a	a	X
ejde-741	66	3	)	)	PUNCT
ejde-741	66	4	for	for	ADP
ejde-741	66	5	the	the	DET
ejde-741	66	6	(	(	PUNCT
ejde-741	66	7	wdc	wdc	PROPN
ejde-741	66	8	)	)	PUNCT
ejde-741	66	9	,	,	PUNCT
ejde-741	66	10	we	we	PRON
ejde-741	66	11	set	set	VERB
ejde-741	66	12	h1	h1	VERB
ejde-741	66	13	a	a	DET
ejde-741	66	14	:	:	PUNCT
ejde-741	66	15	=	=	SYM
ejde-741	66	16	{	{	PUNCT
ejde-741	66	17	u	u	PROPN
ejde-741	66	18	∈	∈	PROPN
ejde-741	66	19	l2(0	l2(0	NOUN
ejde-741	66	20	,	,	PUNCT
ejde-741	66	21	1	1	NUM
ejde-741	66	22	)	)	PUNCT
ejde-741	66	23	such	such	ADJ
ejde-741	66	24	that	that	SCONJ
ejde-741	66	25	u	u	NOUN
ejde-741	66	26	is	be	AUX
ejde-741	66	27	absolutely	absolutely	ADV
ejde-741	66	28	continuous	continuous	ADJ
ejde-741	66	29	in	in	ADP
ejde-741	66	30	[	[	X
ejde-741	66	31	0	0	NUM
ejde-741	66	32	,	,	PUNCT
ejde-741	66	33	1	1	NUM
ejde-741	66	34	]	]	PUNCT
ejde-741	66	35	,	,	PUNCT
ejde-741	66	36	√	√	PROPN
ejde-741	66	37	aux	aux	PROPN
ejde-741	66	38	∈	∈	PROPN
ejde-741	66	39	l2(0	l2(0	NOUN
ejde-741	66	40	,	,	PUNCT
ejde-741	66	41	1	1	NUM
ejde-741	66	42	)	)	PUNCT
ejde-741	66	43	,	,	PUNCT
ejde-741	66	44	and	and	CCONJ
ejde-741	66	45	u(1	u(1	PROPN
ejde-741	66	46	)	)	PUNCT
ejde-741	66	47	=	=	SYM
ejde-741	66	48	u(0	u(0	NOUN
ejde-741	66	49	)	)	PUNCT
ejde-741	66	50	=	=	SYM
ejde-741	67	1	0	0	PUNCT
ejde-741	67	2	}	}	PUNCT
ejde-741	67	3	,	,	PUNCT
ejde-741	67	4	equipped	equip	VERB
ejde-741	67	5	with	with	ADP
ejde-741	67	6	the	the	DET
ejde-741	67	7	natural	natural	ADJ
ejde-741	67	8	norm	norm	NOUN
ejde-741	67	9	∥u∥h1	∥u∥h1	PROPN
ejde-741	67	10	a	a	PRON
ejde-741	67	11	:	:	PUNCT
ejde-741	67	12	=	=	SYM
ejde-741	67	13	(	(	PUNCT
ejde-741	67	14	∥u∥2l2(0,1	∥u∥2l2(0,1	NOUN
ejde-741	67	15	)	)	PUNCT
ejde-741	67	16	+	+	CCONJ
ejde-741	67	17	∥	∥	NOUN
ejde-741	67	18	√	√	NUM
ejde-741	67	19	aux∥2l2(0,1	aux∥2l2(0,1	NOUN
ejde-741	67	20	)	)	PUNCT
ejde-741	67	21	)	)	PUNCT
ejde-741	68	1	1/2	1/2	NUM
ejde-741	68	2	.	.	PUNCT
ejde-741	69	1	(	(	PUNCT
ejde-741	69	2	b	b	X
ejde-741	69	3	)	)	PUNCT
ejde-741	69	4	for	for	ADP
ejde-741	69	5	the	the	DET
ejde-741	69	6	(	(	PUNCT
ejde-741	69	7	sdc	sdc	NOUN
ejde-741	69	8	)	)	PUNCT
ejde-741	69	9	,	,	PUNCT
ejde-741	69	10	we	we	PRON
ejde-741	69	11	set	set	VERB
ejde-741	69	12	h1	h1	VERB
ejde-741	69	13	a	a	DET
ejde-741	69	14	:	:	PUNCT
ejde-741	69	15	=	=	SYM
ejde-741	69	16	{	{	PUNCT
ejde-741	69	17	u	u	PROPN
ejde-741	69	18	∈	∈	PROPN
ejde-741	69	19	l2(0	l2(0	NOUN
ejde-741	69	20	,	,	PUNCT
ejde-741	69	21	1	1	NUM
ejde-741	69	22	)	)	PUNCT
ejde-741	69	23	such	such	ADJ
ejde-741	69	24	that	that	SCONJ
ejde-741	69	25	u	u	NOUN
ejde-741	69	26	is	be	AUX
ejde-741	69	27	absolutely	absolutely	ADV
ejde-741	69	28	continuous	continuous	ADJ
ejde-741	69	29	in	in	ADP
ejde-741	69	30	(	(	PUNCT
ejde-741	69	31	0	0	NUM
ejde-741	69	32	,	,	PUNCT
ejde-741	69	33	1	1	NUM
ejde-741	69	34	]	]	PUNCT
ejde-741	69	35	,	,	PUNCT
ejde-741	69	36	√	√	PROPN
ejde-741	69	37	aux	aux	PROPN
ejde-741	69	38	∈	∈	PROPN
ejde-741	69	39	l2(0	l2(0	NOUN
ejde-741	69	40	,	,	PUNCT
ejde-741	69	41	1	1	NUM
ejde-741	69	42	)	)	PUNCT
ejde-741	69	43	,	,	PUNCT
ejde-741	69	44	and	and	CCONJ
ejde-741	69	45	u(1	u(1	PROPN
ejde-741	69	46	)	)	PUNCT
ejde-741	69	47	=	=	SYM
ejde-741	70	1	0	0	NUM
ejde-741	70	2	}	}	PUNCT
ejde-741	70	3	,	,	PUNCT
ejde-741	70	4	with	with	ADP
ejde-741	70	5	the	the	DET
ejde-741	70	6	same	same	ADJ
ejde-741	70	7	norm	norm	NOUN
ejde-741	70	8	taken	take	VERB
ejde-741	70	9	for	for	ADP
ejde-741	70	10	the	the	DET
ejde-741	70	11	(	(	PUNCT
ejde-741	70	12	wdc	wdc	PROPN
ejde-741	70	13	)	)	PUNCT
ejde-741	70	14	;	;	PUNCT
ejde-741	70	15	(	(	PUNCT
ejde-741	70	16	c	c	X
ejde-741	70	17	)	)	PUNCT
ejde-741	70	18	in	in	ADP
ejde-741	70	19	both	both	DET
ejde-741	70	20	situations	situation	NOUN
ejde-741	70	21	,	,	PUNCT
ejde-741	70	22	the	the	DET
ejde-741	70	23	(	(	PUNCT
ejde-741	70	24	wdc	wdc	PROPN
ejde-741	70	25	)	)	PUNCT
ejde-741	70	26	and	and	CCONJ
ejde-741	70	27	the	the	DET
ejde-741	70	28	(	(	PUNCT
ejde-741	70	29	sdc	sdc	NOUN
ejde-741	70	30	)	)	PUNCT
ejde-741	70	31	,	,	PUNCT
ejde-741	70	32	h2	h2	NOUN
ejde-741	70	33	a	a	PRON
ejde-741	70	34	:	:	PUNCT
ejde-741	70	35	=	=	SYM
ejde-741	70	36	{	{	PUNCT
ejde-741	70	37	u	u	NOUN
ejde-741	70	38	∈	∈	PROPN
ejde-741	70	39	h1	h1	VERB
ejde-741	70	40	a	a	DET
ejde-741	70	41	:	:	PUNCT
ejde-741	70	42	aux	aux	PROPN
ejde-741	70	43	∈	∈	PROPN
ejde-741	70	44	h1(0	h1(0	PROPN
ejde-741	70	45	,	,	PUNCT
ejde-741	70	46	1	1	NUM
ejde-741	70	47	)	)	PUNCT
ejde-741	70	48	}	}	PUNCT
ejde-741	70	49	with	with	ADP
ejde-741	70	50	the	the	DET
ejde-741	70	51	norm	norm	NOUN
ejde-741	70	52	∥u∥h2	∥u∥h2	PROPN
ejde-741	70	53	a	a	X
ejde-741	70	54	:	:	PUNCT
ejde-741	70	55	=	=	SYM
ejde-741	70	56	(	(	PUNCT
ejde-741	70	57	∥u∥2h1	∥u∥2h1	NUM
ejde-741	70	58	a	a	DET
ejde-741	70	59	+	+	X
ejde-741	70	60	∥(aux)x∥2l2(0,1	∥(aux)x∥2l2(0,1	NOUN
ejde-741	70	61	)	)	PUNCT
ejde-741	70	62	)	)	PUNCT
ejde-741	71	1	1/2	1/2	NUM
ejde-741	71	2	.	.	PUNCT
ejde-741	72	1	now	now	ADV
ejde-741	72	2	,	,	PUNCT
ejde-741	72	3	let	let	VERB
ejde-741	72	4	us	we	PRON
ejde-741	72	5	state	state	VERB
ejde-741	72	6	the	the	DET
ejde-741	72	7	properties	property	NOUN
ejde-741	72	8	of	of	ADP
ejde-741	72	9	the	the	DET
ejde-741	72	10	functions	function	NOUN
ejde-741	72	11	ℓ	ℓ	NOUN
ejde-741	72	12	:	:	PUNCT
ejde-741	73	1	r	r	NOUN
ejde-741	73	2	→	→	SYM
ejde-741	73	3	r	r	NOUN
ejde-741	73	4	and	and	CCONJ
ejde-741	73	5	f	f	NOUN
ejde-741	73	6	:	:	PUNCT
ejde-741	74	1	[	[	X
ejde-741	74	2	0	0	NUM
ejde-741	74	3	,	,	PUNCT
ejde-741	74	4	t	t	X
ejde-741	74	5	]	]	X
ejde-741	74	6	×	×	NOUN
ejde-741	75	1	[	[	X
ejde-741	75	2	0	0	NUM
ejde-741	75	3	,	,	PUNCT
ejde-741	75	4	1]×	1]×	NUM
ejde-741	75	5	r	r	NOUN
ejde-741	75	6	→	→	SYM
ejde-741	75	7	r	r	NOUN
ejde-741	75	8	,	,	PUNCT
ejde-741	75	9	both	both	PRON
ejde-741	75	10	mentioned	mention	VERB
ejde-741	75	11	in	in	ADP
ejde-741	75	12	(	(	PUNCT
ejde-741	75	13	1.1	1.1	NUM
ejde-741	75	14	)	)	PUNCT
ejde-741	75	15	.	.	PUNCT
ejde-741	76	1	assumption	assumption	NOUN
ejde-741	76	2	1.3	1.3	NUM
ejde-741	76	3	.	.	PUNCT
ejde-741	77	1	let	let	VERB
ejde-741	77	2	ℓ	ℓ	NOUN
ejde-741	77	3	:	:	PUNCT
ejde-741	78	1	r	r	NOUN
ejde-741	78	2	→	→	SYM
ejde-741	78	3	r	r	NOUN
ejde-741	78	4	be	be	AUX
ejde-741	78	5	a	a	DET
ejde-741	78	6	c1	c1	NOUN
ejde-741	78	7	function	function	NOUN
ejde-741	78	8	with	with	ADP
ejde-741	78	9	bounded	bounded	ADJ
ejde-741	78	10	derivative	derivative	NOUN
ejde-741	78	11	and	and	CCONJ
ejde-741	78	12	suppose	suppose	VERB
ejde-741	78	13	that	that	SCONJ
ejde-741	78	14	ℓ(0	ℓ(0	NOUN
ejde-741	78	15	)	)	PUNCT
ejde-741	78	16	=	=	SYM
ejde-741	79	1	1	1	X
ejde-741	79	2	.	.	X
ejde-741	79	3	we	we	PRON
ejde-741	79	4	should	should	AUX
ejde-741	79	5	observe	observe	VERB
ejde-741	79	6	that	that	SCONJ
ejde-741	79	7	our	our	PRON
ejde-741	79	8	results	result	NOUN
ejde-741	79	9	remain	remain	VERB
ejde-741	79	10	the	the	DET
ejde-741	79	11	same	same	ADJ
ejde-741	79	12	if	if	SCONJ
ejde-741	79	13	we	we	PRON
ejde-741	79	14	just	just	ADV
ejde-741	79	15	suppose	suppose	VERB
ejde-741	79	16	that	that	SCONJ
ejde-741	79	17	ℓ(0	ℓ(0	NOUN
ejde-741	79	18	)	)	PUNCT
ejde-741	79	19	>	>	X
ejde-741	79	20	0	0	X
ejde-741	79	21	.	.	PUNCT
ejde-741	79	22	assumption	assumption	NOUN
ejde-741	79	23	1.4	1.4	NUM
ejde-741	79	24	.	.	PUNCT
ejde-741	80	1	we	we	PRON
ejde-741	80	2	assume	assume	VERB
ejde-741	80	3	that	that	SCONJ
ejde-741	80	4	f	f	X
ejde-741	80	5	:	:	PUNCT
ejde-741	81	1	[	[	X
ejde-741	81	2	0	0	NUM
ejde-741	81	3	,	,	PUNCT
ejde-741	81	4	t	t	X
ejde-741	81	5	]	]	PUNCT
ejde-741	81	6	×	×	NOUN
ejde-741	82	1	[	[	X
ejde-741	82	2	0	0	NUM
ejde-741	82	3	,	,	PUNCT
ejde-741	82	4	1	1	NUM
ejde-741	82	5	]	]	SYM
ejde-741	82	6	×	×	NOUN
ejde-741	82	7	r	r	NOUN
ejde-741	82	8	→	→	SYM
ejde-741	82	9	r	r	NOUN
ejde-741	82	10	is	be	AUX
ejde-741	82	11	a	a	DET
ejde-741	82	12	c1	c1	NOUN
ejde-741	82	13	function	function	NOUN
ejde-741	82	14	,	,	PUNCT
ejde-741	82	15	with	with	ADP
ejde-741	82	16	bounded	bounded	ADJ
ejde-741	82	17	derivatives	derivative	NOUN
ejde-741	82	18	,	,	PUNCT
ejde-741	82	19	such	such	ADJ
ejde-741	82	20	that	that	SCONJ
ejde-741	82	21	f(t	f(t	NOUN
ejde-741	82	22	,	,	PUNCT
ejde-741	82	23	x	x	NOUN
ejde-741	82	24	,	,	PUNCT
ejde-741	82	25	0	0	NUM
ejde-741	82	26	)	)	PUNCT
ejde-741	82	27	≡	≡	PROPN
ejde-741	82	28	0	0	NUM
ejde-741	82	29	and	and	CCONJ
ejde-741	82	30	c(t	c(t	PROPN
ejde-741	82	31	,	,	PUNCT
ejde-741	82	32	x	x	NOUN
ejde-741	82	33	)	)	PUNCT
ejde-741	82	34	=	=	SYM
ejde-741	82	35	∂3f(t	∂3f(t	NOUN
ejde-741	82	36	,	,	PUNCT
ejde-741	82	37	x	x	X
ejde-741	82	38	,	,	PUNCT
ejde-741	82	39	0	0	NUM
ejde-741	82	40	)	)	PUNCT
ejde-741	82	41	belongs	belong	VERB
ejde-741	82	42	to	to	ADP
ejde-741	82	43	l∞(q	l∞(q	NOUN
ejde-741	82	44	)	)	PUNCT
ejde-741	82	45	,	,	PUNCT
ejde-741	82	46	where	where	SCONJ
ejde-741	82	47	(	(	PUNCT
ejde-741	82	48	t	t	PROPN
ejde-741	82	49	,	,	PUNCT
ejde-741	82	50	x	x	NOUN
ejde-741	82	51	)	)	PUNCT
ejde-741	82	52	∈	∈	PROPN
ejde-741	83	1	[	[	X
ejde-741	83	2	0	0	NUM
ejde-741	83	3	,	,	PUNCT
ejde-741	83	4	t	t	X
ejde-741	83	5	]	]	X
ejde-741	83	6	×	×	NOUN
ejde-741	84	1	[	[	X
ejde-741	84	2	0	0	NUM
ejde-741	84	3	,	,	PUNCT
ejde-741	84	4	1	1	NUM
ejde-741	84	5	]	]	PUNCT
ejde-741	84	6	.	.	PUNCT
ejde-741	85	1	to	to	PART
ejde-741	85	2	state	state	VERB
ejde-741	85	3	our	our	PRON
ejde-741	85	4	main	main	ADJ
ejde-741	85	5	result	result	NOUN
ejde-741	85	6	,	,	PUNCT
ejde-741	85	7	we	we	PRON
ejde-741	85	8	recall	recall	VERB
ejde-741	85	9	the	the	DET
ejde-741	85	10	important	important	ADJ
ejde-741	85	11	concept	concept	NOUN
ejde-741	85	12	below	below	ADP
ejde-741	85	13	:	:	PUNCT
ejde-741	85	14	4	4	NUM
ejde-741	85	15	p.	p.	NOUN
ejde-741	85	16	p.	p.	NOUN
ejde-741	85	17	de	de	PROPN
ejde-741	85	18	carvalho	carvalho	PROPN
ejde-741	85	19	,	,	PUNCT
ejde-741	85	20	r.	r.	PROPN
ejde-741	85	21	demarque	demarque	PROPN
ejde-741	85	22	,	,	PUNCT
ejde-741	85	23	j.	j.	PROPN
ejde-741	85	24	límaco	límaco	PROPN
ejde-741	85	25	,	,	PUNCT
ejde-741	85	26	l.	l.	PROPN
ejde-741	85	27	viana	viana	PROPN
ejde-741	85	28	ejde-2025/15	ejde-2025/15	PROPN
ejde-741	85	29	definition	definition	NOUN
ejde-741	85	30	1.5	1.5	NUM
ejde-741	85	31	.	.	PUNCT
ejde-741	86	1	system	system	NOUN
ejde-741	86	2	(	(	PUNCT
ejde-741	86	3	1.1	1.1	NUM
ejde-741	86	4	)	)	PUNCT
ejde-741	86	5	is	be	AUX
ejde-741	86	6	said	say	VERB
ejde-741	86	7	locally	locally	ADV
ejde-741	86	8	null	null	NOUN
ejde-741	86	9	-	-	PUNCT
ejde-741	86	10	controllable	controllable	ADJ
ejde-741	86	11	at	at	ADP
ejde-741	86	12	a	a	DET
ejde-741	86	13	given	give	VERB
ejde-741	86	14	time	time	NOUN
ejde-741	86	15	t	t	PROPN
ejde-741	86	16	>	>	X
ejde-741	86	17	0	0	PUNCT
ejde-741	87	1	if	if	SCONJ
ejde-741	87	2	there	there	PRON
ejde-741	87	3	exists	exist	VERB
ejde-741	87	4	ε	ε	PROPN
ejde-741	87	5	>	>	X
ejde-741	87	6	0	0	PUNCT
ejde-741	87	7	with	with	ADP
ejde-741	87	8	the	the	DET
ejde-741	87	9	following	follow	VERB
ejde-741	87	10	property	property	NOUN
ejde-741	87	11	:	:	PUNCT
ejde-741	87	12	whenever	whenever	SCONJ
ejde-741	87	13	u0	u0	PROPN
ejde-741	87	14	∈	∈	PROPN
ejde-741	87	15	h1	h1	VERB
ejde-741	87	16	a	a	PRON
ejde-741	87	17	and	and	CCONJ
ejde-741	87	18	∥u0∥h1	∥u0∥h1	PUNCT
ejde-741	87	19	a	a	DET
ejde-741	87	20	≤	≤	NUM
ejde-741	87	21	ε	ε	PROPN
ejde-741	87	22	,	,	PUNCT
ejde-741	87	23	we	we	PRON
ejde-741	87	24	can	can	AUX
ejde-741	87	25	find	find	VERB
ejde-741	87	26	a	a	DET
ejde-741	87	27	control	control	NOUN
ejde-741	87	28	function	function	NOUN
ejde-741	87	29	h	h	PROPN
ejde-741	87	30	∈	∈	PROPN
ejde-741	87	31	l2(qω	l2(qω	PROPN
ejde-741	87	32	)	)	PUNCT
ejde-741	87	33	,	,	PUNCT
ejde-741	87	34	associated	associate	VERB
ejde-741	87	35	with	with	ADP
ejde-741	87	36	a	a	DET
ejde-741	87	37	state	state	NOUN
ejde-741	87	38	u	u	NOUN
ejde-741	87	39	,	,	PUNCT
ejde-741	87	40	such	such	ADJ
ejde-741	87	41	that	that	SCONJ
ejde-741	87	42	u(t	u(t	NOUN
ejde-741	87	43	,	,	PUNCT
ejde-741	87	44	x	x	NOUN
ejde-741	87	45	)	)	PUNCT
ejde-741	87	46	=	=	SYM
ejde-741	87	47	0	0	NUM
ejde-741	87	48	,	,	PUNCT
ejde-741	87	49	for	for	ADP
ejde-741	87	50	every	every	DET
ejde-741	87	51	x	x	SYM
ejde-741	87	52	∈	∈	PROPN
ejde-741	88	1	[	[	X
ejde-741	88	2	0	0	NUM
ejde-741	88	3	,	,	PUNCT
ejde-741	88	4	1	1	NUM
ejde-741	88	5	]	]	PUNCT
ejde-741	88	6	.	.	PUNCT
ejde-741	89	1	having	have	VERB
ejde-741	89	2	in	in	ADP
ejde-741	89	3	mind	mind	NOUN
ejde-741	89	4	the	the	DET
ejde-741	89	5	considerations	consideration	NOUN
ejde-741	89	6	above	above	ADV
ejde-741	89	7	,	,	PUNCT
ejde-741	89	8	we	we	PRON
ejde-741	89	9	state	state	VERB
ejde-741	89	10	our	our	PRON
ejde-741	89	11	main	main	ADJ
ejde-741	89	12	result	result	NOUN
ejde-741	89	13	.	.	PUNCT
ejde-741	90	1	theorem	theorem	VERB
ejde-741	90	2	1.6	1.6	NUM
ejde-741	90	3	(	(	PUNCT
ejde-741	90	4	local	local	ADJ
ejde-741	90	5	null	null	NOUN
ejde-741	90	6	-	-	PUNCT
ejde-741	90	7	controllability	controllability	NOUN
ejde-741	90	8	)	)	PUNCT
ejde-741	90	9	.	.	PUNCT
ejde-741	91	1	under	under	ADP
ejde-741	91	2	assumptions	assumption	NOUN
ejde-741	91	3	1.1	1.1	NUM
ejde-741	91	4	,	,	PUNCT
ejde-741	91	5	1.3	1.3	NUM
ejde-741	91	6	and	and	CCONJ
ejde-741	91	7	1.4	1.4	NUM
ejde-741	91	8	,	,	PUNCT
ejde-741	91	9	the	the	DET
ejde-741	91	10	nonlinear	nonlinear	ADJ
ejde-741	91	11	system	system	NOUN
ejde-741	91	12	(	(	PUNCT
ejde-741	91	13	1.1	1.1	NUM
ejde-741	91	14	)	)	PUNCT
ejde-741	91	15	is	be	AUX
ejde-741	91	16	locally	locally	ADV
ejde-741	91	17	null	null	ADJ
ejde-741	91	18	-	-	PUNCT
ejde-741	91	19	controllable	controllable	NOUN
ejde-741	91	20	at	at	ADP
ejde-741	91	21	any	any	DET
ejde-741	91	22	time	time	NOUN
ejde-741	91	23	t	t	X
ejde-741	91	24	>	>	X
ejde-741	91	25	0	0	NUM
ejde-741	91	26	,	,	PUNCT
ejde-741	91	27	in	in	ADP
ejde-741	91	28	the	the	DET
ejde-741	91	29	sense	sense	NOUN
ejde-741	91	30	of	of	ADP
ejde-741	91	31	definition	definition	NOUN
ejde-741	91	32	1.5	1.5	NUM
ejde-741	91	33	,	,	PUNCT
ejde-741	91	34	provided	provide	VERB
ejde-741	91	35	that	that	SCONJ
ejde-741	91	36	one	one	NUM
ejde-741	91	37	of	of	ADP
ejde-741	91	38	the	the	DET
ejde-741	91	39	following	follow	VERB
ejde-741	91	40	conditions	condition	NOUN
ejde-741	91	41	holds	hold	VERB
ejde-741	91	42	:	:	PUNCT
ejde-741	91	43	(	(	PUNCT
ejde-741	91	44	a	a	X
ejde-741	91	45	)	)	PUNCT
ejde-741	91	46	k	k	PROPN
ejde-741	91	47	̸=	̸=	PROPN
ejde-741	91	48	1	1	NUM
ejde-741	91	49	;	;	PUNCT
ejde-741	91	50	(	(	PUNCT
ejde-741	91	51	b	b	X
ejde-741	91	52	)	)	PUNCT
ejde-741	91	53	k	k	NOUN
ejde-741	92	1	=	=	SYM
ejde-741	92	2	1	1	NUM
ejde-741	92	3	and	and	CCONJ
ejde-741	92	4	θ	θ	PROPN
ejde-741	92	5	≥	≥	NUM
ejde-741	92	6	1/2	1/2	NUM
ejde-741	92	7	.	.	PUNCT
ejde-741	93	1	conditions	condition	NOUN
ejde-741	93	2	(	(	PUNCT
ejde-741	93	3	a	a	X
ejde-741	93	4	)	)	PUNCT
ejde-741	93	5	and	and	CCONJ
ejde-741	93	6	(	(	PUNCT
ejde-741	93	7	b	b	NOUN
ejde-741	93	8	)	)	PUNCT
ejde-741	93	9	in	in	ADP
ejde-741	93	10	theorem	theorem	ADJ
ejde-741	93	11	1.6	1.6	NUM
ejde-741	93	12	are	be	AUX
ejde-741	93	13	both	both	ADV
ejde-741	93	14	sufficient	sufficient	ADJ
ejde-741	93	15	to	to	PART
ejde-741	93	16	assure	assure	VERB
ejde-741	93	17	that	that	SCONJ
ejde-741	93	18	au	au	PROPN
ejde-741	93	19	∈	∈	PROPN
ejde-741	93	20	l∞(0	l∞(0	PRON
ejde-741	93	21	,	,	PUNCT
ejde-741	93	22	1	1	NUM
ejde-741	93	23	)	)	PUNCT
ejde-741	93	24	,	,	PUNCT
ejde-741	93	25	for	for	SCONJ
ejde-741	93	26	any	any	DET
ejde-741	93	27	u	u	PROPN
ejde-741	93	28	∈	∈	PROPN
ejde-741	93	29	h1	h1	VERB
ejde-741	93	30	a	a	PRON
ejde-741	93	31	.	.	PUNCT
ejde-741	94	1	this	this	PRON
ejde-741	94	2	will	will	AUX
ejde-741	94	3	play	play	VERB
ejde-741	94	4	a	a	DET
ejde-741	94	5	very	very	ADV
ejde-741	94	6	important	important	ADJ
ejde-741	94	7	role	role	NOUN
ejde-741	94	8	in	in	ADP
ejde-741	94	9	the	the	DET
ejde-741	94	10	proofs	proof	NOUN
ejde-741	94	11	of	of	ADP
ejde-741	94	12	lemma	lemma	PROPN
ejde-741	94	13	3.2	3.2	NUM
ejde-741	94	14	and	and	CCONJ
ejde-741	94	15	proposition	proposition	NOUN
ejde-741	94	16	3.4	3.4	NUM
ejde-741	94	17	.	.	PUNCT
ejde-741	95	1	a	a	DET
ejde-741	95	2	complete	complete	ADJ
ejde-741	95	3	explanation	explanation	NOUN
ejde-741	95	4	about	about	ADP
ejde-741	95	5	this	this	PRON
ejde-741	95	6	will	will	AUX
ejde-741	95	7	be	be	AUX
ejde-741	95	8	given	give	VERB
ejde-741	95	9	in	in	ADP
ejde-741	95	10	appendix	appendix	NOUN
ejde-741	95	11	5	5	NUM
ejde-741	95	12	.	.	PUNCT
ejde-741	96	1	the	the	DET
ejde-741	96	2	remainder	remainder	NOUN
ejde-741	96	3	of	of	ADP
ejde-741	96	4	this	this	DET
ejde-741	96	5	paper	paper	NOUN
ejde-741	96	6	is	be	AUX
ejde-741	96	7	organized	organize	VERB
ejde-741	96	8	as	as	SCONJ
ejde-741	96	9	follows	follow	VERB
ejde-741	96	10	:	:	PUNCT
ejde-741	96	11	in	in	ADP
ejde-741	96	12	section	section	NOUN
ejde-741	96	13	2	2	NUM
ejde-741	96	14	,	,	PUNCT
ejde-741	96	15	we	we	PRON
ejde-741	96	16	present	present	VERB
ejde-741	96	17	useful	useful	ADJ
ejde-741	96	18	notation	notation	NOUN
ejde-741	96	19	and	and	CCONJ
ejde-741	96	20	preliminary	preliminary	ADJ
ejde-741	96	21	results	result	NOUN
ejde-741	96	22	.	.	PUNCT
ejde-741	97	1	the	the	DET
ejde-741	97	2	first	first	ADJ
ejde-741	97	3	part	part	NOUN
ejde-741	97	4	brings	bring	VERB
ejde-741	97	5	some	some	DET
ejde-741	97	6	explanation	explanation	NOUN
ejde-741	97	7	about	about	ADP
ejde-741	97	8	a	a	DET
ejde-741	97	9	local	local	ADJ
ejde-741	97	10	inversion	inversion	NOUN
ejde-741	97	11	argument	argument	NOUN
ejde-741	97	12	,	,	PUNCT
ejde-741	97	13	while	while	SCONJ
ejde-741	97	14	the	the	DET
ejde-741	97	15	second	second	ADJ
ejde-741	97	16	one	one	NOUN
ejde-741	97	17	is	be	AUX
ejde-741	97	18	concerned	concern	VERB
ejde-741	97	19	with	with	ADP
ejde-741	97	20	carleman	carleman	ADJ
ejde-741	97	21	and	and	CCONJ
ejde-741	97	22	observability	observability	NOUN
ejde-741	97	23	estimates	estimate	NOUN
ejde-741	97	24	,	,	PUNCT
ejde-741	97	25	valid	valid	ADJ
ejde-741	97	26	for	for	ADP
ejde-741	97	27	both	both	CCONJ
ejde-741	97	28	the	the	DET
ejde-741	97	29	(	(	PUNCT
ejde-741	97	30	wdc	wdc	PROPN
ejde-741	97	31	)	)	PUNCT
ejde-741	97	32	and	and	CCONJ
ejde-741	97	33	the	the	DET
ejde-741	97	34	(	(	PUNCT
ejde-741	97	35	sdc	sdc	NOUN
ejde-741	97	36	)	)	PUNCT
ejde-741	97	37	.	.	PUNCT
ejde-741	98	1	in	in	ADP
ejde-741	98	2	section	section	NOUN
ejde-741	98	3	3	3	NUM
ejde-741	98	4	,	,	PUNCT
ejde-741	98	5	we	we	PRON
ejde-741	98	6	verify	verify	VERB
ejde-741	98	7	some	some	DET
ejde-741	98	8	properties	property	NOUN
ejde-741	98	9	of	of	ADP
ejde-741	98	10	a	a	DET
ejde-741	98	11	mapping	mapping	NOUN
ejde-741	98	12	h	h	NOUN
ejde-741	98	13	:	:	PUNCT
ejde-741	98	14	e	e	X
ejde-741	98	15	→	→	SYM
ejde-741	98	16	f	f	PROPN
ejde-741	98	17	,	,	PUNCT
ejde-741	98	18	set	set	VERB
ejde-741	98	19	in	in	ADP
ejde-741	98	20	(	(	PUNCT
ejde-741	98	21	2.1	2.1	NUM
ejde-741	98	22	)	)	PUNCT
ejde-741	98	23	,	,	PUNCT
ejde-741	98	24	which	which	PRON
ejde-741	98	25	will	will	AUX
ejde-741	98	26	allow	allow	VERB
ejde-741	98	27	us	we	PRON
ejde-741	98	28	to	to	PART
ejde-741	98	29	apply	apply	VERB
ejde-741	98	30	lyusternik	lyusternik	PROPN
ejde-741	98	31	’s	’s	PART
ejde-741	98	32	theorem	theorem	NOUN
ejde-741	98	33	(	(	PUNCT
ejde-741	98	34	stated	state	VERB
ejde-741	98	35	as	as	ADP
ejde-741	98	36	theorem	theorem	VERB
ejde-741	98	37	2.1	2.1	NUM
ejde-741	98	38	)	)	PUNCT
ejde-741	98	39	to	to	PART
ejde-741	98	40	achieve	achieve	VERB
ejde-741	98	41	the	the	DET
ejde-741	98	42	local	local	ADJ
ejde-741	98	43	null	null	NOUN
ejde-741	98	44	-	-	PUNCT
ejde-741	98	45	controllability	controllability	NOUN
ejde-741	98	46	of	of	ADP
ejde-741	98	47	(	(	PUNCT
ejde-741	98	48	1.1	1.1	NUM
ejde-741	98	49	)	)	PUNCT
ejde-741	98	50	.	.	PUNCT
ejde-741	99	1	at	at	ADP
ejde-741	99	2	this	this	DET
ejde-741	99	3	point	point	NOUN
ejde-741	99	4	,	,	PUNCT
ejde-741	99	5	the	the	DET
ejde-741	99	6	key	key	ADJ
ejde-741	99	7	information	information	NOUN
ejde-741	99	8	comes	come	VERB
ejde-741	99	9	from	from	ADP
ejde-741	99	10	the	the	DET
ejde-741	99	11	global	global	ADJ
ejde-741	99	12	-	-	PUNCT
ejde-741	99	13	null	null	ADJ
ejde-741	99	14	controllability	controllability	NOUN
ejde-741	99	15	of	of	ADP
ejde-741	99	16	the	the	DET
ejde-741	99	17	linearization	linearization	NOUN
ejde-741	99	18	of	of	ADP
ejde-741	99	19	(	(	PUNCT
ejde-741	99	20	1.1	1.1	NUM
ejde-741	99	21	)	)	PUNCT
ejde-741	99	22	,	,	PUNCT
ejde-741	99	23	given	give	VERB
ejde-741	99	24	in	in	ADP
ejde-741	99	25	(	(	PUNCT
ejde-741	99	26	2.2	2.2	NUM
ejde-741	99	27	)	)	PUNCT
ejde-741	99	28	,	,	PUNCT
ejde-741	99	29	as	as	ADV
ejde-741	99	30	well	well	ADV
ejde-741	99	31	as	as	ADP
ejde-741	99	32	from	from	ADP
ejde-741	99	33	some	some	DET
ejde-741	99	34	additional	additional	ADJ
ejde-741	99	35	regularity	regularity	NOUN
ejde-741	99	36	results	result	NOUN
ejde-741	99	37	.	.	PUNCT
ejde-741	100	1	in	in	ADP
ejde-741	100	2	section	section	NOUN
ejde-741	100	3	4	4	NUM
ejde-741	100	4	,	,	PUNCT
ejde-741	100	5	we	we	PRON
ejde-741	100	6	prove	prove	VERB
ejde-741	100	7	the	the	DET
ejde-741	100	8	main	main	ADJ
ejde-741	100	9	result	result	NOUN
ejde-741	100	10	of	of	ADP
ejde-741	100	11	this	this	DET
ejde-741	100	12	paper	paper	NOUN
ejde-741	100	13	(	(	PUNCT
ejde-741	100	14	theorem	theorem	VERB
ejde-741	100	15	1.6	1.6	NUM
ejde-741	100	16	)	)	PUNCT
ejde-741	100	17	,	,	PUNCT
ejde-741	100	18	where	where	SCONJ
ejde-741	100	19	the	the	DET
ejde-741	100	20	local	local	ADJ
ejde-741	100	21	null	null	NOUN
ejde-741	100	22	-	-	PUNCT
ejde-741	100	23	controllability	controllability	NOUN
ejde-741	100	24	of	of	ADP
ejde-741	100	25	(	(	PUNCT
ejde-741	100	26	1.1	1.1	NUM
ejde-741	100	27	)	)	PUNCT
ejde-741	100	28	is	be	AUX
ejde-741	100	29	obtained	obtain	VERB
ejde-741	100	30	.	.	PUNCT
ejde-741	101	1	additionally	additionally	ADV
ejde-741	101	2	,	,	PUNCT
ejde-741	101	3	we	we	PRON
ejde-741	101	4	include	include	VERB
ejde-741	101	5	some	some	DET
ejde-741	101	6	further	further	ADJ
ejde-741	101	7	comments	comment	NOUN
ejde-741	101	8	related	relate	VERB
ejde-741	101	9	possible	possible	ADJ
ejde-741	101	10	future	future	ADJ
ejde-741	101	11	directions	direction	NOUN
ejde-741	101	12	.	.	PUNCT
ejde-741	102	1	the	the	DET
ejde-741	102	2	last	last	ADJ
ejde-741	102	3	section	section	NOUN
ejde-741	102	4	is	be	AUX
ejde-741	102	5	appendix	appendix	ADJ
ejde-741	102	6	5	5	NUM
ejde-741	102	7	,	,	PUNCT
ejde-741	102	8	which	which	PRON
ejde-741	102	9	complements	complement	VERB
ejde-741	102	10	the	the	DET
ejde-741	102	11	content	content	NOUN
ejde-741	102	12	studied	study	VERB
ejde-741	102	13	in	in	ADP
ejde-741	102	14	section	section	NOUN
ejde-741	102	15	3	3	NUM
ejde-741	102	16	.	.	NOUN
ejde-741	102	17	2	2	NUM
ejde-741	102	18	.	.	X
ejde-741	102	19	preliminary	preliminary	ADJ
ejde-741	102	20	results	result	NOUN
ejde-741	102	21	in	in	ADP
ejde-741	102	22	this	this	DET
ejde-741	102	23	section	section	NOUN
ejde-741	102	24	,	,	PUNCT
ejde-741	102	25	we	we	PRON
ejde-741	102	26	introduce	introduce	VERB
ejde-741	102	27	some	some	DET
ejde-741	102	28	notation	notation	NOUN
ejde-741	102	29	and	and	CCONJ
ejde-741	102	30	useful	useful	ADJ
ejde-741	102	31	auxiliary	auxiliary	ADJ
ejde-741	102	32	results	result	NOUN
ejde-741	102	33	,	,	PUNCT
ejde-741	102	34	which	which	PRON
ejde-741	102	35	will	will	AUX
ejde-741	102	36	help	help	VERB
ejde-741	102	37	us	we	PRON
ejde-741	102	38	to	to	PART
ejde-741	102	39	prove	prove	VERB
ejde-741	102	40	our	our	PRON
ejde-741	102	41	main	main	ADJ
ejde-741	102	42	result	result	NOUN
ejde-741	102	43	.	.	PUNCT
ejde-741	103	1	2.1	2.1	NUM
ejde-741	103	2	.	.	PUNCT
ejde-741	103	3	notation	notation	NOUN
ejde-741	103	4	and	and	CCONJ
ejde-741	103	5	results	result	NOUN
ejde-741	103	6	related	relate	VERB
ejde-741	103	7	to	to	ADP
ejde-741	103	8	the	the	DET
ejde-741	103	9	local	local	ADJ
ejde-741	103	10	inversion	inversion	NOUN
ejde-741	103	11	argument	argument	NOUN
ejde-741	103	12	.	.	PUNCT
ejde-741	104	1	the	the	DET
ejde-741	104	2	first	first	ADJ
ejde-741	104	3	part	part	NOUN
ejde-741	104	4	of	of	ADP
ejde-741	104	5	this	this	DET
ejde-741	104	6	section	section	NOUN
ejde-741	104	7	is	be	AUX
ejde-741	104	8	devoted	devote	VERB
ejde-741	104	9	to	to	ADP
ejde-741	104	10	a	a	DET
ejde-741	104	11	brief	brief	ADJ
ejde-741	104	12	explanation	explanation	NOUN
ejde-741	104	13	about	about	ADP
ejde-741	104	14	the	the	DET
ejde-741	104	15	most	most	ADV
ejde-741	104	16	important	important	ADJ
ejde-741	104	17	strategies	strategy	NOUN
ejde-741	104	18	for	for	ADP
ejde-741	104	19	proving	prove	VERB
ejde-741	104	20	our	our	PRON
ejde-741	104	21	main	main	ADJ
ejde-741	104	22	results	result	NOUN
ejde-741	104	23	.	.	PUNCT
ejde-741	105	1	our	our	PRON
ejde-741	105	2	approach	approach	NOUN
ejde-741	105	3	relies	rely	VERB
ejde-741	105	4	on	on	ADP
ejde-741	105	5	a	a	DET
ejde-741	105	6	version	version	NOUN
ejde-741	105	7	of	of	ADP
ejde-741	105	8	lyusternik	lyusternik	PROPN
ejde-741	105	9	’s	’s	PART
ejde-741	105	10	inverse	inverse	NOUN
ejde-741	105	11	mapping	mapping	NOUN
ejde-741	105	12	theorem	theorem	NOUN
ejde-741	105	13	(	(	PUNCT
ejde-741	105	14	see	see	VERB
ejde-741	105	15	[	[	X
ejde-741	105	16	2	2	NUM
ejde-741	105	17	,	,	PUNCT
ejde-741	105	18	30	30	NUM
ejde-741	105	19	]	]	PUNCT
ejde-741	105	20	,	,	PUNCT
ejde-741	105	21	for	for	ADP
ejde-741	105	22	instance	instance	NOUN
ejde-741	105	23	)	)	PUNCT
ejde-741	105	24	,	,	PUNCT
ejde-741	105	25	whose	whose	DET
ejde-741	105	26	statement	statement	NOUN
ejde-741	105	27	is	be	AUX
ejde-741	105	28	given	give	VERB
ejde-741	105	29	below	below	ADV
ejde-741	105	30	.	.	PUNCT
ejde-741	106	1	theorem	theorem	VERB
ejde-741	106	2	2.1	2.1	NUM
ejde-741	106	3	(	(	PUNCT
ejde-741	106	4	lyusternik	lyusternik	PROPN
ejde-741	106	5	)	)	PUNCT
ejde-741	106	6	.	.	PUNCT
ejde-741	107	1	let	let	VERB
ejde-741	107	2	e	e	NOUN
ejde-741	107	3	and	and	CCONJ
ejde-741	107	4	f	f	PROPN
ejde-741	107	5	be	be	AUX
ejde-741	107	6	two	two	NUM
ejde-741	107	7	banach	banach	NOUN
ejde-741	107	8	spaces	space	NOUN
ejde-741	107	9	,	,	PUNCT
ejde-741	107	10	consider	consider	VERB
ejde-741	107	11	h	h	NOUN
ejde-741	107	12	∈	∈	PROPN
ejde-741	108	1	c1(e	c1(e	PROPN
ejde-741	108	2	,	,	PUNCT
ejde-741	108	3	f	f	PROPN
ejde-741	108	4	)	)	PUNCT
ejde-741	108	5	and	and	CCONJ
ejde-741	108	6	put	put	VERB
ejde-741	108	7	η0	η0	NOUN
ejde-741	108	8	=	=	PUNCT
ejde-741	108	9	h(0	h(0	PROPN
ejde-741	108	10	)	)	PUNCT
ejde-741	108	11	.	.	PUNCT
ejde-741	109	1	if	if	SCONJ
ejde-741	109	2	h	h	NOUN
ejde-741	109	3	′(0	′(0	NOUN
ejde-741	109	4	)	)	PUNCT
ejde-741	109	5	∈	∈	PROPN
ejde-741	109	6	l(e	l(e	NOUN
ejde-741	109	7	,	,	PUNCT
ejde-741	109	8	f	f	PROPN
ejde-741	109	9	)	)	PUNCT
ejde-741	109	10	is	be	AUX
ejde-741	109	11	onto	onto	ADP
ejde-741	109	12	,	,	PUNCT
ejde-741	109	13	then	then	ADV
ejde-741	109	14	there	there	PRON
ejde-741	109	15	exist	exist	VERB
ejde-741	109	16	r	r	NOUN
ejde-741	109	17	>	>	X
ejde-741	109	18	0	0	PUNCT
ejde-741	110	1	and	and	CCONJ
ejde-741	110	2	h̃	h̃	PROPN
ejde-741	110	3	:	:	PUNCT
ejde-741	111	1	br(η0	br(η0	X
ejde-741	111	2	)	)	PUNCT
ejde-741	111	3	⊂	⊂	PROPN
ejde-741	111	4	f	f	X
ejde-741	111	5	→	→	PUNCT
ejde-741	111	6	e	e	X
ejde-741	111	7	such	such	ADJ
ejde-741	111	8	that	that	DET
ejde-741	111	9	h(h̃(ξ	h(h̃(ξ	NOUN
ejde-741	111	10	)	)	PUNCT
ejde-741	111	11	)	)	PUNCT
ejde-741	112	1	=	=	SYM
ejde-741	112	2	ξ	ξ	NOUN
ejde-741	112	3	,	,	PUNCT
ejde-741	112	4	∀ξ	∀ξ	ADJ
ejde-741	112	5	∈	∈	PROPN
ejde-741	112	6	br(η0	br(η0	NOUN
ejde-741	112	7	)	)	PUNCT
ejde-741	112	8	,	,	PUNCT
ejde-741	112	9	which	which	PRON
ejde-741	112	10	means	mean	VERB
ejde-741	112	11	that	that	SCONJ
ejde-741	112	12	h̃	h̃	PROPN
ejde-741	112	13	is	be	AUX
ejde-741	112	14	a	a	DET
ejde-741	112	15	right	right	ADJ
ejde-741	112	16	inverse	inverse	NOUN
ejde-741	112	17	of	of	ADP
ejde-741	112	18	h	h	NOUN
ejde-741	112	19	in	in	ADP
ejde-741	112	20	br(η0	br(η0	NOUN
ejde-741	112	21	)	)	PUNCT
ejde-741	112	22	.	.	PUNCT
ejde-741	113	1	in	in	ADP
ejde-741	113	2	addition	addition	NOUN
ejde-741	113	3	,	,	PUNCT
ejde-741	113	4	there	there	PRON
ejde-741	113	5	exists	exist	VERB
ejde-741	113	6	k	k	PROPN
ejde-741	113	7	>	>	X
ejde-741	113	8	0	0	NUM
ejde-741	113	9	such	such	ADJ
ejde-741	113	10	that	that	SCONJ
ejde-741	113	11	∥h̃(ξ)∥e	∥h̃(ξ)∥e	PROPN
ejde-741	113	12	≤	≤	X
ejde-741	113	13	k∥ξ	k∥ξ	NOUN
ejde-741	113	14	−	−	PROPN
ejde-741	113	15	η0∥f	η0∥f	NOUN
ejde-741	113	16	,	,	PUNCT
ejde-741	113	17	∀ξ	∀ξ	NOUN
ejde-741	113	18	∈	∈	PROPN
ejde-741	113	19	br(η0	br(η0	NOUN
ejde-741	113	20	)	)	PUNCT
ejde-741	113	21	.	.	PUNCT
ejde-741	114	1	ejde-2025/15	ejde-2025/15	NOUN
ejde-741	114	2	null	null	ADJ
ejde-741	114	3	-	-	PUNCT
ejde-741	114	4	controllability	controllability	NOUN
ejde-741	114	5	degenerate	degenerate	ADJ
ejde-741	114	6	quasilinear	quasilinear	NOUN
ejde-741	114	7	equations	equation	NOUN
ejde-741	114	8	5	5	NUM
ejde-741	114	9	next	next	ADV
ejde-741	114	10	,	,	PUNCT
ejde-741	114	11	let	let	VERB
ejde-741	114	12	us	we	PRON
ejde-741	114	13	indicate	indicate	VERB
ejde-741	114	14	how	how	SCONJ
ejde-741	114	15	the	the	DET
ejde-741	114	16	proof	proof	NOUN
ejde-741	114	17	of	of	ADP
ejde-741	114	18	theorem	theorem	ADJ
ejde-741	114	19	1.6	1.6	NUM
ejde-741	114	20	can	can	AUX
ejde-741	114	21	be	be	AUX
ejde-741	114	22	seen	see	VERB
ejde-741	114	23	as	as	ADP
ejde-741	114	24	an	an	DET
ejde-741	114	25	application	application	NOUN
ejde-741	114	26	of	of	ADP
ejde-741	114	27	theorem	theorem	NOUN
ejde-741	114	28	2.1	2.1	NUM
ejde-741	114	29	.	.	PUNCT
ejde-741	115	1	even	even	ADV
ejde-741	115	2	though	though	SCONJ
ejde-741	115	3	we	we	PRON
ejde-741	115	4	have	have	AUX
ejde-741	115	5	not	not	PART
ejde-741	115	6	set	set	VERB
ejde-741	115	7	the	the	DET
ejde-741	115	8	desired	desire	VERB
ejde-741	115	9	hilbert	hilbert	NOUN
ejde-741	115	10	spaces	space	NOUN
ejde-741	115	11	e	e	NOUN
ejde-741	115	12	and	and	CCONJ
ejde-741	115	13	f	f	PROPN
ejde-741	115	14	yet	yet	ADV
ejde-741	115	15	,	,	PUNCT
ejde-741	115	16	let	let	VERB
ejde-741	115	17	us	we	PRON
ejde-741	115	18	put	put	VERB
ejde-741	115	19	h(u	h(u	PROPN
ejde-741	115	20	,	,	PUNCT
ejde-741	115	21	h	h	NOUN
ejde-741	115	22	)	)	PUNCT
ejde-741	115	23	=	=	SYM
ejde-741	115	24	(	(	PUNCT
ejde-741	115	25	h1(u	h1(u	PROPN
ejde-741	115	26	,	,	PUNCT
ejde-741	115	27	h),h2(u	h),h2(u	PROPN
ejde-741	115	28	,	,	PUNCT
ejde-741	115	29	h	h	NOUN
ejde-741	115	30	)	)	PUNCT
ejde-741	115	31	)	)	PUNCT
ejde-741	115	32	,	,	PUNCT
ejde-741	115	33	(	(	PUNCT
ejde-741	115	34	2.1	2.1	NUM
ejde-741	115	35	)	)	PUNCT
ejde-741	116	1	where	where	SCONJ
ejde-741	116	2	h1(u	h1(u	PROPN
ejde-741	116	3	,	,	PUNCT
ejde-741	116	4	h	h	NOUN
ejde-741	116	5	)	)	PUNCT
ejde-741	116	6	:	:	PUNCT
ejde-741	117	1	=	=	PUNCT
ejde-741	117	2	ut	ut	PROPN
ejde-741	117	3	−	−	PROPN
ejde-741	117	4	ℓ(au)(aux)x	ℓ(au)(aux)x	NUM
ejde-741	117	5	+	+	NUM
ejde-741	117	6	f(t	f(t	NOUN
ejde-741	117	7	,	,	PUNCT
ejde-741	117	8	x	x	PRON
ejde-741	117	9	,	,	PUNCT
ejde-741	117	10	u)−	u)−	PROPN
ejde-741	117	11	hχω	hχω	PROPN
ejde-741	117	12	,	,	PUNCT
ejde-741	117	13	h2(u	h2(u	PROPN
ejde-741	117	14	,	,	PUNCT
ejde-741	117	15	h	h	NOUN
ejde-741	117	16	)	)	PUNCT
ejde-741	117	17	:	:	PUNCT
ejde-741	117	18	=	=	SYM
ejde-741	117	19	u(0	u(0	PROPN
ejde-741	117	20	,	,	PUNCT
ejde-741	117	21	·	·	PUNCT
ejde-741	117	22	)	)	PUNCT
ejde-741	117	23	.	.	PUNCT
ejde-741	118	1	notice	notice	VERB
ejde-741	118	2	that	that	SCONJ
ejde-741	118	3	for	for	ADP
ejde-741	118	4	u0	u0	ADJ
ejde-741	118	5	∈	∈	PROPN
ejde-741	118	6	h1	h1	PROPN
ejde-741	118	7	a	a	PRON
ejde-741	118	8	,	,	PUNCT
ejde-741	118	9	the	the	DET
ejde-741	118	10	first	first	ADJ
ejde-741	118	11	and	and	CCONJ
ejde-741	118	12	the	the	DET
ejde-741	118	13	second	second	ADJ
ejde-741	118	14	relations	relation	NOUN
ejde-741	118	15	in	in	ADP
ejde-741	118	16	(	(	PUNCT
ejde-741	118	17	1.1	1.1	NUM
ejde-741	118	18	)	)	PUNCT
ejde-741	118	19	are	be	AUX
ejde-741	118	20	satisfied	satisfied	ADJ
ejde-741	118	21	if	if	SCONJ
ejde-741	118	22	,	,	PUNCT
ejde-741	118	23	and	and	CCONJ
ejde-741	118	24	only	only	ADV
ejde-741	118	25	if	if	SCONJ
ejde-741	118	26	,	,	PUNCT
ejde-741	118	27	there	there	PRON
ejde-741	118	28	exists	exist	VERB
ejde-741	118	29	(	(	PUNCT
ejde-741	118	30	u	u	NOUN
ejde-741	118	31	,	,	PUNCT
ejde-741	118	32	h	h	NOUN
ejde-741	118	33	)	)	PUNCT
ejde-741	118	34	∈	∈	NOUN
ejde-741	118	35	e	e	NOUN
ejde-741	118	36	solving	solve	VERB
ejde-741	118	37	h(u	h(u	PROPN
ejde-741	118	38	,	,	PUNCT
ejde-741	118	39	h	h	NOUN
ejde-741	118	40	)	)	PUNCT
ejde-741	118	41	=	=	SYM
ejde-741	118	42	(	(	PUNCT
ejde-741	118	43	0	0	NUM
ejde-741	118	44	,	,	PUNCT
ejde-741	118	45	u0	u0	ADJ
ejde-741	118	46	)	)	PUNCT
ejde-741	118	47	.	.	PUNCT
ejde-741	119	1	from	from	ADP
ejde-741	119	2	this	this	DET
ejde-741	119	3	point	point	NOUN
ejde-741	119	4	,	,	PUNCT
ejde-741	119	5	we	we	PRON
ejde-741	119	6	realize	realize	VERB
ejde-741	119	7	that	that	SCONJ
ejde-741	119	8	,	,	PUNCT
ejde-741	119	9	among	among	ADP
ejde-741	119	10	other	other	ADJ
ejde-741	119	11	properties	property	NOUN
ejde-741	119	12	,	,	PUNCT
ejde-741	119	13	e	e	PROPN
ejde-741	119	14	and	and	CCONJ
ejde-741	119	15	f	f	PROPN
ejde-741	119	16	must	must	AUX
ejde-741	119	17	be	be	AUX
ejde-741	119	18	built	build	VERB
ejde-741	119	19	:	:	PUNCT
ejde-741	119	20	•	•	ADP
ejde-741	119	21	considering	consider	VERB
ejde-741	119	22	the	the	DET
ejde-741	119	23	boundary	boundary	ADJ
ejde-741	119	24	conditions	condition	NOUN
ejde-741	119	25	mentioned	mention	VERB
ejde-741	119	26	in	in	ADP
ejde-741	119	27	(	(	PUNCT
ejde-741	119	28	1.1	1.1	NUM
ejde-741	119	29	)	)	PUNCT
ejde-741	119	30	;	;	PUNCT
ejde-741	119	31	•	•	ADP
ejde-741	119	32	having	have	VERB
ejde-741	119	33	some	some	DET
ejde-741	119	34	imposition	imposition	NOUN
ejde-741	119	35	on	on	ADP
ejde-741	119	36	its	its	PRON
ejde-741	119	37	elements	element	NOUN
ejde-741	119	38	,	,	PUNCT
ejde-741	119	39	assuring	assure	VERB
ejde-741	119	40	that	that	SCONJ
ejde-741	119	41	u(t	u(t	NOUN
ejde-741	119	42	,	,	PUNCT
ejde-741	119	43	·	·	PUNCT
ejde-741	119	44	)	)	PUNCT
ejde-741	119	45	≡	≡	PROPN
ejde-741	119	46	0	0	NUM
ejde-741	119	47	.	.	PUNCT
ejde-741	120	1	it	it	PRON
ejde-741	120	2	will	will	AUX
ejde-741	120	3	be	be	AUX
ejde-741	120	4	done	do	VERB
ejde-741	120	5	having	have	VERB
ejde-741	120	6	in	in	ADP
ejde-741	120	7	mind	mind	NOUN
ejde-741	120	8	some	some	DET
ejde-741	120	9	weights	weight	NOUN
ejde-741	120	10	which	which	PRON
ejde-741	120	11	appear	appear	VERB
ejde-741	120	12	in	in	ADP
ejde-741	120	13	(	(	PUNCT
ejde-741	120	14	2.13	2.13	NUM
ejde-741	120	15	)	)	PUNCT
ejde-741	120	16	;	;	PUNCT
ejde-741	120	17	•	•	ADP
ejde-741	121	1	having	have	VERB
ejde-741	121	2	in	in	ADP
ejde-741	121	3	mind	mind	NOUN
ejde-741	121	4	that	that	SCONJ
ejde-741	121	5	we	we	PRON
ejde-741	121	6	want	want	VERB
ejde-741	121	7	h′(0	h′(0	NOUN
ejde-741	121	8	,	,	PUNCT
ejde-741	121	9	0	0	NUM
ejde-741	121	10	)	)	PUNCT
ejde-741	121	11	∈	∈	PROPN
ejde-741	122	1	l(e	l(e	NOUN
ejde-741	122	2	,	,	PUNCT
ejde-741	122	3	f	f	PROPN
ejde-741	122	4	)	)	PUNCT
ejde-741	122	5	to	to	PART
ejde-741	122	6	be	be	AUX
ejde-741	122	7	onto	onto	ADP
ejde-741	122	8	.	.	PUNCT
ejde-741	123	1	in	in	ADP
ejde-741	123	2	fact	fact	NOUN
ejde-741	123	3	,	,	PUNCT
ejde-741	123	4	it	it	PRON
ejde-741	123	5	is	be	AUX
ejde-741	123	6	equivalent	equivalent	ADJ
ejde-741	123	7	to	to	PART
ejde-741	123	8	say	say	VERB
ejde-741	123	9	that	that	SCONJ
ejde-741	123	10	,	,	PUNCT
ejde-741	123	11	given	give	VERB
ejde-741	123	12	any	any	DET
ejde-741	123	13	(	(	PUNCT
ejde-741	123	14	g	g	NOUN
ejde-741	123	15	,	,	PUNCT
ejde-741	123	16	u0	u0	ADJ
ejde-741	123	17	)	)	PUNCT
ejde-741	123	18	∈	∈	PROPN
ejde-741	123	19	f	f	PROPN
ejde-741	123	20	,	,	PUNCT
ejde-741	123	21	the	the	DET
ejde-741	123	22	linear	linear	ADJ
ejde-741	123	23	system	system	NOUN
ejde-741	123	24	ut	ut	PROPN
ejde-741	123	25	−	−	PROPN
ejde-741	123	26	(	(	PUNCT
ejde-741	123	27	a(x)ux)x	a(x)ux)x	VERB
ejde-741	123	28	+	+	CCONJ
ejde-741	123	29	c(t	c(t	PROPN
ejde-741	123	30	,	,	PUNCT
ejde-741	123	31	x)u	x)u	PUNCT
ejde-741	123	32	=	=	PUNCT
ejde-741	124	1	hχω	hχω	PROPN
ejde-741	124	2	+	+	CCONJ
ejde-741	124	3	g	g	PROPN
ejde-741	124	4	,	,	PUNCT
ejde-741	124	5	(	(	PUNCT
ejde-741	124	6	t	t	PROPN
ejde-741	124	7	,	,	PUNCT
ejde-741	124	8	x	x	X
ejde-741	124	9	)	)	PUNCT
ejde-741	124	10	∈	∈	PROPN
ejde-741	125	1	q	q	NOUN
ejde-741	125	2	;	;	PUNCT
ejde-741	125	3	u(t	u(t	NOUN
ejde-741	125	4	,	,	PUNCT
ejde-741	125	5	1	1	NUM
ejde-741	125	6	)	)	PUNCT
ejde-741	125	7	=	=	SYM
ejde-741	125	8	0	0	NUM
ejde-741	125	9	,	,	PUNCT
ejde-741	125	10	in	in	ADP
ejde-741	125	11	(	(	PUNCT
ejde-741	125	12	0	0	NUM
ejde-741	125	13	,	,	PUNCT
ejde-741	125	14	t	t	NOUN
ejde-741	125	15	)	)	PUNCT
ejde-741	125	16	,	,	PUNCT
ejde-741	125	17			PROPN
ejde-741	125	18	u(t	u(t	NOUN
ejde-741	125	19	,	,	PUNCT
ejde-741	125	20	0	0	NUM
ejde-741	125	21	)	)	PUNCT
ejde-741	125	22	=	=	SYM
ejde-741	125	23	0	0	NUM
ejde-741	125	24	,	,	PUNCT
ejde-741	125	25	(	(	PUNCT
ejde-741	125	26	weak	weak	ADJ
ejde-741	125	27	)	)	PUNCT
ejde-741	125	28	,	,	PUNCT
ejde-741	125	29	t	t	PROPN
ejde-741	125	30	∈	∈	PROPN
ejde-741	125	31	(	(	PUNCT
ejde-741	125	32	0	0	NUM
ejde-741	125	33	,	,	PUNCT
ejde-741	125	34	t	t	NOUN
ejde-741	125	35	)	)	PUNCT
ejde-741	125	36	or	or	CCONJ
ejde-741	125	37	(	(	PUNCT
ejde-741	125	38	aux)(t	aux)(t	NOUN
ejde-741	125	39	,	,	PUNCT
ejde-741	125	40	0	0	NUM
ejde-741	125	41	)	)	PUNCT
ejde-741	125	42	=	=	SYM
ejde-741	125	43	0	0	NUM
ejde-741	125	44	,	,	PUNCT
ejde-741	125	45	(	(	PUNCT
ejde-741	125	46	strong	strong	ADJ
ejde-741	125	47	)	)	PUNCT
ejde-741	125	48	,	,	PUNCT
ejde-741	125	49	t	t	PROPN
ejde-741	125	50	∈	∈	PROPN
ejde-741	125	51	(	(	PUNCT
ejde-741	125	52	0	0	NUM
ejde-741	125	53	,	,	PUNCT
ejde-741	125	54	t	t	NOUN
ejde-741	125	55	)	)	PUNCT
ejde-741	126	1	u(0	u(0	PROPN
ejde-741	126	2	,	,	PUNCT
ejde-741	126	3	x	x	NOUN
ejde-741	126	4	)	)	PUNCT
ejde-741	126	5	=	=	SYM
ejde-741	126	6	u0(x	u0(x	NUM
ejde-741	126	7	)	)	PUNCT
ejde-741	126	8	,	,	PUNCT
ejde-741	126	9	x	x	PUNCT
ejde-741	126	10	∈	∈	PROPN
ejde-741	126	11	(	(	PUNCT
ejde-741	126	12	0	0	NUM
ejde-741	126	13	,	,	PUNCT
ejde-741	126	14	1	1	NUM
ejde-741	126	15	)	)	PUNCT
ejde-741	126	16	,	,	PUNCT
ejde-741	126	17	(	(	PUNCT
ejde-741	126	18	2.2	2.2	NUM
ejde-741	126	19	)	)	PUNCT
ejde-741	126	20	is	be	AUX
ejde-741	126	21	globally	globally	ADV
ejde-741	126	22	null	null	ADJ
ejde-741	126	23	-	-	PUNCT
ejde-741	126	24	controllable	controllable	NOUN
ejde-741	126	25	at	at	ADP
ejde-741	126	26	the	the	DET
ejde-741	126	27	time	time	NOUN
ejde-741	126	28	t	t	PROPN
ejde-741	126	29	>	>	X
ejde-741	126	30	0	0	PROPN
ejde-741	126	31	,	,	PUNCT
ejde-741	126	32	where	where	SCONJ
ejde-741	126	33	h	h	PROPN
ejde-741	126	34	∈	∈	PROPN
ejde-741	126	35	l2(qω	l2(qω	PROPN
ejde-741	126	36	)	)	PUNCT
ejde-741	126	37	is	be	AUX
ejde-741	126	38	the	the	DET
ejde-741	126	39	control	control	NOUN
ejde-741	126	40	function	function	NOUN
ejde-741	126	41	and	and	CCONJ
ejde-741	126	42	a	a	DET
ejde-741	126	43	satisfies	satisfie	NOUN
ejde-741	126	44	assumption	assumption	NOUN
ejde-741	126	45	1.1	1.1	NUM
ejde-741	126	46	.	.	PUNCT
ejde-741	127	1	hence	hence	ADV
ejde-741	127	2	,	,	PUNCT
ejde-741	127	3	it	it	PRON
ejde-741	127	4	seems	seem	VERB
ejde-741	127	5	that	that	SCONJ
ejde-741	127	6	e	e	PRON
ejde-741	127	7	should	should	AUX
ejde-741	127	8	contain	contain	VERB
ejde-741	127	9	some	some	DET
ejde-741	127	10	information	information	NOUN
ejde-741	127	11	involving	involve	VERB
ejde-741	127	12	the	the	DET
ejde-741	127	13	well	well	NOUN
ejde-741	127	14	-	-	PUNCT
ejde-741	127	15	posedness	posedness	NOUN
ejde-741	127	16	(	(	PUNCT
ejde-741	127	17	and	and	CCONJ
ejde-741	127	18	additional	additional	ADJ
ejde-741	127	19	regularity	regularity	NOUN
ejde-741	127	20	)	)	PUNCT
ejde-741	127	21	of	of	ADP
ejde-741	127	22	the	the	DET
ejde-741	127	23	linear	linear	ADJ
ejde-741	127	24	system	system	NOUN
ejde-741	127	25	(	(	PUNCT
ejde-741	127	26	2.2	2.2	NUM
ejde-741	127	27	)	)	PUNCT
ejde-741	127	28	.	.	PUNCT
ejde-741	128	1	the	the	DET
ejde-741	128	2	well	well	ADV
ejde-741	128	3	-	-	PUNCT
ejde-741	128	4	posedness	posedness	NOUN
ejde-741	128	5	of	of	ADP
ejde-741	128	6	(	(	PUNCT
ejde-741	128	7	2.2	2.2	NUM
ejde-741	128	8	)	)	PUNCT
ejde-741	128	9	was	be	AUX
ejde-741	128	10	proved	prove	VERB
ejde-741	128	11	in	in	ADP
ejde-741	128	12	[	[	X
ejde-741	128	13	1	1	NUM
ejde-741	128	14	]	]	PUNCT
ejde-741	128	15	,	,	PUNCT
ejde-741	128	16	with	with	ADP
ejde-741	128	17	the	the	DET
ejde-741	128	18	following	follow	VERB
ejde-741	128	19	statement	statement	NOUN
ejde-741	128	20	.	.	PUNCT
ejde-741	129	1	proposition	proposition	NOUN
ejde-741	129	2	2.2	2.2	NUM
ejde-741	129	3	.	.	PUNCT
ejde-741	130	1	for	for	ADP
ejde-741	130	2	each	each	DET
ejde-741	130	3	g	g	PROPN
ejde-741	130	4	∈	∈	PROPN
ejde-741	130	5	l2(q	l2(q	PROPN
ejde-741	130	6	)	)	PUNCT
ejde-741	130	7	,	,	PUNCT
ejde-741	130	8	h	h	PROPN
ejde-741	130	9	∈	∈	PROPN
ejde-741	130	10	l2(qω	l2(qω	PROPN
ejde-741	130	11	)	)	PUNCT
ejde-741	130	12	and	and	CCONJ
ejde-741	130	13	u0	u0	PROPN
ejde-741	130	14	∈	∈	PROPN
ejde-741	130	15	l2(0	l2(0	NOUN
ejde-741	130	16	,	,	PUNCT
ejde-741	130	17	1	1	NUM
ejde-741	130	18	)	)	PUNCT
ejde-741	130	19	,	,	PUNCT
ejde-741	130	20	there	there	PRON
ejde-741	130	21	exists	exist	VERB
ejde-741	130	22	a	a	DET
ejde-741	130	23	unique	unique	ADJ
ejde-741	130	24	weak	weak	ADJ
ejde-741	130	25	solution	solution	NOUN
ejde-741	130	26	u	u	NOUN
ejde-741	130	27	∈	∈	PROPN
ejde-741	130	28	c0([0	c0([0	PROPN
ejde-741	130	29	,	,	PUNCT
ejde-741	130	30	t	t	X
ejde-741	130	31	]	]	PUNCT
ejde-741	130	32	;	;	PUNCT
ejde-741	130	33	l2(0	l2(0	NOUN
ejde-741	130	34	,	,	PUNCT
ejde-741	130	35	1	1	NUM
ejde-741	130	36	)	)	PUNCT
ejde-741	130	37	)	)	PUNCT
ejde-741	130	38	∩	∩	ADJ
ejde-741	130	39	l2(0	l2(0	NOUN
ejde-741	130	40	,	,	PUNCT
ejde-741	130	41	t	t	NOUN
ejde-741	130	42	;	;	PUNCT
ejde-741	130	43	h1	h1	VERB
ejde-741	130	44	a	a	PRON
ejde-741	130	45	)	)	PUNCT
ejde-741	130	46	of	of	ADP
ejde-741	130	47	(	(	PUNCT
ejde-741	130	48	2.2	2.2	NUM
ejde-741	130	49	)	)	PUNCT
ejde-741	130	50	.	.	PUNCT
ejde-741	131	1	moreover	moreover	ADV
ejde-741	131	2	,	,	PUNCT
ejde-741	131	3	if	if	SCONJ
ejde-741	131	4	u0	u0	PROPN
ejde-741	131	5	∈	∈	PROPN
ejde-741	131	6	h1	h1	VERB
ejde-741	131	7	a	a	DET
ejde-741	131	8	,	,	PUNCT
ejde-741	131	9	then	then	ADV
ejde-741	131	10	u	u	PROPN
ejde-741	131	11	∈	∈	PROPN
ejde-741	131	12	u	u	NOUN
ejde-741	131	13	:	:	PUNCT
ejde-741	131	14	=	=	SYM
ejde-741	131	15	h1(0	h1(0	PROPN
ejde-741	131	16	,	,	PUNCT
ejde-741	131	17	t	t	PROPN
ejde-741	131	18	;	;	PUNCT
ejde-741	131	19	l2(0	l2(0	NOUN
ejde-741	131	20	,	,	PUNCT
ejde-741	131	21	1	1	NUM
ejde-741	131	22	)	)	PUNCT
ejde-741	131	23	)	)	PUNCT
ejde-741	131	24	∩	∩	ADJ
ejde-741	131	25	l2(0	l2(0	NOUN
ejde-741	131	26	,	,	PUNCT
ejde-741	131	27	t	t	NOUN
ejde-741	131	28	;	;	PUNCT
ejde-741	131	29	h2	h2	PROPN
ejde-741	131	30	a	a	PRON
ejde-741	131	31	)	)	PUNCT
ejde-741	131	32	∩	∩	NOUN
ejde-741	131	33	c0([0	c0([0	PROPN
ejde-741	131	34	,	,	PUNCT
ejde-741	131	35	t	t	X
ejde-741	131	36	]	]	PUNCT
ejde-741	131	37	;	;	PUNCT
ejde-741	131	38	h1	h1	VERB
ejde-741	131	39	a	a	PRON
ejde-741	131	40	)	)	PUNCT
ejde-741	131	41	,	,	PUNCT
ejde-741	131	42	and	and	CCONJ
ejde-741	131	43	there	there	PRON
ejde-741	131	44	exists	exist	VERB
ejde-741	131	45	a	a	DET
ejde-741	131	46	constant	constant	ADJ
ejde-741	131	47	ct	ct	NOUN
ejde-741	131	48	>	>	X
ejde-741	131	49	0	0	NUM
ejde-741	132	1	such	such	ADJ
ejde-741	132	2	that	that	DET
ejde-741	132	3	sup	sup	NOUN
ejde-741	132	4	t∈[0,t	t∈[0,t	NOUN
ejde-741	132	5	]	]	PUNCT
ejde-741	132	6	(	(	PUNCT
ejde-741	132	7	∥u(t)∥2h1	∥u(t)∥2h1	NUM
ejde-741	132	8	a	a	X
ejde-741	132	9	)	)	PUNCT
ejde-741	133	1	+	+	CCONJ
ejde-741	133	2	∫	∫	PROPN
ejde-741	133	3	t	t	PROPN
ejde-741	133	4	0	0	NUM
ejde-741	133	5	(	(	PUNCT
ejde-741	133	6	∥ut|2l2(0,1	∥ut|2l2(0,1	NOUN
ejde-741	133	7	)	)	PUNCT
ejde-741	133	8	+	+	CCONJ
ejde-741	133	9	∥(aux)x∥2l2(0,1	∥(aux)x∥2l2(0,1	NOUN
ejde-741	133	10	)	)	PUNCT
ejde-741	133	11	)	)	PUNCT
ejde-741	134	1	≤	≤	NUM
ejde-741	134	2	ct	ct	INTJ
ejde-741	134	3	(	(	PUNCT
ejde-741	134	4	∥u0∥2h1	∥u0∥2h1	VERB
ejde-741	134	5	a	a	DET
ejde-741	134	6	+	+	NUM
ejde-741	134	7	∥g∥2l2(q	∥g∥2l2(q	NOUN
ejde-741	134	8	)	)	PUNCT
ejde-741	134	9	+	+	PUNCT
ejde-741	134	10	∥h∥2l2(qω	∥h∥2l2(qω	NUM
ejde-741	134	11	)	)	PUNCT
ejde-741	134	12	)	)	PUNCT
ejde-741	134	13	.	.	PUNCT
ejde-741	135	1	(	(	PUNCT
ejde-741	135	2	2.3	2.3	NUM
ejde-741	135	3	)	)	PUNCT
ejde-741	135	4	definition	definition	NOUN
ejde-741	135	5	2.3	2.3	NUM
ejde-741	135	6	.	.	PUNCT
ejde-741	136	1	let	let	VERB
ejde-741	136	2	δ	δ	PROPN
ejde-741	136	3	=	=	SYM
ejde-741	136	4	δ(t	δ(t	PROPN
ejde-741	136	5	,	,	PUNCT
ejde-741	136	6	x	x	NOUN
ejde-741	136	7	)	)	PUNCT
ejde-741	136	8	and	and	CCONJ
ejde-741	136	9	f	f	NOUN
ejde-741	136	10	=	=	PUNCT
ejde-741	136	11	f(t	f(t	PROPN
ejde-741	136	12	,	,	PUNCT
ejde-741	136	13	x	x	X
ejde-741	136	14	)	)	PUNCT
ejde-741	136	15	be	be	VERB
ejde-741	136	16	two	two	NUM
ejde-741	136	17	real	real	ADV
ejde-741	136	18	-	-	PUNCT
ejde-741	136	19	valued	value	VERB
ejde-741	136	20	measurable	measurable	ADJ
ejde-741	136	21	functions	function	NOUN
ejde-741	136	22	defined	define	VERB
ejde-741	136	23	in	in	ADP
ejde-741	136	24	q	q	NOUN
ejde-741	136	25	,	,	PUNCT
ejde-741	136	26	where	where	SCONJ
ejde-741	136	27	δ	δ	PROPN
ejde-741	136	28	is	be	AUX
ejde-741	136	29	non	non	ADJ
ejde-741	136	30	-	-	ADJ
ejde-741	136	31	negative	negative	ADJ
ejde-741	136	32	.	.	PUNCT
ejde-741	137	1	we	we	PRON
ejde-741	137	2	say	say	VERB
ejde-741	137	3	that	that	SCONJ
ejde-741	137	4	f	f	PROPN
ejde-741	137	5	belongs	belong	VERB
ejde-741	137	6	to	to	ADP
ejde-741	137	7	l2(q	l2(q	PROPN
ejde-741	137	8	;	;	PUNCT
ejde-741	137	9	δ	δ	X
ejde-741	137	10	)	)	PUNCT
ejde-741	137	11	if	if	SCONJ
ejde-741	137	12	√	√	ADV
ejde-741	137	13	δf	δf	VERB
ejde-741	137	14	∈	∈	PROPN
ejde-741	137	15	l2(q	l2(q	PROPN
ejde-741	137	16	)	)	PUNCT
ejde-741	137	17	.	.	PUNCT
ejde-741	138	1	moreover	moreover	ADV
ejde-741	138	2	,	,	PUNCT
ejde-741	138	3	the	the	DET
ejde-741	138	4	natural	natural	ADJ
ejde-741	138	5	norm	norm	NOUN
ejde-741	138	6	in	in	ADP
ejde-741	138	7	l2(q	l2(q	PROPN
ejde-741	138	8	;	;	PUNCT
ejde-741	138	9	δ	δ	PROPN
ejde-741	138	10	)	)	PUNCT
ejde-741	138	11	will	will	AUX
ejde-741	138	12	be	be	AUX
ejde-741	138	13	denoted	denote	VERB
ejde-741	138	14	by	by	ADP
ejde-741	138	15	∥	∥	X
ejde-741	139	1	·	·	PUNCT
ejde-741	139	2	∥δ	∥δ	NOUN
ejde-741	139	3	,	,	PUNCT
ejde-741	139	4	that	that	ADV
ejde-741	139	5	is	is	ADV
ejde-741	139	6	,	,	PUNCT
ejde-741	139	7	∥f∥δ	∥f∥δ	X
ejde-741	139	8	=	=	SYM
ejde-741	140	1	(	(	PUNCT
ejde-741	140	2	∫	∫	PROPN
ejde-741	140	3	t	t	PROPN
ejde-741	140	4	0	0	NUM
ejde-741	140	5	∫	∫	PROPN
ejde-741	140	6	1	1	NUM
ejde-741	140	7	0	0	NUM
ejde-741	140	8	δf2	δf2	NOUN
ejde-741	140	9	dx	dx	PROPN
ejde-741	140	10	dt	dt	PROPN
ejde-741	140	11	)	)	PUNCT
ejde-741	140	12	1/2	1/2	NUM
ejde-741	140	13	for	for	ADP
ejde-741	140	14	each	each	DET
ejde-741	140	15	f	f	PROPN
ejde-741	140	16	∈	∈	PROPN
ejde-741	140	17	l2(q	l2(q	PROPN
ejde-741	140	18	;	;	PUNCT
ejde-741	140	19	δ	δ	PROPN
ejde-741	140	20	)	)	PUNCT
ejde-741	140	21	.	.	PUNCT
ejde-741	141	1	6	6	NUM
ejde-741	141	2	p.	p.	NOUN
ejde-741	141	3	p.	p.	NOUN
ejde-741	141	4	de	de	PROPN
ejde-741	141	5	carvalho	carvalho	PROPN
ejde-741	141	6	,	,	PUNCT
ejde-741	141	7	r.	r.	PROPN
ejde-741	141	8	demarque	demarque	PROPN
ejde-741	141	9	,	,	PUNCT
ejde-741	141	10	j.	j.	PROPN
ejde-741	141	11	límaco	límaco	PROPN
ejde-741	141	12	,	,	PUNCT
ejde-741	141	13	l.	l.	PROPN
ejde-741	141	14	viana	viana	PROPN
ejde-741	141	15	ejde-2025/15	ejde-2025/15	PROPN
ejde-741	141	16	let	let	VERB
ejde-741	141	17	us	we	PRON
ejde-741	141	18	take	take	VERB
ejde-741	141	19	ω′	ω′	PUNCT
ejde-741	141	20	=	=	PUNCT
ejde-741	141	21	(	(	PUNCT
ejde-741	141	22	α′	α′	NUM
ejde-741	141	23	,	,	PUNCT
ejde-741	141	24	β′	β′	NUM
ejde-741	141	25	)	)	PUNCT
ejde-741	142	1	⊂⊂	⊂⊂	PROPN
ejde-741	142	2	ω	ω	NOUN
ejde-741	142	3	and	and	CCONJ
ejde-741	142	4	consider	consider	VERB
ejde-741	142	5	ψ	ψ	X
ejde-741	142	6	∈	∈	PROPN
ejde-741	142	7	c2([0	c2([0	PROPN
ejde-741	142	8	,	,	PUNCT
ejde-741	142	9	1];r	1];r	NUM
ejde-741	142	10	)	)	PUNCT
ejde-741	142	11	satisfying	satisfy	VERB
ejde-741	142	12	ψ(x	ψ(x	NOUN
ejde-741	142	13	)	)	PUNCT
ejde-741	142	14	:	:	PUNCT
ejde-741	143	1	=	=	SYM
ejde-741	143	2	{	{	PUNCT
ejde-741	143	3	∫	∫	PROPN
ejde-741	143	4	x	x	PROPN
ejde-741	143	5	0	0	PROPN
ejde-741	143	6	y	y	PROPN
ejde-741	143	7	a(y)dy	a(y)dy	PROPN
ejde-741	143	8	,	,	PUNCT
ejde-741	143	9	x	x	SYM
ejde-741	143	10	∈	∈	PROPN
ejde-741	144	1	[	[	X
ejde-741	144	2	0	0	NUM
ejde-741	144	3	,	,	PUNCT
ejde-741	144	4	α′	α′	NUM
ejde-741	144	5	)	)	PUNCT
ejde-741	144	6	;	;	PUNCT
ejde-741	145	1	−	−	PROPN
ejde-741	145	2	∫	∫	PROPN
ejde-741	145	3	x	x	X
ejde-741	145	4	β′	β′	NUM
ejde-741	145	5	y	y	PROPN
ejde-741	145	6	a(y)dy	a(y)dy	PROPN
ejde-741	145	7	,	,	PUNCT
ejde-741	145	8	x	x	SYM
ejde-741	145	9	∈	∈	PROPN
ejde-741	146	1	[	[	X
ejde-741	146	2	β′	β′	NUM
ejde-741	146	3	,	,	PUNCT
ejde-741	146	4	1	1	NUM
ejde-741	146	5	]	]	PUNCT
ejde-741	146	6	.	.	PUNCT
ejde-741	147	1	(	(	PUNCT
ejde-741	147	2	2.4	2.4	NUM
ejde-741	147	3	)	)	PUNCT
ejde-741	147	4	also	also	ADV
ejde-741	147	5	,	,	PUNCT
ejde-741	147	6	let	let	VERB
ejde-741	147	7	us	we	PRON
ejde-741	147	8	set	set	VERB
ejde-741	147	9	the	the	DET
ejde-741	147	10	functions	function	NOUN
ejde-741	147	11	η(x	η(x	NOUN
ejde-741	147	12	)	)	PUNCT
ejde-741	147	13	:	:	PUNCT
ejde-741	147	14	=	=	PUNCT
ejde-741	147	15	eλ(|ψ|∞+ψ	eλ(|ψ|∞+ψ	NOUN
ejde-741	147	16	)	)	PUNCT
ejde-741	147	17	,	,	PUNCT
ejde-741	147	18	ηr(x	ηr(x	X
ejde-741	147	19	)	)	PUNCT
ejde-741	147	20	:	:	PUNCT
ejde-741	148	1	=	=	PUNCT
ejde-741	148	2	eλ(|ψ|∞+ψ	eλ(|ψ|∞+ψ	NOUN
ejde-741	148	3	)	)	PUNCT
ejde-741	148	4	−	−	PROPN
ejde-741	148	5	eλr|ψ|∞	eλr|ψ|∞	PROPN
ejde-741	148	6	.	.	PUNCT
ejde-741	149	1	(	(	PUNCT
ejde-741	149	2	2.5	2.5	NUM
ejde-741	149	3	)	)	PUNCT
ejde-741	149	4	where	where	SCONJ
ejde-741	149	5	(	(	PUNCT
ejde-741	149	6	t	t	PROPN
ejde-741	149	7	,	,	PUNCT
ejde-741	149	8	x	x	NOUN
ejde-741	149	9	)	)	PUNCT
ejde-741	149	10	∈	∈	PROPN
ejde-741	149	11	(	(	PUNCT
ejde-741	149	12	0	0	NUM
ejde-741	149	13	,	,	PUNCT
ejde-741	149	14	t	t	NOUN
ejde-741	149	15	)	)	PUNCT
ejde-741	149	16	×	×	NOUN
ejde-741	150	1	[	[	X
ejde-741	150	2	0	0	NUM
ejde-741	150	3	,	,	PUNCT
ejde-741	150	4	1	1	NUM
ejde-741	150	5	]	]	PUNCT
ejde-741	150	6	and	and	CCONJ
ejde-741	150	7	λ	λ	PROPN
ejde-741	150	8	,	,	PUNCT
ejde-741	150	9	r	r	NOUN
ejde-741	150	10	∈	∈	PROPN
ejde-741	150	11	(	(	PUNCT
ejde-741	150	12	0,+∞	0,+∞	NUM
ejde-741	150	13	)	)	PUNCT
ejde-741	150	14	.	.	PUNCT
ejde-741	151	1	in	in	ADP
ejde-741	151	2	addition	addition	NOUN
ejde-741	151	3	,	,	PUNCT
ejde-741	151	4	consider	consider	VERB
ejde-741	151	5	the	the	DET
ejde-741	151	6	constants	constant	NOUN
ejde-741	151	7	:	:	PUNCT
ejde-741	151	8	η̂	η̂	NUM
ejde-741	151	9	=	=	SYM
ejde-741	151	10	min	min	NOUN
ejde-741	151	11	x∈[0,1	x∈[0,1	NOUN
ejde-741	151	12	]	]	X
ejde-741	151	13	η(x	η(x	NOUN
ejde-741	151	14	)	)	PUNCT
ejde-741	151	15	,	,	PUNCT
ejde-741	151	16	η∗	η∗	NOUN
ejde-741	151	17	:	:	PUNCT
ejde-741	151	18	=	=	SYM
ejde-741	151	19	max	max	PROPN
ejde-741	151	20	x∈[0,1	x∈[0,1	X
ejde-741	151	21	]	]	X
ejde-741	151	22	η(x	η(x	NOUN
ejde-741	151	23	)	)	PUNCT
ejde-741	151	24	,	,	PUNCT
ejde-741	151	25	η̂r	η̂r	NUM
ejde-741	151	26	=	=	SYM
ejde-741	151	27	min	min	PROPN
ejde-741	151	28	x∈[0,1	x∈[0,1	PROPN
ejde-741	151	29	]	]	X
ejde-741	151	30	ηr(x	ηr(x	X
ejde-741	151	31	)	)	PUNCT
ejde-741	151	32	,	,	PUNCT
ejde-741	151	33	η∗r	η∗r	NOUN
ejde-741	151	34	:	:	PUNCT
ejde-741	151	35	=	=	PUNCT
ejde-741	151	36	max	max	PROPN
ejde-741	151	37	x∈[0,1	x∈[0,1	SYM
ejde-741	151	38	]	]	X
ejde-741	151	39	ηr(x	ηr(x	X
ejde-741	151	40	)	)	PUNCT
ejde-741	151	41	.	.	PUNCT
ejde-741	152	1	(	(	PUNCT
ejde-741	152	2	2.6	2.6	NUM
ejde-741	152	3	)	)	PUNCT
ejde-741	152	4	it	it	PRON
ejde-741	152	5	is	be	AUX
ejde-741	152	6	important	important	ADJ
ejde-741	152	7	to	to	PART
ejde-741	152	8	notice	notice	VERB
ejde-741	152	9	that	that	SCONJ
ejde-741	152	10	,	,	PUNCT
ejde-741	152	11	if	if	SCONJ
ejde-741	152	12	r	r	NOUN
ejde-741	152	13	∈	∈	PROPN
ejde-741	152	14	(	(	PUNCT
ejde-741	152	15	3,+∞	3,+∞	NUM
ejde-741	152	16	)	)	PUNCT
ejde-741	152	17	is	be	AUX
ejde-741	152	18	sufficiently	sufficiently	ADV
ejde-741	152	19	large	large	ADJ
ejde-741	152	20	,	,	PUNCT
ejde-741	152	21	then	then	ADV
ejde-741	152	22	ηr(x	ηr(x	X
ejde-741	152	23	)	)	PUNCT
ejde-741	152	24	<	<	X
ejde-741	152	25	0	0	NUM
ejde-741	152	26	,	,	PUNCT
ejde-741	152	27	for	for	ADP
ejde-741	152	28	any	any	DET
ejde-741	152	29	x	x	SYM
ejde-741	152	30	∈	∈	PROPN
ejde-741	153	1	[	[	X
ejde-741	153	2	0	0	NUM
ejde-741	153	3	,	,	PUNCT
ejde-741	153	4	1	1	NUM
ejde-741	153	5	]	]	PUNCT
ejde-741	153	6	,	,	PUNCT
ejde-741	153	7	and	and	CCONJ
ejde-741	153	8	3η∗r	3η∗r	NUM
ejde-741	153	9	<	<	X
ejde-741	153	10	2η̂r	2η̂r	NUM
ejde-741	153	11	.	.	PUNCT
ejde-741	154	1	in	in	ADP
ejde-741	154	2	this	this	DET
ejde-741	154	3	case	case	NOUN
ejde-741	154	4	,	,	PUNCT
ejde-741	154	5	putting	put	VERB
ejde-741	154	6	η̄r	η̄r	NOUN
ejde-741	154	7	:	:	PUNCT
ejde-741	154	8	=	=	SYM
ejde-741	154	9	3η∗r	3η∗r	NUM
ejde-741	154	10	−	−	PROPN
ejde-741	154	11	2η̂r	2η̂r	PROPN
ejde-741	154	12	,	,	PUNCT
ejde-741	154	13	we	we	PRON
ejde-741	154	14	can	can	AUX
ejde-741	154	15	see	see	VERB
ejde-741	154	16	that	that	DET
ejde-741	154	17	ηr(x	ηr(x	X
ejde-741	154	18	)	)	PUNCT
ejde-741	154	19	≤	≤	NUM
ejde-741	155	1	η̄r	η̄r	X
ejde-741	155	2	<	<	X
ejde-741	155	3	0	0	NUM
ejde-741	155	4	,	,	PUNCT
ejde-741	155	5	for	for	ADP
ejde-741	155	6	any	any	DET
ejde-741	155	7	x	x	SYM
ejde-741	155	8	∈	∈	PROPN
ejde-741	156	1	[	[	X
ejde-741	156	2	0	0	NUM
ejde-741	156	3	,	,	PUNCT
ejde-741	156	4	1	1	NUM
ejde-741	156	5	]	]	PUNCT
ejde-741	156	6	.	.	PUNCT
ejde-741	157	1	next	next	ADV
ejde-741	157	2	,	,	PUNCT
ejde-741	157	3	we	we	PRON
ejde-741	157	4	consider	consider	VERB
ejde-741	157	5	m	m	PRON
ejde-741	157	6	∈	∈	NOUN
ejde-741	157	7	c∞([0	c∞([0	NOUN
ejde-741	157	8	,	,	PUNCT
ejde-741	157	9	t	t	X
ejde-741	157	10	]	]	PUNCT
ejde-741	157	11	;	;	PUNCT
ejde-741	157	12	r	r	X
ejde-741	157	13	)	)	PUNCT
ejde-741	157	14	satisfying	satisfy	VERB
ejde-741	157	15	m(t	m(t	NOUN
ejde-741	157	16	)	)	PUNCT
ejde-741	157	17	≥	≥	NOUN
ejde-741	157	18	t4(t	t4(t	X
ejde-741	157	19	−	−	PROPN
ejde-741	157	20	t)4	t)4	PROPN
ejde-741	157	21	,	,	PUNCT
ejde-741	157	22	t	t	PROPN
ejde-741	157	23	∈	∈	PROPN
ejde-741	157	24	(	(	PUNCT
ejde-741	157	25	0	0	NUM
ejde-741	157	26	,	,	PUNCT
ejde-741	157	27	t/2	t/2	NUM
ejde-741	157	28	]	]	PUNCT
ejde-741	157	29	;	;	PUNCT
ejde-741	157	30	m(t	m(t	X
ejde-741	157	31	)	)	PUNCT
ejde-741	157	32	=	=	PUNCT
ejde-741	158	1	t4(t	t4(t	NUM
ejde-741	159	1	−	−	PROPN
ejde-741	159	2	t)4	t)4	PROPN
ejde-741	159	3	,	,	PUNCT
ejde-741	159	4	t	t	PROPN
ejde-741	159	5	∈	∈	PROPN
ejde-741	160	1	[	[	X
ejde-741	160	2	t/2	t/2	NUM
ejde-741	160	3	,	,	PUNCT
ejde-741	160	4	t	t	X
ejde-741	160	5	]	]	PUNCT
ejde-741	160	6	;	;	PUNCT
ejde-741	160	7	m(0	m(0	PROPN
ejde-741	160	8	)	)	PUNCT
ejde-741	160	9	>	>	X
ejde-741	160	10	0	0	NUM
ejde-741	160	11	,	,	PUNCT
ejde-741	160	12	to	to	PART
ejde-741	160	13	define	define	VERB
ejde-741	160	14	τ(t	τ(t	NOUN
ejde-741	160	15	)	)	PUNCT
ejde-741	160	16	:	:	PUNCT
ejde-741	160	17	=	=	SYM
ejde-741	160	18	1	1	NUM
ejde-741	160	19	m(t	m(t	NOUN
ejde-741	160	20	)	)	PUNCT
ejde-741	160	21	,	,	PUNCT
ejde-741	160	22	ζ(x	ζ(x	PROPN
ejde-741	160	23	,	,	PUNCT
ejde-741	160	24	t	t	PROPN
ejde-741	160	25	)	)	PUNCT
ejde-741	160	26	:	:	PUNCT
ejde-741	160	27	=	=	SYM
ejde-741	160	28	τ(t)η(x	τ(t)η(x	NOUN
ejde-741	160	29	)	)	PUNCT
ejde-741	160	30	,	,	PUNCT
ejde-741	160	31	ζ∗(t	ζ∗(t	NOUN
ejde-741	160	32	)	)	PUNCT
ejde-741	160	33	:	:	PUNCT
ejde-741	160	34	=	=	SYM
ejde-741	160	35	τ(t)η∗	τ(t)η∗	PROPN
ejde-741	160	36	,	,	PUNCT
ejde-741	160	37	a(t	a(t	PROPN
ejde-741	160	38	,	,	PUNCT
ejde-741	160	39	x	x	NOUN
ejde-741	160	40	)	)	PUNCT
ejde-741	160	41	:	:	PUNCT
ejde-741	160	42	=	=	PUNCT
ejde-741	160	43	τ(t)ηr(x	τ(t)ηr(x	X
ejde-741	160	44	)	)	PUNCT
ejde-741	160	45	,	,	PUNCT
ejde-741	160	46	ā(t	ā(t	PROPN
ejde-741	160	47	)	)	PUNCT
ejde-741	160	48	:	:	PUNCT
ejde-741	160	49	=	=	SYM
ejde-741	160	50	τ(t)η̄r	τ(t)η̄r	NOUN
ejde-741	160	51	,	,	PUNCT
ejde-741	160	52	(	(	PUNCT
ejde-741	160	53	2.7	2.7	NUM
ejde-741	160	54	)	)	PUNCT
ejde-741	160	55	where	where	SCONJ
ejde-741	160	56	(	(	PUNCT
ejde-741	160	57	t	t	PROPN
ejde-741	160	58	,	,	PUNCT
ejde-741	160	59	x	x	NOUN
ejde-741	160	60	)	)	PUNCT
ejde-741	160	61	∈	∈	PROPN
ejde-741	161	1	[	[	X
ejde-741	161	2	0	0	NUM
ejde-741	161	3	,	,	PUNCT
ejde-741	161	4	t	t	PROPN
ejde-741	161	5	)	)	PUNCT
ejde-741	161	6	×	×	NOUN
ejde-741	162	1	[	[	X
ejde-741	162	2	0	0	NUM
ejde-741	162	3	,	,	PUNCT
ejde-741	162	4	1	1	NUM
ejde-741	162	5	]	]	PUNCT
ejde-741	162	6	(	(	PUNCT
ejde-741	162	7	see	see	VERB
ejde-741	162	8	remark	remark	NOUN
ejde-741	162	9	2.4	2.4	NUM
ejde-741	162	10	)	)	PUNCT
ejde-741	162	11	.	.	PUNCT
ejde-741	163	1	finally	finally	ADV
ejde-741	163	2	,	,	PUNCT
ejde-741	163	3	we	we	PRON
ejde-741	163	4	can	can	AUX
ejde-741	163	5	mention	mention	VERB
ejde-741	163	6	the	the	DET
ejde-741	163	7	weight	weight	NOUN
ejde-741	163	8	functions	function	NOUN
ejde-741	163	9	ρ0	ρ0	X
ejde-741	163	10	=	=	SYM
ejde-741	163	11	e−saζ−5/6	e−saζ−5/6	PROPN
ejde-741	163	12	,	,	PUNCT
ejde-741	163	13	ρ̂	ρ̂	NUM
ejde-741	163	14	=	=	SYM
ejde-741	163	15	e−s(a+ā)/2(ζ∗)−11/6	e−s(a+ā)/2(ζ∗)−11/6	PROPN
ejde-741	163	16	,	,	PUNCT
ejde-741	163	17	ρ∗	ρ∗	PROPN
ejde-741	163	18	=	=	SYM
ejde-741	163	19	e−sā(ζ∗)−17/6	e−sā(ζ∗)−17/6	PROPN
ejde-741	163	20	,	,	PUNCT
ejde-741	163	21	(	(	PUNCT
ejde-741	163	22	2.8	2.8	NUM
ejde-741	163	23	)	)	PUNCT
ejde-741	163	24	associated	associate	VERB
ejde-741	163	25	with	with	ADP
ejde-741	163	26	the	the	DET
ejde-741	163	27	desired	desire	VERB
ejde-741	163	28	spaces	space	NOUN
ejde-741	163	29	e	e	NOUN
ejde-741	163	30	and	and	CCONJ
ejde-741	163	31	f	f	PROPN
ejde-741	163	32	.	.	PUNCT
ejde-741	164	1	we	we	PRON
ejde-741	164	2	observe	observe	VERB
ejde-741	164	3	that	that	SCONJ
ejde-741	164	4	ρ̂2	ρ̂2	PROPN
ejde-741	164	5	≤	≤	NUM
ejde-741	164	6	ρ0ρ∗	ρ0ρ∗	PROPN
ejde-741	164	7	and	and	CCONJ
ejde-741	164	8	that	that	SCONJ
ejde-741	164	9	there	there	PRON
ejde-741	164	10	exists	exist	VERB
ejde-741	164	11	a	a	DET
ejde-741	164	12	constant	constant	ADJ
ejde-741	164	13	ct	ct	NOUN
ejde-741	164	14	>	>	X
ejde-741	164	15	0	0	NUM
ejde-741	164	16	,	,	PUNCT
ejde-741	164	17	only	only	ADV
ejde-741	164	18	depending	depend	VERB
ejde-741	164	19	on	on	ADP
ejde-741	164	20	t	t	PROPN
ejde-741	164	21	,	,	PUNCT
ejde-741	164	22	such	such	ADJ
ejde-741	164	23	that	that	SCONJ
ejde-741	164	24	0	0	NUM
ejde-741	164	25	<	<	X
ejde-741	164	26	ct	ct	PROPN
ejde-741	164	27	≤	≤	PROPN
ejde-741	164	28	ρ∗	ρ∗	PROPN
ejde-741	164	29	≤	≤	PROPN
ejde-741	164	30	cρ̂	cρ̂	PROPN
ejde-741	164	31	≤	≤	PROPN
ejde-741	164	32	cρ0	cρ0	NOUN
ejde-741	164	33	.	.	PUNCT
ejde-741	165	1	thus	thus	ADV
ejde-741	165	2	,	,	PUNCT
ejde-741	165	3	l2(q	l2(q	PROPN
ejde-741	165	4	;	;	PUNCT
ejde-741	165	5	ρ20	ρ20	NUM
ejde-741	165	6	)	)	PUNCT
ejde-741	165	7	↪	↪	PROPN
ejde-741	165	8	→	→	SYM
ejde-741	165	9	l2(q	l2(q	PROPN
ejde-741	165	10	;	;	PUNCT
ejde-741	165	11	ρ̂2	ρ̂2	X
ejde-741	165	12	)	)	PUNCT
ejde-741	165	13	↪	↪	PROPN
ejde-741	165	14	→	→	SYM
ejde-741	165	15	l2(q	l2(q	PROPN
ejde-741	165	16	;	;	PUNCT
ejde-741	165	17	ρ2∗	ρ2∗	NUM
ejde-741	165	18	)	)	PUNCT
ejde-741	165	19	↪	↪	PROPN
ejde-741	165	20	→	→	SYM
ejde-741	165	21	l2(q	l2(q	PROPN
ejde-741	165	22	)	)	PUNCT
ejde-741	165	23	.	.	PUNCT
ejde-741	166	1	remark	remark	VERB
ejde-741	166	2	2.4	2.4	NUM
ejde-741	166	3	.	.	PUNCT
ejde-741	167	1	in	in	ADP
ejde-741	167	2	(	(	PUNCT
ejde-741	167	3	2.7	2.7	NUM
ejde-741	167	4	)	)	PUNCT
ejde-741	167	5	,	,	PUNCT
ejde-741	167	6	we	we	PRON
ejde-741	167	7	define	define	VERB
ejde-741	167	8	τ	τ	X
ejde-741	167	9	=	=	SYM
ejde-741	167	10	τ(t	τ(t	PROPN
ejde-741	167	11	)	)	PUNCT
ejde-741	167	12	satisfying	satisfy	VERB
ejde-741	167	13	limt→0	limt→0	PROPN
ejde-741	167	14	+	+	CCONJ
ejde-741	167	15	τ(t	τ(t	NOUN
ejde-741	167	16	)	)	PUNCT
ejde-741	167	17	=	=	SYM
ejde-741	167	18	τ(0	τ(0	PROPN
ejde-741	167	19	)	)	PUNCT
ejde-741	167	20	>	>	X
ejde-741	168	1	0	0	X
ejde-741	168	2	.	.	PUNCT
ejde-741	169	1	it	it	PRON
ejde-741	169	2	plays	play	VERB
ejde-741	169	3	a	a	DET
ejde-741	169	4	crucial	crucial	ADJ
ejde-741	169	5	role	role	NOUN
ejde-741	169	6	in	in	ADP
ejde-741	169	7	order	order	NOUN
ejde-741	169	8	to	to	PART
ejde-741	169	9	guarantee	guarantee	VERB
ejde-741	169	10	that	that	SCONJ
ejde-741	169	11	(	(	PUNCT
ejde-741	169	12	1.1	1.1	NUM
ejde-741	169	13	)	)	PUNCT
ejde-741	169	14	is	be	AUX
ejde-741	169	15	locally	locally	ADV
ejde-741	169	16	null	null	ADJ
ejde-741	169	17	-	-	PUNCT
ejde-741	169	18	controllable	controllable	NOUN
ejde-741	169	19	at	at	ADP
ejde-741	169	20	the	the	DET
ejde-741	169	21	time	time	NOUN
ejde-741	169	22	t	t	PROPN
ejde-741	169	23	>	>	X
ejde-741	169	24	0	0	NUM
ejde-741	169	25	,	,	PUNCT
ejde-741	169	26	as	as	SCONJ
ejde-741	169	27	stated	state	VERB
ejde-741	169	28	in	in	ADP
ejde-741	169	29	theorem	theorem	ADJ
ejde-741	169	30	1.6	1.6	NUM
ejde-741	169	31	.	.	PUNCT
ejde-741	170	1	precisely	precisely	ADV
ejde-741	170	2	,	,	PUNCT
ejde-741	170	3	each	each	DET
ejde-741	170	4	function	function	NOUN
ejde-741	170	5	given	give	VERB
ejde-741	170	6	in	in	ADP
ejde-741	170	7	(	(	PUNCT
ejde-741	170	8	2.7	2.7	NUM
ejde-741	170	9	)	)	PUNCT
ejde-741	170	10	is	be	AUX
ejde-741	170	11	based	base	VERB
ejde-741	170	12	on	on	ADP
ejde-741	170	13	the	the	DET
ejde-741	170	14	weights	weight	NOUN
ejde-741	170	15	which	which	PRON
ejde-741	170	16	will	will	AUX
ejde-741	170	17	appear	appear	VERB
ejde-741	170	18	in	in	ADP
ejde-741	170	19	(	(	PUNCT
ejde-741	170	20	2.13	2.13	NUM
ejde-741	170	21	)	)	PUNCT
ejde-741	170	22	.	.	PUNCT
ejde-741	171	1	as	as	ADP
ejde-741	171	2	a	a	DET
ejde-741	171	3	result	result	NOUN
ejde-741	171	4	,	,	PUNCT
ejde-741	171	5	since	since	SCONJ
ejde-741	171	6	ρ0(t	ρ0(t	NUM
ejde-741	171	7	)	)	PUNCT
ejde-741	171	8	→	→	PUNCT
ejde-741	171	9	+	+	NUM
ejde-741	171	10	∞	∞	PROPN
ejde-741	171	11	,	,	PUNCT
ejde-741	171	12	as	as	ADP
ejde-741	171	13	t→	t→	X
ejde-741	171	14	t−	t−	PROPN
ejde-741	171	15	,	,	PUNCT
ejde-741	171	16	and	and	CCONJ
ejde-741	171	17	ρ0(0	ρ0(0	NOUN
ejde-741	171	18	)	)	PUNCT
ejde-741	171	19	>	>	X
ejde-741	171	20	0	0	PUNCT
ejde-741	171	21	(	(	PUNCT
ejde-741	171	22	since	since	SCONJ
ejde-741	171	23	m(0	m(0	PROPN
ejde-741	171	24	)	)	PUNCT
ejde-741	171	25	>	>	X
ejde-741	171	26	0	0	NUM
ejde-741	171	27	)	)	PUNCT
ejde-741	171	28	,	,	PUNCT
ejde-741	171	29	it	it	PRON
ejde-741	171	30	is	be	AUX
ejde-741	171	31	possible	possible	ADJ
ejde-741	171	32	to	to	PART
ejde-741	171	33	conclude	conclude	VERB
ejde-741	171	34	that	that	SCONJ
ejde-741	171	35	u(t	u(t	NOUN
ejde-741	171	36	,	,	PUNCT
ejde-741	171	37	x	x	NOUN
ejde-741	171	38	)	)	PUNCT
ejde-741	171	39	=	=	SYM
ejde-741	171	40	0	0	NUM
ejde-741	171	41	for	for	ADP
ejde-741	171	42	any	any	PRON
ejde-741	171	43	for	for	ADP
ejde-741	171	44	u	u	PROPN
ejde-741	171	45	∈	∈	PROPN
ejde-741	171	46	l2(q	l2(q	PROPN
ejde-741	171	47	;	;	PUNCT
ejde-741	171	48	ρ20	ρ20	NUM
ejde-741	171	49	)	)	PUNCT
ejde-741	171	50	.	.	PUNCT
ejde-741	172	1	hence	hence	ADV
ejde-741	172	2	,	,	PUNCT
ejde-741	172	3	it	it	PRON
ejde-741	172	4	seems	seem	VERB
ejde-741	172	5	reasonable	reasonable	ADJ
ejde-741	172	6	to	to	PART
ejde-741	172	7	require	require	VERB
ejde-741	172	8	that	that	SCONJ
ejde-741	172	9	,	,	PUNCT
ejde-741	172	10	if	if	SCONJ
ejde-741	172	11	(	(	PUNCT
ejde-741	172	12	u	u	NOUN
ejde-741	172	13	,	,	PUNCT
ejde-741	172	14	h	h	NOUN
ejde-741	172	15	)	)	PUNCT
ejde-741	172	16	∈	∈	PROPN
ejde-741	172	17	e	e	NOUN
ejde-741	172	18	,	,	PUNCT
ejde-741	172	19	then	then	ADV
ejde-741	172	20	u	u	PRON
ejde-741	172	21	must	must	AUX
ejde-741	172	22	belong	belong	VERB
ejde-741	172	23	to	to	ADP
ejde-741	172	24	l2(q	l2(q	PROPN
ejde-741	172	25	;	;	PUNCT
ejde-741	172	26	ρ20	ρ20	NUM
ejde-741	172	27	)	)	PUNCT
ejde-741	172	28	.	.	PUNCT
ejde-741	173	1	now	now	ADV
ejde-741	173	2	,	,	PUNCT
ejde-741	173	3	we	we	PRON
ejde-741	173	4	are	be	AUX
ejde-741	173	5	ready	ready	ADJ
ejde-741	173	6	to	to	PART
ejde-741	173	7	define	define	VERB
ejde-741	173	8	e	e	NOUN
ejde-741	173	9	and	and	CCONJ
ejde-741	173	10	f	f	PROPN
ejde-741	173	11	.	.	PUNCT
ejde-741	174	1	let	let	VERB
ejde-741	174	2	us	we	PRON
ejde-741	174	3	consider	consider	VERB
ejde-741	174	4	u	u	PRON
ejde-741	174	5	:	:	PUNCT
ejde-741	174	6	=	=	SYM
ejde-741	174	7	h1(0	h1(0	PROPN
ejde-741	174	8	,	,	PUNCT
ejde-741	174	9	t	t	PROPN
ejde-741	174	10	;	;	PUNCT
ejde-741	174	11	l2(0	l2(0	NOUN
ejde-741	174	12	,	,	PUNCT
ejde-741	174	13	1	1	NUM
ejde-741	174	14	)	)	PUNCT
ejde-741	174	15	)	)	PUNCT
ejde-741	175	1	∩	∩	ADJ
ejde-741	175	2	l2(0	l2(0	NOUN
ejde-741	175	3	,	,	PUNCT
ejde-741	175	4	t	t	NOUN
ejde-741	175	5	;	;	PUNCT
ejde-741	175	6	h2	h2	PROPN
ejde-741	175	7	a	a	PRON
ejde-741	175	8	)	)	PUNCT
ejde-741	175	9	∩	∩	NOUN
ejde-741	175	10	c0([0	c0([0	PROPN
ejde-741	175	11	,	,	PUNCT
ejde-741	175	12	t	t	X
ejde-741	175	13	]	]	PUNCT
ejde-741	175	14	;	;	PUNCT
ejde-741	175	15	h1	h1	VERB
ejde-741	175	16	a	a	PRON
ejde-741	175	17	)	)	PUNCT
ejde-741	175	18	,	,	PUNCT
ejde-741	175	19	as	as	ADP
ejde-741	175	20	in	in	ADP
ejde-741	175	21	proposition	proposition	NOUN
ejde-741	175	22	2.2	2.2	NUM
ejde-741	175	23	,	,	PUNCT
ejde-741	175	24	and	and	CCONJ
ejde-741	175	25	put	put	VERB
ejde-741	175	26	lu	lu	NOUN
ejde-741	175	27	:	:	PUNCT
ejde-741	175	28	=	=	PUNCT
ejde-741	175	29	ut	ut	INTJ
ejde-741	175	30	−	−	PROPN
ejde-741	175	31	(	(	PUNCT
ejde-741	175	32	aux)x	aux)x	PROPN
ejde-741	175	33	for	for	ADP
ejde-741	175	34	each	each	DET
ejde-741	175	35	u	u	PROPN
ejde-741	175	36	∈	∈	PROPN
ejde-741	175	37	u	u	NOUN
ejde-741	175	38	.	.	PUNCT
ejde-741	176	1	under	under	ADP
ejde-741	176	2	all	all	DET
ejde-741	176	3	these	these	DET
ejde-741	176	4	notations	notation	NOUN
ejde-741	176	5	,	,	PUNCT
ejde-741	176	6	we	we	PRON
ejde-741	176	7	set	set	VERB
ejde-741	176	8	,	,	PUNCT
ejde-741	176	9	for	for	ADP
ejde-741	176	10	the	the	DET
ejde-741	176	11	(	(	PUNCT
ejde-741	176	12	wdc	wdc	PROPN
ejde-741	176	13	)	)	PUNCT
ejde-741	176	14	,	,	PUNCT
ejde-741	176	15	the	the	DET
ejde-741	176	16	hilbert	hilbert	NOUN
ejde-741	176	17	spaces	space	NOUN
ejde-741	176	18	e	e	NOUN
ejde-741	176	19	:	:	PUNCT
ejde-741	176	20	=	=	SYM
ejde-741	176	21	{	{	PUNCT
ejde-741	176	22	(	(	PUNCT
ejde-741	176	23	u	u	NOUN
ejde-741	176	24	,	,	PUNCT
ejde-741	176	25	h	h	NOUN
ejde-741	176	26	)	)	PUNCT
ejde-741	176	27	∈	∈	PROPN
ejde-741	176	28	u	u	NOUN
ejde-741	176	29	×	×	PROPN
ejde-741	176	30	l2(qω	l2(qω	PROPN
ejde-741	176	31	;	;	PUNCT
ejde-741	176	32	ρ	ρ	NUM
ejde-741	176	33	2	2	NUM
ejde-741	176	34	∗	∗	NOUN
ejde-741	176	35	)	)	PUNCT
ejde-741	176	36	:	:	PUNCT
ejde-741	177	1	ρ0u	ρ0u	NOUN
ejde-741	177	2	,	,	PUNCT
ejde-741	177	3	ρ0(lu−	ρ0(lu−	PROPN
ejde-741	177	4	hχω	hχω	NOUN
ejde-741	177	5	)	)	PUNCT
ejde-741	177	6	∈	∈	PROPN
ejde-741	177	7	l2((0	l2((0	PROPN
ejde-741	177	8	,	,	PUNCT
ejde-741	177	9	t	t	NOUN
ejde-741	177	10	)	)	PUNCT
ejde-741	177	11	×	×	NOUN
ejde-741	177	12	(	(	PUNCT
ejde-741	177	13	0	0	NUM
ejde-741	177	14	,	,	PUNCT
ejde-741	177	15	1	1	NUM
ejde-741	177	16	)	)	PUNCT
ejde-741	177	17	)	)	PUNCT
ejde-741	177	18	}	}	PUNCT
ejde-741	177	19	,	,	PUNCT
ejde-741	177	20	(	(	PUNCT
ejde-741	177	21	2.9	2.9	NUM
ejde-741	177	22	)	)	PUNCT
ejde-741	177	23	and	and	CCONJ
ejde-741	177	24	f	f	X
ejde-741	177	25	:	:	PUNCT
ejde-741	177	26	=	=	SYM
ejde-741	177	27	l2(q	l2(q	PROPN
ejde-741	177	28	;	;	PUNCT
ejde-741	177	29	ρ20)×h1	ρ20)×h1	NOUN
ejde-741	177	30	a	a	PRON
ejde-741	177	31	,	,	PUNCT
ejde-741	177	32	(	(	PUNCT
ejde-741	177	33	2.10	2.10	NUM
ejde-741	177	34	)	)	PUNCT
ejde-741	177	35	ejde-2025/15	ejde-2025/15	NOUN
ejde-741	177	36	null	null	ADJ
ejde-741	177	37	-	-	PUNCT
ejde-741	177	38	controllability	controllability	NOUN
ejde-741	177	39	degenerate	degenerate	ADJ
ejde-741	177	40	quasilinear	quasilinear	NOUN
ejde-741	177	41	equations	equation	NOUN
ejde-741	177	42	7	7	NUM
ejde-741	177	43	equipped	equip	VERB
ejde-741	177	44	with	with	ADP
ejde-741	177	45	the	the	DET
ejde-741	177	46	norms	norm	NOUN
ejde-741	177	47	∥(u	∥(u	NOUN
ejde-741	177	48	,	,	PUNCT
ejde-741	177	49	h)∥e	h)∥e	X
ejde-741	177	50	:	:	PUNCT
ejde-741	177	51	=	=	SYM
ejde-741	177	52	(	(	PUNCT
ejde-741	177	53	∥u∥2ρ20	∥u∥2ρ20	X
ejde-741	177	54	+	+	CCONJ
ejde-741	177	55	∥hχω∥2ρ2∗	∥hχω∥2ρ2∗	PROPN
ejde-741	177	56	+	+	CCONJ
ejde-741	177	57	∥lu−	∥lu−	PROPN
ejde-741	177	58	hχω∥2ρ20	hχω∥2ρ20	PROPN
ejde-741	177	59	+	+	CCONJ
ejde-741	177	60	∥u(0	∥u(0	NOUN
ejde-741	177	61	,	,	PUNCT
ejde-741	177	62	·	·	PUNCT
ejde-741	177	63	)	)	PUNCT
ejde-741	177	64	∥2h1	∥2h1	NOUN
ejde-741	177	65	a	a	DET
ejde-741	177	66	)	)	PUNCT
ejde-741	177	67	1/2	1/2	NUM
ejde-741	177	68	,	,	PUNCT
ejde-741	177	69	and	and	CCONJ
ejde-741	177	70	∥(g	∥(g	NOUN
ejde-741	177	71	,	,	PUNCT
ejde-741	177	72	v)∥f	v)∥f	NOUN
ejde-741	177	73	:	:	PUNCT
ejde-741	177	74	=	=	SYM
ejde-741	177	75	(	(	PUNCT
ejde-741	177	76	∥g∥2ρ20	∥g∥2ρ20	X
ejde-741	177	77	+	+	CCONJ
ejde-741	177	78	∥v∥2h1	∥v∥2h1	NUM
ejde-741	177	79	a	a	PRON
ejde-741	177	80	)	)	PUNCT
ejde-741	177	81	1/2	1/2	NUM
ejde-741	177	82	,	,	PUNCT
ejde-741	177	83	respectively	respectively	ADV
ejde-741	177	84	.	.	PUNCT
ejde-741	178	1	we	we	PRON
ejde-741	178	2	observe	observe	VERB
ejde-741	178	3	that	that	SCONJ
ejde-741	178	4	,	,	PUNCT
ejde-741	178	5	for	for	ADP
ejde-741	178	6	the	the	DET
ejde-741	178	7	(	(	PUNCT
ejde-741	178	8	sdp	sdp	NOUN
ejde-741	178	9	)	)	PUNCT
ejde-741	178	10	,	,	PUNCT
ejde-741	178	11	the	the	DET
ejde-741	178	12	definition	definition	NOUN
ejde-741	178	13	of	of	ADP
ejde-741	178	14	e	e	NOUN
ejde-741	178	15	must	must	AUX
ejde-741	178	16	also	also	ADV
ejde-741	178	17	contain	contain	VERB
ejde-741	178	18	the	the	DET
ejde-741	178	19	condition	condition	NOUN
ejde-741	178	20	aux(t	aux(t	PROPN
ejde-741	178	21	,	,	PUNCT
ejde-741	178	22	0	0	NUM
ejde-741	178	23	)	)	PUNCT
ejde-741	178	24	≡	≡	PROPN
ejde-741	178	25	0	0	NUM
ejde-741	178	26	,	,	PUNCT
ejde-741	178	27	a.e	a.e	PROPN
ejde-741	178	28	.	.	PROPN
ejde-741	178	29	in	in	ADP
ejde-741	178	30	[	[	X
ejde-741	178	31	0	0	NUM
ejde-741	178	32	,	,	PUNCT
ejde-741	178	33	t	t	X
ejde-741	178	34	]	]	PUNCT
ejde-741	178	35	,	,	PUNCT
ejde-741	178	36	while	while	SCONJ
ejde-741	178	37	the	the	DET
ejde-741	178	38	definition	definition	NOUN
ejde-741	178	39	of	of	ADP
ejde-741	178	40	f	f	PROPN
ejde-741	178	41	remains	remain	VERB
ejde-741	178	42	the	the	DET
ejde-741	178	43	same	same	ADJ
ejde-741	178	44	.	.	PUNCT
ejde-741	179	1	since	since	SCONJ
ejde-741	179	2	we	we	PRON
ejde-741	179	3	have	have	AUX
ejde-741	179	4	already	already	ADV
ejde-741	179	5	defined	define	VERB
ejde-741	179	6	the	the	DET
ejde-741	179	7	weight	weight	NOUN
ejde-741	179	8	functions	function	NOUN
ejde-741	179	9	,	,	PUNCT
ejde-741	179	10	and	and	CCONJ
ejde-741	179	11	identified	identify	VERB
ejde-741	179	12	the	the	DET
ejde-741	179	13	hilbert	hilbert	NOUN
ejde-741	179	14	spaces	space	NOUN
ejde-741	179	15	e	e	NOUN
ejde-741	179	16	and	and	CCONJ
ejde-741	179	17	f	f	PROPN
ejde-741	179	18	,	,	PUNCT
ejde-741	179	19	as	as	ADV
ejde-741	179	20	well	well	ADV
ejde-741	179	21	as	as	ADP
ejde-741	179	22	the	the	DET
ejde-741	179	23	mapping	mapping	NOUN
ejde-741	179	24	h	h	NOUN
ejde-741	179	25	:	:	PUNCT
ejde-741	179	26	e	e	X
ejde-741	179	27	→	→	SYM
ejde-741	179	28	f	f	PROPN
ejde-741	179	29	,	,	PUNCT
ejde-741	179	30	given	give	VERB
ejde-741	179	31	in	in	ADP
ejde-741	179	32	(	(	PUNCT
ejde-741	179	33	2.1	2.1	NUM
ejde-741	179	34	)	)	PUNCT
ejde-741	179	35	,	,	PUNCT
ejde-741	179	36	we	we	PRON
ejde-741	179	37	are	be	AUX
ejde-741	179	38	supposed	suppose	VERB
ejde-741	179	39	to	to	PART
ejde-741	179	40	verify	verify	VERB
ejde-741	179	41	that	that	SCONJ
ejde-741	179	42	h	h	NOUN
ejde-741	179	43	satisfies	satisfy	VERB
ejde-741	179	44	the	the	DET
ejde-741	179	45	hypotheses	hypothesis	NOUN
ejde-741	179	46	of	of	ADP
ejde-741	179	47	theorem	theorem	ADJ
ejde-741	179	48	2.1	2.1	NUM
ejde-741	179	49	(	(	PUNCT
ejde-741	179	50	section	section	NOUN
ejde-741	179	51	3	3	NUM
ejde-741	179	52	)	)	PUNCT
ejde-741	179	53	.	.	PUNCT
ejde-741	180	1	to	to	PART
ejde-741	180	2	do	do	VERB
ejde-741	180	3	that	that	PRON
ejde-741	180	4	,	,	PUNCT
ejde-741	180	5	we	we	PRON
ejde-741	180	6	need	need	VERB
ejde-741	180	7	to	to	PART
ejde-741	180	8	establish	establish	VERB
ejde-741	180	9	a	a	DET
ejde-741	180	10	carleman	carleman	ADJ
ejde-741	180	11	estimate	estimate	NOUN
ejde-741	180	12	that	that	PRON
ejde-741	180	13	will	will	AUX
ejde-741	180	14	guarantee	guarantee	VERB
ejde-741	180	15	those	those	DET
ejde-741	180	16	hypotheses	hypothesis	NOUN
ejde-741	180	17	.	.	PUNCT
ejde-741	181	1	2.2	2.2	NUM
ejde-741	181	2	.	.	PUNCT
ejde-741	181	3	carleman	carleman	ADJ
ejde-741	181	4	inequality	inequality	NOUN
ejde-741	181	5	.	.	PUNCT
ejde-741	182	1	in	in	ADP
ejde-741	182	2	this	this	DET
ejde-741	182	3	second	second	ADJ
ejde-741	182	4	part	part	NOUN
ejde-741	182	5	of	of	ADP
ejde-741	182	6	section	section	NOUN
ejde-741	182	7	2	2	NUM
ejde-741	182	8	,	,	PUNCT
ejde-741	182	9	we	we	PRON
ejde-741	182	10	present	present	VERB
ejde-741	182	11	a	a	DET
ejde-741	182	12	key	key	ADJ
ejde-741	182	13	carleman	carleman	ADJ
ejde-741	182	14	estimate	estimate	NOUN
ejde-741	182	15	,	,	PUNCT
ejde-741	182	16	closely	closely	ADV
ejde-741	182	17	related	relate	VERB
ejde-741	182	18	to	to	ADP
ejde-741	182	19	those	those	DET
ejde-741	182	20	properties	property	NOUN
ejde-741	182	21	of	of	ADP
ejde-741	182	22	h	h	NOUN
ejde-741	182	23	:	:	PUNCT
ejde-741	182	24	e	e	X
ejde-741	182	25	→	→	SYM
ejde-741	182	26	f	f	PROPN
ejde-741	182	27	that	that	PRON
ejde-741	182	28	we	we	PRON
ejde-741	182	29	expect	expect	VERB
ejde-741	182	30	to	to	PART
ejde-741	182	31	prove	prove	VERB
ejde-741	182	32	.	.	PUNCT
ejde-741	183	1	we	we	PRON
ejde-741	183	2	start	start	VERB
ejde-741	183	3	taking	take	VERB
ejde-741	183	4	into	into	ADP
ejde-741	183	5	consideration	consideration	NOUN
ejde-741	183	6	the	the	DET
ejde-741	183	7	adjoint	adjoint	NOUN
ejde-741	183	8	system	system	NOUN
ejde-741	183	9	associated	associate	VERB
ejde-741	183	10	with	with	ADP
ejde-741	183	11	(	(	PUNCT
ejde-741	183	12	2.2	2.2	NUM
ejde-741	183	13	)	)	PUNCT
ejde-741	183	14	,	,	PUNCT
ejde-741	183	15	given	give	VERB
ejde-741	183	16	by	by	ADP
ejde-741	183	17	−vt	−vt	NUM
ejde-741	183	18	−	−	PROPN
ejde-741	183	19	(	(	PUNCT
ejde-741	183	20	a(x)vx)x	a(x)vx)x	PROPN
ejde-741	183	21	+	+	CCONJ
ejde-741	183	22	c(t	c(t	PROPN
ejde-741	183	23	,	,	PUNCT
ejde-741	183	24	x)v	x)v	PUNCT
ejde-741	184	1	=	=	SYM
ejde-741	184	2	f	f	X
ejde-741	184	3	,	,	PUNCT
ejde-741	184	4	(	(	PUNCT
ejde-741	184	5	t	t	PROPN
ejde-741	184	6	,	,	PUNCT
ejde-741	184	7	x	x	X
ejde-741	184	8	)	)	PUNCT
ejde-741	184	9	∈	∈	PROPN
ejde-741	184	10	q,	q,	X
ejde-741	184	11	v(t	v(t	NOUN
ejde-741	184	12	,	,	PUNCT
ejde-741	184	13	0	0	NUM
ejde-741	184	14	)	)	PUNCT
ejde-741	184	15	=	=	SYM
ejde-741	184	16	0	0	NUM
ejde-741	184	17	,	,	PUNCT
ejde-741	184	18	t	t	PROPN
ejde-741	184	19	∈	∈	PROPN
ejde-741	184	20	(	(	PUNCT
ejde-741	184	21	0	0	NUM
ejde-741	184	22	,	,	PUNCT
ejde-741	184	23	t	t	PROPN
ejde-741	184	24	)	)	PUNCT
ejde-741	184	25	or	or	CCONJ
ejde-741	184	26	(	(	PUNCT
ejde-741	184	27	avx)(t	avx)(t	PROPN
ejde-741	184	28	,	,	PUNCT
ejde-741	184	29	0	0	NUM
ejde-741	184	30	)	)	PUNCT
ejde-741	184	31	=	=	SYM
ejde-741	184	32	0	0	NUM
ejde-741	184	33	,	,	PUNCT
ejde-741	184	34	t	t	PROPN
ejde-741	184	35	∈	∈	PROPN
ejde-741	184	36	(	(	PUNCT
ejde-741	184	37	0	0	NUM
ejde-741	184	38	,	,	PUNCT
ejde-741	184	39	t	t	NOUN
ejde-741	184	40	)	)	PUNCT
ejde-741	184	41	v(t	v(t	PROPN
ejde-741	184	42	,	,	PUNCT
ejde-741	184	43	x	x	NOUN
ejde-741	184	44	)	)	PUNCT
ejde-741	184	45	=	=	SYM
ejde-741	184	46	vt	vt	PROPN
ejde-741	184	47	(	(	PUNCT
ejde-741	184	48	x	x	NOUN
ejde-741	184	49	)	)	PUNCT
ejde-741	184	50	,	,	PUNCT
ejde-741	184	51	x	x	PUNCT
ejde-741	184	52	∈	∈	PROPN
ejde-741	184	53	(	(	PUNCT
ejde-741	184	54	0	0	NUM
ejde-741	184	55	,	,	PUNCT
ejde-741	184	56	1	1	NUM
ejde-741	184	57	)	)	PUNCT
ejde-741	184	58	,	,	PUNCT
ejde-741	184	59	(	(	PUNCT
ejde-741	184	60	2.11	2.11	NUM
ejde-741	184	61	)	)	PUNCT
ejde-741	184	62	where	where	SCONJ
ejde-741	184	63	f	f	PROPN
ejde-741	184	64	∈	∈	PROPN
ejde-741	184	65	l2(q	l2(q	PROPN
ejde-741	184	66	)	)	PUNCT
ejde-741	184	67	and	and	CCONJ
ejde-741	184	68	vt	vt	PROPN
ejde-741	184	69	∈	∈	PROPN
ejde-741	184	70	l2(0	l2(0	NOUN
ejde-741	184	71	,	,	PUNCT
ejde-741	184	72	1	1	NUM
ejde-741	184	73	)	)	PUNCT
ejde-741	184	74	.	.	PUNCT
ejde-741	185	1	now	now	ADV
ejde-741	185	2	,	,	PUNCT
ejde-741	185	3	we	we	PRON
ejde-741	185	4	consider	consider	VERB
ejde-741	185	5	the	the	DET
ejde-741	185	6	functions	function	NOUN
ejde-741	185	7	and	and	CCONJ
ejde-741	185	8	the	the	DET
ejde-741	185	9	constants	constant	NOUN
ejde-741	185	10	given	give	VERB
ejde-741	185	11	in	in	ADP
ejde-741	185	12	(	(	PUNCT
ejde-741	185	13	2.5	2.5	NUM
ejde-741	185	14	)	)	PUNCT
ejde-741	185	15	and	and	CCONJ
ejde-741	185	16	(	(	PUNCT
ejde-741	185	17	2.6	2.6	NUM
ejde-741	185	18	)	)	PUNCT
ejde-741	185	19	,	,	PUNCT
ejde-741	185	20	and	and	CCONJ
ejde-741	185	21	define	define	VERB
ejde-741	185	22	σ(x	σ(x	PROPN
ejde-741	185	23	,	,	PUNCT
ejde-741	185	24	t	t	PROPN
ejde-741	185	25	)	)	PUNCT
ejde-741	185	26	:	:	PUNCT
ejde-741	185	27	=	=	SYM
ejde-741	185	28	η(x	η(x	X
ejde-741	185	29	)	)	PUNCT
ejde-741	186	1	[	[	X
ejde-741	186	2	t(t	t(t	NOUN
ejde-741	186	3	−	−	PROPN
ejde-741	186	4	t)]4	t)]4	NOUN
ejde-741	186	5	,	,	PUNCT
ejde-741	186	6	and	and	CCONJ
ejde-741	186	7	φ(x	φ(x	PROPN
ejde-741	186	8	,	,	PUNCT
ejde-741	186	9	t	t	PROPN
ejde-741	186	10	)	)	PUNCT
ejde-741	186	11	:	:	PUNCT
ejde-741	186	12	=	=	SYM
ejde-741	186	13	ηr(x	ηr(x	X
ejde-741	186	14	)	)	PUNCT
ejde-741	187	1	[	[	X
ejde-741	187	2	t(t	t(t	NOUN
ejde-741	187	3	−	−	PROPN
ejde-741	187	4	t)]4	t)]4	NOUN
ejde-741	187	5	,	,	PUNCT
ejde-741	187	6	where	where	SCONJ
ejde-741	187	7	(	(	PUNCT
ejde-741	187	8	t	t	PROPN
ejde-741	187	9	,	,	PUNCT
ejde-741	187	10	x	x	NOUN
ejde-741	187	11	)	)	PUNCT
ejde-741	187	12	∈	∈	PROPN
ejde-741	187	13	(	(	PUNCT
ejde-741	187	14	0	0	NUM
ejde-741	187	15	,	,	PUNCT
ejde-741	187	16	t	t	PROPN
ejde-741	187	17	)	)	PUNCT
ejde-741	187	18	×	×	NOUN
ejde-741	188	1	[	[	X
ejde-741	188	2	0	0	NUM
ejde-741	188	3	,	,	PUNCT
ejde-741	188	4	1	1	NUM
ejde-741	188	5	]	]	PUNCT
ejde-741	188	6	.	.	PUNCT
ejde-741	189	1	our	our	PRON
ejde-741	189	2	desired	desire	VERB
ejde-741	189	3	carleman	carleman	NOUN
ejde-741	189	4	and	and	CCONJ
ejde-741	189	5	observability	observability	NOUN
ejde-741	189	6	inequalities	inequality	NOUN
ejde-741	189	7	,	,	PUNCT
ejde-741	189	8	mentioned	mention	VERB
ejde-741	189	9	above	above	ADV
ejde-741	189	10	,	,	PUNCT
ejde-741	189	11	will	will	AUX
ejde-741	189	12	be	be	AUX
ejde-741	189	13	achieved	achieve	VERB
ejde-741	189	14	as	as	ADP
ejde-741	189	15	consequences	consequence	NOUN
ejde-741	189	16	of	of	ADP
ejde-741	189	17	the	the	DET
ejde-741	189	18	next	next	ADJ
ejde-741	189	19	lemma	lemma	PROPN
ejde-741	189	20	,	,	PUNCT
ejde-741	189	21	whose	whose	DET
ejde-741	189	22	proof	proof	NOUN
ejde-741	189	23	can	can	AUX
ejde-741	189	24	be	be	AUX
ejde-741	189	25	found	find	VERB
ejde-741	189	26	in	in	ADP
ejde-741	189	27	[	[	X
ejde-741	189	28	20	20	NUM
ejde-741	189	29	]	]	PUNCT
ejde-741	189	30	,	,	PUNCT
ejde-741	189	31	for	for	ADP
ejde-741	189	32	the	the	DET
ejde-741	189	33	(	(	PUNCT
ejde-741	189	34	wdc	wdc	PROPN
ejde-741	189	35	)	)	PUNCT
ejde-741	189	36	,	,	PUNCT
ejde-741	189	37	and	and	CCONJ
ejde-741	189	38	in	in	ADP
ejde-741	189	39	[	[	X
ejde-741	189	40	19	19	NUM
ejde-741	189	41	]	]	PUNCT
ejde-741	189	42	,	,	PUNCT
ejde-741	189	43	for	for	ADP
ejde-741	189	44	the	the	DET
ejde-741	189	45	(	(	PUNCT
ejde-741	189	46	sdc	sdc	NOUN
ejde-741	189	47	)	)	PUNCT
ejde-741	189	48	.	.	PUNCT
ejde-741	190	1	lemma	lemma	PROPN
ejde-741	190	2	2.5	2.5	NUM
ejde-741	190	3	.	.	PUNCT
ejde-741	191	1	there	there	PRON
ejde-741	191	2	exist	exist	VERB
ejde-741	191	3	c	c	PROPN
ejde-741	191	4	>	>	X
ejde-741	191	5	0	0	PUNCT
ejde-741	191	6	and	and	CCONJ
ejde-741	191	7	λ0	λ0	NOUN
ejde-741	191	8	,	,	PUNCT
ejde-741	191	9	s0	s0	PROPN
ejde-741	191	10	>	>	X
ejde-741	191	11	0	0	NUM
ejde-741	192	1	such	such	ADJ
ejde-741	192	2	that	that	SCONJ
ejde-741	192	3	every	every	DET
ejde-741	192	4	solution	solution	NOUN
ejde-741	192	5	v	v	ADP
ejde-741	192	6	of	of	ADP
ejde-741	192	7	(	(	PUNCT
ejde-741	192	8	2.11	2.11	NUM
ejde-741	192	9	)	)	PUNCT
ejde-741	192	10	satisfies	satisfie	NOUN
ejde-741	192	11	,	,	PUNCT
ejde-741	192	12	for	for	ADP
ejde-741	192	13	all	all	DET
ejde-741	192	14	s	s	PART
ejde-741	192	15	≥	≥	NOUN
ejde-741	192	16	s0	s0	NOUN
ejde-741	192	17	and	and	CCONJ
ejde-741	192	18	λ	λ	PROPN
ejde-741	192	19	≥	≥	NOUN
ejde-741	192	20	λ0	λ0	NOUN
ejde-741	192	21	,	,	PUNCT
ejde-741	192	22	the	the	DET
ejde-741	192	23	estimate∫	estimate∫	PROPN
ejde-741	192	24	t	t	PROPN
ejde-741	192	25	0	0	NUM
ejde-741	192	26	∫	∫	PROPN
ejde-741	192	27	1	1	NUM
ejde-741	192	28	0	0	NUM
ejde-741	192	29	e2sφ	e2sφ	PUNCT
ejde-741	192	30	(	(	PUNCT
ejde-741	192	31	(	(	PUNCT
ejde-741	192	32	sλ)σav2x	sλ)σav2x	VERB
ejde-741	192	33	+	+	CCONJ
ejde-741	192	34	(	(	PUNCT
ejde-741	192	35	sλ)5/3σ5/3v2	sλ)5/3σ5/3v2	ADJ
ejde-741	192	36	)	)	PUNCT
ejde-741	192	37	≤	≤	NUM
ejde-741	192	38	c	c	X
ejde-741	192	39	(	(	PUNCT
ejde-741	192	40	∫	∫	PROPN
ejde-741	192	41	t	t	PROPN
ejde-741	192	42	0	0	NUM
ejde-741	192	43	∫	∫	PROPN
ejde-741	192	44	1	1	NUM
ejde-741	192	45	0	0	NUM
ejde-741	192	46	e2sφ|f	e2sφ|f	NOUN
ejde-741	192	47	|2	|2	NUM
ejde-741	192	48	+	+	CCONJ
ejde-741	192	49	(	(	PUNCT
ejde-741	192	50	λs)17/3	λs)17/3	PROPN
ejde-741	192	51	∫	∫	PROPN
ejde-741	193	1	t	t	PROPN
ejde-741	193	2	0	0	NUM
ejde-741	193	3	∫	∫	PROPN
ejde-741	193	4	ω	ω	NUM
ejde-741	193	5	e2sφσ17/3v2	e2sφσ17/3v2	PROPN
ejde-741	193	6	)	)	PUNCT
ejde-741	193	7	,	,	PUNCT
ejde-741	193	8	(	(	PUNCT
ejde-741	193	9	2.12	2.12	NUM
ejde-741	193	10	)	)	PUNCT
ejde-741	193	11	where	where	SCONJ
ejde-741	193	12	the	the	DET
ejde-741	193	13	constants	constant	NOUN
ejde-741	193	14	c	c	NOUN
ejde-741	193	15	,	,	PUNCT
ejde-741	193	16	λ0	λ0	NOUN
ejde-741	193	17	and	and	CCONJ
ejde-741	193	18	s0	s0	NOUN
ejde-741	193	19	only	only	ADV
ejde-741	193	20	depend	depend	VERB
ejde-741	193	21	on	on	ADP
ejde-741	193	22	ω	ω	PROPN
ejde-741	193	23	,	,	PUNCT
ejde-741	193	24	a	a	PRON
ejde-741	193	25	,	,	PUNCT
ejde-741	193	26	∥c∥l∞(q	∥c∥l∞(q	ADJ
ejde-741	193	27	)	)	PUNCT
ejde-741	193	28	and	and	CCONJ
ejde-741	193	29	t	t	PROPN
ejde-741	193	30	.	.	PUNCT
ejde-741	194	1	the	the	DET
ejde-741	194	2	functions	function	NOUN
ejde-741	194	3	in	in	ADP
ejde-741	194	4	(	(	PUNCT
ejde-741	194	5	2.8	2.8	NUM
ejde-741	194	6	)	)	PUNCT
ejde-741	194	7	were	be	AUX
ejde-741	194	8	not	not	PART
ejde-741	194	9	directly	directly	ADV
ejde-741	194	10	based	base	VERB
ejde-741	194	11	on	on	ADP
ejde-741	194	12	the	the	DET
ejde-741	194	13	weights	weight	NOUN
ejde-741	194	14	which	which	PRON
ejde-741	194	15	appear	appear	VERB
ejde-741	194	16	in	in	ADP
ejde-741	194	17	(	(	PUNCT
ejde-741	194	18	2.12	2.12	NUM
ejde-741	194	19	)	)	PUNCT
ejde-741	194	20	,	,	PUNCT
ejde-741	194	21	because	because	SCONJ
ejde-741	194	22	lim	lim	PROPN
ejde-741	194	23	t→0	t→0	PROPN
ejde-741	194	24	+	+	PROPN
ejde-741	195	1	1	1	NUM
ejde-741	195	2	[	[	X
ejde-741	195	3	t(t	t(t	NOUN
ejde-741	195	4	−	−	NOUN
ejde-741	195	5	t)]4	t)]4	NOUN
ejde-741	195	6	=	=	PUNCT
ejde-741	196	1	+	+	NUM
ejde-741	196	2	∞.	∞.	PROPN
ejde-741	196	3	instead	instead	ADV
ejde-741	196	4	of	of	ADP
ejde-741	196	5	that	that	PRON
ejde-741	196	6	,	,	PUNCT
ejde-741	196	7	we	we	PRON
ejde-741	196	8	have	have	AUX
ejde-741	196	9	taken	take	VERB
ejde-741	196	10	τ	τ	X
ejde-741	196	11	=	=	SYM
ejde-741	196	12	τ(t	τ(t	PROPN
ejde-741	196	13	)	)	PUNCT
ejde-741	196	14	in	in	ADP
ejde-741	196	15	order	order	NOUN
ejde-741	196	16	to	to	PART
ejde-741	196	17	build	build	VERB
ejde-741	196	18	ρ0	ρ0	PROPN
ejde-741	196	19	,	,	PUNCT
ejde-741	196	20	ρ̂	ρ̂	NUM
ejde-741	196	21	and	and	CCONJ
ejde-741	196	22	ρ∗	ρ∗	PROPN
ejde-741	196	23	(	(	PUNCT
ejde-741	196	24	recall	recall	PROPN
ejde-741	196	25	remark	remark	NOUN
ejde-741	196	26	2.4	2.4	NUM
ejde-741	196	27	)	)	PUNCT
ejde-741	196	28	.	.	PUNCT
ejde-741	197	1	8	8	NUM
ejde-741	198	1	p.	p.	NOUN
ejde-741	198	2	p.	p.	NOUN
ejde-741	198	3	de	de	PROPN
ejde-741	198	4	carvalho	carvalho	PROPN
ejde-741	198	5	,	,	PUNCT
ejde-741	198	6	r.	r.	PROPN
ejde-741	198	7	demarque	demarque	PROPN
ejde-741	198	8	,	,	PUNCT
ejde-741	198	9	j.	j.	PROPN
ejde-741	198	10	límaco	límaco	PROPN
ejde-741	198	11	,	,	PUNCT
ejde-741	198	12	l.	l.	PROPN
ejde-741	198	13	viana	viana	PROPN
ejde-741	198	14	ejde-2025/15	ejde-2025/15	PROPN
ejde-741	198	15	next	next	ADV
ejde-741	198	16	,	,	PUNCT
ejde-741	198	17	we	we	PRON
ejde-741	198	18	will	will	AUX
ejde-741	198	19	state	state	VERB
ejde-741	198	20	a	a	DET
ejde-741	198	21	new	new	ADJ
ejde-741	198	22	version	version	NOUN
ejde-741	198	23	of	of	ADP
ejde-741	198	24	(	(	PUNCT
ejde-741	198	25	2.12	2.12	NUM
ejde-741	198	26	)	)	PUNCT
ejde-741	198	27	,	,	PUNCT
ejde-741	198	28	involving	involve	VERB
ejde-741	198	29	the	the	DET
ejde-741	198	30	weights	weight	NOUN
ejde-741	198	31	given	give	VERB
ejde-741	198	32	in	in	ADP
ejde-741	198	33	(	(	PUNCT
ejde-741	198	34	2.8	2.8	NUM
ejde-741	198	35	)	)	PUNCT
ejde-741	198	36	,	,	PUNCT
ejde-741	198	37	whose	whose	DET
ejde-741	198	38	proof	proof	NOUN
ejde-741	198	39	can	can	AUX
ejde-741	198	40	be	be	AUX
ejde-741	198	41	done	do	VERB
ejde-741	198	42	following	follow	VERB
ejde-741	198	43	the	the	DET
ejde-741	198	44	same	same	ADJ
ejde-741	198	45	steps	step	NOUN
ejde-741	198	46	of	of	ADP
ejde-741	198	47	[	[	X
ejde-741	198	48	19	19	NUM
ejde-741	198	49	,	,	PUNCT
ejde-741	198	50	prop	prop	NOUN
ejde-741	198	51	.	.	PUNCT
ejde-741	198	52	3.6	3.6	NUM
ejde-741	198	53	]	]	PUNCT
ejde-741	198	54	.	.	PUNCT
ejde-741	199	1	proposition	proposition	NOUN
ejde-741	199	2	2.6	2.6	NUM
ejde-741	199	3	(	(	PUNCT
ejde-741	199	4	carleman	carleman	ADJ
ejde-741	199	5	inequality	inequality	NOUN
ejde-741	199	6	)	)	PUNCT
ejde-741	199	7	.	.	PUNCT
ejde-741	200	1	there	there	PRON
ejde-741	200	2	exist	exist	VERB
ejde-741	200	3	c	c	PROPN
ejde-741	200	4	>	>	X
ejde-741	200	5	0	0	PUNCT
ejde-741	200	6	and	and	CCONJ
ejde-741	200	7	λ0	λ0	NOUN
ejde-741	200	8	,	,	PUNCT
ejde-741	200	9	s0	s0	PROPN
ejde-741	200	10	>	>	X
ejde-741	200	11	0	0	NUM
ejde-741	201	1	such	such	ADJ
ejde-741	201	2	that	that	SCONJ
ejde-741	201	3	every	every	DET
ejde-741	201	4	solution	solution	NOUN
ejde-741	201	5	v	v	ADP
ejde-741	201	6	of	of	ADP
ejde-741	201	7	(	(	PUNCT
ejde-741	201	8	2.11	2.11	NUM
ejde-741	201	9	)	)	PUNCT
ejde-741	201	10	satisfies	satisfie	NOUN
ejde-741	201	11	,	,	PUNCT
ejde-741	201	12	for	for	ADP
ejde-741	201	13	all	all	DET
ejde-741	201	14	s	s	PART
ejde-741	201	15	≥	≥	NOUN
ejde-741	201	16	s0	s0	NOUN
ejde-741	201	17	and	and	CCONJ
ejde-741	201	18	λ	λ	PROPN
ejde-741	201	19	≥	≥	NOUN
ejde-741	201	20	λ0	λ0	NOUN
ejde-741	201	21	,	,	PUNCT
ejde-741	201	22	the	the	DET
ejde-741	201	23	estimate∫	estimate∫	PROPN
ejde-741	201	24	t	t	PROPN
ejde-741	201	25	0	0	NUM
ejde-741	201	26	∫	∫	PROPN
ejde-741	201	27	1	1	NUM
ejde-741	201	28	0	0	NUM
ejde-741	201	29	e2sa	e2sa	PUNCT
ejde-741	202	1	[	[	PUNCT
ejde-741	202	2	sλζa|vx|2	sλζa|vx|2	PROPN
ejde-741	202	3	+	+	CCONJ
ejde-741	202	4	(	(	PUNCT
ejde-741	202	5	sλ)5/3ζ5/3|v|2	sλ)5/3ζ5/3|v|2	X
ejde-741	202	6	]	]	PUNCT
ejde-741	202	7	≤	≤	NUM
ejde-741	202	8	c	c	X
ejde-741	202	9	(	(	PUNCT
ejde-741	202	10	∫	∫	PROPN
ejde-741	202	11	t	t	PROPN
ejde-741	202	12	0	0	NUM
ejde-741	202	13	∫	∫	PROPN
ejde-741	202	14	1	1	NUM
ejde-741	202	15	0	0	NUM
ejde-741	202	16	e2sa|f	e2sa|f	NOUN
ejde-741	202	17	|2	|2	NUM
ejde-741	203	1	+	+	CCONJ
ejde-741	203	2	(	(	PUNCT
ejde-741	203	3	sλ)17/3	sλ)17/3	NUM
ejde-741	204	1	∫	∫	PROPN
ejde-741	204	2	t	t	PROPN
ejde-741	204	3	0	0	NUM
ejde-741	205	1	∫	∫	PROPN
ejde-741	206	1	ω	ω	PROPN
ejde-741	206	2	e2sa(ζ∗)17/3|v|2	e2sa(ζ∗)17/3|v|2	PROPN
ejde-741	206	3	)	)	PUNCT
ejde-741	206	4	,	,	PUNCT
ejde-741	206	5	(	(	PUNCT
ejde-741	206	6	2.13	2.13	NUM
ejde-741	206	7	)	)	PUNCT
ejde-741	206	8	where	where	SCONJ
ejde-741	206	9	the	the	DET
ejde-741	206	10	constants	constant	NOUN
ejde-741	206	11	c	c	NOUN
ejde-741	206	12	,	,	PUNCT
ejde-741	206	13	λ0	λ0	NOUN
ejde-741	206	14	and	and	CCONJ
ejde-741	206	15	s0	s0	NOUN
ejde-741	206	16	only	only	ADV
ejde-741	206	17	depend	depend	VERB
ejde-741	206	18	on	on	ADP
ejde-741	206	19	ω	ω	PROPN
ejde-741	206	20	,	,	PUNCT
ejde-741	206	21	a	a	PRON
ejde-741	206	22	,	,	PUNCT
ejde-741	206	23	∥c∥l∞(q	∥c∥l∞(q	ADJ
ejde-741	206	24	)	)	PUNCT
ejde-741	206	25	and	and	CCONJ
ejde-741	206	26	t	t	PROPN
ejde-741	206	27	.	.	PUNCT
ejde-741	207	1	we	we	PRON
ejde-741	207	2	would	would	AUX
ejde-741	207	3	like	like	VERB
ejde-741	207	4	to	to	PART
ejde-741	207	5	complete	complete	VERB
ejde-741	207	6	this	this	DET
ejde-741	207	7	section	section	NOUN
ejde-741	207	8	making	make	VERB
ejde-741	207	9	some	some	DET
ejde-741	207	10	brief	brief	ADJ
ejde-741	207	11	comments	comment	NOUN
ejde-741	207	12	about	about	ADP
ejde-741	207	13	lemma	lemma	PROPN
ejde-741	207	14	2.5	2.5	NUM
ejde-741	207	15	and	and	CCONJ
ejde-741	207	16	proposition	proposition	NOUN
ejde-741	207	17	2.6	2.6	NUM
ejde-741	207	18	.	.	PUNCT
ejde-741	208	1	(	(	PUNCT
ejde-741	208	2	a	a	X
ejde-741	208	3	)	)	PUNCT
ejde-741	208	4	for	for	ADP
ejde-741	208	5	the	the	DET
ejde-741	208	6	(	(	PUNCT
ejde-741	208	7	wdc	wdc	PROPN
ejde-741	208	8	)	)	PUNCT
ejde-741	208	9	,	,	PUNCT
ejde-741	208	10	lemma	lemma	PROPN
ejde-741	208	11	2.5	2.5	NUM
ejde-741	208	12	and	and	CCONJ
ejde-741	208	13	proposition	proposition	NOUN
ejde-741	208	14	2.6	2.6	NUM
ejde-741	208	15	are	be	AUX
ejde-741	208	16	both	both	PRON
ejde-741	208	17	detailed	detailed	ADJ
ejde-741	208	18	in	in	ADP
ejde-741	208	19	[	[	X
ejde-741	208	20	20	20	NUM
ejde-741	208	21	]	]	PUNCT
ejde-741	208	22	.	.	PUNCT
ejde-741	209	1	in	in	ADP
ejde-741	209	2	that	that	DET
ejde-741	209	3	paper	paper	NOUN
ejde-741	209	4	,	,	PUNCT
ejde-741	209	5	the	the	DET
ejde-741	209	6	discussion	discussion	NOUN
ejde-741	209	7	is	be	AUX
ejde-741	209	8	organized	organize	VERB
ejde-741	209	9	under	under	ADP
ejde-741	209	10	the	the	DET
ejde-741	209	11	presentation	presentation	NOUN
ejde-741	209	12	of	of	ADP
ejde-741	209	13	several	several	ADJ
ejde-741	209	14	technical	technical	ADJ
ejde-741	209	15	lemmas	lemma	NOUN
ejde-741	209	16	,	,	PUNCT
ejde-741	209	17	whose	whose	DET
ejde-741	209	18	proofs	proof	NOUN
ejde-741	209	19	rely	rely	VERB
ejde-741	209	20	on	on	ADP
ejde-741	209	21	energy	energy	NOUN
ejde-741	209	22	estimates	estimate	NOUN
ejde-741	209	23	,	,	PUNCT
ejde-741	209	24	as	as	ADV
ejde-741	209	25	well	well	ADV
ejde-741	209	26	as	as	ADP
ejde-741	209	27	on	on	ADP
ejde-741	209	28	the	the	DET
ejde-741	209	29	crucial	crucial	ADJ
ejde-741	209	30	hardy	hardy	ADJ
ejde-741	209	31	-	-	PUNCT
ejde-741	209	32	poincaré	poincaré	ADJ
ejde-741	209	33	inequality	inequality	NOUN
ejde-741	209	34	proved	prove	VERB
ejde-741	209	35	in	in	ADP
ejde-741	209	36	[	[	X
ejde-741	209	37	1	1	NUM
ejde-741	209	38	]	]	PUNCT
ejde-741	209	39	;	;	PUNCT
ejde-741	209	40	(	(	PUNCT
ejde-741	209	41	b	b	X
ejde-741	209	42	)	)	PUNCT
ejde-741	209	43	for	for	ADP
ejde-741	209	44	the	the	DET
ejde-741	209	45	(	(	PUNCT
ejde-741	209	46	sdc	sdc	NOUN
ejde-741	209	47	)	)	PUNCT
ejde-741	209	48	,	,	PUNCT
ejde-741	209	49	we	we	PRON
ejde-741	209	50	follow	follow	VERB
ejde-741	209	51	the	the	DET
ejde-741	209	52	same	same	ADJ
ejde-741	209	53	strategy	strategy	NOUN
ejde-741	209	54	used	use	VERB
ejde-741	209	55	for	for	ADP
ejde-741	209	56	the	the	DET
ejde-741	209	57	(	(	PUNCT
ejde-741	209	58	wdc	wdc	PROPN
ejde-741	209	59	)	)	PUNCT
ejde-741	209	60	.	.	PUNCT
ejde-741	210	1	however	however	ADV
ejde-741	210	2	,	,	PUNCT
ejde-741	210	3	considering	consider	VERB
ejde-741	210	4	k	k	PROPN
ejde-741	210	5	∈	∈	PROPN
ejde-741	211	1	[	[	X
ejde-741	211	2	1	1	NUM
ejde-741	211	3	,	,	PUNCT
ejde-741	211	4	2	2	NUM
ejde-741	211	5	)	)	PUNCT
ejde-741	211	6	,	,	PUNCT
ejde-741	211	7	as	as	ADP
ejde-741	211	8	in	in	ADP
ejde-741	211	9	assumption	assumption	NOUN
ejde-741	211	10	1.1	1.1	NUM
ejde-741	211	11	,	,	PUNCT
ejde-741	211	12	the	the	DET
ejde-741	211	13	case	case	NOUN
ejde-741	211	14	k	k	NOUN
ejde-741	211	15	=	=	SYM
ejde-741	211	16	1	1	NUM
ejde-741	211	17	deserves	deserve	VERB
ejde-741	211	18	a	a	DET
ejde-741	211	19	special	special	ADJ
ejde-741	211	20	attention	attention	NOUN
ejde-741	211	21	(	(	PUNCT
ejde-741	211	22	precisely	precisely	ADV
ejde-741	211	23	,	,	PUNCT
ejde-741	211	24	see	see	VERB
ejde-741	211	25	[	[	X
ejde-741	211	26	19	19	NUM
ejde-741	211	27	,	,	PUNCT
ejde-741	211	28	lemma	lemma	PROPN
ejde-741	211	29	3.2	3.2	NUM
ejde-741	211	30	]	]	PUNCT
ejde-741	211	31	)	)	PUNCT
ejde-741	211	32	;	;	PUNCT
ejde-741	211	33	(	(	PUNCT
ejde-741	211	34	c	c	X
ejde-741	211	35	)	)	PUNCT
ejde-741	211	36	the	the	DET
ejde-741	211	37	definition	definition	NOUN
ejde-741	211	38	of	of	ADP
ejde-741	211	39	ρ0	ρ0	PROPN
ejde-741	211	40	in	in	ADV
ejde-741	211	41	(	(	PUNCT
ejde-741	211	42	2.8	2.8	NUM
ejde-741	211	43	)	)	PUNCT
ejde-741	211	44	is	be	AUX
ejde-741	211	45	inspired	inspire	VERB
ejde-741	211	46	by	by	ADP
ejde-741	211	47	the	the	DET
ejde-741	211	48	integral	integral	ADJ
ejde-741	211	49	(	(	PUNCT
ejde-741	211	50	sλ)5/3	sλ)5/3	ADJ
ejde-741	211	51	∫	∫	PROPN
ejde-741	211	52	t	t	PROPN
ejde-741	211	53	0	0	NUM
ejde-741	211	54	∫	∫	PROPN
ejde-741	211	55	1	1	NUM
ejde-741	211	56	0	0	NUM
ejde-741	211	57	e2saζ5/3|v|2	e2saζ5/3|v|2	NOUN
ejde-741	211	58	,	,	PUNCT
ejde-741	211	59	which	which	PRON
ejde-741	211	60	appears	appear	VERB
ejde-741	211	61	in	in	ADP
ejde-741	211	62	(	(	PUNCT
ejde-741	211	63	2.13	2.13	NUM
ejde-741	211	64	)	)	PUNCT
ejde-741	211	65	,	,	PUNCT
ejde-741	211	66	according	accord	VERB
ejde-741	211	67	to	to	ADP
ejde-741	211	68	a	a	DET
ejde-741	211	69	standard	standard	ADJ
ejde-741	211	70	argument	argument	NOUN
ejde-741	211	71	.	.	PUNCT
ejde-741	212	1	likewise	likewise	ADV
ejde-741	212	2	,	,	PUNCT
ejde-741	212	3	ρ̂	ρ̂	NUM
ejde-741	212	4	and	and	CCONJ
ejde-741	212	5	ρ∗	ρ∗	PROPN
ejde-741	212	6	are	be	AUX
ejde-741	212	7	set	set	VERB
ejde-741	212	8	having	have	VERB
ejde-741	212	9	in	in	ADP
ejde-741	212	10	mind	mind	NOUN
ejde-741	212	11	the	the	DET
ejde-741	212	12	definition	definition	NOUN
ejde-741	212	13	of	of	ADP
ejde-741	212	14	ρ0	ρ0	PROPN
ejde-741	212	15	,	,	PUNCT
ejde-741	212	16	however	however	ADV
ejde-741	212	17	,	,	PUNCT
ejde-741	212	18	there	there	PRON
ejde-741	212	19	are	be	VERB
ejde-741	212	20	technical	technical	ADJ
ejde-741	212	21	reasons	reason	NOUN
ejde-741	212	22	to	to	PART
ejde-741	212	23	consider	consider	VERB
ejde-741	212	24	ā	ā	ADJ
ejde-741	212	25	=	=	SYM
ejde-741	212	26	ā(t	ā(t	PROPN
ejde-741	212	27	)	)	PUNCT
ejde-741	212	28	and	and	CCONJ
ejde-741	212	29	ζ∗	ζ∗	PROPN
ejde-741	212	30	=	=	SYM
ejde-741	212	31	ζ∗(t	ζ∗(t	NOUN
ejde-741	212	32	)	)	PUNCT
ejde-741	212	33	in	in	ADP
ejde-741	212	34	their	their	PRON
ejde-741	212	35	expressions	expression	NOUN
ejde-741	212	36	;	;	PUNCT
ejde-741	212	37	(	(	PUNCT
ejde-741	212	38	d	d	X
ejde-741	212	39	)	)	PUNCT
ejde-741	212	40	it	it	PRON
ejde-741	212	41	is	be	AUX
ejde-741	212	42	well	well	ADV
ejde-741	212	43	-	-	PUNCT
ejde-741	212	44	known	know	VERB
ejde-741	212	45	that	that	SCONJ
ejde-741	212	46	(	(	PUNCT
ejde-741	212	47	2.13	2.13	NUM
ejde-741	212	48	)	)	PUNCT
ejde-741	212	49	implies	imply	VERB
ejde-741	212	50	the	the	DET
ejde-741	212	51	observability	observability	NOUN
ejde-741	212	52	inequality	inequality	NOUN
ejde-741	212	53	∥v(0)∥2l2(0,1	∥v(0)∥2l2(0,1	NOUN
ejde-741	212	54	)	)	PUNCT
ejde-741	212	55	≤	≤	NUM
ejde-741	213	1	c	c	X
ejde-741	213	2	∫	∫	PROPN
ejde-741	213	3	t	t	PROPN
ejde-741	213	4	0	0	NUM
ejde-741	213	5	∫	∫	PROPN
ejde-741	213	6	ω	ω	NUM
ejde-741	213	7	e2sa(sλ)17/3(ζ∗	e2sa(sλ)17/3(ζ∗	PROPN
ejde-741	213	8	)	)	PUNCT
ejde-741	213	9	17/3|v|2	17/3|v|2	PROPN
ejde-741	213	10	,	,	PUNCT
ejde-741	213	11	(	(	PUNCT
ejde-741	213	12	2.14	2.14	NUM
ejde-741	213	13	)	)	PUNCT
ejde-741	213	14	valid	valid	NOUN
ejde-741	213	15	for	for	ADP
ejde-741	213	16	any	any	DET
ejde-741	213	17	solution	solution	NOUN
ejde-741	213	18	v	v	ADP
ejde-741	213	19	of	of	ADP
ejde-741	213	20	(	(	PUNCT
ejde-741	213	21	2.11	2.11	NUM
ejde-741	213	22	)	)	PUNCT
ejde-741	213	23	,	,	PUNCT
ejde-741	213	24	with	with	ADP
ejde-741	213	25	f	f	PROPN
ejde-741	213	26	≡	≡	PROPN
ejde-741	213	27	0	0	PROPN
ejde-741	213	28	.	.	PUNCT
ejde-741	214	1	in	in	ADP
ejde-741	214	2	fact	fact	NOUN
ejde-741	214	3	,	,	PUNCT
ejde-741	214	4	this	this	DET
ejde-741	214	5	inequality	inequality	NOUN
ejde-741	214	6	holds	hold	VERB
ejde-741	214	7	if	if	SCONJ
ejde-741	214	8	λ	λ	PROPN
ejde-741	214	9	>	>	X
ejde-741	214	10	0	0	PUNCT
ejde-741	214	11	and	and	CCONJ
ejde-741	214	12	s	s	X
ejde-741	214	13	>	>	X
ejde-741	214	14	0	0	NUM
ejde-741	214	15	are	be	AUX
ejde-741	214	16	sufficiently	sufficiently	ADV
ejde-741	214	17	large	large	ADJ
ejde-741	214	18	.	.	PUNCT
ejde-741	215	1	3	3	X
ejde-741	215	2	.	.	X
ejde-741	215	3	properties	property	NOUN
ejde-741	215	4	of	of	ADP
ejde-741	215	5	the	the	DET
ejde-741	215	6	mapping	mapping	NOUN
ejde-741	215	7	h	h	NOUN
ejde-741	215	8	this	this	DET
ejde-741	215	9	section	section	NOUN
ejde-741	215	10	we	we	PRON
ejde-741	215	11	prove	prove	VERB
ejde-741	215	12	the	the	DET
ejde-741	215	13	properties	property	NOUN
ejde-741	215	14	defined	define	VERB
ejde-741	215	15	in	in	ADP
ejde-741	215	16	(	(	PUNCT
ejde-741	215	17	2.1	2.1	NUM
ejde-741	215	18	)	)	PUNCT
ejde-741	215	19	,	,	PUNCT
ejde-741	215	20	which	which	PRON
ejde-741	215	21	required	require	VERB
ejde-741	215	22	to	to	PART
ejde-741	215	23	apply	apply	VERB
ejde-741	215	24	lyusternik	lyusternik	PROPN
ejde-741	215	25	’s	’s	PART
ejde-741	215	26	theorem	theorem	ADJ
ejde-741	215	27	,	,	PUNCT
ejde-741	215	28	namely	namely	ADV
ejde-741	215	29	:	:	PUNCT
ejde-741	215	30	h′(0	h′(0	ADJ
ejde-741	215	31	,	,	PUNCT
ejde-741	215	32	0	0	NUM
ejde-741	215	33	)	)	PUNCT
ejde-741	215	34	must	must	AUX
ejde-741	215	35	be	be	AUX
ejde-741	215	36	onto	onto	ADP
ejde-741	215	37	and	and	CCONJ
ejde-741	215	38	h	h	NOUN
ejde-741	215	39	must	must	AUX
ejde-741	215	40	belong	belong	VERB
ejde-741	215	41	to	to	ADP
ejde-741	215	42	c1(e	c1(e	PROPN
ejde-741	215	43	,	,	PUNCT
ejde-741	215	44	f	f	PROPN
ejde-741	215	45	)	)	PUNCT
ejde-741	215	46	.	.	PUNCT
ejde-741	216	1	these	these	DET
ejde-741	216	2	tasks	task	NOUN
ejde-741	216	3	will	will	AUX
ejde-741	216	4	be	be	AUX
ejde-741	216	5	done	do	VERB
ejde-741	216	6	in	in	ADP
ejde-741	216	7	the	the	DET
ejde-741	216	8	next	next	ADJ
ejde-741	216	9	two	two	NUM
ejde-741	216	10	subsections	subsection	NOUN
ejde-741	216	11	.	.	PUNCT
ejde-741	217	1	however	however	ADV
ejde-741	217	2	,	,	PUNCT
ejde-741	217	3	before	before	ADP
ejde-741	217	4	presenting	present	VERB
ejde-741	217	5	them	they	PRON
ejde-741	217	6	,	,	PUNCT
ejde-741	217	7	let	let	VERB
ejde-741	217	8	us	we	PRON
ejde-741	217	9	state	state	VERB
ejde-741	217	10	a	a	DET
ejde-741	217	11	global	global	ADJ
ejde-741	217	12	null	null	ADJ
ejde-741	217	13	-	-	PUNCT
ejde-741	217	14	controllability	controllability	NOUN
ejde-741	217	15	result	result	NOUN
ejde-741	217	16	for	for	ADP
ejde-741	217	17	the	the	DET
ejde-741	217	18	linearized	linearize	VERB
ejde-741	217	19	system	system	NOUN
ejde-741	217	20	(	(	PUNCT
ejde-741	217	21	2.2	2.2	NUM
ejde-741	217	22	)	)	PUNCT
ejde-741	217	23	,	,	PUNCT
ejde-741	217	24	as	as	ADV
ejde-741	217	25	well	well	ADV
ejde-741	217	26	as	as	ADP
ejde-741	217	27	some	some	DET
ejde-741	217	28	additional	additional	ADJ
ejde-741	217	29	regularity	regularity	NOUN
ejde-741	217	30	of	of	ADP
ejde-741	217	31	this	this	DET
ejde-741	217	32	system	system	NOUN
ejde-741	217	33	that	that	PRON
ejde-741	217	34	,	,	PUNCT
ejde-741	217	35	as	as	SCONJ
ejde-741	217	36	we	we	PRON
ejde-741	217	37	have	have	AUX
ejde-741	217	38	already	already	ADV
ejde-741	217	39	pointed	point	VERB
ejde-741	217	40	out	out	ADP
ejde-741	217	41	in	in	ADP
ejde-741	217	42	section	section	NOUN
ejde-741	217	43	2	2	NUM
ejde-741	217	44	,	,	PUNCT
ejde-741	217	45	will	will	AUX
ejde-741	217	46	be	be	AUX
ejde-741	217	47	necessary	necessary	ADJ
ejde-741	217	48	to	to	PART
ejde-741	217	49	check	check	VERB
ejde-741	217	50	the	the	DET
ejde-741	217	51	required	require	VERB
ejde-741	217	52	hypotheses	hypothesis	NOUN
ejde-741	217	53	over	over	ADP
ejde-741	217	54	h.	h.	NOUN
ejde-741	217	55	its	its	PRON
ejde-741	217	56	proof	proof	NOUN
ejde-741	217	57	will	will	AUX
ejde-741	217	58	be	be	AUX
ejde-741	217	59	given	give	VERB
ejde-741	217	60	at	at	ADP
ejde-741	217	61	the	the	DET
ejde-741	217	62	end	end	NOUN
ejde-741	217	63	of	of	ADP
ejde-741	217	64	this	this	DET
ejde-741	217	65	section	section	NOUN
ejde-741	217	66	.	.	PUNCT
ejde-741	218	1	proposition	proposition	NOUN
ejde-741	218	2	3.1	3.1	NUM
ejde-741	218	3	.	.	PUNCT
ejde-741	219	1	if	if	SCONJ
ejde-741	219	2	t	t	PROPN
ejde-741	219	3	>	>	X
ejde-741	219	4	0	0	PUNCT
ejde-741	220	1	and	and	CCONJ
ejde-741	220	2	(	(	PUNCT
ejde-741	220	3	u0	u0	ADJ
ejde-741	220	4	,	,	PUNCT
ejde-741	220	5	g	g	NOUN
ejde-741	220	6	)	)	PUNCT
ejde-741	220	7	∈	∈	PROPN
ejde-741	220	8	h1	h1	VERB
ejde-741	220	9	a	a	DET
ejde-741	220	10	×	×	PROPN
ejde-741	220	11	l2(q	l2(q	PROPN
ejde-741	220	12	;	;	PUNCT
ejde-741	220	13	ρ20	ρ20	NUM
ejde-741	220	14	)	)	PUNCT
ejde-741	220	15	,	,	PUNCT
ejde-741	220	16	then	then	ADV
ejde-741	220	17	there	there	PRON
ejde-741	220	18	exists	exist	VERB
ejde-741	220	19	a	a	DET
ejde-741	220	20	state	state	NOUN
ejde-741	220	21	-	-	PUNCT
ejde-741	220	22	control	control	NOUN
ejde-741	220	23	pair	pair	NOUN
ejde-741	220	24	(	(	PUNCT
ejde-741	220	25	u	u	NOUN
ejde-741	220	26	,	,	PUNCT
ejde-741	220	27	h	h	NOUN
ejde-741	220	28	)	)	PUNCT
ejde-741	220	29	∈	∈	PROPN
ejde-741	220	30	l2(q	l2(q	PROPN
ejde-741	220	31	;	;	PUNCT
ejde-741	220	32	ρ20	ρ20	NUM
ejde-741	220	33	)	)	PUNCT
ejde-741	220	34	×	×	PROPN
ejde-741	220	35	l2(qω	l2(qω	PROPN
ejde-741	220	36	;	;	PUNCT
ejde-741	220	37	ρ	ρ	NUM
ejde-741	220	38	2	2	NUM
ejde-741	220	39	∗	∗	NOUN
ejde-741	220	40	)	)	PUNCT
ejde-741	220	41	such	such	ADJ
ejde-741	220	42	that	that	SCONJ
ejde-741	220	43	the	the	DET
ejde-741	220	44	null	null	NOUN
ejde-741	220	45	-	-	PUNCT
ejde-741	220	46	controllability	controllability	NOUN
ejde-741	220	47	of	of	ADP
ejde-741	220	48	(	(	PUNCT
ejde-741	220	49	2.2	2.2	NUM
ejde-741	220	50	)	)	PUNCT
ejde-741	220	51	,	,	PUNCT
ejde-741	220	52	at	at	ADP
ejde-741	220	53	time	time	NOUN
ejde-741	220	54	t	t	PROPN
ejde-741	220	55	>	>	X
ejde-741	220	56	0	0	PROPN
ejde-741	220	57	,	,	PUNCT
ejde-741	220	58	holds	hold	VERB
ejde-741	220	59	.	.	PUNCT
ejde-741	221	1	furthermore	furthermore	ADV
ejde-741	221	2	,	,	PUNCT
ejde-741	221	3	we	we	PRON
ejde-741	221	4	have	have	VERB
ejde-741	221	5	√	√	NUM
ejde-741	221	6	aux	aux	PROPN
ejde-741	221	7	∈	∈	PROPN
ejde-741	221	8	l2(q	l2(q	PROPN
ejde-741	221	9	;	;	PUNCT
ejde-741	221	10	ρ̂2	ρ̂2	X
ejde-741	221	11	)	)	PUNCT
ejde-741	221	12	:	:	PUNCT
ejde-741	222	1	ut	ut	PROPN
ejde-741	222	2	,	,	PUNCT
ejde-741	222	3	(	(	PUNCT
ejde-741	222	4	aux)x	aux)x	PROPN
ejde-741	222	5	∈	∈	PROPN
ejde-741	222	6	l2(q	l2(q	PROPN
ejde-741	222	7	;	;	PUNCT
ejde-741	222	8	ρ̂2	ρ̂2	PROPN
ejde-741	222	9	)	)	PUNCT
ejde-741	222	10	,	,	PUNCT
ejde-741	222	11	ρ̂u	ρ̂u	NOUN
ejde-741	222	12	,	,	PUNCT
ejde-741	222	13	ρ∗	ρ∗	PROPN
ejde-741	222	14	√	√	NUM
ejde-741	222	15	aux	aux	PROPN
ejde-741	222	16	∈	∈	PROPN
ejde-741	222	17	l∞(0	l∞(0	PROPN
ejde-741	222	18	,	,	PUNCT
ejde-741	222	19	t	t	PROPN
ejde-741	222	20	;	;	PUNCT
ejde-741	222	21	l2(0	l2(0	NOUN
ejde-741	222	22	,	,	PUNCT
ejde-741	222	23	1	1	NUM
ejde-741	222	24	)	)	PUNCT
ejde-741	222	25	)	)	PUNCT
ejde-741	222	26	,	,	PUNCT
ejde-741	222	27	ejde-2025/15	ejde-2025/15	NOUN
ejde-741	222	28	null	null	ADJ
ejde-741	222	29	-	-	PUNCT
ejde-741	222	30	controllability	controllability	NOUN
ejde-741	222	31	degenerate	degenerate	ADJ
ejde-741	222	32	quasilinear	quasilinear	NOUN
ejde-741	222	33	equations	equation	NOUN
ejde-741	222	34	9	9	NUM
ejde-741	222	35	and	and	CCONJ
ejde-741	222	36	there	there	PRON
ejde-741	222	37	exists	exist	VERB
ejde-741	222	38	c	c	NOUN
ejde-741	222	39	>	>	X
ejde-741	222	40	0	0	NUM
ejde-741	222	41	such	such	ADJ
ejde-741	222	42	that	that	DET
ejde-741	222	43	sup	sup	NOUN
ejde-741	223	1	[	[	X
ejde-741	223	2	0,t	0,t	X
ejde-741	223	3	]	]	X
ejde-741	223	4	∥ρ̂u(t	∥ρ̂u(t	PROPN
ejde-741	223	5	,	,	PUNCT
ejde-741	223	6	·	·	PUNCT
ejde-741	223	7	)	)	PUNCT
ejde-741	223	8	∥2l2(0,1	∥2l2(0,1	ADV
ejde-741	223	9	)	)	PUNCT
ejde-741	224	1	+	+	CCONJ
ejde-741	224	2	sup	sup	NOUN
ejde-741	224	3	[	[	X
ejde-741	224	4	0,t	0,t	X
ejde-741	224	5	]	]	PUNCT
ejde-741	224	6	∥ρ∗	∥ρ∗	PROPN
ejde-741	224	7	√	√	PROPN
ejde-741	224	8	aux(t	aux(t	PROPN
ejde-741	224	9	,	,	PUNCT
ejde-741	224	10	·	·	PUNCT
ejde-741	224	11	)	)	PUNCT
ejde-741	224	12	∥2l2(0,1	∥2l2(0,1	ADV
ejde-741	224	13	)	)	PUNCT
ejde-741	225	1	+	+	CCONJ
ejde-741	225	2	∥	∥	NOUN
ejde-741	225	3	√	√	VERB
ejde-741	225	4	aux∥2ρ̂2	aux∥2ρ̂2	VERB
ejde-741	226	1	+	+	CCONJ
ejde-741	226	2	∥ut∥2ρ2∗	∥ut∥2ρ2∗	PROPN
ejde-741	226	3	+	+	NUM
ejde-741	226	4	∥(aux)x∥2ρ2∗	∥(aux)x∥2ρ2∗	NOUN
ejde-741	226	5	≤	≤	NUM
ejde-741	226	6	c(∥u∥2ρ20	c(∥u∥2ρ20	X
ejde-741	227	1	+	+	CCONJ
ejde-741	227	2	∥hχω∥2ρ2∗	∥hχω∥2ρ2∗	PROPN
ejde-741	227	3	+	+	CCONJ
ejde-741	227	4	∥g∥2ρ20	∥g∥2ρ20	X
ejde-741	227	5	+	+	CCONJ
ejde-741	227	6	∥u0∥2h1	∥u0∥2h1	NOUN
ejde-741	227	7	a	a	PRON
ejde-741	227	8	)	)	PUNCT
ejde-741	227	9	.	.	PUNCT
ejde-741	228	1	(	(	PUNCT
ejde-741	228	2	3.1	3.1	NUM
ejde-741	228	3	)	)	PUNCT
ejde-741	228	4	3.1	3.1	NUM
ejde-741	228	5	.	.	PUNCT
ejde-741	228	6	surjectiveness	surjectiveness	NOUN
ejde-741	228	7	of	of	ADP
ejde-741	228	8	h′(0,0	h′(0,0	NOUN
ejde-741	228	9	)	)	PUNCT
ejde-741	228	10	.	.	PUNCT
ejde-741	229	1	before	before	SCONJ
ejde-741	229	2	we	we	PRON
ejde-741	229	3	establish	establish	VERB
ejde-741	229	4	the	the	DET
ejde-741	229	5	properties	property	NOUN
ejde-741	229	6	of	of	ADP
ejde-741	229	7	h	h	NOUN
ejde-741	229	8	,	,	PUNCT
ejde-741	229	9	let	let	VERB
ejde-741	229	10	us	we	PRON
ejde-741	229	11	verify	verify	VERB
ejde-741	229	12	that	that	SCONJ
ejde-741	229	13	h	h	NOUN
ejde-741	229	14	is	be	AUX
ejde-741	229	15	well	well	ADV
ejde-741	229	16	defined	define	VERB
ejde-741	229	17	.	.	PUNCT
ejde-741	230	1	to	to	PART
ejde-741	230	2	check	check	VERB
ejde-741	230	3	that	that	PRON
ejde-741	230	4	,	,	PUNCT
ejde-741	230	5	it	it	PRON
ejde-741	230	6	will	will	AUX
ejde-741	230	7	be	be	AUX
ejde-741	230	8	essential	essential	ADJ
ejde-741	230	9	to	to	PART
ejde-741	230	10	know	know	VERB
ejde-741	230	11	that	that	SCONJ
ejde-741	230	12	au	au	ADV
ejde-741	230	13	∈	∈	PROPN
ejde-741	230	14	l∞(0	l∞(0	PRON
ejde-741	230	15	,	,	PUNCT
ejde-741	230	16	1	1	NUM
ejde-741	230	17	)	)	PUNCT
ejde-741	230	18	,	,	PUNCT
ejde-741	230	19	for	for	SCONJ
ejde-741	230	20	each	each	DET
ejde-741	230	21	u	u	PROPN
ejde-741	230	22	∈	∈	PROPN
ejde-741	230	23	h1	h1	VERB
ejde-741	230	24	a	a	PRON
ejde-741	230	25	.	.	PUNCT
ejde-741	231	1	it	it	PRON
ejde-741	231	2	is	be	AUX
ejde-741	231	3	always	always	ADV
ejde-741	231	4	true	true	ADJ
ejde-741	231	5	for	for	ADP
ejde-741	231	6	k	k	PROPN
ejde-741	231	7	̸=	̸=	PROPN
ejde-741	231	8	1	1	NUM
ejde-741	231	9	,	,	PUNCT
ejde-741	231	10	where	where	SCONJ
ejde-741	231	11	k	k	PROPN
ejde-741	231	12	∈	∈	PROPN
ejde-741	232	1	[	[	X
ejde-741	232	2	0	0	NUM
ejde-741	232	3	,	,	PUNCT
ejde-741	232	4	2	2	NUM
ejde-741	232	5	)	)	PUNCT
ejde-741	232	6	is	be	AUX
ejde-741	232	7	mentioned	mention	VERB
ejde-741	232	8	in	in	ADP
ejde-741	232	9	hypothesis	hypothesis	NOUN
ejde-741	232	10	1.1	1.1	NUM
ejde-741	232	11	.	.	PUNCT
ejde-741	233	1	for	for	ADP
ejde-741	233	2	the	the	DET
ejde-741	233	3	(	(	PUNCT
ejde-741	233	4	wdc	wdc	PROPN
ejde-741	233	5	)	)	PUNCT
ejde-741	233	6	,	,	PUNCT
ejde-741	233	7	it	it	PRON
ejde-741	233	8	comes	come	VERB
ejde-741	233	9	from	from	ADP
ejde-741	233	10	the	the	DET
ejde-741	233	11	continuous	continuous	ADJ
ejde-741	233	12	embedding	embed	VERB
ejde-741	233	13	h1	h1	PROPN
ejde-741	233	14	a	a	DET
ejde-741	233	15	↪	↪	PROPN
ejde-741	233	16	→	→	SYM
ejde-741	233	17	l∞(0	l∞(0	ADJ
ejde-741	233	18	,	,	PUNCT
ejde-741	233	19	1	1	NUM
ejde-741	233	20	)	)	PUNCT
ejde-741	233	21	.	.	PUNCT
ejde-741	234	1	for	for	ADP
ejde-741	234	2	the	the	DET
ejde-741	234	3	(	(	PUNCT
ejde-741	234	4	sdc	sdc	NOUN
ejde-741	234	5	)	)	PUNCT
ejde-741	234	6	it	it	PRON
ejde-741	234	7	comes	come	VERB
ejde-741	234	8	from	from	ADP
ejde-741	234	9	the	the	DET
ejde-741	234	10	fact	fact	NOUN
ejde-741	234	11	that	that	SCONJ
ejde-741	234	12	a	a	DET
ejde-741	234	13	∈	∈	PROPN
ejde-741	234	14	w	w	PROPN
ejde-741	234	15	1,∞(0	1,∞(0	NOUN
ejde-741	234	16	,	,	PUNCT
ejde-741	234	17	1	1	NUM
ejde-741	234	18	)	)	PUNCT
ejde-741	234	19	.	.	PUNCT
ejde-741	235	1	for	for	ADP
ejde-741	235	2	the	the	DET
ejde-741	235	3	casek	casek	NOUN
ejde-741	235	4	=	=	SYM
ejde-741	235	5	1	1	NUM
ejde-741	235	6	,	,	PUNCT
ejde-741	235	7	we	we	PRON
ejde-741	235	8	could	could	AUX
ejde-741	235	9	just	just	ADV
ejde-741	235	10	prove	prove	VERB
ejde-741	235	11	it	it	PRON
ejde-741	235	12	for	for	ADP
ejde-741	235	13	θ	θ	PROPN
ejde-741	235	14	≥	≥	NUM
ejde-741	235	15	1/2	1/2	NUM
ejde-741	235	16	,	,	PUNCT
ejde-741	235	17	where	where	SCONJ
ejde-741	235	18	θ	θ	PROPN
ejde-741	235	19	is	be	AUX
ejde-741	235	20	given	give	VERB
ejde-741	235	21	in	in	ADP
ejde-741	235	22	(	(	PUNCT
ejde-741	235	23	1.3	1.3	NUM
ejde-741	235	24	)	)	PUNCT
ejde-741	235	25	.	.	PUNCT
ejde-741	236	1	the	the	DET
ejde-741	236	2	case	case	NOUN
ejde-741	236	3	0	0	PUNCT
ejde-741	236	4	<	<	X
ejde-741	236	5	θ	θ	X
ejde-741	236	6	<	<	X
ejde-741	236	7	1/2	1/2	NUM
ejde-741	236	8	remains	remain	VERB
ejde-741	236	9	open	open	ADJ
ejde-741	236	10	.	.	PUNCT
ejde-741	237	1	a	a	DET
ejde-741	237	2	detailed	detailed	ADJ
ejde-741	237	3	proof	proof	NOUN
ejde-741	237	4	of	of	ADP
ejde-741	237	5	these	these	DET
ejde-741	237	6	facts	fact	NOUN
ejde-741	237	7	will	will	AUX
ejde-741	237	8	be	be	AUX
ejde-741	237	9	presented	present	VERB
ejde-741	237	10	in	in	ADP
ejde-741	237	11	appendix	appendix	NOUN
ejde-741	237	12	5	5	NUM
ejde-741	237	13	(	(	PUNCT
ejde-741	237	14	see	see	VERB
ejde-741	237	15	propositions	proposition	NOUN
ejde-741	237	16	5.2	5.2	NUM
ejde-741	237	17	and	and	CCONJ
ejde-741	237	18	5.1	5.1	NUM
ejde-741	237	19	)	)	PUNCT
ejde-741	237	20	.	.	PUNCT
ejde-741	238	1	lemma	lemma	PROPN
ejde-741	238	2	3.2	3.2	NUM
ejde-741	238	3	.	.	PUNCT
ejde-741	239	1	the	the	DET
ejde-741	239	2	mapping	mapping	NOUN
ejde-741	239	3	h	h	NOUN
ejde-741	239	4	:	:	PUNCT
ejde-741	239	5	e	e	X
ejde-741	239	6	→	→	SYM
ejde-741	239	7	f	f	PROPN
ejde-741	239	8	,	,	PUNCT
ejde-741	239	9	given	give	VERB
ejde-741	239	10	in	in	ADP
ejde-741	239	11	(	(	PUNCT
ejde-741	239	12	2.1	2.1	NUM
ejde-741	239	13	)	)	PUNCT
ejde-741	239	14	,	,	PUNCT
ejde-741	239	15	is	be	AUX
ejde-741	239	16	well	well	ADV
ejde-741	239	17	defined	define	VERB
ejde-741	239	18	,	,	PUNCT
ejde-741	239	19	recalling	recall	VERB
ejde-741	239	20	that	that	SCONJ
ejde-741	239	21	the	the	DET
ejde-741	239	22	spaces	space	NOUN
ejde-741	239	23	e	e	NOUN
ejde-741	239	24	and	and	CCONJ
ejde-741	239	25	f	f	PROPN
ejde-741	239	26	are	be	AUX
ejde-741	239	27	defined	define	VERB
ejde-741	239	28	in	in	ADP
ejde-741	239	29	(	(	PUNCT
ejde-741	239	30	2.9	2.9	NUM
ejde-741	239	31	)	)	PUNCT
ejde-741	239	32	and	and	CCONJ
ejde-741	239	33	(	(	PUNCT
ejde-741	239	34	2.10	2.10	NUM
ejde-741	239	35	)	)	PUNCT
ejde-741	239	36	,	,	PUNCT
ejde-741	239	37	respectively	respectively	ADV
ejde-741	239	38	.	.	PUNCT
ejde-741	240	1	proof	proof	NOUN
ejde-741	240	2	.	.	PUNCT
ejde-741	241	1	for	for	ADP
ejde-741	241	2	each	each	DET
ejde-741	241	3	(	(	PUNCT
ejde-741	241	4	u	u	NOUN
ejde-741	241	5	,	,	PUNCT
ejde-741	241	6	h	h	NOUN
ejde-741	241	7	)	)	PUNCT
ejde-741	241	8	∈	∈	PROPN
ejde-741	241	9	e	e	NOUN
ejde-741	241	10	,	,	PUNCT
ejde-741	241	11	let	let	VERB
ejde-741	241	12	us	we	PRON
ejde-741	241	13	check	check	VERB
ejde-741	241	14	that	that	SCONJ
ejde-741	241	15	h(u	h(u	PROPN
ejde-741	241	16	,	,	PUNCT
ejde-741	241	17	h	h	NOUN
ejde-741	241	18	)	)	PUNCT
ejde-741	241	19	∈	∈	PROPN
ejde-741	241	20	f	f	PROPN
ejde-741	241	21	.	.	PUNCT
ejde-741	242	1	clearly	clearly	ADV
ejde-741	242	2	,	,	PUNCT
ejde-741	242	3	h2(u	h2(u	PROPN
ejde-741	242	4	,	,	PUNCT
ejde-741	242	5	h	h	NOUN
ejde-741	242	6	)	)	PUNCT
ejde-741	242	7	=	=	SYM
ejde-741	242	8	u	u	NOUN
ejde-741	242	9	(	(	PUNCT
ejde-741	242	10	·	·	PUNCT
ejde-741	242	11	,	,	PUNCT
ejde-741	242	12	0	0	NUM
ejde-741	242	13	)	)	PUNCT
ejde-741	242	14	∈	∈	PROPN
ejde-741	242	15	h1	h1	VERB
ejde-741	242	16	a	a	PRON
ejde-741	242	17	.	.	PUNCT
ejde-741	243	1	also	also	ADV
ejde-741	243	2	,	,	PUNCT
ejde-741	243	3	recalling	recall	VERB
ejde-741	243	4	assumptions	assumption	NOUN
ejde-741	243	5	1.1	1.1	NUM
ejde-741	243	6	,	,	PUNCT
ejde-741	243	7	1.3	1.3	NUM
ejde-741	243	8	and	and	CCONJ
ejde-741	243	9	1.4	1.4	NUM
ejde-741	243	10	,	,	PUNCT
ejde-741	243	11	we	we	PRON
ejde-741	243	12	have∫	have∫	VERB
ejde-741	243	13	t	t	NOUN
ejde-741	243	14	0	0	NUM
ejde-741	243	15	∫	∫	PROPN
ejde-741	243	16	1	1	NUM
ejde-741	243	17	0	0	NUM
ejde-741	243	18	ρ20|h1(u	ρ20|h1(u	NOUN
ejde-741	243	19	,	,	PUNCT
ejde-741	243	20	v	v	NOUN
ejde-741	243	21	,	,	PUNCT
ejde-741	243	22	h)|2	h)|2	NOUN
ejde-741	243	23	=	=	SYM
ejde-741	243	24	∫	∫	PROPN
ejde-741	243	25	t	t	PROPN
ejde-741	243	26	0	0	NUM
ejde-741	244	1	∫	∫	PROPN
ejde-741	244	2	1	1	NUM
ejde-741	244	3	0	0	NUM
ejde-741	244	4	ρ20	ρ20	NOUN
ejde-741	244	5	|ut	|ut	ADP
ejde-741	244	6	−	−	PROPN
ejde-741	244	7	ℓ(au)(aux)x	ℓ(au)(aux)x	NUM
ejde-741	244	8	+	+	NUM
ejde-741	244	9	f(t	f(t	NOUN
ejde-741	244	10	,	,	PUNCT
ejde-741	244	11	x	x	PRON
ejde-741	244	12	,	,	PUNCT
ejde-741	244	13	u)−	u)−	PROPN
ejde-741	244	14	hχω|2	hχω|2	NOUN
ejde-741	244	15	≤	≤	ADV
ejde-741	244	16	3	3	NUM
ejde-741	244	17	∫	∫	NOUN
ejde-741	244	18	t	t	PROPN
ejde-741	244	19	0	0	NUM
ejde-741	245	1	∫	∫	PROPN
ejde-741	245	2	1	1	NUM
ejde-741	245	3	0	0	NUM
ejde-741	245	4	ρ20|lu−	ρ20|lu−	PROPN
ejde-741	245	5	hχω|2	hχω|2	PROPN
ejde-741	245	6	+	+	CCONJ
ejde-741	246	1	3	3	NUM
ejde-741	246	2	∫	∫	NOUN
ejde-741	246	3	t	t	PROPN
ejde-741	246	4	0	0	NUM
ejde-741	246	5	∫	∫	PROPN
ejde-741	246	6	1	1	NUM
ejde-741	246	7	0	0	NUM
ejde-741	246	8	ρ20	ρ20	NOUN
ejde-741	246	9	|ℓ(au)−	|ℓ(au)−	NOUN
ejde-741	246	10	ℓ(0)|2	ℓ(0)|2	NOUN
ejde-741	246	11	|(aux)x|2	|(aux)x|2	ADV
ejde-741	247	1	+	+	CCONJ
ejde-741	247	2	3	3	NUM
ejde-741	247	3	∫	∫	NOUN
ejde-741	247	4	t	t	PROPN
ejde-741	247	5	0	0	NUM
ejde-741	247	6	∫	∫	PROPN
ejde-741	247	7	1	1	NUM
ejde-741	247	8	0	0	NUM
ejde-741	247	9	ρ20|f(t	ρ20|f(t	NUM
ejde-741	247	10	,	,	PUNCT
ejde-741	247	11	x	x	PRON
ejde-741	247	12	,	,	PUNCT
ejde-741	247	13	u)−	u)−	PROPN
ejde-741	247	14	f(t	f(t	NOUN
ejde-741	247	15	,	,	PUNCT
ejde-741	247	16	x	x	PRON
ejde-741	247	17	,	,	PUNCT
ejde-741	247	18	0)|2	0)|2	NUM
ejde-741	247	19	≤	≤	NUM
ejde-741	247	20	3∥(u	3∥(u	NUM
ejde-741	247	21	,	,	PUNCT
ejde-741	247	22	h)∥2e	h)∥2e	PROPN
ejde-741	248	1	+	+	NUM
ejde-741	248	2	c	c	NOUN
ejde-741	248	3	∫	∫	PROPN
ejde-741	248	4	t	t	PROPN
ejde-741	248	5	0	0	NUM
ejde-741	248	6	∫	∫	PROPN
ejde-741	248	7	1	1	NUM
ejde-741	248	8	0	0	NUM
ejde-741	248	9	ρ20|au|2|(aux)x|2	ρ20|au|2|(aux)x|2	NUM
ejde-741	249	1	+	+	CCONJ
ejde-741	249	2	c	c	NOUN
ejde-741	249	3	∫	∫	PROPN
ejde-741	249	4	t	t	PROPN
ejde-741	249	5	0	0	NUM
ejde-741	249	6	∫	∫	PROPN
ejde-741	249	7	1	1	NUM
ejde-741	249	8	0	0	NUM
ejde-741	249	9	ρ20|u|2	ρ20|u|2	NOUN
ejde-741	249	10	≤	≤	NOUN
ejde-741	249	11	c∥(u	c∥(u	NOUN
ejde-741	249	12	,	,	PUNCT
ejde-741	249	13	h)∥2e	h)∥2e	PROPN
ejde-741	250	1	+	+	NUM
ejde-741	250	2	c	c	NOUN
ejde-741	250	3	∫	∫	PROPN
ejde-741	250	4	t	t	PROPN
ejde-741	250	5	0	0	NUM
ejde-741	250	6	∫	∫	PROPN
ejde-741	250	7	1	1	NUM
ejde-741	250	8	0	0	NUM
ejde-741	250	9	ρ20|au|2|(aux)x|2	ρ20|au|2|(aux)x|2	NOUN
ejde-741	250	10	.	.	PUNCT
ejde-741	251	1	at	at	ADP
ejde-741	251	2	this	this	DET
ejde-741	251	3	point	point	NOUN
ejde-741	251	4	,	,	PUNCT
ejde-741	251	5	we	we	PRON
ejde-741	251	6	must	must	AUX
ejde-741	251	7	estimate	estimate	VERB
ejde-741	251	8	i	i	PRON
ejde-741	251	9	:	:	PUNCT
ejde-741	252	1	=	=	SYM
ejde-741	252	2	∫	∫	PROPN
ejde-741	252	3	t	t	PROPN
ejde-741	252	4	0	0	NUM
ejde-741	252	5	∫	∫	PROPN
ejde-741	252	6	1	1	NUM
ejde-741	252	7	0	0	NUM
ejde-741	252	8	ρ20|au|2|(aux)x|2	ρ20|au|2|(aux)x|2	NOUN
ejde-741	252	9	.	.	PUNCT
ejde-741	253	1	we	we	PRON
ejde-741	253	2	start	start	VERB
ejde-741	253	3	recalling	recall	VERB
ejde-741	253	4	that	that	SCONJ
ejde-741	253	5	a	a	DET
ejde-741	253	6	=	=	SYM
ejde-741	253	7	τ(t)ηr(x	τ(t)ηr(x	X
ejde-741	253	8	)	)	PUNCT
ejde-741	253	9	≥	≥	NOUN
ejde-741	253	10	τ(t)η̂r	τ(t)η̂r	PROPN
ejde-741	253	11	and	and	CCONJ
ejde-741	253	12	au	au	ADP
ejde-741	253	13	∈	∈	PROPN
ejde-741	253	14	l∞(0	l∞(0	PRON
ejde-741	253	15	,	,	PUNCT
ejde-741	253	16	1	1	NUM
ejde-741	253	17	)	)	PUNCT
ejde-741	253	18	to	to	PART
ejde-741	253	19	obtain	obtain	VERB
ejde-741	253	20	i	i	PRON
ejde-741	253	21	=	=	PUNCT
ejde-741	254	1	∫	∫	PROPN
ejde-741	254	2	t	t	PROPN
ejde-741	254	3	0	0	NUM
ejde-741	255	1	∫	∫	PROPN
ejde-741	256	1	1	1	NUM
ejde-741	256	2	0	0	NUM
ejde-741	256	3	e−2saζ−5/3|au|2|(aux)x|2	e−2saζ−5/3|au|2|(aux)x|2	PROPN
ejde-741	256	4	≤	≤	NUM
ejde-741	256	5	∫	∫	PROPN
ejde-741	256	6	t	t	PROPN
ejde-741	256	7	0	0	NUM
ejde-741	256	8	∫	∫	PROPN
ejde-741	256	9	1	1	NUM
ejde-741	256	10	0	0	NUM
ejde-741	256	11	e−2sτη̂rη−5/3τ−5/3|au|2|(aux)x|2	e−2sτη̂rη−5/3τ−5/3|au|2|(aux)x|2	NOUN
ejde-741	256	12	≤	≤	NOUN
ejde-741	256	13	c	c	X
ejde-741	256	14	∫	∫	PROPN
ejde-741	256	15	t	t	PROPN
ejde-741	256	16	0	0	NUM
ejde-741	257	1	e−2sτη̂rτ−5/3	e−2sτη̂rτ−5/3	PROPN
ejde-741	257	2	∫	∫	PROPN
ejde-741	257	3	1	1	NUM
ejde-741	257	4	0	0	NUM
ejde-741	257	5	|au|2|(aux)x|2	|au|2|(aux)x|2	PROPN
ejde-741	257	6	≤	≤	PROPN
ejde-741	258	1	c	c	PROPN
ejde-741	258	2	∫	∫	PROPN
ejde-741	258	3	t	t	PROPN
ejde-741	258	4	0	0	NUM
ejde-741	258	5	e−2sτη̂rτ−5/3(∥u∥2l2(0,1	e−2sτη̂rτ−5/3(∥u∥2l2(0,1	NOUN
ejde-741	258	6	)	)	PUNCT
ejde-741	259	1	+	+	CCONJ
ejde-741	259	2	∥	∥	NOUN
ejde-741	259	3	√	√	NUM
ejde-741	259	4	aux∥2l2(0,1	aux∥2l2(0,1	NOUN
ejde-741	259	5	)	)	PUNCT
ejde-741	259	6	)	)	PUNCT
ejde-741	259	7	∫	∫	PROPN
ejde-741	260	1	1	1	NUM
ejde-741	260	2	0	0	NUM
ejde-741	260	3	|(aux)x|2	|(aux)x|2	NUM
ejde-741	261	1	=	=	PUNCT
ejde-741	261	2	c	c	X
ejde-741	261	3	(	(	PUNCT
ejde-741	261	4	∫	∫	PROPN
ejde-741	261	5	t	t	PROPN
ejde-741	261	6	0	0	NUM
ejde-741	261	7	e−2sτη̂rτ−5/3∥u∥2l2(0,1)∥(aux)x∥	e−2sτη̂rτ−5/3∥u∥2l2(0,1)∥(aux)x∥	PROPN
ejde-741	261	8	2	2	NUM
ejde-741	261	9	l2(0,1	l2(0,1	ADJ
ejde-741	261	10	)	)	PUNCT
ejde-741	261	11	10	10	NUM
ejde-741	262	1	p.	p.	NOUN
ejde-741	262	2	p.	p.	NOUN
ejde-741	262	3	de	de	PROPN
ejde-741	262	4	carvalho	carvalho	PROPN
ejde-741	262	5	,	,	PUNCT
ejde-741	262	6	r.	r.	PROPN
ejde-741	262	7	demarque	demarque	PROPN
ejde-741	262	8	,	,	PUNCT
ejde-741	262	9	j.	j.	PROPN
ejde-741	262	10	límaco	límaco	PROPN
ejde-741	262	11	,	,	PUNCT
ejde-741	262	12	l.	l.	PROPN
ejde-741	262	13	viana	viana	PROPN
ejde-741	262	14	ejde-2025/15	ejde-2025/15	PROPN
ejde-741	263	1	+	+	CCONJ
ejde-741	264	1	∫	∫	PROPN
ejde-741	264	2	t	t	PROPN
ejde-741	264	3	0	0	NUM
ejde-741	264	4	e−2sτη̂rτ−5/3∥	e−2sτη̂rτ−5/3∥	PROPN
ejde-741	264	5	√	√	PROPN
ejde-741	264	6	aux∥2l2(0,1)∥(aux)x∥	aux∥2l2(0,1)∥(aux)x∥	PROPN
ejde-741	264	7	2	2	NUM
ejde-741	264	8	l2(0,1	l2(0,1	ADV
ejde-741	264	9	)	)	PUNCT
ejde-741	264	10	)	)	PUNCT
ejde-741	265	1	=	=	PRON
ejde-741	265	2	:	:	PUNCT
ejde-741	265	3	i1	i1	PROPN
ejde-741	265	4	+	+	CCONJ
ejde-741	265	5	i2	i2	PROPN
ejde-741	265	6	.	.	PUNCT
ejde-741	266	1	(	(	PUNCT
ejde-741	266	2	3.2	3.2	NUM
ejde-741	266	3	)	)	PUNCT
ejde-741	266	4	recall	recall	NOUN
ejde-741	266	5	that	that	DET
ejde-741	266	6	η̄r	η̄r	NOUN
ejde-741	266	7	=	=	PUNCT
ejde-741	267	1	3η∗r−2η̂r	3η∗r−2η̂r	PROPN
ejde-741	267	2	<	<	X
ejde-741	267	3	0	0	NUM
ejde-741	267	4	which	which	PRON
ejde-741	267	5	implies	imply	VERB
ejde-741	267	6	κ	κ	X
ejde-741	267	7	:	:	PUNCT
ejde-741	267	8	=	=	SYM
ejde-741	267	9	η̂r−2η̄r	η̂r−2η̄r	PROPN
ejde-741	267	10	>	>	PUNCT
ejde-741	267	11	0	0	PUNCT
ejde-741	268	1	and	and	CCONJ
ejde-741	268	2	,	,	PUNCT
ejde-741	268	3	consequently	consequently	ADV
ejde-741	268	4	,	,	PUNCT
ejde-741	268	5	e−2sτη̂rτ−5/3ρ−4	e−2sτη̂rτ−5/3ρ−4	ADV
ejde-741	268	6	∗	∗	NOUN
ejde-741	268	7	=	=	PUNCT
ejde-741	269	1	e−2sτη̂rτ−5/3e4sā(ζ∗)34/3	e−2sτη̂rτ−5/3e4sā(ζ∗)34/3	PROPN
ejde-741	269	2	=	=	SYM
ejde-741	269	3	e−2sτ(η̂r−2η̄r)τ29/3	e−2sτ(η̂r−2η̄r)τ29/3	PROPN
ejde-741	269	4	=	=	PUNCT
ejde-741	269	5	e−2sκττ29/3	e−2sκττ29/3	PROPN
ejde-741	269	6	≤	≤	ADJ
ejde-741	269	7	c.	c.	NOUN
ejde-741	269	8	(	(	PUNCT
ejde-741	269	9	3.3	3.3	NUM
ejde-741	269	10	)	)	PUNCT
ejde-741	269	11	since	since	SCONJ
ejde-741	269	12	ρ∗	ρ∗	PROPN
ejde-741	269	13	≤	≤	PROPN
ejde-741	269	14	cρ̂	cρ̂	PROPN
ejde-741	269	15	,	,	PUNCT
ejde-741	269	16	from	from	ADP
ejde-741	269	17	inequality	inequality	NOUN
ejde-741	269	18	(	(	PUNCT
ejde-741	269	19	3.1	3.1	NUM
ejde-741	269	20	)	)	PUNCT
ejde-741	269	21	,	,	PUNCT
ejde-741	269	22	we	we	PRON
ejde-741	269	23	have	have	VERB
ejde-741	269	24	i1	i1	PROPN
ejde-741	269	25	=	=	SYM
ejde-741	270	1	∫	∫	PROPN
ejde-741	270	2	t	t	PROPN
ejde-741	270	3	0	0	NUM
ejde-741	270	4	(	(	PUNCT
ejde-741	270	5	e−2sτη̂rττ−5/3ρ−4	e−2sτη̂rττ−5/3ρ−4	ADV
ejde-741	270	6	∗	∗	NOUN
ejde-741	270	7	)	)	PUNCT
ejde-741	270	8	(	(	PUNCT
ejde-741	270	9	ρ2∗∥u∥2l2(0,1))(ρ	ρ2∗∥u∥2l2(0,1))(ρ	X
ejde-741	270	10	2	2	NUM
ejde-741	270	11	∗∥(aux)x∥2l2(0,1	∗∥(aux)x∥2l2(0,1	NOUN
ejde-741	270	12	)	)	PUNCT
ejde-741	270	13	)	)	PUNCT
ejde-741	271	1	≤	≤	NUM
ejde-741	272	1	c	c	NOUN
ejde-741	272	2	sup	sup	NOUN
ejde-741	272	3	t∈[0,t	t∈[0,t	NOUN
ejde-741	272	4	]	]	PUNCT
ejde-741	272	5	∥ρ̂u(t	∥ρ̂u(t	PROPN
ejde-741	272	6	,	,	PUNCT
ejde-741	272	7	·	·	PUNCT
ejde-741	272	8	)	)	PUNCT
ejde-741	272	9	∥2l2(0,1)∥(aux)x∥	∥2l2(0,1)∥(aux)x∥	VERB
ejde-741	272	10	2	2	NUM
ejde-741	272	11	ρ2∗	ρ2∗	NUM
ejde-741	272	12	≤	≤	NOUN
ejde-741	272	13	c∥(u	c∥(u	NOUN
ejde-741	272	14	,	,	PUNCT
ejde-741	272	15	h)∥4e	h)∥4e	X
ejde-741	272	16	(	(	PUNCT
ejde-741	272	17	3.4	3.4	NUM
ejde-741	272	18	)	)	PUNCT
ejde-741	272	19	and	and	CCONJ
ejde-741	272	20	i2	i2	PROPN
ejde-741	272	21	=	=	SYM
ejde-741	273	1	∫	∫	PROPN
ejde-741	273	2	t	t	PROPN
ejde-741	273	3	0	0	NUM
ejde-741	273	4	(	(	PUNCT
ejde-741	273	5	e−2sτη̂rττ−5/3ρ−4	e−2sτη̂rττ−5/3ρ−4	ADV
ejde-741	273	6	∗	∗	NOUN
ejde-741	273	7	)	)	PUNCT
ejde-741	273	8	(	(	PUNCT
ejde-741	273	9	ρ2∗∥	ρ2∗∥	ADJ
ejde-741	273	10	√	√	INTJ
ejde-741	273	11	aux∥2l2(0,1))(ρ	aux∥2l2(0,1))(ρ	PRON
ejde-741	273	12	2	2	NUM
ejde-741	273	13	∗∥(aux)x∥2l2(0,1	∗∥(aux)x∥2l2(0,1	NOUN
ejde-741	273	14	)	)	PUNCT
ejde-741	273	15	)	)	PUNCT
ejde-741	274	1	≤	≤	NUM
ejde-741	275	1	c	c	NOUN
ejde-741	275	2	sup	sup	NOUN
ejde-741	275	3	t∈[0,t	t∈[0,t	NOUN
ejde-741	275	4	]	]	PUNCT
ejde-741	275	5	∥ρ∗	∥ρ∗	PROPN
ejde-741	275	6	√	√	PROPN
ejde-741	275	7	aux(t	aux(t	PROPN
ejde-741	275	8	,	,	PUNCT
ejde-741	275	9	·	·	PUNCT
ejde-741	275	10	)	)	PUNCT
ejde-741	275	11	∥2l2(0,1)∥(aux)x∥	∥2l2(0,1)∥(aux)x∥	VERB
ejde-741	275	12	2	2	NUM
ejde-741	275	13	ρ2∗	ρ2∗	NUM
ejde-741	275	14	≤	≤	NOUN
ejde-741	275	15	c∥(u	c∥(u	NOUN
ejde-741	275	16	,	,	PUNCT
ejde-741	275	17	h)∥4e	h)∥4e	NOUN
ejde-741	275	18	.	.	PUNCT
ejde-741	276	1	(	(	PUNCT
ejde-741	276	2	3.5	3.5	NUM
ejde-741	276	3	)	)	PUNCT
ejde-741	276	4	as	as	ADP
ejde-741	276	5	a	a	DET
ejde-741	276	6	conclusion	conclusion	NOUN
ejde-741	276	7	,	,	PUNCT
ejde-741	276	8	h1(u	h1(u	PROPN
ejde-741	276	9	,	,	PUNCT
ejde-741	276	10	v	v	NOUN
ejde-741	276	11	)	)	PUNCT
ejde-741	276	12	∈	∈	PROPN
ejde-741	276	13	l2(q	l2(q	PROPN
ejde-741	276	14	,	,	PUNCT
ejde-741	276	15	ρ20	ρ20	NUM
ejde-741	276	16	)	)	PUNCT
ejde-741	276	17	and	and	CCONJ
ejde-741	276	18	the	the	DET
ejde-741	276	19	proof	proof	NOUN
ejde-741	276	20	is	be	AUX
ejde-741	276	21	complete	complete	ADJ
ejde-741	276	22	.	.	PUNCT
ejde-741	277	1	□	□	PUNCT
ejde-741	277	2	proposition	proposition	NOUN
ejde-741	277	3	3.3	3.3	NUM
ejde-741	277	4	.	.	PUNCT
ejde-741	278	1	h′(0	h′(0	NOUN
ejde-741	278	2	,	,	PUNCT
ejde-741	278	3	0	0	X
ejde-741	278	4	)	)	PUNCT
ejde-741	278	5	∈	∈	PROPN
ejde-741	278	6	l(e;f	l(e;f	PROPN
ejde-741	278	7	)	)	PUNCT
ejde-741	278	8	is	be	AUX
ejde-741	278	9	onto	onto	ADP
ejde-741	278	10	.	.	PUNCT
ejde-741	279	1	proof	proof	NOUN
ejde-741	279	2	.	.	PUNCT
ejde-741	280	1	take	take	VERB
ejde-741	280	2	(	(	PUNCT
ejde-741	280	3	g	g	NOUN
ejde-741	280	4	,	,	PUNCT
ejde-741	280	5	u0	u0	ADJ
ejde-741	280	6	)	)	PUNCT
ejde-741	280	7	∈	∈	PROPN
ejde-741	280	8	f	f	PROPN
ejde-741	280	9	.	.	PUNCT
ejde-741	281	1	by	by	ADP
ejde-741	281	2	propositions	proposition	NOUN
ejde-741	281	3	3.1	3.1	NUM
ejde-741	281	4	and	and	CCONJ
ejde-741	281	5	2.2	2.2	NUM
ejde-741	281	6	,	,	PUNCT
ejde-741	281	7	there	there	PRON
ejde-741	281	8	exists	exist	VERB
ejde-741	281	9	(	(	PUNCT
ejde-741	281	10	u	u	NOUN
ejde-741	281	11	,	,	PUNCT
ejde-741	281	12	h	h	NOUN
ejde-741	281	13	)	)	PUNCT
ejde-741	281	14	∈	∈	PROPN
ejde-741	281	15	e	e	NOUN
ejde-741	281	16	that	that	PRON
ejde-741	281	17	solves	solve	VERB
ejde-741	281	18	(	(	PUNCT
ejde-741	281	19	2.2	2.2	NUM
ejde-741	281	20	)	)	PUNCT
ejde-741	281	21	.	.	PUNCT
ejde-741	282	1	in	in	ADP
ejde-741	282	2	other	other	ADJ
ejde-741	282	3	words	word	NOUN
ejde-741	282	4	,	,	PUNCT
ejde-741	282	5	h′(0	h′(0	NOUN
ejde-741	282	6	,	,	PUNCT
ejde-741	282	7	0)(u	0)(u	ADJ
ejde-741	282	8	,	,	PUNCT
ejde-741	282	9	h	h	NOUN
ejde-741	282	10	)	)	PUNCT
ejde-741	282	11	=	=	SYM
ejde-741	282	12	(	(	PUNCT
ejde-741	282	13	h′	h′	PROPN
ejde-741	282	14	1(0	1(0	NUM
ejde-741	282	15	,	,	PUNCT
ejde-741	282	16	0)(u	0)(u	PROPN
ejde-741	282	17	,	,	PUNCT
ejde-741	282	18	h),h′	h),h′	PROPN
ejde-741	282	19	2(0	2(0	NUM
ejde-741	282	20	,	,	PUNCT
ejde-741	282	21	0)(u	0)(u	ADJ
ejde-741	282	22	,	,	PUNCT
ejde-741	282	23	h	h	NOUN
ejde-741	282	24	)	)	PUNCT
ejde-741	282	25	)	)	PUNCT
ejde-741	283	1	=	=	SYM
ejde-741	283	2	(	(	PUNCT
ejde-741	283	3	ut	ut	PROPN
ejde-741	283	4	−	−	PROPN
ejde-741	283	5	(	(	PUNCT
ejde-741	283	6	a(x)ux)x	a(x)ux)x	VERB
ejde-741	283	7	+	+	CCONJ
ejde-741	283	8	c(t	c(t	PROPN
ejde-741	283	9	,	,	PUNCT
ejde-741	283	10	x)u−	x)u−	PROPN
ejde-741	283	11	hχw	hχw	PROPN
ejde-741	283	12	,	,	PUNCT
ejde-741	283	13	u	u	NOUN
ejde-741	283	14	(	(	PUNCT
ejde-741	283	15	·	·	PUNCT
ejde-741	283	16	,	,	PUNCT
ejde-741	283	17	0	0	NUM
ejde-741	283	18	)	)	PUNCT
ejde-741	283	19	)	)	PUNCT
ejde-741	284	1	=	=	PRON
ejde-741	284	2	(	(	PUNCT
ejde-741	284	3	g	g	NOUN
ejde-741	284	4	,	,	PUNCT
ejde-741	284	5	u0	u0	ADJ
ejde-741	284	6	)	)	PUNCT
ejde-741	284	7	.	.	PUNCT
ejde-741	285	1	this	this	PRON
ejde-741	285	2	completes	complete	VERB
ejde-741	285	3	the	the	DET
ejde-741	285	4	proof	proof	NOUN
ejde-741	285	5	.	.	PUNCT
ejde-741	286	1	□	□	PUNCT
ejde-741	286	2	3.2	3.2	NUM
ejde-741	286	3	.	.	PUNCT
ejde-741	287	1	h	h	NOUN
ejde-741	287	2	is	be	AUX
ejde-741	287	3	continuously	continuously	ADV
ejde-741	287	4	differentiable	differentiable	ADJ
ejde-741	287	5	.	.	PUNCT
ejde-741	288	1	in	in	ADP
ejde-741	288	2	this	this	DET
ejde-741	288	3	subsection	subsection	NOUN
ejde-741	288	4	,	,	PUNCT
ejde-741	288	5	we	we	PRON
ejde-741	288	6	will	will	AUX
ejde-741	288	7	prove	prove	VERB
ejde-741	288	8	that	that	SCONJ
ejde-741	288	9	h	h	NOUN
ejde-741	288	10	∈	∈	PROPN
ejde-741	289	1	c1(e	c1(e	PROPN
ejde-741	289	2	,	,	PUNCT
ejde-741	289	3	f	f	PROPN
ejde-741	289	4	)	)	PUNCT
ejde-741	289	5	.	.	PUNCT
ejde-741	290	1	the	the	DET
ejde-741	290	2	proof	proof	NOUN
ejde-741	290	3	will	will	AUX
ejde-741	290	4	rely	rely	VERB
ejde-741	290	5	on	on	ADP
ejde-741	290	6	the	the	DET
ejde-741	290	7	additional	additional	ADJ
ejde-741	290	8	regularity	regularity	NOUN
ejde-741	290	9	described	describe	VERB
ejde-741	290	10	in	in	ADP
ejde-741	290	11	(	(	PUNCT
ejde-741	290	12	3.1	3.1	NUM
ejde-741	290	13	)	)	PUNCT
ejde-741	290	14	.	.	PUNCT
ejde-741	291	1	proposition	proposition	NOUN
ejde-741	291	2	3.4	3.4	NUM
ejde-741	291	3	.	.	PUNCT
ejde-741	292	1	the	the	DET
ejde-741	292	2	mapping	mapping	NOUN
ejde-741	292	3	h	h	NOUN
ejde-741	292	4	is	be	AUX
ejde-741	292	5	continuously	continuously	ADV
ejde-741	292	6	differentiable	differentiable	ADJ
ejde-741	292	7	.	.	PUNCT
ejde-741	293	1	proof	proof	NOUN
ejde-741	293	2	.	.	PUNCT
ejde-741	294	1	it	it	PRON
ejde-741	294	2	is	be	AUX
ejde-741	294	3	clear	clear	ADJ
ejde-741	294	4	that	that	SCONJ
ejde-741	294	5	h2	h2	PROPN
ejde-741	294	6	∈	∈	PROPN
ejde-741	294	7	c1(e	c1(e	PROPN
ejde-741	294	8	,	,	PUNCT
ejde-741	294	9	f	f	PROPN
ejde-741	294	10	)	)	PUNCT
ejde-741	294	11	.	.	PUNCT
ejde-741	295	1	so	so	ADV
ejde-741	295	2	that	that	SCONJ
ejde-741	295	3	,	,	PUNCT
ejde-741	295	4	the	the	DET
ejde-741	295	5	proof	proof	NOUN
ejde-741	295	6	is	be	AUX
ejde-741	295	7	focused	focus	VERB
ejde-741	295	8	on	on	ADP
ejde-741	295	9	checking	check	VERB
ejde-741	295	10	that	that	SCONJ
ejde-741	295	11	h1	h1	PROPN
ejde-741	295	12	has	have	VERB
ejde-741	295	13	a	a	DET
ejde-741	295	14	continuous	continuous	ADJ
ejde-741	295	15	gateaux	gateaux	ADV
ejde-741	295	16	derivative	derivative	NOUN
ejde-741	295	17	on	on	ADP
ejde-741	295	18	e.	e.	PROPN
ejde-741	295	19	for	for	ADP
ejde-741	295	20	(	(	PUNCT
ejde-741	295	21	u	u	NOUN
ejde-741	295	22	,	,	PUNCT
ejde-741	295	23	h	h	NOUN
ejde-741	295	24	)	)	PUNCT
ejde-741	295	25	,	,	PUNCT
ejde-741	295	26	(	(	PUNCT
ejde-741	295	27	ū	ū	NOUN
ejde-741	295	28	,	,	PUNCT
ejde-741	295	29	h̄	h̄	NOUN
ejde-741	295	30	)	)	PUNCT
ejde-741	295	31	∈	∈	PROPN
ejde-741	295	32	e	e	NOUN
ejde-741	295	33	and	and	CCONJ
ejde-741	295	34	λ	λ	X
ejde-741	295	35	>	>	X
ejde-741	295	36	0	0	PROPN
ejde-741	295	37	,	,	PUNCT
ejde-741	295	38	set	set	VERB
ejde-741	295	39	b	b	NOUN
ejde-741	295	40	:	:	PUNCT
ejde-741	296	1	=	=	SYM
ejde-741	296	2	ℓ(au)a(x	ℓ(au)a(x	NOUN
ejde-741	296	3	)	)	PUNCT
ejde-741	296	4	,	,	PUNCT
ejde-741	296	5	bλ	bλ	INTJ
ejde-741	296	6	:	:	PUNCT
ejde-741	296	7	=	=	SYM
ejde-741	296	8	ℓ(a(u+	ℓ(a(u+	PROPN
ejde-741	296	9	λū))a(x	λū))a(x	PROPN
ejde-741	296	10	)	)	PUNCT
ejde-741	296	11	,	,	PUNCT
ejde-741	296	12	f	f	X
ejde-741	296	13	:	:	PUNCT
ejde-741	296	14	=	=	SYM
ejde-741	296	15	f(t	f(t	NOUN
ejde-741	296	16	,	,	PUNCT
ejde-741	296	17	x	x	NOUN
ejde-741	296	18	,	,	PUNCT
ejde-741	296	19	u	u	NOUN
ejde-741	296	20	)	)	PUNCT
ejde-741	296	21	,	,	PUNCT
ejde-741	296	22	fλ	fλ	ADJ
ejde-741	296	23	:	:	PUNCT
ejde-741	296	24	=	=	SYM
ejde-741	296	25	f(t	f(t	NOUN
ejde-741	296	26	,	,	PUNCT
ejde-741	296	27	x	x	PRON
ejde-741	296	28	,	,	PUNCT
ejde-741	296	29	u+	u+	ADJ
ejde-741	296	30	λū	λū	ADV
ejde-741	296	31	)	)	PUNCT
ejde-741	296	32	,	,	PUNCT
ejde-741	296	33	f3	f3	ADJ
ejde-741	296	34	:	:	PUNCT
ejde-741	296	35	=	=	SYM
ejde-741	296	36	d3f(t	d3f(t	NOUN
ejde-741	296	37	,	,	PUNCT
ejde-741	296	38	x	x	X
ejde-741	296	39	,	,	PUNCT
ejde-741	296	40	u	u	NOUN
ejde-741	296	41	)	)	PUNCT
ejde-741	296	42	.	.	PUNCT
ejde-741	297	1	claim	claim	NOUN
ejde-741	297	2	1	1	NUM
ejde-741	297	3	:	:	PUNCT
ejde-741	297	4	given	give	VERB
ejde-741	297	5	(	(	PUNCT
ejde-741	297	6	u	u	NOUN
ejde-741	297	7	,	,	PUNCT
ejde-741	297	8	h	h	NOUN
ejde-741	297	9	)	)	PUNCT
ejde-741	297	10	∈	∈	PROPN
ejde-741	297	11	e	e	NOUN
ejde-741	297	12	,	,	PUNCT
ejde-741	297	13	the	the	DET
ejde-741	297	14	linear	linear	ADJ
ejde-741	297	15	mapping	mapping	NOUN
ejde-741	297	16	l	l	NOUN
ejde-741	297	17	:	:	PUNCT
ejde-741	297	18	e	e	X
ejde-741	297	19	→	→	SYM
ejde-741	297	20	l2(q	l2(q	PROPN
ejde-741	297	21	;	;	PUNCT
ejde-741	297	22	ρ20	ρ20	NUM
ejde-741	297	23	)	)	PUNCT
ejde-741	297	24	,	,	PUNCT
ejde-741	297	25	defined	define	VERB
ejde-741	297	26	by	by	ADP
ejde-741	297	27	l(ū	l(ū	PROPN
ejde-741	297	28	,	,	PUNCT
ejde-741	297	29	h̄	h̄	NOUN
ejde-741	297	30	)	)	PUNCT
ejde-741	297	31	:	:	PUNCT
ejde-741	298	1	=	=	PUNCT
ejde-741	298	2	ūt	ūt	ADJ
ejde-741	298	3	−	−	PROPN
ejde-741	298	4	ℓ′(au)aū(aux)x	ℓ′(au)aū(aux)x	PROPN
ejde-741	298	5	+	+	NUM
ejde-741	298	6	ℓ(au)(aūx)x	ℓ(au)(aūx)x	NOUN
ejde-741	298	7	+	+	CCONJ
ejde-741	298	8	f3ū−	f3ū−	PROPN
ejde-741	298	9	h̄χω	h̄χω	PROPN
ejde-741	298	10	,	,	PUNCT
ejde-741	298	11	is	be	AUX
ejde-741	298	12	the	the	DET
ejde-741	298	13	gateaux	gateaux	ADV
ejde-741	298	14	derivative	derivative	NOUN
ejde-741	298	15	of	of	ADP
ejde-741	298	16	h1	h1	NOUN
ejde-741	298	17	at	at	ADP
ejde-741	298	18	(	(	PUNCT
ejde-741	298	19	u	u	NOUN
ejde-741	298	20	,	,	PUNCT
ejde-741	298	21	h	h	NOUN
ejde-741	298	22	)	)	PUNCT
ejde-741	298	23	∈	∈	PROPN
ejde-741	298	24	e.	e.	PROPN
ejde-741	298	25	indeed	indeed	ADV
ejde-741	298	26	,	,	PUNCT
ejde-741	298	27	for	for	ADP
ejde-741	298	28	each	each	DET
ejde-741	298	29	(	(	PUNCT
ejde-741	298	30	ū	ū	NOUN
ejde-741	298	31	,	,	PUNCT
ejde-741	298	32	h̄	h̄	NOUN
ejde-741	298	33	)	)	PUNCT
ejde-741	298	34	∈	∈	PROPN
ejde-741	298	35	e	e	NOUN
ejde-741	298	36	,	,	PUNCT
ejde-741	298	37	we	we	PRON
ejde-741	298	38	have∥∥	have∥∥	VERB
ejde-741	298	39	1	1	NUM
ejde-741	298	40	λ	λ	NOUN
ejde-741	298	41	(	(	PUNCT
ejde-741	298	42	h1(u+	h1(u+	PROPN
ejde-741	298	43	λū	λū	ADV
ejde-741	298	44	,	,	PUNCT
ejde-741	298	45	h+	h+	X
ejde-741	298	46	λh̄)−h1(u	λh̄)−h1(u	X
ejde-741	298	47	,	,	PUNCT
ejde-741	298	48	h))−	h))−	NOUN
ejde-741	298	49	l(ū	l(ū	PROPN
ejde-741	298	50	,	,	PUNCT
ejde-741	298	51	h̄	h̄	NOUN
ejde-741	298	52	)	)	PUNCT
ejde-741	298	53	∥∥	∥∥	PRON
ejde-741	298	54	ejde-2025/15	ejde-2025/15	VERB
ejde-741	298	55	null	null	ADJ
ejde-741	298	56	-	-	PUNCT
ejde-741	298	57	controllability	controllability	NOUN
ejde-741	298	58	degenerate	degenerate	ADJ
ejde-741	298	59	quasilinear	quasilinear	NOUN
ejde-741	298	60	equations	equation	NOUN
ejde-741	298	61	11	11	NUM
ejde-741	298	62	=	=	NOUN
ejde-741	298	63	∥∥∥ūt	∥∥∥ūt	ADV
ejde-741	298	64	−	−	VERB
ejde-741	299	1	[	[	X
ejde-741	299	2	ℓ(a(u+	ℓ(a(u+	PROPN
ejde-741	299	3	λū))(a(u+	λū))(a(u+	NOUN
ejde-741	299	4	λū)x)x	λū)x)x	ADP
ejde-741	299	5	−	−	PROPN
ejde-741	299	6	ℓ(au)(aux)x	ℓ(au)(aux)x	NUM
ejde-741	299	7	λ	λ	X
ejde-741	299	8	]	]	PUNCT
ejde-741	300	1	+	+	CCONJ
ejde-741	300	2	1	1	NUM
ejde-741	300	3	λ	λ	NOUN
ejde-741	300	4	(	(	PUNCT
ejde-741	300	5	fλ	fλ	NOUN
ejde-741	300	6	−	−	PROPN
ejde-741	300	7	f)−	f)−	PROPN
ejde-741	300	8	h̄χω	h̄χω	PROPN
ejde-741	300	9	−	−	PROPN
ejde-741	300	10	l(ū	l(ū	PROPN
ejde-741	300	11	,	,	PUNCT
ejde-741	300	12	h̄	h̄	NOUN
ejde-741	300	13	)	)	PUNCT
ejde-741	300	14	∥∥∥	∥∥∥	PROPN
ejde-741	300	15	≤	≤	NUM
ejde-741	300	16	∥∥∥[ℓ(a(u+	∥∥∥[ℓ(a(u+	PROPN
ejde-741	300	17	λū))−	λū))−	PROPN
ejde-741	300	18	ℓ(au	ℓ(au	PROPN
ejde-741	300	19	)	)	PUNCT
ejde-741	300	20	λ	λ	NOUN
ejde-741	300	21	−	−	NOUN
ejde-741	300	22	ℓ′(au)aū	ℓ′(au)aū	NOUN
ejde-741	300	23	]	]	PUNCT
ejde-741	300	24	(	(	PUNCT
ejde-741	300	25	aux)x	aux)x	PROPN
ejde-741	300	26	∥∥∥	∥∥∥	PROPN
ejde-741	300	27	+	+	CCONJ
ejde-741	300	28	∥[ℓ(a(u+	∥[ℓ(a(u+	PUNCT
ejde-741	300	29	λū	λū	ADV
ejde-741	300	30	)	)	PUNCT
ejde-741	300	31	)	)	PUNCT
ejde-741	301	1	+	+	CCONJ
ejde-741	301	2	ℓ(au)](aūx)x∥ρ20	ℓ(au)](aūx)x∥ρ20	X
ejde-741	301	3	+	+	CCONJ
ejde-741	301	4	∥	∥	NUM
ejde-741	301	5	1	1	NUM
ejde-741	301	6	λ	λ	NOUN
ejde-741	301	7	(	(	PUNCT
ejde-741	301	8	fλ	fλ	ADJ
ejde-741	301	9	−	−	PROPN
ejde-741	301	10	f)−	f)−	PROPN
ejde-741	301	11	f3ū∥ρ20	f3ū∥ρ20	PROPN
ejde-741	301	12	=	=	PRON
ejde-741	301	13	:	:	PUNCT
ejde-741	301	14	b1	b1	NOUN
ejde-741	301	15	+	+	SYM
ejde-741	301	16	b2	b2	NOUN
ejde-741	301	17	+	+	SYM
ejde-741	301	18	b3	b3	NOUN
ejde-741	301	19	.	.	PUNCT
ejde-741	302	1	we	we	PRON
ejde-741	302	2	will	will	AUX
ejde-741	302	3	see	see	VERB
ejde-741	302	4	that	that	DET
ejde-741	302	5	bi	bi	NOUN
ejde-741	302	6	→	→	PROPN
ejde-741	302	7	0	0	NUM
ejde-741	302	8	,	,	PUNCT
ejde-741	302	9	as	as	ADP
ejde-741	302	10	λ→	λ→	PROPN
ejde-741	302	11	0	0	NUM
ejde-741	302	12	,	,	PUNCT
ejde-741	302	13	for	for	ADP
ejde-741	302	14	any	any	DET
ejde-741	302	15	i	i	NOUN
ejde-741	302	16	=	=	NOUN
ejde-741	302	17	1	1	NUM
ejde-741	302	18	,	,	PUNCT
ejde-741	302	19	2	2	NUM
ejde-741	302	20	,	,	PUNCT
ejde-741	302	21	3	3	NUM
ejde-741	302	22	.	.	PUNCT
ejde-741	303	1	firstly	firstly	ADV
ejde-741	303	2	,	,	PUNCT
ejde-741	303	3	from	from	ADP
ejde-741	303	4	assumption	assumption	NOUN
ejde-741	303	5	1.4	1.4	NUM
ejde-741	303	6	,	,	PUNCT
ejde-741	303	7	for	for	ADP
ejde-741	303	8	each	each	PRON
ejde-741	303	9	(	(	PUNCT
ejde-741	303	10	t	t	PROPN
ejde-741	303	11	,	,	PUNCT
ejde-741	303	12	x	x	NOUN
ejde-741	303	13	)	)	PUNCT
ejde-741	303	14	∈	∈	PROPN
ejde-741	303	15	(	(	PUNCT
ejde-741	303	16	0	0	NUM
ejde-741	303	17	,	,	PUNCT
ejde-741	303	18	1)×(0	1)×(0	NUM
ejde-741	303	19	,	,	PUNCT
ejde-741	303	20	t	t	PROPN
ejde-741	303	21	)	)	PUNCT
ejde-741	303	22	,	,	PUNCT
ejde-741	303	23	we	we	PRON
ejde-741	303	24	apply	apply	VERB
ejde-741	303	25	mean	mean	NOUN
ejde-741	303	26	value	value	NOUN
ejde-741	303	27	theorem	theorem	VERB
ejde-741	303	28	to	to	PART
ejde-741	303	29	obtain	obtain	VERB
ejde-741	303	30	u∗λ	u∗λ	NOUN
ejde-741	303	31	=	=	SYM
ejde-741	303	32	u∗λ(t	u∗λ(t	PROPN
ejde-741	303	33	,	,	PUNCT
ejde-741	303	34	x	x	X
ejde-741	303	35	)	)	PUNCT
ejde-741	303	36	∈	∈	NOUN
ejde-741	303	37	r	r	NOUN
ejde-741	303	38	such	such	ADJ
ejde-741	303	39	that	that	DET
ejde-741	303	40	b2	b2	NOUN
ejde-741	303	41	3	3	NUM
ejde-741	303	42	≤	≤	NUM
ejde-741	303	43	∫	∫	PROPN
ejde-741	303	44	t	t	PROPN
ejde-741	303	45	0	0	NUM
ejde-741	304	1	∫	∫	PROPN
ejde-741	304	2	1	1	NUM
ejde-741	304	3	0	0	NUM
ejde-741	304	4	ρ20	ρ20	NUM
ejde-741	304	5	|(d3f(t	|(d3f(t	NOUN
ejde-741	304	6	,	,	PUNCT
ejde-741	304	7	x	x	PRON
ejde-741	304	8	,	,	PUNCT
ejde-741	304	9	u	u	NOUN
ejde-741	304	10	∗	∗	NOUN
ejde-741	304	11	λ)−d3f(t	λ)−d3f(t	NOUN
ejde-741	304	12	,	,	PUNCT
ejde-741	304	13	x	x	PRON
ejde-741	304	14	,	,	PUNCT
ejde-741	304	15	u))ū|2	u))ū|2	PROPN
ejde-741	304	16	→	→	X
ejde-741	304	17	0	0	NUM
ejde-741	304	18	,	,	PUNCT
ejde-741	304	19	as	as	ADP
ejde-741	304	20	λ→	λ→	PROPN
ejde-741	304	21	0	0	NUM
ejde-741	304	22	,	,	PUNCT
ejde-741	304	23	where	where	SCONJ
ejde-741	304	24	this	this	DET
ejde-741	304	25	convergence	convergence	NOUN
ejde-741	304	26	comes	come	VERB
ejde-741	304	27	from	from	ADP
ejde-741	304	28	lebesgue	lebesgue	PROPN
ejde-741	304	29	’s	’s	PART
ejde-741	304	30	theorem	theorem	PROPN
ejde-741	304	31	.	.	PUNCT
ejde-741	305	1	secondly	secondly	ADV
ejde-741	305	2	,	,	PUNCT
ejde-741	305	3	applying	apply	VERB
ejde-741	305	4	assumption	assumption	NOUN
ejde-741	305	5	1.3	1.3	NUM
ejde-741	305	6	and	and	CCONJ
ejde-741	305	7	the	the	DET
ejde-741	305	8	mean	mean	ADJ
ejde-741	305	9	value	value	NOUN
ejde-741	305	10	theorem	theorem	VERB
ejde-741	305	11	again	again	ADV
ejde-741	305	12	,	,	PUNCT
ejde-741	305	13	there	there	PRON
ejde-741	305	14	exists	exist	VERB
ejde-741	305	15	sλ	sλ	NOUN
ejde-741	305	16	=	=	SYM
ejde-741	305	17	sλ(t	sλ(t	X
ejde-741	305	18	,	,	PUNCT
ejde-741	305	19	x	x	X
ejde-741	305	20	)	)	PUNCT
ejde-741	305	21	∈	∈	NOUN
ejde-741	305	22	r	r	NOUN
ejde-741	305	23	such	such	ADJ
ejde-741	305	24	that	that	DET
ejde-741	305	25	b2	b2	NOUN
ejde-741	305	26	1	1	NUM
ejde-741	305	27	=	=	SYM
ejde-741	305	28	∫	∫	PROPN
ejde-741	305	29	t	t	PROPN
ejde-741	305	30	0	0	NUM
ejde-741	306	1	∫	∫	PROPN
ejde-741	306	2	1	1	NUM
ejde-741	306	3	0	0	NUM
ejde-741	306	4	ρ20	ρ20	NOUN
ejde-741	306	5	∣∣∣∣ℓ(a(u+	∣∣∣∣ℓ(a(u+	NOUN
ejde-741	306	6	λū))−	λū))−	PROPN
ejde-741	306	7	ℓ(au	ℓ(au	NUM
ejde-741	306	8	)	)	PUNCT
ejde-741	306	9	λ	λ	NOUN
ejde-741	306	10	−	−	PROPN
ejde-741	306	11	ℓ′(au)aū	ℓ′(au)aū	NOUN
ejde-741	306	12	∣∣∣∣2	∣∣∣∣2	NOUN
ejde-741	306	13	|(aux)x|2	|(aux)x|2	PUNCT
ejde-741	307	1	=	=	PUNCT
ejde-741	308	1	∫	∫	PROPN
ejde-741	308	2	t	t	PROPN
ejde-741	308	3	0	0	NUM
ejde-741	308	4	∫	∫	PROPN
ejde-741	309	1	1	1	NUM
ejde-741	309	2	0	0	NUM
ejde-741	310	1	ρ20|ℓ′(sλ)−	ρ20|ℓ′(sλ)−	NUM
ejde-741	310	2	ℓ′(au)|2|aū|2|(aux)x|2	ℓ′(au)|2|aū|2|(aux)x|2	NUM
ejde-741	310	3	→	→	SYM
ejde-741	310	4	0	0	NUM
ejde-741	310	5	,	,	PUNCT
ejde-741	310	6	as	as	ADP
ejde-741	310	7	λ→	λ→	PROPN
ejde-741	310	8	0	0	NUM
ejde-741	310	9	.	.	PUNCT
ejde-741	311	1	since	since	SCONJ
ejde-741	311	2	we	we	PRON
ejde-741	311	3	can	can	AUX
ejde-741	311	4	argue	argue	VERB
ejde-741	311	5	as	as	ADP
ejde-741	311	6	in	in	ADP
ejde-741	311	7	(	(	PUNCT
ejde-741	311	8	3.2	3.2	NUM
ejde-741	311	9	)	)	PUNCT
ejde-741	311	10	,	,	PUNCT
ejde-741	311	11	(	(	PUNCT
ejde-741	311	12	3.4	3.4	NUM
ejde-741	311	13	)	)	PUNCT
ejde-741	311	14	and	and	CCONJ
ejde-741	311	15	(	(	PUNCT
ejde-741	311	16	3.5	3.5	NUM
ejde-741	311	17	)	)	PUNCT
ejde-741	311	18	we	we	PRON
ejde-741	311	19	obtain∫	obtain∫	VERB
ejde-741	311	20	t	t	PROPN
ejde-741	311	21	0	0	NUM
ejde-741	311	22	∫	∫	PROPN
ejde-741	312	1	1	1	NUM
ejde-741	312	2	0	0	NUM
ejde-741	313	1	ρ20|ℓ′(sλ)−	ρ20|ℓ′(sλ)−	NUM
ejde-741	313	2	ℓ′(au)|2|aū|2|(aux)x|2	ℓ′(au)|2|aū|2|(aux)x|2	PROPN
ejde-741	313	3	≤	≤	NUM
ejde-741	314	1	∫	∫	PROPN
ejde-741	314	2	t	t	PROPN
ejde-741	314	3	0	0	NUM
ejde-741	314	4	∫	∫	PROPN
ejde-741	314	5	1	1	NUM
ejde-741	314	6	0	0	NUM
ejde-741	314	7	ρ20|aū|2|(aux)x|2	ρ20|aū|2|(aux)x|2	NUM
ejde-741	314	8	≤	≤	NUM
ejde-741	314	9	c	c	NOUN
ejde-741	314	10	(	(	PUNCT
ejde-741	314	11	sup	sup	NOUN
ejde-741	314	12	t∈[0,t	t∈[0,t	NOUN
ejde-741	314	13	]	]	X
ejde-741	314	14	∥ρ̂ū(t	∥ρ̂ū(t	NOUN
ejde-741	314	15	,	,	PUNCT
ejde-741	314	16	·	·	PUNCT
ejde-741	314	17	)	)	PUNCT
ejde-741	314	18	∥2l2(0,1)∥(aux)x∥	∥2l2(0,1)∥(aux)x∥	VERB
ejde-741	314	19	2	2	NUM
ejde-741	314	20	ρ2∗	ρ2∗	NUM
ejde-741	314	21	+	+	CCONJ
ejde-741	314	22	sup	sup	NOUN
ejde-741	314	23	t∈[0,t	t∈[0,t	NOUN
ejde-741	314	24	]	]	PUNCT
ejde-741	314	25	∥ρ∗	∥ρ∗	PROPN
ejde-741	314	26	√	√	PROPN
ejde-741	314	27	aūx(t	aūx(t	PROPN
ejde-741	314	28	,	,	PUNCT
ejde-741	314	29	·	·	PUNCT
ejde-741	314	30	)	)	PUNCT
ejde-741	314	31	∥2l2(0,1)∥(aux)x∥	∥2l2(0,1)∥(aux)x∥	NOUN
ejde-741	314	32	2	2	NUM
ejde-741	314	33	ρ2∗	ρ2∗	NUM
ejde-741	314	34	)	)	PUNCT
ejde-741	314	35	≤	≤	NOUN
ejde-741	314	36	c∥(u	c∥(u	NOUN
ejde-741	314	37	,	,	PUNCT
ejde-741	314	38	h)∥2e∥(ū	h)∥2e∥(ū	PROPN
ejde-741	314	39	,	,	PUNCT
ejde-741	314	40	h̄)∥2e	h̄)∥2e	PROPN
ejde-741	314	41	.	.	PUNCT
ejde-741	315	1	analogously	analogously	ADV
ejde-741	315	2	,	,	PUNCT
ejde-741	315	3	there	there	PRON
ejde-741	315	4	exists	exist	VERB
ejde-741	315	5	uλ	uλ	ADP
ejde-741	315	6	=	=	X
ejde-741	315	7	uλ(t	uλ(t	X
ejde-741	315	8	,	,	PUNCT
ejde-741	315	9	x	x	X
ejde-741	315	10	)	)	PUNCT
ejde-741	315	11	∈	∈	NOUN
ejde-741	315	12	r	r	NOUN
ejde-741	315	13	such	such	ADJ
ejde-741	315	14	that	that	DET
ejde-741	315	15	b2	b2	NOUN
ejde-741	315	16	2	2	NUM
ejde-741	315	17	=	=	SYM
ejde-741	315	18	∫	∫	PROPN
ejde-741	315	19	t	t	PROPN
ejde-741	315	20	0	0	NUM
ejde-741	316	1	∫	∫	PROPN
ejde-741	316	2	1	1	NUM
ejde-741	316	3	0	0	NUM
ejde-741	316	4	ρ20|ℓ(a(u+	ρ20|ℓ(a(u+	NOUN
ejde-741	316	5	λū	λū	ADV
ejde-741	316	6	)	)	PUNCT
ejde-741	316	7	)	)	PUNCT
ejde-741	317	1	+	+	CCONJ
ejde-741	317	2	ℓ(au)|2|(aux)x|2	ℓ(au)|2|(aux)x|2	PROPN
ejde-741	318	1	=	=	SYM
ejde-741	318	2	∫	∫	PROPN
ejde-741	318	3	t	t	PROPN
ejde-741	318	4	0	0	NUM
ejde-741	319	1	∫	∫	PROPN
ejde-741	319	2	1	1	NUM
ejde-741	319	3	0	0	NUM
ejde-741	319	4	ρ20|ℓ′(uλ)|2|aλū|2|(aūx)x|2	ρ20|ℓ′(uλ)|2|aλū|2|(aūx)x|2	NUM
ejde-741	319	5	≤	≤	PROPN
ejde-741	319	6	cλ2	cλ2	PROPN
ejde-741	319	7	∫	∫	PROPN
ejde-741	319	8	t	t	PROPN
ejde-741	319	9	0	0	NUM
ejde-741	320	1	∫	∫	PROPN
ejde-741	320	2	1	1	NUM
ejde-741	320	3	0	0	NUM
ejde-741	320	4	ρ20|aū|2|(aūx)x|2	ρ20|aū|2|(aūx)x|2	VERB
ejde-741	320	5	≤	≤	NOUN
ejde-741	320	6	cλ2∥(ū	cλ2∥(ū	PROPN
ejde-741	320	7	,	,	PUNCT
ejde-741	320	8	h̄)∥2e	h̄)∥2e	PROPN
ejde-741	320	9	→	→	SYM
ejde-741	320	10	0	0	NUM
ejde-741	320	11	,	,	PUNCT
ejde-741	320	12	as	as	ADP
ejde-741	320	13	λ→	λ→	PROPN
ejde-741	320	14	0	0	NUM
ejde-741	320	15	.	.	PUNCT
ejde-741	321	1	thus	thus	ADV
ejde-741	321	2	,	,	PUNCT
ejde-741	321	3	the	the	DET
ejde-741	321	4	claim	claim	NOUN
ejde-741	321	5	1	1	NUM
ejde-741	321	6	is	be	AUX
ejde-741	321	7	concluded	conclude	VERB
ejde-741	321	8	.	.	PUNCT
ejde-741	322	1	claim	claim	NOUN
ejde-741	322	2	2	2	NUM
ejde-741	322	3	:	:	PUNCT
ejde-741	322	4	the	the	DET
ejde-741	322	5	gateaux	gateaux	ADV
ejde-741	322	6	derivative	derivative	ADJ
ejde-741	322	7	h′	h′	PROPN
ejde-741	322	8	1	1	NUM
ejde-741	322	9	:	:	PUNCT
ejde-741	322	10	e	e	X
ejde-741	322	11	→	→	SYM
ejde-741	322	12	l(e;l2(q	l(e;l2(q	PROPN
ejde-741	322	13	;	;	PUNCT
ejde-741	322	14	ρ20	ρ20	NUM
ejde-741	322	15	)	)	PUNCT
ejde-741	322	16	)	)	PUNCT
ejde-741	322	17	is	be	AUX
ejde-741	322	18	continuous	continuous	ADJ
ejde-741	322	19	.	.	PUNCT
ejde-741	323	1	take	take	VERB
ejde-741	323	2	(	(	PUNCT
ejde-741	323	3	u	u	NOUN
ejde-741	323	4	,	,	PUNCT
ejde-741	323	5	h	h	NOUN
ejde-741	323	6	)	)	PUNCT
ejde-741	323	7	∈	∈	NOUN
ejde-741	323	8	e	e	NOUN
ejde-741	323	9	and	and	CCONJ
ejde-741	323	10	let	let	VERB
ejde-741	323	11	(	(	PUNCT
ejde-741	323	12	(	(	PUNCT
ejde-741	323	13	un	un	PROPN
ejde-741	323	14	,	,	PUNCT
ejde-741	323	15	hn))∞n=1	hn))∞n=1	PROPN
ejde-741	323	16	be	be	VERB
ejde-741	323	17	a	a	DET
ejde-741	323	18	sequence	sequence	NOUN
ejde-741	323	19	such	such	ADJ
ejde-741	323	20	that	that	SCONJ
ejde-741	323	21	∥(un	∥(un	PROPN
ejde-741	323	22	,	,	PUNCT
ejde-741	323	23	hn)−	hn)−	X
ejde-741	323	24	(	(	PUNCT
ejde-741	323	25	u	u	NOUN
ejde-741	323	26	,	,	PUNCT
ejde-741	323	27	h)∥e	h)∥e	X
ejde-741	323	28	→	→	SYM
ejde-741	323	29	0	0	NUM
ejde-741	323	30	.	.	PROPN
ejde-741	323	31	12	12	NUM
ejde-741	324	1	p.	p.	NOUN
ejde-741	324	2	p.	p.	NOUN
ejde-741	324	3	de	de	PROPN
ejde-741	324	4	carvalho	carvalho	PROPN
ejde-741	324	5	,	,	PUNCT
ejde-741	324	6	r.	r.	PROPN
ejde-741	324	7	demarque	demarque	PROPN
ejde-741	324	8	,	,	PUNCT
ejde-741	324	9	j.	j.	PROPN
ejde-741	324	10	límaco	límaco	PROPN
ejde-741	324	11	,	,	PUNCT
ejde-741	324	12	l.	l.	PROPN
ejde-741	324	13	viana	viana	PROPN
ejde-741	324	14	ejde-2025/15	ejde-2025/15	PROPN
ejde-741	324	15	we	we	PRON
ejde-741	324	16	will	will	AUX
ejde-741	324	17	prove	prove	VERB
ejde-741	324	18	that	that	SCONJ
ejde-741	324	19	∥h′	∥h′	NOUN
ejde-741	324	20	1(u	1(u	NUM
ejde-741	324	21	n	n	CCONJ
ejde-741	324	22	,	,	PUNCT
ejde-741	324	23	hn	hn	PROPN
ejde-741	324	24	)	)	PUNCT
ejde-741	324	25	−h′	−h′	PROPN
ejde-741	324	26	1(u	1(u	NUM
ejde-741	324	27	,	,	PUNCT
ejde-741	324	28	h)∥l(e;l2(q;ρ20	h)∥l(e;l2(q;ρ20	NOUN
ejde-741	324	29	)	)	PUNCT
ejde-741	324	30	)	)	PUNCT
ejde-741	325	1	→	→	SYM
ejde-741	325	2	0	0	X
ejde-741	325	3	.	.	PUNCT
ejde-741	326	1	in	in	ADP
ejde-741	326	2	fact	fact	NOUN
ejde-741	326	3	,	,	PUNCT
ejde-741	326	4	we	we	PRON
ejde-741	326	5	consider	consider	VERB
ejde-741	326	6	(	(	PUNCT
ejde-741	326	7	ū	ū	NOUN
ejde-741	326	8	,	,	PUNCT
ejde-741	326	9	h̄	h̄	NOUN
ejde-741	326	10	)	)	PUNCT
ejde-741	326	11	on	on	ADP
ejde-741	326	12	the	the	DET
ejde-741	326	13	unit	unit	NOUN
ejde-741	326	14	sphere	sphere	ADV
ejde-741	326	15	of	of	ADP
ejde-741	326	16	e.	e.	PROPN
ejde-741	326	17	since	since	SCONJ
ejde-741	326	18	h′	h′	PROPN
ejde-741	326	19	1(u	1(u	NUM
ejde-741	326	20	,	,	PUNCT
ejde-741	326	21	h)(ū	h)(ū	PRON
ejde-741	326	22	,	,	PUNCT
ejde-741	326	23	h̄	h̄	NOUN
ejde-741	326	24	)	)	PUNCT
ejde-741	326	25	=	=	PUNCT
ejde-741	327	1	ūt	ūt	ADJ
ejde-741	327	2	−	−	PROPN
ejde-741	327	3	ℓ′(au)aū(aux)x	ℓ′(au)aū(aux)x	PROPN
ejde-741	327	4	+	+	NUM
ejde-741	327	5	ℓ(au)(aūx)x	ℓ(au)(aūx)x	NOUN
ejde-741	327	6	+	+	CCONJ
ejde-741	327	7	f3ū−	f3ū−	PROPN
ejde-741	327	8	h̄χω	h̄χω	PROPN
ejde-741	327	9	and	and	CCONJ
ejde-741	327	10	h′	h′	PROPN
ejde-741	327	11	1(u	1(u	NUM
ejde-741	327	12	n	n	CCONJ
ejde-741	327	13	,	,	PUNCT
ejde-741	327	14	hn)(ū	hn)(ū	NOUN
ejde-741	327	15	,	,	PUNCT
ejde-741	327	16	h̄	h̄	NOUN
ejde-741	327	17	)	)	PUNCT
ejde-741	327	18	=	=	PUNCT
ejde-741	327	19	ūt	ūt	ADJ
ejde-741	327	20	−	−	PROPN
ejde-741	327	21	ℓ′(aun)aū(aunx)x	ℓ′(aun)aū(aunx)x	NOUN
ejde-741	327	22	+	+	CCONJ
ejde-741	327	23	ℓ(aun)(aūx)x	ℓ(aun)(aūx)x	NUM
ejde-741	327	24	+	+	ADJ
ejde-741	327	25	d3f(t	d3f(t	NOUN
ejde-741	327	26	,	,	PUNCT
ejde-741	327	27	x	x	X
ejde-741	327	28	,	,	PUNCT
ejde-741	327	29	u	u	PROPN
ejde-741	327	30	n)ū−	n)ū−	PROPN
ejde-741	327	31	h̄χω	h̄χω	PROPN
ejde-741	327	32	,	,	PUNCT
ejde-741	327	33	we	we	PRON
ejde-741	327	34	obtain	obtain	VERB
ejde-741	327	35	∥(h	∥(h	NOUN
ejde-741	327	36	′	′	NUM
ejde-741	327	37	1(u	1(u	NUM
ejde-741	327	38	n	n	CCONJ
ejde-741	327	39	,	,	PUNCT
ejde-741	327	40	hn)−h	hn)−h	PROPN
ejde-741	327	41	′	′	NUM
ejde-741	327	42	1(u	1(u	NUM
ejde-741	327	43	,	,	PUNCT
ejde-741	327	44	h))(ū	h))(ū	PRON
ejde-741	327	45	,	,	PUNCT
ejde-741	327	46	h̄)∥2ρ20	h̄)∥2ρ20	NOUN
ejde-741	327	47	≤	≤	NUM
ejde-741	328	1	c	c	PROPN
ejde-741	328	2	∫	∫	PROPN
ejde-741	328	3	t	t	PROPN
ejde-741	328	4	0	0	NUM
ejde-741	328	5	∫	∫	PROPN
ejde-741	328	6	1	1	NUM
ejde-741	328	7	0	0	NUM
ejde-741	329	1	ρ20|ℓ′(aun)|2|aū|2|[a(u−	ρ20|ℓ′(aun)|2|aū|2|[a(u−	PRON
ejde-741	329	2	un)x]x|2	un)x]x|2	PROPN
ejde-741	330	1	+	+	NUM
ejde-741	330	2	c	c	NOUN
ejde-741	330	3	∫	∫	PROPN
ejde-741	330	4	t	t	PROPN
ejde-741	330	5	0	0	NUM
ejde-741	330	6	∫	∫	PROPN
ejde-741	330	7	1	1	NUM
ejde-741	330	8	0	0	NUM
ejde-741	330	9	ρ20|ℓ′(aun)−	ρ20|ℓ′(aun)−	PRON
ejde-741	330	10	ℓ′(au)|2|aū|2|(aux)x|2	ℓ′(au)|2|aū|2|(aux)x|2	PROPN
ejde-741	331	1	+	+	CCONJ
ejde-741	331	2	c	c	NOUN
ejde-741	331	3	∫	∫	PROPN
ejde-741	331	4	t	t	PROPN
ejde-741	331	5	0	0	NUM
ejde-741	331	6	∫	∫	PROPN
ejde-741	331	7	1	1	NUM
ejde-741	331	8	0	0	NUM
ejde-741	331	9	ρ20|ℓ(aun)−	ρ20|ℓ(aun)−	NOUN
ejde-741	331	10	ℓ(au)|2|(aūx)x|2	ℓ(au)|2|(aūx)x|2	PUNCT
ejde-741	332	1	+	+	CCONJ
ejde-741	332	2	c	c	NOUN
ejde-741	332	3	∫	∫	PROPN
ejde-741	332	4	t	t	PROPN
ejde-741	332	5	0	0	NUM
ejde-741	332	6	∫	∫	PROPN
ejde-741	332	7	1	1	NUM
ejde-741	332	8	0	0	NUM
ejde-741	332	9	ρ20|ū|2|d3f(t	ρ20|ū|2|d3f(t	NOUN
ejde-741	332	10	,	,	PUNCT
ejde-741	332	11	x	x	PRON
ejde-741	332	12	,	,	PUNCT
ejde-741	332	13	u	u	NOUN
ejde-741	332	14	n)−d3f(t	n)−d3f(t	NOUN
ejde-741	332	15	,	,	PUNCT
ejde-741	332	16	x	x	X
ejde-741	332	17	,	,	PUNCT
ejde-741	332	18	u)|2	u)|2	NOUN
ejde-741	332	19	=	=	NOUN
ejde-741	332	20	:	:	PUNCT
ejde-741	332	21	c(j1	c(j1	VERB
ejde-741	333	1	+	+	CCONJ
ejde-741	333	2	j2	j2	PROPN
ejde-741	333	3	+	+	CCONJ
ejde-741	333	4	j3	j3	PROPN
ejde-741	333	5	+	+	CCONJ
ejde-741	333	6	j4	j4	PROPN
ejde-741	333	7	)	)	PUNCT
ejde-741	333	8	.	.	PUNCT
ejde-741	334	1	once	once	ADV
ejde-741	334	2	again	again	ADV
ejde-741	334	3	,	,	PUNCT
ejde-741	334	4	arguing	argue	VERB
ejde-741	334	5	as	as	ADP
ejde-741	334	6	in	in	ADP
ejde-741	334	7	(	(	PUNCT
ejde-741	334	8	3.2	3.2	NUM
ejde-741	334	9	)	)	PUNCT
ejde-741	334	10	,	,	PUNCT
ejde-741	334	11	(	(	PUNCT
ejde-741	334	12	3.4	3.4	NUM
ejde-741	334	13	)	)	PUNCT
ejde-741	334	14	and	and	CCONJ
ejde-741	334	15	(	(	PUNCT
ejde-741	334	16	3.5	3.5	NUM
ejde-741	334	17	)	)	PUNCT
ejde-741	334	18	,	,	PUNCT
ejde-741	334	19	we	we	PRON
ejde-741	334	20	obtain	obtain	VERB
ejde-741	334	21	j1	j1	PROPN
ejde-741	334	22	≤	≤	NUM
ejde-741	334	23	∫	∫	PROPN
ejde-741	334	24	t	t	PROPN
ejde-741	334	25	0	0	NUM
ejde-741	334	26	∫	∫	PROPN
ejde-741	334	27	1	1	NUM
ejde-741	334	28	0	0	NUM
ejde-741	335	1	ρ20|aū|2|(a(u−	ρ20|aū|2|(a(u−	NUM
ejde-741	335	2	un)x)x|2	un)x)x|2	PROPN
ejde-741	335	3	≤	≤	PROPN
ejde-741	335	4	c	c	NOUN
ejde-741	335	5	(	(	PUNCT
ejde-741	335	6	sup	sup	NOUN
ejde-741	335	7	t∈[0,t	t∈[0,t	NOUN
ejde-741	335	8	]	]	X
ejde-741	335	9	∥ρ̂ū(t	∥ρ̂ū(t	NOUN
ejde-741	335	10	,	,	PUNCT
ejde-741	335	11	·	·	PUNCT
ejde-741	335	12	)	)	PUNCT
ejde-741	335	13	∥2l2(0,1)∥[a(u−	∥2l2(0,1)∥[a(u−	PUNCT
ejde-741	335	14	un)x]x∥2ρ2∗	un)x]x∥2ρ2∗	PRON
ejde-741	336	1	+	+	CCONJ
ejde-741	336	2	sup	sup	NOUN
ejde-741	336	3	t∈[0,t	t∈[0,t	NOUN
ejde-741	336	4	]	]	PUNCT
ejde-741	336	5	∥ρ∗	∥ρ∗	PROPN
ejde-741	336	6	√	√	PROPN
ejde-741	336	7	aūx(t	aūx(t	PROPN
ejde-741	336	8	,	,	PUNCT
ejde-741	336	9	·	·	PUNCT
ejde-741	336	10	)	)	PUNCT
ejde-741	336	11	∥2l2(0,1)∥[a(u−	∥2l2(0,1)∥[a(u−	PROPN
ejde-741	336	12	un)x]x∥2ρ2∗	un)x]x∥2ρ2∗	SYM
ejde-741	336	13	)	)	PUNCT
ejde-741	336	14	≤	≤	NOUN
ejde-741	336	15	c∥(ū	c∥(ū	NOUN
ejde-741	336	16	,	,	PUNCT
ejde-741	336	17	h̄)∥2e	h̄)∥2e	PROPN
ejde-741	336	18	∥(un	∥(un	PROPN
ejde-741	336	19	,	,	PUNCT
ejde-741	336	20	hn)−	hn)−	X
ejde-741	336	21	(	(	PUNCT
ejde-741	336	22	u	u	NOUN
ejde-741	336	23	,	,	PUNCT
ejde-741	336	24	h)∥2e	h)∥2e	PROPN
ejde-741	336	25	.	.	PUNCT
ejde-741	337	1	next	next	ADJ
ejde-741	337	2	,	,	PUNCT
ejde-741	337	3	applying	apply	VERB
ejde-741	337	4	relation	relation	NOUN
ejde-741	337	5	(	(	PUNCT
ejde-741	337	6	3.3	3.3	NUM
ejde-741	337	7	)	)	PUNCT
ejde-741	337	8	,	,	PUNCT
ejde-741	337	9	we	we	PRON
ejde-741	337	10	have	have	VERB
ejde-741	337	11	j2	j2	PROPN
ejde-741	337	12	≤	≤	PROPN
ejde-741	337	13	∫	∫	PROPN
ejde-741	337	14	t	t	PROPN
ejde-741	337	15	0	0	NUM
ejde-741	337	16	(	(	PUNCT
ejde-741	337	17	e−2sτη̂rτ−5/3ρ−4	e−2sτη̂rτ−5/3ρ−4	PUNCT
ejde-741	337	18	∗	∗	NOUN
ejde-741	337	19	)	)	PUNCT
ejde-741	337	20	(	(	PUNCT
ejde-741	337	21	ρ2∗∥aū∥2∞	ρ2∗∥aū∥2∞	PROPN
ejde-741	337	22	)	)	PUNCT
ejde-741	337	23	∫	∫	PROPN
ejde-741	337	24	1	1	NUM
ejde-741	337	25	0	0	NUM
ejde-741	338	1	η−5/3ρ2∗|ℓ′(aun)−	η−5/3ρ2∗|ℓ′(aun)−	NOUN
ejde-741	339	1	ℓ′(au)|2|(aux)x|2	ℓ′(au)|2|(aux)x|2	PROPN
ejde-741	339	2	≤	≤	PROPN
ejde-741	339	3	c	c	X
ejde-741	339	4	[	[	PUNCT
ejde-741	339	5	∫	∫	PROPN
ejde-741	339	6	1	1	NUM
ejde-741	339	7	0	0	NUM
ejde-741	339	8	ρ2∗(∥ū∥2l2(0,1	ρ2∗(∥ū∥2l2(0,1	NOUN
ejde-741	339	9	)	)	PUNCT
ejde-741	340	1	+	+	CCONJ
ejde-741	340	2	∥	∥	NOUN
ejde-741	340	3	√	√	NUM
ejde-741	340	4	aūx∥2l2(0,1	aūx∥2l2(0,1	NUM
ejde-741	340	5	)	)	PUNCT
ejde-741	340	6	)	)	PUNCT
ejde-741	340	7	∫	∫	PROPN
ejde-741	340	8	1	1	NUM
ejde-741	340	9	0	0	NUM
ejde-741	340	10	ρ2∗|ℓ′(aun)−	ρ2∗|ℓ′(aun)−	PROPN
ejde-741	340	11	ℓ′(au)|2|(aux)x|2	ℓ′(au)|2|(aux)x|2	PROPN
ejde-741	340	12	]	]	PUNCT
ejde-741	341	1	≤	≤	NUM
ejde-741	341	2	c	c	NOUN
ejde-741	341	3	[	[	PUNCT
ejde-741	341	4	sup	sup	NOUN
ejde-741	341	5	t∈[0,t	t∈[0,t	NOUN
ejde-741	341	6	]	]	X
ejde-741	341	7	∥ρ̂ū(t	∥ρ̂ū(t	NOUN
ejde-741	341	8	,	,	PUNCT
ejde-741	341	9	·	·	PUNCT
ejde-741	341	10	)	)	PUNCT
ejde-741	341	11	∥2l2(0,1	∥2l2(0,1	ADV
ejde-741	341	12	)	)	PUNCT
ejde-741	342	1	+	+	CCONJ
ejde-741	342	2	sup	sup	NOUN
ejde-741	342	3	t∈[0,t	t∈[0,t	NOUN
ejde-741	342	4	]	]	PUNCT
ejde-741	342	5	∥ρ∗	∥ρ∗	PROPN
ejde-741	342	6	√	√	PROPN
ejde-741	342	7	aūx(t	aūx(t	PROPN
ejde-741	342	8	,	,	PUNCT
ejde-741	342	9	·	·	PUNCT
ejde-741	342	10	)	)	PUNCT
ejde-741	342	11	∥2l2(0,1	∥2l2(0,1	ADV
ejde-741	342	12	)	)	PUNCT
ejde-741	342	13	]	]	PUNCT
ejde-741	343	1	×	×	NOUN
ejde-741	343	2	∫	∫	PROPN
ejde-741	343	3	t	t	PROPN
ejde-741	343	4	0	0	NUM
ejde-741	344	1	∫	∫	PROPN
ejde-741	344	2	1	1	NUM
ejde-741	344	3	0	0	NUM
ejde-741	345	1	ρ2∗|ℓ′(aun)−	ρ2∗|ℓ′(aun)−	PROPN
ejde-741	345	2	ℓ′(au)|2|(aux)x|2	ℓ′(au)|2|(aux)x|2	PROPN
ejde-741	345	3	≤	≤	NUM
ejde-741	345	4	c∥(ū	c∥(ū	VERB
ejde-741	345	5	,	,	PUNCT
ejde-741	346	1	h̄)∥2e	h̄)∥2e	PROPN
ejde-741	346	2	∫	∫	PROPN
ejde-741	346	3	t	t	PROPN
ejde-741	346	4	0	0	NUM
ejde-741	346	5	∫	∫	PROPN
ejde-741	346	6	1	1	NUM
ejde-741	346	7	0	0	NUM
ejde-741	346	8	ρ2∗|ℓ′(aun)−	ρ2∗|ℓ′(aun)−	PROPN
ejde-741	346	9	ℓ′(au)|2|(aux)x|2	ℓ′(au)|2|(aux)x|2	PROPN
ejde-741	346	10	→	→	SYM
ejde-741	346	11	0	0	NUM
ejde-741	346	12	,	,	PUNCT
ejde-741	346	13	as	as	ADP
ejde-741	346	14	n	n	PROPN
ejde-741	346	15	→	→	SYM
ejde-741	346	16	+	+	PROPN
ejde-741	346	17	∞	∞	PROPN
ejde-741	346	18	,	,	PUNCT
ejde-741	346	19	where	where	SCONJ
ejde-741	346	20	the	the	DET
ejde-741	346	21	convergence	convergence	NOUN
ejde-741	346	22	is	be	AUX
ejde-741	346	23	a	a	DET
ejde-741	346	24	consequence	consequence	NOUN
ejde-741	346	25	of	of	ADP
ejde-741	346	26	lebesgue	lebesgue	PROPN
ejde-741	346	27	’s	’s	PART
ejde-741	346	28	theorem	theorem	PROPN
ejde-741	346	29	.	.	PUNCT
ejde-741	347	1	in	in	ADP
ejde-741	347	2	a	a	DET
ejde-741	347	3	very	very	ADV
ejde-741	347	4	similar	similar	ADJ
ejde-741	347	5	way	way	NOUN
ejde-741	347	6	,	,	PUNCT
ejde-741	347	7	j3	j3	PROPN
ejde-741	347	8	≤	≤	PROPN
ejde-741	347	9	∫	∫	PROPN
ejde-741	347	10	t	t	PROPN
ejde-741	347	11	0	0	NUM
ejde-741	347	12	(	(	PUNCT
ejde-741	347	13	e−2sτη̂rτ−5/3ρ−4	e−2sτη̂rτ−5/3ρ−4	PUNCT
ejde-741	347	14	∗	∗	NOUN
ejde-741	347	15	)	)	PUNCT
ejde-741	347	16	(	(	PUNCT
ejde-741	347	17	ρ2∗∥a(un	ρ2∗∥a(un	NOUN
ejde-741	347	18	−	−	PROPN
ejde-741	347	19	u)∥2∞	u)∥2∞	PROPN
ejde-741	347	20	)	)	PUNCT
ejde-741	347	21	∫	∫	PROPN
ejde-741	348	1	1	1	NUM
ejde-741	348	2	0	0	NUM
ejde-741	348	3	η−5/3ρ2∗|(aūx)x|2	η−5/3ρ2∗|(aūx)x|2	ADJ
ejde-741	348	4	ejde-2025/15	ejde-2025/15	NOUN
ejde-741	348	5	null	null	ADJ
ejde-741	348	6	-	-	PUNCT
ejde-741	348	7	controllability	controllability	NOUN
ejde-741	348	8	degenerate	degenerate	ADJ
ejde-741	348	9	quasilinear	quasilinear	NOUN
ejde-741	348	10	equations	equation	NOUN
ejde-741	348	11	13	13	NUM
ejde-741	348	12	≤	≤	NOUN
ejde-741	349	1	c	c	NOUN
ejde-741	349	2	[	[	PUNCT
ejde-741	349	3	sup	sup	NOUN
ejde-741	349	4	t∈[0,t	t∈[0,t	NOUN
ejde-741	349	5	]	]	PUNCT
ejde-741	349	6	∥ρ̂(un	∥ρ̂(un	PROPN
ejde-741	349	7	−	−	PROPN
ejde-741	349	8	u)(t	u)(t	PROPN
ejde-741	349	9	,	,	PUNCT
ejde-741	349	10	·	·	PUNCT
ejde-741	349	11	)	)	PUNCT
ejde-741	349	12	∥2l2(0,1	∥2l2(0,1	ADV
ejde-741	349	13	)	)	PUNCT
ejde-741	350	1	+	+	CCONJ
ejde-741	350	2	sup	sup	NOUN
ejde-741	350	3	t∈[0,t	t∈[0,t	NOUN
ejde-741	350	4	]	]	PUNCT
ejde-741	350	5	∥ρ∗	∥ρ∗	PROPN
ejde-741	350	6	√	√	PROPN
ejde-741	350	7	a(un	a(un	PROPN
ejde-741	350	8	−	−	PROPN
ejde-741	350	9	u)x(t	u)x(t	PROPN
ejde-741	350	10	,	,	PUNCT
ejde-741	350	11	·	·	PUNCT
ejde-741	350	12	)	)	PUNCT
ejde-741	350	13	∥2l2(0,1	∥2l2(0,1	ADV
ejde-741	350	14	)	)	PUNCT
ejde-741	350	15	]	]	PUNCT
ejde-741	351	1	×	×	NOUN
ejde-741	351	2	∫	∫	PROPN
ejde-741	351	3	t	t	PROPN
ejde-741	351	4	0	0	NUM
ejde-741	352	1	∫	∫	PROPN
ejde-741	352	2	1	1	NUM
ejde-741	352	3	0	0	NUM
ejde-741	352	4	ρ2∗|(aūx)x|2	ρ2∗|(aūx)x|2	ADJ
ejde-741	352	5	≤	≤	PUNCT
ejde-741	352	6	c∥(un	c∥(un	PROPN
ejde-741	352	7	,	,	PUNCT
ejde-741	352	8	hn)−	hn)−	X
ejde-741	352	9	(	(	PUNCT
ejde-741	352	10	u	u	NOUN
ejde-741	352	11	,	,	PUNCT
ejde-741	352	12	h)∥2e∥(ū	h)∥2e∥(ū	PROPN
ejde-741	352	13	,	,	PUNCT
ejde-741	352	14	h̄)∥2e	h̄)∥2e	PROPN
ejde-741	352	15	.	.	PUNCT
ejde-741	353	1	at	at	ADP
ejde-741	353	2	last	last	ADJ
ejde-741	353	3	,	,	PUNCT
ejde-741	353	4	applying	apply	VERB
ejde-741	353	5	hypothesis	hypothesis	NOUN
ejde-741	353	6	1.3	1.3	NUM
ejde-741	353	7	and	and	CCONJ
ejde-741	353	8	(	(	PUNCT
ejde-741	353	9	3.1	3.1	NUM
ejde-741	353	10	)	)	PUNCT
ejde-741	353	11	,	,	PUNCT
ejde-741	353	12	we	we	PRON
ejde-741	353	13	obtain	obtain	VERB
ejde-741	353	14	j4	j4	PROPN
ejde-741	353	15	=	=	SYM
ejde-741	353	16	(	(	PUNCT
ejde-741	353	17	∫	∫	PROPN
ejde-741	353	18	t	t	PROPN
ejde-741	353	19	0	0	NUM
ejde-741	353	20	∫	∫	PROPN
ejde-741	353	21	1	1	NUM
ejde-741	353	22	0	0	NUM
ejde-741	353	23	ρ20|ū|2|d3f(t	ρ20|ū|2|d3f(t	NOUN
ejde-741	353	24	,	,	PUNCT
ejde-741	353	25	x	x	PRON
ejde-741	353	26	,	,	PUNCT
ejde-741	353	27	u	u	NOUN
ejde-741	353	28	n)−d3f(t	n)−d3f(t	NOUN
ejde-741	353	29	,	,	PUNCT
ejde-741	353	30	x	x	SYM
ejde-741	353	31	,	,	PUNCT
ejde-741	353	32	u)|2	u)|2	NOUN
ejde-741	353	33	)	)	PUNCT
ejde-741	353	34	≤	≤	NUM
ejde-741	353	35	sup	sup	NOUN
ejde-741	353	36	(	(	PUNCT
ejde-741	353	37	t	t	PROPN
ejde-741	353	38	,	,	PUNCT
ejde-741	353	39	x)∈q	x)∈q	PROPN
ejde-741	353	40	|d3f(t	|d3f(t	PROPN
ejde-741	353	41	,	,	PUNCT
ejde-741	353	42	x	x	PRON
ejde-741	353	43	,	,	PUNCT
ejde-741	353	44	u	u	NOUN
ejde-741	353	45	n)−d3f(t	n)−d3f(t	NOUN
ejde-741	353	46	,	,	PUNCT
ejde-741	353	47	x	x	PRON
ejde-741	353	48	,	,	PUNCT
ejde-741	353	49	u)|2	u)|2	PROPN
ejde-741	353	50	∫	∫	PROPN
ejde-741	353	51	t	t	PROPN
ejde-741	353	52	0	0	NUM
ejde-741	353	53	∫	∫	PROPN
ejde-741	353	54	1	1	NUM
ejde-741	353	55	0	0	NUM
ejde-741	353	56	ρ20|ū|2	ρ20|ū|2	NOUN
ejde-741	353	57	≤	≤	NOUN
ejde-741	353	58	∥(un	∥(un	PROPN
ejde-741	353	59	,	,	PUNCT
ejde-741	353	60	hn)−	hn)−	X
ejde-741	353	61	(	(	PUNCT
ejde-741	353	62	u	u	NOUN
ejde-741	353	63	,	,	PUNCT
ejde-741	353	64	h)∥2e∥(ū	h)∥2e∥(ū	PROPN
ejde-741	353	65	,	,	PUNCT
ejde-741	353	66	h̄)∥2e	h̄)∥2e	PROPN
ejde-741	353	67	,	,	PUNCT
ejde-741	353	68	where	where	SCONJ
ejde-741	353	69	we	we	PRON
ejde-741	353	70	have	have	AUX
ejde-741	353	71	also	also	ADV
ejde-741	353	72	used	use	VERB
ejde-741	353	73	the	the	DET
ejde-741	353	74	continuous	continuous	ADJ
ejde-741	353	75	embedding	embed	VERB
ejde-741	353	76	c([0	c([0	NOUN
ejde-741	353	77	,	,	PUNCT
ejde-741	353	78	t	t	X
ejde-741	353	79	]	]	PUNCT
ejde-741	353	80	;	;	PUNCT
ejde-741	353	81	h1	h1	VERB
ejde-741	353	82	a	a	PRON
ejde-741	353	83	)	)	PUNCT
ejde-741	353	84	↪	↪	PROPN
ejde-741	353	85	→	→	SYM
ejde-741	353	86	c(q	c(q	PROPN
ejde-741	353	87	)	)	PUNCT
ejde-741	353	88	.	.	PUNCT
ejde-741	354	1	therefore	therefore	ADV
ejde-741	354	2	,	,	PUNCT
ejde-741	354	3	h′	h′	PROPN
ejde-741	354	4	1(u	1(u	NUM
ejde-741	354	5	n	n	CCONJ
ejde-741	354	6	,	,	PUNCT
ejde-741	354	7	hn	hn	PROPN
ejde-741	354	8	)	)	PUNCT
ejde-741	354	9	→	→	SYM
ejde-741	354	10	h′	h′	PROPN
ejde-741	354	11	1(u	1(u	NUM
ejde-741	354	12	,	,	PUNCT
ejde-741	354	13	h	h	NOUN
ejde-741	354	14	)	)	PUNCT
ejde-741	354	15	in	in	ADP
ejde-741	354	16	l(e;l2(q	l(e;l2(q	PROPN
ejde-741	354	17	;	;	PUNCT
ejde-741	354	18	ρ20	ρ20	NUM
ejde-741	354	19	)	)	PUNCT
ejde-741	354	20	)	)	PUNCT
ejde-741	354	21	,	,	PUNCT
ejde-741	354	22	which	which	PRON
ejde-741	354	23	means	mean	VERB
ejde-741	354	24	that	that	SCONJ
ejde-741	354	25	h′	h′	PROPN
ejde-741	354	26	1	1	NUM
ejde-741	354	27	:	:	PUNCT
ejde-741	354	28	e	e	X
ejde-741	354	29	→	→	SYM
ejde-741	354	30	l(e;l2(q	l(e;l2(q	PROPN
ejde-741	354	31	;	;	PUNCT
ejde-741	354	32	ρ20	ρ20	NUM
ejde-741	354	33	)	)	PUNCT
ejde-741	354	34	)	)	PUNCT
ejde-741	354	35	is	be	AUX
ejde-741	354	36	a	a	DET
ejde-741	354	37	continuous	continuous	ADJ
ejde-741	354	38	mapping	mapping	NOUN
ejde-741	354	39	,	,	PUNCT
ejde-741	354	40	as	as	SCONJ
ejde-741	354	41	stated	state	VERB
ejde-741	354	42	in	in	ADP
ejde-741	354	43	claim	claim	NOUN
ejde-741	354	44	2	2	X
ejde-741	354	45	.	.	PUNCT
ejde-741	355	1	this	this	PRON
ejde-741	355	2	completes	complete	VERB
ejde-741	355	3	the	the	DET
ejde-741	355	4	proof	proof	NOUN
ejde-741	355	5	.	.	PUNCT
ejde-741	356	1	□	□	PUNCT
ejde-741	356	2	3.3	3.3	NUM
ejde-741	356	3	.	.	PUNCT
ejde-741	357	1	proof	proof	NOUN
ejde-741	357	2	of	of	ADP
ejde-741	357	3	proposition	proposition	NOUN
ejde-741	357	4	3.1	3.1	NUM
ejde-741	357	5	.	.	PUNCT
ejde-741	358	1	proof	proof	NOUN
ejde-741	358	2	.	.	PUNCT
ejde-741	359	1	given	give	VERB
ejde-741	359	2	t	t	PROPN
ejde-741	359	3	>	>	X
ejde-741	359	4	0	0	PUNCT
ejde-741	360	1	and	and	CCONJ
ejde-741	360	2	(	(	PUNCT
ejde-741	360	3	u0	u0	ADJ
ejde-741	360	4	,	,	PUNCT
ejde-741	360	5	g	g	NOUN
ejde-741	360	6	)	)	PUNCT
ejde-741	360	7	∈	∈	PROPN
ejde-741	360	8	h1	h1	VERB
ejde-741	360	9	a	a	DET
ejde-741	360	10	×	×	PROPN
ejde-741	360	11	l2(q	l2(q	PROPN
ejde-741	360	12	;	;	PUNCT
ejde-741	360	13	ρ20	ρ20	NUM
ejde-741	360	14	)	)	PUNCT
ejde-741	360	15	,	,	PUNCT
ejde-741	360	16	let	let	VERB
ejde-741	360	17	us	we	PRON
ejde-741	360	18	consider	consider	VERB
ejde-741	360	19	the	the	DET
ejde-741	360	20	problem	problem	NOUN
ejde-741	360	21	ut	ut	PROPN
ejde-741	361	1	−	−	PROPN
ejde-741	362	1	(	(	PUNCT
ejde-741	362	2	a(x)ux)x	a(x)ux)x	VERB
ejde-741	362	3	+	+	CCONJ
ejde-741	362	4	c(t	c(t	PROPN
ejde-741	362	5	,	,	PUNCT
ejde-741	362	6	x)u	x)u	PUNCT
ejde-741	362	7	=	=	SYM
ejde-741	362	8	h+	h+	X
ejde-741	362	9	g	g	PROPN
ejde-741	362	10	,	,	PUNCT
ejde-741	362	11	(	(	PUNCT
ejde-741	362	12	t	t	PROPN
ejde-741	362	13	,	,	PUNCT
ejde-741	362	14	x	x	NOUN
ejde-741	362	15	)	)	PUNCT
ejde-741	362	16	in	in	ADP
ejde-741	362	17	q	q	NOUN
ejde-741	362	18	,	,	PUNCT
ejde-741	362	19	u(t	u(t	NOUN
ejde-741	362	20	,	,	PUNCT
ejde-741	362	21	1	1	NUM
ejde-741	362	22	)	)	PUNCT
ejde-741	362	23	=	=	SYM
ejde-741	362	24	0	0	NUM
ejde-741	362	25	,	,	PUNCT
ejde-741	362	26	t	t	PROPN
ejde-741	362	27	∈	∈	PROPN
ejde-741	362	28	(	(	PUNCT
ejde-741	362	29	0	0	NUM
ejde-741	362	30	,	,	PUNCT
ejde-741	362	31	t	t	NOUN
ejde-741	362	32	)	)	PUNCT
ejde-741	362	33	,	,	PUNCT
ejde-741	362	34			PROPN
ejde-741	362	35	u(t	u(t	NOUN
ejde-741	362	36	,	,	PUNCT
ejde-741	362	37	0	0	NUM
ejde-741	362	38	)	)	PUNCT
ejde-741	362	39	=	=	SYM
ejde-741	362	40	0	0	NUM
ejde-741	362	41	,	,	PUNCT
ejde-741	362	42	(	(	PUNCT
ejde-741	362	43	weak	weak	ADJ
ejde-741	362	44	)	)	PUNCT
ejde-741	362	45	,	,	PUNCT
ejde-741	362	46	t	t	PROPN
ejde-741	362	47	∈	∈	PROPN
ejde-741	362	48	(	(	PUNCT
ejde-741	362	49	0	0	NUM
ejde-741	362	50	,	,	PUNCT
ejde-741	362	51	t	t	NOUN
ejde-741	362	52	)	)	PUNCT
ejde-741	362	53	or	or	CCONJ
ejde-741	362	54	(	(	PUNCT
ejde-741	362	55	aux)(t	aux)(t	NOUN
ejde-741	362	56	,	,	PUNCT
ejde-741	362	57	0	0	NUM
ejde-741	362	58	)	)	PUNCT
ejde-741	362	59	=	=	SYM
ejde-741	362	60	0	0	NUM
ejde-741	362	61	,	,	PUNCT
ejde-741	362	62	(	(	PUNCT
ejde-741	362	63	strong	strong	ADJ
ejde-741	362	64	)	)	PUNCT
ejde-741	362	65	,	,	PUNCT
ejde-741	362	66	t	t	PROPN
ejde-741	362	67	∈	∈	PROPN
ejde-741	362	68	(	(	PUNCT
ejde-741	362	69	0	0	NUM
ejde-741	362	70	,	,	PUNCT
ejde-741	362	71	t	t	NOUN
ejde-741	362	72	)	)	PUNCT
ejde-741	363	1	u(0	u(0	PROPN
ejde-741	363	2	,	,	PUNCT
ejde-741	363	3	x	x	NOUN
ejde-741	363	4	)	)	PUNCT
ejde-741	363	5	=	=	SYM
ejde-741	363	6	u0(x	u0(x	NUM
ejde-741	363	7	)	)	PUNCT
ejde-741	363	8	,	,	PUNCT
ejde-741	363	9	xin	xin	PROPN
ejde-741	363	10	(	(	PUNCT
ejde-741	363	11	0	0	NUM
ejde-741	363	12	,	,	PUNCT
ejde-741	363	13	1	1	NUM
ejde-741	363	14	)	)	PUNCT
ejde-741	363	15	,	,	PUNCT
ejde-741	363	16	(	(	PUNCT
ejde-741	363	17	3.6	3.6	NUM
ejde-741	363	18	)	)	PUNCT
ejde-741	363	19	where	where	SCONJ
ejde-741	363	20	h	h	PROPN
ejde-741	363	21	∈	∈	PROPN
ejde-741	363	22	l2(q	l2(q	PROPN
ejde-741	363	23	)	)	PUNCT
ejde-741	363	24	.	.	PUNCT
ejde-741	364	1	observe	observe	VERB
ejde-741	364	2	that	that	SCONJ
ejde-741	364	3	(	(	PUNCT
ejde-741	364	4	3.6	3.6	NUM
ejde-741	364	5	)	)	PUNCT
ejde-741	364	6	is	be	AUX
ejde-741	364	7	similar	similar	ADJ
ejde-741	364	8	to	to	ADP
ejde-741	364	9	(	(	PUNCT
ejde-741	364	10	2.2	2.2	NUM
ejde-741	364	11	)	)	PUNCT
ejde-741	364	12	,	,	PUNCT
ejde-741	364	13	where	where	SCONJ
ejde-741	364	14	we	we	PRON
ejde-741	364	15	are	be	AUX
ejde-741	364	16	replacing	replace	VERB
ejde-741	364	17	hχω	hχω	PROPN
ejde-741	364	18	,	,	PUNCT
ejde-741	364	19	with	with	ADP
ejde-741	364	20	support	support	NOUN
ejde-741	364	21	in	in	ADP
ejde-741	364	22	qω	qω	PROPN
ejde-741	364	23	,	,	PUNCT
ejde-741	364	24	just	just	ADV
ejde-741	364	25	by	by	ADP
ejde-741	364	26	h.	h.	PROPN
ejde-741	364	27	our	our	PRON
ejde-741	364	28	aim	aim	NOUN
ejde-741	364	29	is	be	AUX
ejde-741	364	30	to	to	PART
ejde-741	364	31	define	define	VERB
ejde-741	364	32	,	,	PUNCT
ejde-741	364	33	for	for	ADP
ejde-741	364	34	each	each	DET
ejde-741	364	35	n	n	PRON
ejde-741	364	36	∈	∈	PROPN
ejde-741	364	37	n∗	n∗	PROPN
ejde-741	364	38	,	,	PUNCT
ejde-741	364	39	a	a	DET
ejde-741	364	40	functional	functional	ADJ
ejde-741	364	41	jn	jn	X
ejde-741	364	42	:	:	PUNCT
ejde-741	364	43	[	[	PUNCT
ejde-741	364	44	l2(q	l2(q	NOUN
ejde-741	364	45	)	)	PUNCT
ejde-741	364	46	]	]	PUNCT
ejde-741	364	47	2	2	NUM
ejde-741	364	48	→	→	SYM
ejde-741	364	49	r	r	NOUN
ejde-741	364	50	,	,	PUNCT
ejde-741	364	51	minimizing	minimize	VERB
ejde-741	364	52	each	each	DET
ejde-741	364	53	one	one	NUM
ejde-741	364	54	of	of	ADP
ejde-741	364	55	them	they	PRON
ejde-741	364	56	subject	subject	ADJ
ejde-741	364	57	to	to	ADP
ejde-741	364	58	the	the	DET
ejde-741	364	59	natural	natural	ADJ
ejde-741	364	60	constraint	constraint	NOUN
ejde-741	364	61	determined	determine	VERB
ejde-741	364	62	by	by	ADP
ejde-741	364	63	(	(	PUNCT
ejde-741	364	64	2.2	2.2	NUM
ejde-741	364	65	)	)	PUNCT
ejde-741	364	66	.	.	PUNCT
ejde-741	365	1	it	it	PRON
ejde-741	365	2	will	will	AUX
ejde-741	365	3	allow	allow	VERB
ejde-741	365	4	us	we	PRON
ejde-741	365	5	to	to	PART
ejde-741	365	6	obtain	obtain	VERB
ejde-741	365	7	a	a	DET
ejde-741	365	8	sequence	sequence	NOUN
ejde-741	365	9	(	(	PUNCT
ejde-741	365	10	(	(	PUNCT
ejde-741	365	11	un	un	PROPN
ejde-741	365	12	,	,	PUNCT
ejde-741	365	13	hn	hn	NOUN
ejde-741	365	14	)	)	PUNCT
ejde-741	365	15	)	)	PUNCT
ejde-741	366	1	∞	∞	PROPN
ejde-741	366	2	n=1	n=1	PROPN
ejde-741	366	3	of	of	ADP
ejde-741	366	4	solutions	solution	NOUN
ejde-741	366	5	to	to	ADP
ejde-741	366	6	(	(	PUNCT
ejde-741	366	7	2.2	2.2	NUM
ejde-741	366	8	)	)	PUNCT
ejde-741	366	9	converging	converge	VERB
ejde-741	366	10	,	,	PUNCT
ejde-741	366	11	in	in	ADP
ejde-741	366	12	some	some	DET
ejde-741	366	13	sense	sense	NOUN
ejde-741	366	14	,	,	PUNCT
ejde-741	366	15	to	to	ADP
ejde-741	366	16	(	(	PUNCT
ejde-741	366	17	u	u	NOUN
ejde-741	366	18	,	,	PUNCT
ejde-741	366	19	h	h	NOUN
ejde-741	366	20	)	)	PUNCT
ejde-741	366	21	∈	∈	PROPN
ejde-741	366	22	l2(q	l2(q	PROPN
ejde-741	366	23	;	;	PUNCT
ejde-741	366	24	ρ20)×	ρ20)×	NUM
ejde-741	366	25	l2(qω	l2(qω	PROPN
ejde-741	366	26	;	;	PUNCT
ejde-741	366	27	ρ	ρ	NOUN
ejde-741	366	28	2	2	NUM
ejde-741	366	29	∗	∗	NOUN
ejde-741	366	30	)	)	PUNCT
ejde-741	366	31	,	,	PUNCT
ejde-741	366	32	which	which	PRON
ejde-741	366	33	is	be	AUX
ejde-741	366	34	also	also	ADV
ejde-741	366	35	a	a	DET
ejde-741	366	36	solution	solution	NOUN
ejde-741	366	37	to	to	ADP
ejde-741	366	38	(	(	PUNCT
ejde-741	366	39	2.2	2.2	NUM
ejde-741	366	40	)	)	PUNCT
ejde-741	366	41	.	.	PUNCT
ejde-741	367	1	to	to	PART
ejde-741	367	2	do	do	VERB
ejde-741	367	3	so	so	ADV
ejde-741	367	4	,	,	PUNCT
ejde-741	367	5	for	for	ADP
ejde-741	367	6	each	each	DET
ejde-741	367	7	n	n	PRON
ejde-741	367	8	∈	∈	PROPN
ejde-741	367	9	n∗	n∗	NOUN
ejde-741	367	10	,	,	PUNCT
ejde-741	367	11	let	let	VERB
ejde-741	367	12	us	we	PRON
ejde-741	367	13	define	define	VERB
ejde-741	367	14	an(t	an(t	PUNCT
ejde-741	367	15	,	,	PUNCT
ejde-741	367	16	x	x	X
ejde-741	367	17	)	)	PUNCT
ejde-741	367	18	=	=	PUNCT
ejde-741	368	1	a(t	a(t	NOUN
ejde-741	368	2	−	−	PRON
ejde-741	368	3	t)4	t)4	NOUN
ejde-741	368	4	(	(	PUNCT
ejde-741	368	5	t	t	PROPN
ejde-741	368	6	−	−	PROPN
ejde-741	368	7	t)4	t)4	PROPN
ejde-741	368	8	+	+	CCONJ
ejde-741	368	9	1	1	NUM
ejde-741	368	10	n	n	NOUN
ejde-741	368	11	,	,	PUNCT
ejde-741	368	12	baran(t	baran(t	PROPN
ejde-741	368	13	)	)	PUNCT
ejde-741	368	14	=	=	NOUN
ejde-741	368	15	ā(t	ā(t	NOUN
ejde-741	368	16	−	−	PROPN
ejde-741	368	17	t)4	t)4	PROPN
ejde-741	368	18	(	(	PUNCT
ejde-741	368	19	t	t	PROPN
ejde-741	368	20	−	−	PROPN
ejde-741	368	21	t)4	t)4	PROPN
ejde-741	368	22	+	+	CCONJ
ejde-741	368	23	1	1	NUM
ejde-741	368	24	n	n	NOUN
ejde-741	368	25	,	,	PUNCT
ejde-741	368	26	where	where	SCONJ
ejde-741	368	27	(	(	PUNCT
ejde-741	368	28	t	t	PROPN
ejde-741	368	29	,	,	PUNCT
ejde-741	368	30	x	x	NOUN
ejde-741	368	31	)	)	PUNCT
ejde-741	368	32	∈	∈	PROPN
ejde-741	369	1	[	[	X
ejde-741	369	2	0	0	NUM
ejde-741	369	3	,	,	PUNCT
ejde-741	369	4	t	t	X
ejde-741	369	5	]	]	X
ejde-741	369	6	×	×	NOUN
ejde-741	370	1	[	[	X
ejde-741	370	2	0	0	NUM
ejde-741	370	3	,	,	PUNCT
ejde-741	370	4	1	1	NUM
ejde-741	370	5	]	]	PUNCT
ejde-741	370	6	.	.	PUNCT
ejde-741	371	1	we	we	PRON
ejde-741	371	2	also	also	ADV
ejde-741	371	3	consider	consider	VERB
ejde-741	371	4	ρn	ρn	NOUN
ejde-741	371	5	=	=	PRON
ejde-741	371	6	e−san	e−san	ADJ
ejde-741	371	7	,	,	PUNCT
ejde-741	371	8	ρ̄n	ρ̄n	PROPN
ejde-741	371	9	=	=	PUNCT
ejde-741	371	10	e−sān	e−sān	NOUN
ejde-741	371	11	,	,	PUNCT
ejde-741	371	12	ρ0,n	ρ0,n	PROPN
ejde-741	371	13	=	=	SYM
ejde-741	371	14	ρnζ	ρnζ	NOUN
ejde-741	371	15	−5/6	−5/6	NOUN
ejde-741	371	16	,	,	PUNCT
ejde-741	371	17	ρ∗,n	ρ∗,n	ADJ
ejde-741	371	18	=	=	SYM
ejde-741	371	19	ρ̄nζ	ρ̄nζ	PROPN
ejde-741	371	20	∗−17/6mn	∗−17/6mn	NOUN
ejde-741	371	21	,	,	PUNCT
ejde-741	371	22	where	where	SCONJ
ejde-741	371	23	mn(x	mn(x	X
ejde-741	371	24	)	)	PUNCT
ejde-741	371	25	=	=	NOUN
ejde-741	371	26	{	{	PUNCT
ejde-741	371	27	1	1	NUM
ejde-741	371	28	,	,	PUNCT
ejde-741	371	29	x	x	SYM
ejde-741	371	30	∈	∈	PROPN
ejde-741	371	31	ω	ω	PROPN
ejde-741	371	32	,	,	PUNCT
ejde-741	371	33	n	n	CCONJ
ejde-741	371	34	,	,	PUNCT
ejde-741	371	35	x	x	PROPN
ejde-741	371	36	/∈	/∈	PUNCT
ejde-741	372	1	ω	ω	NUM
ejde-741	372	2	.	.	PROPN
ejde-741	373	1	14	14	NUM
ejde-741	374	1	p.	p.	NOUN
ejde-741	374	2	p.	p.	NOUN
ejde-741	375	1	de	de	PROPN
ejde-741	375	2	carvalho	carvalho	PROPN
ejde-741	375	3	,	,	PUNCT
ejde-741	375	4	r.	r.	PROPN
ejde-741	375	5	demarque	demarque	PROPN
ejde-741	375	6	,	,	PUNCT
ejde-741	375	7	j.	j.	PROPN
ejde-741	375	8	límaco	límaco	PROPN
ejde-741	375	9	,	,	PUNCT
ejde-741	375	10	l.	l.	PROPN
ejde-741	375	11	viana	viana	PROPN
ejde-741	375	12	ejde-2025/15	ejde-2025/15	VERB
ejde-741	375	13	these	these	DET
ejde-741	375	14	weight	weight	NOUN
ejde-741	375	15	functions	function	NOUN
ejde-741	375	16	are	be	AUX
ejde-741	375	17	built	build	VERB
ejde-741	375	18	in	in	ADP
ejde-741	375	19	such	such	DET
ejde-741	375	20	a	a	DET
ejde-741	375	21	way	way	NOUN
ejde-741	375	22	that	that	PRON
ejde-741	375	23	•	•	NUM
ejde-741	375	24	ρ0,n	ρ0,n	PROPN
ejde-741	375	25	and	and	CCONJ
ejde-741	375	26	ρ∗,n	ρ∗,n	PROPN
ejde-741	375	27	are	be	AUX
ejde-741	375	28	bounded	bound	VERB
ejde-741	375	29	from	from	ADP
ejde-741	375	30	below	below	ADV
ejde-741	375	31	by	by	ADP
ejde-741	375	32	a	a	DET
ejde-741	375	33	positive	positive	ADJ
ejde-741	375	34	constant	constant	NOUN
ejde-741	375	35	only	only	ADV
ejde-741	375	36	depending	depend	VERB
ejde-741	375	37	on	on	ADP
ejde-741	375	38	t	t	NOUN
ejde-741	375	39	;	;	PUNCT
ejde-741	375	40	•	•	X
ejde-741	375	41	ρ0,n	ρ0,n	PROPN
ejde-741	375	42	and	and	CCONJ
ejde-741	375	43	ρ∗,n	ρ∗,n	PROPN
ejde-741	375	44	are	be	AUX
ejde-741	375	45	bounded	bound	VERB
ejde-741	375	46	from	from	ADP
ejde-741	375	47	above	above	ADV
ejde-741	375	48	by	by	ADP
ejde-741	375	49	another	another	DET
ejde-741	375	50	positive	positive	ADJ
ejde-741	375	51	constant	constant	ADJ
ejde-741	375	52	depending	depend	VERB
ejde-741	375	53	on	on	ADP
ejde-741	375	54	n	n	PRON
ejde-741	375	55	and	and	CCONJ
ejde-741	375	56	t	t	PROPN
ejde-741	375	57	.	.	PUNCT
ejde-741	376	1	for	for	ADP
ejde-741	376	2	each	each	DET
ejde-741	376	3	n	n	PRON
ejde-741	376	4	∈	∈	PROPN
ejde-741	376	5	n∗	n∗	NOUN
ejde-741	376	6	,	,	PUNCT
ejde-741	376	7	we	we	PRON
ejde-741	376	8	set	set	VERB
ejde-741	376	9	the	the	DET
ejde-741	376	10	functional	functional	ADJ
ejde-741	376	11	jn	jn	NOUN
ejde-741	376	12	:	:	PUNCT
ejde-741	376	13	[	[	PUNCT
ejde-741	376	14	l2(q	l2(q	NOUN
ejde-741	376	15	)	)	PUNCT
ejde-741	376	16	]	]	PUNCT
ejde-741	376	17	2	2	NUM
ejde-741	376	18	→	→	SYM
ejde-741	376	19	r	r	NOUN
ejde-741	376	20	,	,	PUNCT
ejde-741	376	21	given	give	VERB
ejde-741	376	22	by	by	ADP
ejde-741	376	23	jn(u	jn(u	NOUN
ejde-741	376	24	,	,	PUNCT
ejde-741	376	25	h	h	NOUN
ejde-741	376	26	)	)	PUNCT
ejde-741	376	27	=	=	SYM
ejde-741	377	1	1	1	NUM
ejde-741	377	2	2	2	NUM
ejde-741	377	3	∫	∫	NOUN
ejde-741	377	4	t	t	PROPN
ejde-741	377	5	0	0	NUM
ejde-741	377	6	∫	∫	PROPN
ejde-741	377	7	1	1	NUM
ejde-741	377	8	0	0	NUM
ejde-741	377	9	ρ20,n|u|2	ρ20,n|u|2	NOUN
ejde-741	377	10	+	+	SYM
ejde-741	377	11	1	1	NUM
ejde-741	377	12	2	2	NUM
ejde-741	377	13	∫	∫	NOUN
ejde-741	377	14	t	t	PROPN
ejde-741	377	15	0	0	NUM
ejde-741	377	16	∫	∫	PROPN
ejde-741	377	17	1	1	NUM
ejde-741	377	18	0	0	NUM
ejde-741	377	19	ρ2∗,n|h|2	ρ2∗,n|h|2	PROPN
ejde-741	377	20	,	,	PUNCT
ejde-741	377	21	for	for	ADP
ejde-741	377	22	each	each	DET
ejde-741	377	23	(	(	PUNCT
ejde-741	377	24	u	u	NOUN
ejde-741	377	25	,	,	PUNCT
ejde-741	377	26	h	h	NOUN
ejde-741	377	27	)	)	PUNCT
ejde-741	377	28	∈	∈	NOUN
ejde-741	378	1	[	[	X
ejde-741	378	2	l2(q)]2	l2(q)]2	X
ejde-741	378	3	.	.	PUNCT
ejde-741	379	1	since	since	SCONJ
ejde-741	379	2	each	each	DET
ejde-741	379	3	jn	jn	PROPN
ejde-741	379	4	is	be	AUX
ejde-741	379	5	lower	low	ADJ
ejde-741	379	6	semi	semi	ADJ
ejde-741	379	7	-	-	ADJ
ejde-741	379	8	continuous	continuous	ADJ
ejde-741	379	9	,	,	PUNCT
ejde-741	379	10	strictly	strictly	ADV
ejde-741	379	11	convex	convex	ADJ
ejde-741	379	12	and	and	CCONJ
ejde-741	379	13	coercive	coercive	ADJ
ejde-741	379	14	(	(	PUNCT
ejde-741	379	15	see	see	VERB
ejde-741	379	16	[	[	X
ejde-741	379	17	21	21	NUM
ejde-741	379	18	]	]	PUNCT
ejde-741	379	19	)	)	PUNCT
ejde-741	379	20	,	,	PUNCT
ejde-741	379	21	we	we	PRON
ejde-741	379	22	can	can	AUX
ejde-741	379	23	apply	apply	VERB
ejde-741	379	24	[	[	PRON
ejde-741	379	25	23	23	NUM
ejde-741	379	26	,	,	PUNCT
ejde-741	379	27	proposition	proposition	NOUN
ejde-741	379	28	1.2	1.2	NUM
ejde-741	379	29	]	]	PUNCT
ejde-741	379	30	to	to	PART
ejde-741	379	31	obtain	obtain	VERB
ejde-741	379	32	a	a	DET
ejde-741	379	33	unique	unique	ADJ
ejde-741	379	34	(	(	PUNCT
ejde-741	379	35	un	un	PROPN
ejde-741	379	36	,	,	PUNCT
ejde-741	379	37	hn	hn	NOUN
ejde-741	379	38	)	)	PUNCT
ejde-741	379	39	satisfying	satisfy	VERB
ejde-741	379	40	j(un	j(un	PROPN
ejde-741	379	41	,	,	PUNCT
ejde-741	379	42	hn	hn	PROPN
ejde-741	379	43	)	)	PUNCT
ejde-741	379	44	=	=	SYM
ejde-741	379	45	min{j(u	min{j(u	PROPN
ejde-741	379	46	,	,	PUNCT
ejde-741	379	47	h	h	NOUN
ejde-741	379	48	)	)	PUNCT
ejde-741	379	49	;	;	PUNCT
ejde-741	379	50	(	(	PUNCT
ejde-741	379	51	u	u	NOUN
ejde-741	379	52	,	,	PUNCT
ejde-741	379	53	h	h	NOUN
ejde-741	379	54	)	)	PUNCT
ejde-741	379	55	∈	∈	PROPN
ejde-741	380	1	c	c	X
ejde-741	380	2	}	}	PUNCT
ejde-741	380	3	where	where	SCONJ
ejde-741	380	4	c	c	NOUN
ejde-741	380	5	=	=	PRON
ejde-741	380	6	{	{	PUNCT
ejde-741	380	7	(	(	PUNCT
ejde-741	380	8	u	u	NOUN
ejde-741	380	9	,	,	PUNCT
ejde-741	380	10	h	h	NOUN
ejde-741	380	11	)	)	PUNCT
ejde-741	380	12	∈	∈	PROPN
ejde-741	380	13	[	[	PUNCT
ejde-741	380	14	l2(q	l2(q	PROPN
ejde-741	380	15	)	)	PUNCT
ejde-741	380	16	]	]	PUNCT
ejde-741	380	17	2	2	NUM
ejde-741	380	18	;	;	PUNCT
ejde-741	380	19	(	(	PUNCT
ejde-741	380	20	u	u	NOUN
ejde-741	380	21	,	,	PUNCT
ejde-741	380	22	h	h	NOUN
ejde-741	380	23	)	)	PUNCT
ejde-741	380	24	solves	solve	NOUN
ejde-741	380	25	(	(	PUNCT
ejde-741	380	26	2.2	2.2	NUM
ejde-741	380	27	)	)	PUNCT
ejde-741	380	28	}	}	PUNCT
ejde-741	380	29	.	.	PUNCT
ejde-741	381	1	consequently	consequently	ADV
ejde-741	381	2	,	,	PUNCT
ejde-741	381	3	by	by	ADP
ejde-741	381	4	lagrange	lagrange	PROPN
ejde-741	381	5	’s	’s	PART
ejde-741	381	6	principle	principle	NOUN
ejde-741	381	7	,	,	PUNCT
ejde-741	381	8	for	for	ADP
ejde-741	381	9	each	each	DET
ejde-741	381	10	n	n	PRON
ejde-741	381	11	∈	∈	PROPN
ejde-741	381	12	n∗	n∗	NOUN
ejde-741	381	13	,	,	PUNCT
ejde-741	381	14	there	there	PRON
ejde-741	381	15	exists	exist	VERB
ejde-741	381	16	a	a	DET
ejde-741	381	17	function	function	NOUN
ejde-741	381	18	pn	pn	NOUN
ejde-741	381	19	solving	solve	VERB
ejde-741	381	20	the	the	DET
ejde-741	381	21	system	system	NOUN
ejde-741	381	22	−pnt	−pnt	NOUN
ejde-741	381	23	−	−	PROPN
ejde-741	382	1	(	(	PUNCT
ejde-741	382	2	apnx)x	apnx)x	PUNCT
ejde-741	382	3	+	+	CCONJ
ejde-741	382	4	c(t	c(t	PROPN
ejde-741	382	5	,	,	PUNCT
ejde-741	382	6	x)pn	x)pn	PROPN
ejde-741	382	7	=	=	SYM
ejde-741	382	8	−ρ20,nun	−ρ20,nun	PROPN
ejde-741	382	9	,	,	PUNCT
ejde-741	382	10	(	(	PUNCT
ejde-741	382	11	t	t	PROPN
ejde-741	382	12	,	,	PUNCT
ejde-741	382	13	x	x	X
ejde-741	382	14	)	)	PUNCT
ejde-741	382	15	∈	∈	PROPN
ejde-741	382	16	q	q	NOUN
ejde-741	382	17	,	,	PUNCT
ejde-741	382	18	pn(t	pn(t	NUM
ejde-741	382	19	,	,	PUNCT
ejde-741	382	20	1	1	NUM
ejde-741	382	21	)	)	PUNCT
ejde-741	382	22	=	=	SYM
ejde-741	382	23	0	0	NUM
ejde-741	382	24	,	,	PUNCT
ejde-741	382	25	t	t	PROPN
ejde-741	382	26	∈	∈	PROPN
ejde-741	382	27	(	(	PUNCT
ejde-741	382	28	0	0	NUM
ejde-741	382	29	,	,	PUNCT
ejde-741	382	30	t	t	NOUN
ejde-741	382	31	)	)	PUNCT
ejde-741	382	32	,	,	PUNCT
ejde-741	382	33			PRON
ejde-741	382	34	pn(t	pn(t	NUM
ejde-741	382	35	,	,	PUNCT
ejde-741	382	36	0	0	NUM
ejde-741	382	37	)	)	PUNCT
ejde-741	382	38	=	=	SYM
ejde-741	383	1	0	0	NUM
ejde-741	383	2	,	,	PUNCT
ejde-741	383	3	(	(	PUNCT
ejde-741	383	4	weak	weak	ADJ
ejde-741	383	5	)	)	PUNCT
ejde-741	383	6	,	,	PUNCT
ejde-741	383	7	t	t	PROPN
ejde-741	383	8	∈	∈	PROPN
ejde-741	383	9	(	(	PUNCT
ejde-741	383	10	0	0	NUM
ejde-741	383	11	,	,	PUNCT
ejde-741	383	12	t	t	PROPN
ejde-741	383	13	)	)	PUNCT
ejde-741	383	14	or	or	CCONJ
ejde-741	383	15	(	(	PUNCT
ejde-741	383	16	apnx)(t	apnx)(t	ADJ
ejde-741	383	17	,	,	PUNCT
ejde-741	383	18	0	0	NUM
ejde-741	383	19	)	)	PUNCT
ejde-741	383	20	=	=	SYM
ejde-741	383	21	0	0	NUM
ejde-741	383	22	,	,	PUNCT
ejde-741	383	23	(	(	PUNCT
ejde-741	383	24	strong	strong	ADJ
ejde-741	383	25	)	)	PUNCT
ejde-741	383	26	,	,	PUNCT
ejde-741	383	27	t	t	PROPN
ejde-741	383	28	∈	∈	PROPN
ejde-741	383	29	(	(	PUNCT
ejde-741	383	30	0	0	NUM
ejde-741	383	31	,	,	PUNCT
ejde-741	383	32	t	t	PROPN
ejde-741	383	33	)	)	PUNCT
ejde-741	383	34	pn(t	pn(t	SYM
ejde-741	383	35	,	,	PUNCT
ejde-741	383	36	x	x	X
ejde-741	383	37	)	)	PUNCT
ejde-741	383	38	=	=	SYM
ejde-741	383	39	0	0	NUM
ejde-741	383	40	,	,	PUNCT
ejde-741	383	41	x	x	SYM
ejde-741	383	42	∈	∈	PROPN
ejde-741	383	43	(	(	PUNCT
ejde-741	383	44	0	0	NUM
ejde-741	383	45	,	,	PUNCT
ejde-741	383	46	1	1	NUM
ejde-741	383	47	)	)	PUNCT
ejde-741	383	48	,	,	PUNCT
ejde-741	383	49	pn	pn	PROPN
ejde-741	383	50	=	=	SYM
ejde-741	383	51	ρ2∗,nhn	ρ2∗,nhn	PROPN
ejde-741	383	52	,	,	PUNCT
ejde-741	383	53	(	(	PUNCT
ejde-741	383	54	t	t	PROPN
ejde-741	383	55	,	,	PUNCT
ejde-741	383	56	x	x	X
ejde-741	383	57	)	)	PUNCT
ejde-741	383	58	∈	∈	PROPN
ejde-741	383	59	q.	q.	NOUN
ejde-741	383	60	(	(	PUNCT
ejde-741	383	61	3.7	3.7	NUM
ejde-741	383	62	)	)	PUNCT
ejde-741	383	63	by	by	ADP
ejde-741	383	64	standard	standard	ADJ
ejde-741	383	65	arguments	argument	NOUN
ejde-741	383	66	,	,	PUNCT
ejde-741	383	67	(	(	PUNCT
ejde-741	383	68	3.7	3.7	NUM
ejde-741	383	69	)	)	PUNCT
ejde-741	383	70	can	can	AUX
ejde-741	383	71	help	help	VERB
ejde-741	383	72	us	we	PRON
ejde-741	383	73	to	to	PART
ejde-741	383	74	prove	prove	VERB
ejde-741	383	75	that	that	SCONJ
ejde-741	383	76	jn(un	jn(un	PROPN
ejde-741	383	77	,	,	PUNCT
ejde-741	383	78	hn	hn	PROPN
ejde-741	383	79	)	)	PUNCT
ejde-741	383	80	≤	≤	PUNCT
ejde-741	384	1	c	c	NOUN
ejde-741	384	2	√	√	PROPN
ejde-741	384	3	jn(un	jn(un	PROPN
ejde-741	384	4	,	,	PUNCT
ejde-741	384	5	hn	hn	PROPN
ejde-741	384	6	)	)	PUNCT
ejde-741	384	7	for	for	ADP
ejde-741	384	8	all	all	DET
ejde-741	384	9	n	n	PRON
ejde-741	384	10	∈	∈	PROPN
ejde-741	384	11	n∗	n∗	NOUN
ejde-741	384	12	,	,	PUNCT
ejde-741	384	13	i.e.	i.e.	X
ejde-741	384	14	,	,	PUNCT
ejde-741	384	15	(	(	PUNCT
ejde-741	384	16	jn(un	jn(un	PROPN
ejde-741	384	17	,	,	PUNCT
ejde-741	384	18	hn	hn	PROPN
ejde-741	384	19	)	)	PUNCT
ejde-741	384	20	)	)	PUNCT
ejde-741	385	1	∞	∞	PROPN
ejde-741	385	2	n=1	n=1	PROPN
ejde-741	385	3	is	be	AUX
ejde-741	385	4	a	a	DET
ejde-741	385	5	numerical	numerical	ADJ
ejde-741	385	6	bounded	bounded	ADJ
ejde-741	385	7	sequence	sequence	NOUN
ejde-741	385	8	.	.	PUNCT
ejde-741	386	1	since	since	SCONJ
ejde-741	386	2	ρ20,n	ρ20,n	ADJ
ejde-741	386	3	≥	≥	X
ejde-741	386	4	ct	ct	NUM
ejde-741	386	5	and	and	CCONJ
ejde-741	386	6	ρ2∗,n	ρ2∗,n	PROPN
ejde-741	386	7	≥	≥	NOUN
ejde-741	386	8	ctmn	ctmn	NOUN
ejde-741	386	9	,	,	PUNCT
ejde-741	386	10	we	we	PRON
ejde-741	386	11	deduce	deduce	VERB
ejde-741	386	12	that	that	PRON
ejde-741	386	13	∥un∥2l2	∥un∥2l2	VERB
ejde-741	387	1	+	+	NUM
ejde-741	387	2	∫	∫	PROPN
ejde-741	387	3	t	t	PROPN
ejde-741	387	4	0	0	NUM
ejde-741	387	5	∫	∫	PROPN
ejde-741	387	6	ω	ω	PROPN
ejde-741	387	7	|hn|2	|hn|2	PUNCT
ejde-741	387	8	+	+	NUM
ejde-741	387	9	n	n	NUM
ejde-741	387	10	∫	∫	PROPN
ejde-741	387	11	t	t	PROPN
ejde-741	387	12	0	0	NUM
ejde-741	387	13	∫	∫	PROPN
ejde-741	388	1	[	[	X
ejde-741	388	2	0,1]\ω	0,1]\ω	PROPN
ejde-741	388	3	|hn|2	|hn|2	PUNCT
ejde-741	388	4	≤	≤	NUM
ejde-741	388	5	cjn(un	cjn(un	NOUN
ejde-741	388	6	,	,	PUNCT
ejde-741	388	7	hn	hn	PROPN
ejde-741	388	8	)	)	PUNCT
ejde-741	388	9	≤	≤	NOUN
ejde-741	388	10	c	c	X
ejde-741	388	11	,	,	PUNCT
ejde-741	388	12	whence	whence	ADP
ejde-741	388	13	there	there	PRON
ejde-741	388	14	exists	exist	VERB
ejde-741	388	15	(	(	PUNCT
ejde-741	388	16	u	u	NOUN
ejde-741	388	17	,	,	PUNCT
ejde-741	388	18	h	h	NOUN
ejde-741	388	19	)	)	PUNCT
ejde-741	388	20	∈	∈	PROPN
ejde-741	388	21	l2(q)×	l2(q)×	PROPN
ejde-741	388	22	l2(qω	l2(qω	PROPN
ejde-741	388	23	)	)	PUNCT
ejde-741	388	24	,	,	PUNCT
ejde-741	388	25	such	such	ADJ
ejde-741	388	26	that	that	SCONJ
ejde-741	388	27	un	un	PROPN
ejde-741	388	28	⇀	⇀	PROPN
ejde-741	388	29	u	u	PROPN
ejde-741	388	30	,	,	PUNCT
ejde-741	388	31	in	in	ADP
ejde-741	388	32	l2(q	l2(q	PROPN
ejde-741	388	33	)	)	PUNCT
ejde-741	388	34	and	and	CCONJ
ejde-741	388	35	hn	hn	PROPN
ejde-741	388	36	⇀	⇀	PROPN
ejde-741	388	37	hχω	hχω	PROPN
ejde-741	388	38	in	in	ADP
ejde-741	388	39	l2(q	l2(q	PROPN
ejde-741	388	40	)	)	PUNCT
ejde-741	388	41	,	,	PUNCT
ejde-741	388	42	up	up	ADP
ejde-741	388	43	to	to	ADP
ejde-741	388	44	subsequences	subsequence	NOUN
ejde-741	388	45	.	.	PUNCT
ejde-741	389	1	from	from	ADP
ejde-741	389	2	this	this	PRON
ejde-741	389	3	,	,	PUNCT
ejde-741	389	4	we	we	PRON
ejde-741	389	5	have	have	VERB
ejde-741	389	6	ρ0,nun	ρ0,nun	NUM
ejde-741	389	7	⇀	⇀	PUNCT
ejde-741	390	1	ρ0u	ρ0u	PROPN
ejde-741	390	2	and	and	CCONJ
ejde-741	390	3	ρ∗,nhn	ρ∗,nhn	PROPN
ejde-741	390	4	⇀	⇀	NUM
ejde-741	390	5	ρ∗hχω	ρ∗hχω	PROPN
ejde-741	390	6	in	in	ADP
ejde-741	390	7	l2(q	l2(q	PROPN
ejde-741	390	8	)	)	PUNCT
ejde-741	390	9	.	.	PUNCT
ejde-741	391	1	(	(	PUNCT
ejde-741	391	2	3.8	3.8	NUM
ejde-741	391	3	)	)	PUNCT
ejde-741	391	4	consequently	consequently	ADV
ejde-741	391	5	,	,	PUNCT
ejde-741	391	6	u	u	PROPN
ejde-741	391	7	∈	∈	PROPN
ejde-741	391	8	l2(q	l2(q	PROPN
ejde-741	391	9	;	;	PUNCT
ejde-741	391	10	ρ20	ρ20	NUM
ejde-741	391	11	)	)	PUNCT
ejde-741	391	12	and	and	CCONJ
ejde-741	391	13	h	h	NOUN
ejde-741	391	14	∈	∈	PROPN
ejde-741	391	15	l2(qω	l2(qω	PROPN
ejde-741	391	16	;	;	PUNCT
ejde-741	391	17	ρ	ρ	NOUN
ejde-741	391	18	2	2	NUM
ejde-741	391	19	∗	∗	NOUN
ejde-741	391	20	)	)	PUNCT
ejde-741	391	21	.	.	PUNCT
ejde-741	392	1	recalling	recall	VERB
ejde-741	392	2	that	that	PRON
ejde-741	392	3	(	(	PUNCT
ejde-741	392	4	un	un	PROPN
ejde-741	392	5	,	,	PUNCT
ejde-741	392	6	hn	hn	NOUN
ejde-741	392	7	)	)	PUNCT
ejde-741	392	8	is	be	AUX
ejde-741	392	9	a	a	DET
ejde-741	392	10	solution	solution	NOUN
ejde-741	392	11	of	of	ADP
ejde-741	392	12	(	(	PUNCT
ejde-741	392	13	2.2	2.2	NUM
ejde-741	392	14	)	)	PUNCT
ejde-741	392	15	,	,	PUNCT
ejde-741	392	16	for	for	ADP
ejde-741	392	17	each	each	DET
ejde-741	392	18	n	n	PRON
ejde-741	392	19	∈	∈	PROPN
ejde-741	392	20	n∗	n∗	PROPN
ejde-741	392	21	,	,	PUNCT
ejde-741	392	22	a	a	DET
ejde-741	392	23	passing	passing	NOUN
ejde-741	392	24	to	to	ADP
ejde-741	392	25	the	the	DET
ejde-741	392	26	limit	limit	NOUN
ejde-741	392	27	argument	argument	NOUN
ejde-741	392	28	implies	imply	VERB
ejde-741	392	29	that	that	SCONJ
ejde-741	392	30	(	(	PUNCT
ejde-741	392	31	u	u	NOUN
ejde-741	392	32	,	,	PUNCT
ejde-741	392	33	h	h	NOUN
ejde-741	392	34	)	)	PUNCT
ejde-741	392	35	also	also	ADV
ejde-741	392	36	solves	solve	VERB
ejde-741	392	37	(	(	PUNCT
ejde-741	392	38	2.2	2.2	NUM
ejde-741	392	39	)	)	PUNCT
ejde-741	392	40	.	.	PUNCT
ejde-741	393	1	thinking	think	VERB
ejde-741	393	2	about	about	ADP
ejde-741	393	3	a	a	DET
ejde-741	393	4	better	well	ADJ
ejde-741	393	5	presentation	presentation	NOUN
ejde-741	393	6	,	,	PUNCT
ejde-741	393	7	the	the	DET
ejde-741	393	8	estimates	estimate	NOUN
ejde-741	393	9	mentioned	mention	VERB
ejde-741	393	10	in	in	ADP
ejde-741	393	11	(	(	PUNCT
ejde-741	393	12	3.1	3.1	NUM
ejde-741	393	13	)	)	PUNCT
ejde-741	393	14	will	will	AUX
ejde-741	393	15	be	be	AUX
ejde-741	393	16	established	establish	VERB
ejde-741	393	17	in	in	ADP
ejde-741	393	18	two	two	NUM
ejde-741	393	19	subsequent	subsequent	ADJ
ejde-741	393	20	lemmas	lemma	NOUN
ejde-741	393	21	.	.	PUNCT
ejde-741	394	1	□	□	PUNCT
ejde-741	394	2	lemma	lemma	PROPN
ejde-741	394	3	3.5	3.5	NUM
ejde-741	394	4	.	.	PUNCT
ejde-741	395	1	under	under	ADP
ejde-741	395	2	the	the	DET
ejde-741	395	3	assumptions	assumption	NOUN
ejde-741	395	4	of	of	ADP
ejde-741	395	5	proposition	proposition	NOUN
ejde-741	395	6	3.1	3.1	NUM
ejde-741	395	7	,	,	PUNCT
ejde-741	395	8	we	we	PRON
ejde-741	395	9	have	have	VERB
ejde-741	395	10	that	that	DET
ejde-741	395	11	ρ̂u	ρ̂u	NOUN
ejde-741	395	12	∈	∈	PROPN
ejde-741	395	13	l∞(0	l∞(0	PRON
ejde-741	395	14	,	,	PUNCT
ejde-741	395	15	t	t	PROPN
ejde-741	395	16	;	;	PUNCT
ejde-741	395	17	l2(0	l2(0	NOUN
ejde-741	395	18	,	,	PUNCT
ejde-741	395	19	1	1	NUM
ejde-741	395	20	)	)	PUNCT
ejde-741	395	21	)	)	PUNCT
ejde-741	395	22	,	,	PUNCT
ejde-741	395	23	√	√	PROPN
ejde-741	395	24	aux	aux	PROPN
ejde-741	395	25	∈	∈	PROPN
ejde-741	395	26	l2(q	l2(q	PROPN
ejde-741	395	27	;	;	PUNCT
ejde-741	395	28	ρ̂2	ρ̂2	PROPN
ejde-741	395	29	)	)	PUNCT
ejde-741	395	30	,	,	PUNCT
ejde-741	395	31	and	and	CCONJ
ejde-741	395	32	there	there	PRON
ejde-741	395	33	exists	exist	VERB
ejde-741	395	34	c	c	NOUN
ejde-741	395	35	>	>	X
ejde-741	395	36	0	0	NUM
ejde-741	396	1	such	such	ADJ
ejde-741	396	2	that	that	DET
ejde-741	396	3	sup	sup	NOUN
ejde-741	396	4	t∈[0,t	t∈[0,t	NOUN
ejde-741	396	5	]	]	PUNCT
ejde-741	396	6	∥ρ̂u(t	∥ρ̂u(t	PROPN
ejde-741	396	7	,	,	PUNCT
ejde-741	396	8	·	·	PUNCT
ejde-741	396	9	)	)	PUNCT
ejde-741	396	10	∥2l2(0,1	∥2l2(0,1	ADV
ejde-741	396	11	)	)	PUNCT
ejde-741	397	1	+	+	CCONJ
ejde-741	397	2	∥	∥	PRON
ejde-741	397	3	√	√	VERB
ejde-741	397	4	aux∥2ρ̂2	aux∥2ρ̂2	VERB
ejde-741	397	5	≤	≤	NUM
ejde-741	397	6	c(∥u∥2ρ20	c(∥u∥2ρ20	X
ejde-741	398	1	+	+	CCONJ
ejde-741	398	2	∥hχω∥2ρ2∗	∥hχω∥2ρ2∗	PROPN
ejde-741	398	3	+	+	CCONJ
ejde-741	398	4	∥g∥2ρ20	∥g∥2ρ20	X
ejde-741	398	5	+	+	CCONJ
ejde-741	398	6	∥u0∥2h1	∥u0∥2h1	NOUN
ejde-741	398	7	a	a	PRON
ejde-741	398	8	)	)	PUNCT
ejde-741	398	9	.	.	PUNCT
ejde-741	399	1	ejde-2025/15	ejde-2025/15	NOUN
ejde-741	399	2	null	null	ADJ
ejde-741	399	3	-	-	PUNCT
ejde-741	399	4	controllability	controllability	NOUN
ejde-741	399	5	degenerate	degenerate	ADJ
ejde-741	399	6	quasilinear	quasilinear	NOUN
ejde-741	399	7	equations	equation	NOUN
ejde-741	399	8	15	15	NUM
ejde-741	399	9	proof	proof	NOUN
ejde-741	399	10	.	.	PUNCT
ejde-741	400	1	multiplying	multiply	VERB
ejde-741	400	2	the	the	DET
ejde-741	400	3	pde	pde	NOUN
ejde-741	400	4	in	in	ADP
ejde-741	400	5	(	(	PUNCT
ejde-741	400	6	2.2	2.2	NUM
ejde-741	400	7	)	)	PUNCT
ejde-741	400	8	by	by	ADP
ejde-741	400	9	ρ̂2u	ρ̂2u	NUM
ejde-741	400	10	,	,	PUNCT
ejde-741	400	11	integrating	integrate	VERB
ejde-741	400	12	in	in	ADP
ejde-741	400	13	[	[	X
ejde-741	400	14	0	0	NUM
ejde-741	400	15	,	,	PUNCT
ejde-741	400	16	1	1	NUM
ejde-741	400	17	]	]	PUNCT
ejde-741	400	18	and	and	CCONJ
ejde-741	400	19	using	use	VERB
ejde-741	400	20	the	the	DET
ejde-741	400	21	two	two	NUM
ejde-741	400	22	relations	relation	NOUN
ejde-741	400	23	1	1	NUM
ejde-741	400	24	2	2	NUM
ejde-741	400	25	d	d	NOUN
ejde-741	400	26	dt	dt	X
ejde-741	400	27	∫	∫	PROPN
ejde-741	400	28	1	1	NUM
ejde-741	400	29	0	0	NUM
ejde-741	400	30	ρ̂2u2	ρ̂2u2	NOUN
ejde-741	400	31	=	=	SYM
ejde-741	400	32	∫	∫	PROPN
ejde-741	400	33	1	1	NUM
ejde-741	400	34	0	0	NUM
ejde-741	400	35	ρ̂2utu+	ρ̂2utu+	ADJ
ejde-741	400	36	∫	∫	PROPN
ejde-741	400	37	1	1	NUM
ejde-741	400	38	0	0	NUM
ejde-741	400	39	ρ̂ρ̂tu	ρ̂ρ̂tu	NOUN
ejde-741	400	40	2	2	NUM
ejde-741	400	41	and	and	CCONJ
ejde-741	400	42	∫	∫	PROPN
ejde-741	400	43	1	1	NUM
ejde-741	400	44	0	0	NUM
ejde-741	400	45	ρ̂2(aux)xu	ρ̂2(aux)xu	NUM
ejde-741	400	46	=	=	SYM
ejde-741	401	1	−2	−2	PROPN
ejde-741	401	2	∫	∫	PROPN
ejde-741	401	3	1	1	NUM
ejde-741	401	4	0	0	NUM
ejde-741	401	5	ρ̂ρ̂xauux	ρ̂ρ̂xauux	NOUN
ejde-741	401	6	−	−	PROPN
ejde-741	401	7	∫	∫	PROPN
ejde-741	401	8	1	1	NUM
ejde-741	401	9	0	0	NUM
ejde-741	401	10	ρ̂2au2x	ρ̂2au2x	NOUN
ejde-741	401	11	,	,	PUNCT
ejde-741	401	12	we	we	PRON
ejde-741	401	13	obtain	obtain	VERB
ejde-741	401	14	1	1	NUM
ejde-741	401	15	2	2	NUM
ejde-741	401	16	d	d	NOUN
ejde-741	401	17	dt	dt	X
ejde-741	401	18	∫	∫	PROPN
ejde-741	401	19	1	1	NUM
ejde-741	401	20	0	0	NUM
ejde-741	401	21	ρ̂2u2	ρ̂2u2	NOUN
ejde-741	401	22	+	+	CCONJ
ejde-741	401	23	∫	∫	PROPN
ejde-741	401	24	1	1	NUM
ejde-741	401	25	0	0	NUM
ejde-741	401	26	ρ̂2au2x	ρ̂2au2x	NUM
ejde-741	401	27	=	=	SYM
ejde-741	402	1	−	−	NOUN
ejde-741	402	2	∫	∫	NOUN
ejde-741	402	3	1	1	NUM
ejde-741	402	4	0	0	NUM
ejde-741	402	5	ρ̂2cu2	ρ̂2cu2	ADJ
ejde-741	403	1	+	+	CCONJ
ejde-741	403	2	∫	∫	PROPN
ejde-741	403	3	1	1	NUM
ejde-741	403	4	0	0	NUM
ejde-741	404	1	ρ̂2uhχω	ρ̂2uhχω	PROPN
ejde-741	404	2	+	+	NUM
ejde-741	404	3	∫	∫	PROPN
ejde-741	404	4	1	1	NUM
ejde-741	404	5	0	0	NUM
ejde-741	405	1	ρ̂2gu+	ρ̂2gu+	ADJ
ejde-741	405	2	∫	∫	NOUN
ejde-741	405	3	1	1	NUM
ejde-741	405	4	0	0	NUM
ejde-741	405	5	ρ̂ρ̂tu	ρ̂ρ̂tu	NOUN
ejde-741	405	6	2	2	NUM
ejde-741	405	7	−	−	NOUN
ejde-741	405	8	2	2	NUM
ejde-741	405	9	∫	∫	NOUN
ejde-741	405	10	1	1	NUM
ejde-741	405	11	0	0	NUM
ejde-741	405	12	ρ̂ρ̂xauux	ρ̂ρ̂xauux	NOUN
ejde-741	406	1	=	=	NOUN
ejde-741	406	2	:	:	PUNCT
ejde-741	406	3	i1	i1	PROPN
ejde-741	406	4	+	+	CCONJ
ejde-741	406	5	i2	i2	PROPN
ejde-741	406	6	+	+	CCONJ
ejde-741	406	7	i3	i3	NOUN
ejde-741	406	8	+	+	CCONJ
ejde-741	406	9	i4	i4	PROPN
ejde-741	406	10	+	+	CCONJ
ejde-741	406	11	i5	i5	ADJ
ejde-741	406	12	.	.	PUNCT
ejde-741	407	1	(	(	PUNCT
ejde-741	407	2	3.9	3.9	NUM
ejde-741	407	3	)	)	PUNCT
ejde-741	407	4	above	above	ADV
ejde-741	407	5	,	,	PUNCT
ejde-741	407	6	we	we	PRON
ejde-741	407	7	have	have	AUX
ejde-741	407	8	also	also	ADV
ejde-741	407	9	used	use	VERB
ejde-741	407	10	u(t	u(t	NOUN
ejde-741	407	11	,	,	PUNCT
ejde-741	407	12	0	0	NUM
ejde-741	407	13	)	)	PUNCT
ejde-741	407	14	=	=	SYM
ejde-741	407	15	u(t	u(t	NOUN
ejde-741	407	16	,	,	PUNCT
ejde-741	407	17	1	1	NUM
ejde-741	407	18	)	)	PUNCT
ejde-741	407	19	≡	≡	PROPN
ejde-741	407	20	0	0	PUNCT
ejde-741	408	1	for	for	ADP
ejde-741	408	2	(	(	PUNCT
ejde-741	408	3	wdp	wdp	PROPN
ejde-741	408	4	)	)	PUNCT
ejde-741	408	5	,	,	PUNCT
ejde-741	408	6	and	and	CCONJ
ejde-741	408	7	u(t	u(t	NOUN
ejde-741	408	8	,	,	PUNCT
ejde-741	408	9	1	1	NUM
ejde-741	408	10	)	)	PUNCT
ejde-741	408	11	=	=	SYM
ejde-741	408	12	aux(t	aux(t	PROPN
ejde-741	408	13	,	,	PUNCT
ejde-741	408	14	0	0	NUM
ejde-741	408	15	)	)	PUNCT
ejde-741	408	16	≡	≡	PROPN
ejde-741	408	17	0	0	PUNCT
ejde-741	409	1	for	for	ADP
ejde-741	409	2	(	(	PUNCT
ejde-741	409	3	sdp	sdp	NOUN
ejde-741	409	4	)	)	PUNCT
ejde-741	409	5	.	.	PUNCT
ejde-741	410	1	now	now	ADV
ejde-741	410	2	,	,	PUNCT
ejde-741	410	3	since	since	SCONJ
ejde-741	410	4	ρ∗	ρ∗	PROPN
ejde-741	410	5	≤	≤	PROPN
ejde-741	410	6	cρ̂	cρ̂	PROPN
ejde-741	410	7	≤	≤	NOUN
ejde-741	410	8	cρ0	cρ0	NOUN
ejde-741	410	9	and	and	CCONJ
ejde-741	410	10	ρ0ρ∗	ρ0ρ∗	PROPN
ejde-741	410	11	≥	≥	NOUN
ejde-741	410	12	ρ̂2	ρ̂2	NOUN
ejde-741	410	13	,	,	PUNCT
ejde-741	410	14	we	we	PRON
ejde-741	410	15	obtain	obtain	VERB
ejde-741	410	16	i1	i1	PROPN
ejde-741	410	17	≤	≤	PROPN
ejde-741	410	18	c	c	PROPN
ejde-741	410	19	∫	∫	PROPN
ejde-741	410	20	1	1	NUM
ejde-741	410	21	0	0	NUM
ejde-741	410	22	ρ20|u|2	ρ20|u|2	PROPN
ejde-741	410	23	,	,	PUNCT
ejde-741	411	1	i2	i2	PROPN
ejde-741	411	2	≤	≤	PROPN
ejde-741	411	3	c	c	NOUN
ejde-741	411	4	(	(	PUNCT
ejde-741	411	5	1	1	NUM
ejde-741	411	6	2	2	NUM
ejde-741	411	7	∫	∫	NOUN
ejde-741	411	8	1	1	NUM
ejde-741	411	9	0	0	NUM
ejde-741	411	10	ρ2∗|hχω|2	ρ2∗|hχω|2	PROPN
ejde-741	411	11	+	+	CCONJ
ejde-741	411	12	1	1	NUM
ejde-741	411	13	2	2	NUM
ejde-741	411	14	∫	∫	NOUN
ejde-741	411	15	1	1	NUM
ejde-741	411	16	0	0	NUM
ejde-741	411	17	ρ20|u|2	ρ20|u|2	NOUN
ejde-741	411	18	)	)	PUNCT
ejde-741	411	19	,	,	PUNCT
ejde-741	411	20	i3	i3	NOUN
ejde-741	411	21	≤	≤	PUNCT
ejde-741	411	22	c	c	NOUN
ejde-741	411	23	(	(	PUNCT
ejde-741	411	24	1	1	NUM
ejde-741	411	25	2	2	NUM
ejde-741	411	26	∫	∫	NOUN
ejde-741	411	27	1	1	NUM
ejde-741	411	28	0	0	NUM
ejde-741	411	29	ρ20|g|2	ρ20|g|2	NOUN
ejde-741	412	1	+	+	CCONJ
ejde-741	412	2	1	1	NUM
ejde-741	412	3	2	2	NUM
ejde-741	412	4	∫	∫	NOUN
ejde-741	412	5	1	1	NUM
ejde-741	412	6	0	0	NUM
ejde-741	412	7	ρ20|u|2	ρ20|u|2	NOUN
ejde-741	412	8	)	)	PUNCT
ejde-741	412	9	.	.	PUNCT
ejde-741	413	1	let	let	VERB
ejde-741	413	2	us	we	PRON
ejde-741	413	3	estimate	estimate	VERB
ejde-741	413	4	i4	i4	PROPN
ejde-741	413	5	.	.	PUNCT
ejde-741	414	1	firstly	firstly	ADV
ejde-741	414	2	,	,	PUNCT
ejde-741	414	3	rewriting	rewrite	VERB
ejde-741	414	4	a	a	PRON
ejde-741	414	5	and	and	CCONJ
ejde-741	414	6	ā	ā	ADJ
ejde-741	414	7	as	as	ADP
ejde-741	414	8	a(t	a(t	NOUN
ejde-741	414	9	,	,	PUNCT
ejde-741	414	10	x	x	NOUN
ejde-741	414	11	)	)	PUNCT
ejde-741	414	12	=	=	SYM
ejde-741	414	13	ζ(t	ζ(t	PROPN
ejde-741	414	14	,	,	PUNCT
ejde-741	414	15	x)η̃(x	x)η̃(x	PROPN
ejde-741	414	16	)	)	PUNCT
ejde-741	414	17	,	,	PUNCT
ejde-741	414	18	where	where	SCONJ
ejde-741	414	19	η̃	η̃	PROPN
ejde-741	414	20	=	=	SYM
ejde-741	414	21	ηr	ηr	PROPN
ejde-741	414	22	/	/	SYM
ejde-741	414	23	η	η	NOUN
ejde-741	414	24	,	,	PUNCT
ejde-741	414	25	and	and	CCONJ
ejde-741	414	26	ā(t	ā(t	PROPN
ejde-741	414	27	,	,	PUNCT
ejde-741	414	28	x	x	NOUN
ejde-741	414	29	)	)	PUNCT
ejde-741	414	30	=	=	PUNCT
ejde-741	414	31	ζ(t	ζ(t	PROPN
ejde-741	414	32	,	,	PUNCT
ejde-741	414	33	x	x	X
ejde-741	414	34	)	)	PUNCT
ejde-741	414	35	η̄rη	η̄rη	PROPN
ejde-741	414	36	,	,	PUNCT
ejde-741	414	37	we	we	PRON
ejde-741	414	38	have	have	VERB
ejde-741	414	39	|ρ̂t|	|ρ̂t|	PROPN
ejde-741	414	40	=	=	SYM
ejde-741	414	41	∣∣∣−	∣∣∣−	PROPN
ejde-741	414	42	s	s	X
ejde-741	414	43	(	(	PUNCT
ejde-741	414	44	ηr	ηr	PROPN
ejde-741	414	45	+	+	PROPN
ejde-741	414	46	η̄r	η̄r	PROPN
ejde-741	414	47	2η	2η	PROPN
ejde-741	414	48	)	)	PUNCT
ejde-741	414	49	ζte	ζte	VERB
ejde-741	414	50	−s(a+ā	−s(a+ā	PROPN
ejde-741	414	51	2	2	NUM
ejde-741	414	52	)	)	PUNCT
ejde-741	414	53	(	(	PUNCT
ejde-741	414	54	ζ∗)−11/6	ζ∗)−11/6	ADP
ejde-741	414	55	−	−	PROPN
ejde-741	414	56	11	11	NUM
ejde-741	414	57	6	6	NUM
ejde-741	414	58	e−s	e−s	PROPN
ejde-741	414	59	(	(	PUNCT
ejde-741	414	60	a+ā	a+ā	PROPN
ejde-741	414	61	2	2	NUM
ejde-741	414	62	)	)	PUNCT
ejde-741	414	63	(	(	PUNCT
ejde-741	414	64	ζ∗)−11/6ζ∗t	ζ∗)−11/6ζ∗t	X
ejde-741	414	65	∣∣∣	∣∣∣	ADJ
ejde-741	414	66	≤	≤	NUM
ejde-741	414	67	e−sa	e−sa	NOUN
ejde-741	414	68	[	[	PUNCT
ejde-741	414	69	sη̄r(ζ	sη̄r(ζ	ADJ
ejde-741	414	70	∗)−11/6|ζt|+	∗)−11/6|ζt|+	NOUN
ejde-741	414	71	11	11	NUM
ejde-741	414	72	6	6	NUM
ejde-741	414	73	(	(	PUNCT
ejde-741	414	74	ζ∗)−17/6|ζ∗t	ζ∗)−17/6|ζ∗t	VERB
ejde-741	414	75	|	|	ADV
ejde-741	414	76	]	]	PUNCT
ejde-741	414	77	.	.	PUNCT
ejde-741	415	1	secondly	secondly	ADV
ejde-741	415	2	,	,	PUNCT
ejde-741	415	3	we	we	PRON
ejde-741	415	4	obtain	obtain	VERB
ejde-741	415	5	|ρ̂ρ̂t|	|ρ̂ρ̂t|	ADJ
ejde-741	415	6	≤	≤	ADJ
ejde-741	415	7	e−2sa	e−2sa	NOUN
ejde-741	415	8	[	[	PUNCT
ejde-741	415	9	sη̄r(ζ	sη̄r(ζ	ADJ
ejde-741	415	10	∗)−11/6|ζt|+	∗)−11/6|ζt|+	NOUN
ejde-741	415	11	11	11	NUM
ejde-741	415	12	6	6	NUM
ejde-741	415	13	(	(	PUNCT
ejde-741	415	14	ζ∗)−17/6|ζ∗t	ζ∗)−17/6|ζ∗t	NOUN
ejde-741	415	15	|	|	ADV
ejde-741	415	16	]	]	PUNCT
ejde-741	415	17	≤	≤	PROPN
ejde-741	416	1	ce−2sa(ζ−2|ζt|+	ce−2sa(ζ−2|ζt|+	PROPN
ejde-741	416	2	ζ−3|ζt|)ζ−5/3	ζ−3|ζt|)ζ−5/3	PUNCT
ejde-741	416	3	≤	≤	PUNCT
ejde-741	416	4	cρ20	cρ20	PROPN
ejde-741	416	5	,	,	PUNCT
ejde-741	416	6	for	for	ADP
ejde-741	416	7	all	all	DET
ejde-741	416	8	t	t	NOUN
ejde-741	416	9	∈	∈	PROPN
ejde-741	417	1	[	[	X
ejde-741	417	2	0	0	NUM
ejde-741	417	3	,	,	PUNCT
ejde-741	417	4	t	t	X
ejde-741	417	5	]	]	PUNCT
ejde-741	417	6	,	,	PUNCT
ejde-741	417	7	following	follow	VERB
ejde-741	417	8	that	that	SCONJ
ejde-741	417	9	i4	i4	PROPN
ejde-741	417	10	≤	≤	PROPN
ejde-741	417	11	c	c	PROPN
ejde-741	417	12	∫	∫	PROPN
ejde-741	417	13	1	1	NUM
ejde-741	417	14	0	0	NUM
ejde-741	417	15	ρ20|u|2	ρ20|u|2	NOUN
ejde-741	417	16	.	.	PUNCT
ejde-741	418	1	next	next	ADJ
ejde-741	418	2	,	,	PUNCT
ejde-741	418	3	using	use	VERB
ejde-741	418	4	|ρ̂x|	|ρ̂x|	PROPN
ejde-741	418	5	=	=	SYM
ejde-741	418	6	∣∣−	∣∣−	PROPN
ejde-741	418	7	s	s	PART
ejde-741	418	8	(	(	PUNCT
ejde-741	418	9	ax	ax	NOUN
ejde-741	418	10	+	+	X
ejde-741	418	11	āx	āx	ADJ
ejde-741	418	12	2	2	NUM
ejde-741	418	13	)	)	PUNCT
ejde-741	418	14	e	e	NOUN
ejde-741	418	15	−s	−s	NOUN
ejde-741	418	16	(	(	PUNCT
ejde-741	418	17	a+ā	a+ā	PROPN
ejde-741	418	18	2	2	NUM
ejde-741	418	19	)	)	PUNCT
ejde-741	418	20	(	(	PUNCT
ejde-741	418	21	ζ∗)−11/6	ζ∗)−11/6	ADP
ejde-741	418	22	∣∣	∣∣	NUM
ejde-741	418	23	≤	≤	X
ejde-741	418	24	ce−saζe−s	ce−saζe−s	NOUN
ejde-741	418	25	(	(	PUNCT
ejde-741	418	26	a+ā	a+ā	PROPN
ejde-741	418	27	2	2	NUM
ejde-741	418	28	)	)	PUNCT
ejde-741	418	29	(	(	PUNCT
ejde-741	418	30	ζ∗)−11/6	ζ∗)−11/6	ADP
ejde-741	418	31	≤	≤	PROPN
ejde-741	418	32	e−sa(ζ)−5/6	e−sa(ζ)−5/6	PROPN
ejde-741	418	33	=	=	SYM
ejde-741	418	34	ρ0	ρ0	PROPN
ejde-741	418	35	,	,	PUNCT
ejde-741	418	36	we	we	PRON
ejde-741	418	37	obtain	obtain	VERB
ejde-741	418	38	i5	i5	ADJ
ejde-741	418	39	≤	≤	NUM
ejde-741	418	40	1	1	NUM
ejde-741	418	41	2	2	NUM
ejde-741	418	42	∫	∫	NOUN
ejde-741	418	43	1	1	NUM
ejde-741	418	44	0	0	NUM
ejde-741	418	45	ρ̂2au2x	ρ̂2au2x	NOUN
ejde-741	418	46	+	+	CCONJ
ejde-741	418	47	2	2	NUM
ejde-741	418	48	∫	∫	NOUN
ejde-741	418	49	1	1	NUM
ejde-741	418	50	0	0	NUM
ejde-741	418	51	ρ̂2xau	ρ̂2xau	ADP
ejde-741	418	52	2	2	NUM
ejde-741	418	53	≤	≤	NUM
ejde-741	418	54	1	1	NUM
ejde-741	418	55	2	2	NUM
ejde-741	418	56	∫	∫	NOUN
ejde-741	418	57	1	1	NUM
ejde-741	418	58	0	0	NUM
ejde-741	418	59	ρ̂2au2x	ρ̂2au2x	NOUN
ejde-741	419	1	+	+	CCONJ
ejde-741	419	2	2	2	NUM
ejde-741	419	3	∫	∫	NOUN
ejde-741	419	4	1	1	NUM
ejde-741	419	5	0	0	NUM
ejde-741	419	6	ρ20u	ρ20u	NOUN
ejde-741	419	7	2	2	NUM
ejde-741	419	8	.	.	NUM
ejde-741	419	9	16	16	NUM
ejde-741	420	1	p.	p.	NOUN
ejde-741	420	2	p.	p.	NOUN
ejde-741	420	3	de	de	PROPN
ejde-741	420	4	carvalho	carvalho	PROPN
ejde-741	420	5	,	,	PUNCT
ejde-741	420	6	r.	r.	PROPN
ejde-741	420	7	demarque	demarque	PROPN
ejde-741	420	8	,	,	PUNCT
ejde-741	420	9	j.	j.	PROPN
ejde-741	420	10	límaco	límaco	PROPN
ejde-741	420	11	,	,	PUNCT
ejde-741	420	12	l.	l.	PROPN
ejde-741	420	13	viana	viana	PROPN
ejde-741	420	14	ejde-2025/15	ejde-2025/15	PROPN
ejde-741	420	15	hence	hence	ADV
ejde-741	420	16	,	,	PUNCT
ejde-741	420	17	(	(	PUNCT
ejde-741	420	18	3.9	3.9	NUM
ejde-741	420	19	)	)	PUNCT
ejde-741	420	20	gives	give	VERB
ejde-741	420	21	us	we	PRON
ejde-741	421	1	d	d	NOUN
ejde-741	421	2	dt	dt	X
ejde-741	421	3	∫	∫	PROPN
ejde-741	421	4	1	1	NUM
ejde-741	421	5	0	0	NUM
ejde-741	421	6	ρ̂2|u|2	ρ̂2|u|2	NOUN
ejde-741	421	7	+	+	NUM
ejde-741	421	8	∫	∫	PROPN
ejde-741	421	9	1	1	NUM
ejde-741	421	10	0	0	NUM
ejde-741	421	11	ρ̂2a|ux|2	ρ̂2a|ux|2	PROPN
ejde-741	422	1	≤	≤	PROPN
ejde-741	422	2	c	c	NOUN
ejde-741	422	3	(	(	PUNCT
ejde-741	422	4	∫	∫	PROPN
ejde-741	422	5	1	1	NUM
ejde-741	422	6	0	0	NUM
ejde-741	422	7	ρ20|u|2	ρ20|u|2	NOUN
ejde-741	422	8	+	+	CCONJ
ejde-741	422	9	∫	∫	PROPN
ejde-741	422	10	1	1	NUM
ejde-741	422	11	0	0	NUM
ejde-741	422	12	ρ2∗|hχω|2	ρ2∗|hχω|2	PROPN
ejde-741	422	13	+	+	CCONJ
ejde-741	422	14	∫	∫	PROPN
ejde-741	422	15	1	1	NUM
ejde-741	422	16	0	0	NUM
ejde-741	422	17	ρ20|g|2	ρ20|g|2	NOUN
ejde-741	422	18	)	)	PUNCT
ejde-741	422	19	.	.	PUNCT
ejde-741	423	1	integrating	integrate	VERB
ejde-741	423	2	in	in	ADP
ejde-741	423	3	time	time	NOUN
ejde-741	423	4	,	,	PUNCT
ejde-741	423	5	we	we	PRON
ejde-741	423	6	reach	reach	VERB
ejde-741	423	7	the	the	DET
ejde-741	423	8	desired	desire	VERB
ejde-741	423	9	estimate	estimate	NOUN
ejde-741	423	10	.	.	PUNCT
ejde-741	424	1	□	□	PUNCT
ejde-741	424	2	lemma	lemma	PROPN
ejde-741	424	3	3.6	3.6	NUM
ejde-741	424	4	.	.	PUNCT
ejde-741	425	1	under	under	ADP
ejde-741	425	2	the	the	DET
ejde-741	425	3	assumptions	assumption	NOUN
ejde-741	425	4	of	of	ADP
ejde-741	425	5	proposition	proposition	NOUN
ejde-741	425	6	3.1	3.1	NUM
ejde-741	425	7	we	we	PRON
ejde-741	425	8	have	have	VERB
ejde-741	425	9	ρ∗	ρ∗	PROPN
ejde-741	425	10	√	√	PROPN
ejde-741	425	11	aux	aux	PROPN
ejde-741	425	12	∈	∈	PROPN
ejde-741	425	13	l∞(0	l∞(0	PROPN
ejde-741	425	14	,	,	PUNCT
ejde-741	425	15	t	t	PROPN
ejde-741	425	16	;	;	PUNCT
ejde-741	425	17	l2(0	l2(0	PROPN
ejde-741	425	18	,	,	PUNCT
ejde-741	425	19	1));ut	1));ut	NUM
ejde-741	425	20	,	,	PUNCT
ejde-741	425	21	(	(	PUNCT
ejde-741	425	22	aux)x	aux)x	PROPN
ejde-741	425	23	∈	∈	PROPN
ejde-741	425	24	l2(q	l2(q	PROPN
ejde-741	425	25	;	;	PUNCT
ejde-741	425	26	ρ2∗	ρ2∗	NUM
ejde-741	425	27	)	)	PUNCT
ejde-741	425	28	,	,	PUNCT
ejde-741	425	29	and	and	CCONJ
ejde-741	425	30	there	there	PRON
ejde-741	425	31	exists	exist	VERB
ejde-741	425	32	c	c	NOUN
ejde-741	425	33	>	>	X
ejde-741	425	34	0	0	NUM
ejde-741	426	1	such	such	ADJ
ejde-741	426	2	that	that	DET
ejde-741	426	3	sup	sup	NOUN
ejde-741	426	4	t∈[0,t	t∈[0,t	NOUN
ejde-741	426	5	]	]	PUNCT
ejde-741	426	6	∥ρ∗	∥ρ∗	PROPN
ejde-741	426	7	√	√	PROPN
ejde-741	426	8	aux(t	aux(t	PROPN
ejde-741	426	9	,	,	PUNCT
ejde-741	426	10	·	·	PUNCT
ejde-741	426	11	)	)	PUNCT
ejde-741	426	12	∥2l2(0,1	∥2l2(0,1	ADV
ejde-741	426	13	)	)	PUNCT
ejde-741	427	1	+	+	CCONJ
ejde-741	427	2	∥ut∥2ρ2∗	∥ut∥2ρ2∗	PROPN
ejde-741	427	3	+	+	NUM
ejde-741	427	4	∥(aux)x∥2ρ2∗	∥(aux)x∥2ρ2∗	NOUN
ejde-741	427	5	≤	≤	NUM
ejde-741	427	6	c(∥u∥2ρ20	c(∥u∥2ρ20	X
ejde-741	428	1	+	+	CCONJ
ejde-741	428	2	∥hχω∥2ρ2∗	∥hχω∥2ρ2∗	PROPN
ejde-741	428	3	+	+	CCONJ
ejde-741	428	4	∥g∥2ρ20	∥g∥2ρ20	X
ejde-741	428	5	+	+	CCONJ
ejde-741	428	6	∥u0∥2h1	∥u0∥2h1	NOUN
ejde-741	428	7	a	a	PRON
ejde-741	428	8	)	)	PUNCT
ejde-741	428	9	.	.	PUNCT
ejde-741	429	1	proof	proof	NOUN
ejde-741	429	2	.	.	PUNCT
ejde-741	430	1	firstly	firstly	ADV
ejde-741	430	2	,	,	PUNCT
ejde-741	430	3	let	let	VERB
ejde-741	430	4	us	we	PRON
ejde-741	430	5	estimate	estimate	VERB
ejde-741	430	6	the	the	DET
ejde-741	430	7	first	first	ADJ
ejde-741	430	8	and	and	CCONJ
ejde-741	430	9	the	the	DET
ejde-741	430	10	second	second	ADJ
ejde-741	430	11	terms	term	NOUN
ejde-741	430	12	on	on	ADP
ejde-741	430	13	the	the	DET
ejde-741	430	14	left	left	ADJ
ejde-741	430	15	side	side	NOUN
ejde-741	430	16	of	of	ADP
ejde-741	430	17	the	the	DET
ejde-741	430	18	desired	desire	VERB
ejde-741	430	19	inequality	inequality	NOUN
ejde-741	430	20	.	.	PUNCT
ejde-741	431	1	multiplying	multiply	VERB
ejde-741	431	2	the	the	DET
ejde-741	431	3	pde	pde	NOUN
ejde-741	431	4	in	in	ADP
ejde-741	431	5	(	(	PUNCT
ejde-741	431	6	2.2	2.2	NUM
ejde-741	431	7	)	)	PUNCT
ejde-741	431	8	by	by	ADP
ejde-741	431	9	ρ2∗ut	ρ2∗ut	NOUN
ejde-741	431	10	and	and	CCONJ
ejde-741	431	11	integrating	integrate	VERB
ejde-741	431	12	in	in	ADP
ejde-741	431	13	[	[	X
ejde-741	431	14	0	0	NUM
ejde-741	431	15	,	,	PUNCT
ejde-741	431	16	1	1	NUM
ejde-741	431	17	]	]	PUNCT
ejde-741	431	18	,	,	PUNCT
ejde-741	431	19	we	we	PRON
ejde-741	431	20	have∫	have∫	VERB
ejde-741	431	21	1	1	NUM
ejde-741	431	22	0	0	NUM
ejde-741	431	23	ρ2∗u	ρ2∗u	SYM
ejde-741	431	24	2	2	NUM
ejde-741	431	25	t	t	NOUN
ejde-741	431	26	=	=	SYM
ejde-741	431	27	∫	∫	PROPN
ejde-741	432	1	1	1	NUM
ejde-741	432	2	0	0	NUM
ejde-741	432	3	ρ2∗uthχω	ρ2∗uthχω	PROPN
ejde-741	432	4	+	+	NUM
ejde-741	432	5	∫	∫	PROPN
ejde-741	432	6	1	1	NUM
ejde-741	432	7	0	0	NUM
ejde-741	433	1	ρ2∗gut	ρ2∗gut	NUM
ejde-741	433	2	−	−	NOUN
ejde-741	433	3	∫	∫	PROPN
ejde-741	433	4	1	1	NUM
ejde-741	433	5	0	0	NUM
ejde-741	433	6	ρ2∗c(t	ρ2∗c(t	PROPN
ejde-741	433	7	,	,	PUNCT
ejde-741	433	8	x)uut	x)uut	PROPN
ejde-741	434	1	+	+	CCONJ
ejde-741	434	2	∫	∫	PROPN
ejde-741	434	3	1	1	NUM
ejde-741	434	4	0	0	NUM
ejde-741	434	5	ρ2∗(aux)xut	ρ2∗(aux)xut	NOUN
ejde-741	434	6	=	=	SYM
ejde-741	434	7	i1	i1	PROPN
ejde-741	434	8	+	+	CCONJ
ejde-741	434	9	i2	i2	PROPN
ejde-741	434	10	−	−	PROPN
ejde-741	434	11	i3	i3	PROPN
ejde-741	434	12	+	+	CCONJ
ejde-741	434	13	i4	i4	PROPN
ejde-741	434	14	.	.	PUNCT
ejde-741	435	1	(	(	PUNCT
ejde-741	435	2	3.10	3.10	NUM
ejde-741	435	3	)	)	PUNCT
ejde-741	435	4	using	use	VERB
ejde-741	435	5	young	young	PROPN
ejde-741	435	6	’s	’s	PART
ejde-741	435	7	inequality	inequality	NOUN
ejde-741	435	8	with	with	ADP
ejde-741	435	9	ε	ε	PROPN
ejde-741	435	10	and	and	CCONJ
ejde-741	435	11	ρ∗	ρ∗	PROPN
ejde-741	435	12	≤	≤	PROPN
ejde-741	436	1	cρ̂	cρ̂	PROPN
ejde-741	436	2	≤	≤	NOUN
ejde-741	436	3	cρ0	cρ0	NOUN
ejde-741	436	4	≤	≤	NUM
ejde-741	436	5	cρ	cρ	PROPN
ejde-741	436	6	,	,	PUNCT
ejde-741	436	7	we	we	PRON
ejde-741	436	8	obtain	obtain	VERB
ejde-741	436	9	i1	i1	PROPN
ejde-741	436	10	≤	≤	PROPN
ejde-741	436	11	∫	∫	PROPN
ejde-741	437	1	1	1	NUM
ejde-741	437	2	0	0	NUM
ejde-741	437	3	ρ2∗|hχω||ut|	ρ2∗|hχω||ut|	NOUN
ejde-741	437	4	≤	≤	PUNCT
ejde-741	437	5	ε	ε	PROPN
ejde-741	437	6	∫	∫	PROPN
ejde-741	437	7	1	1	NUM
ejde-741	437	8	0	0	NUM
ejde-741	437	9	ρ2∗|ut|2	ρ2∗|ut|2	X
ejde-741	438	1	+	+	CCONJ
ejde-741	438	2	1	1	NUM
ejde-741	438	3	4ε	4ε	NUM
ejde-741	438	4	∫	∫	NOUN
ejde-741	438	5	1	1	NUM
ejde-741	438	6	0	0	NUM
ejde-741	438	7	ρ2∗|hχω|2	ρ2∗|hχω|2	PROPN
ejde-741	438	8	,	,	PUNCT
ejde-741	438	9	i2	i2	PROPN
ejde-741	438	10	≤	≤	NUM
ejde-741	438	11	∫	∫	PROPN
ejde-741	438	12	1	1	NUM
ejde-741	438	13	0	0	NUM
ejde-741	438	14	ρ2∗|gut|	ρ2∗|gut|	PROPN
ejde-741	438	15	≤	≤	NUM
ejde-741	438	16	ε	ε	PROPN
ejde-741	438	17	∫	∫	PROPN
ejde-741	438	18	1	1	NUM
ejde-741	438	19	0	0	NUM
ejde-741	438	20	ρ2∗|ut|2	ρ2∗|ut|2	X
ejde-741	439	1	+	+	CCONJ
ejde-741	439	2	1	1	NUM
ejde-741	439	3	4ε	4ε	NUM
ejde-741	439	4	∫	∫	NOUN
ejde-741	439	5	1	1	NUM
ejde-741	439	6	0	0	NUM
ejde-741	439	7	ρ2∗|g|2	ρ2∗|g|2	PROPN
ejde-741	439	8	≤	≤	PROPN
ejde-741	439	9	ε	ε	PROPN
ejde-741	439	10	∫	∫	PROPN
ejde-741	439	11	1	1	NUM
ejde-741	439	12	0	0	NUM
ejde-741	439	13	ρ2∗|ut|2	ρ2∗|ut|2	PROPN
ejde-741	440	1	+	+	PUNCT
ejde-741	440	2	c	c	NOUN
ejde-741	440	3	4ε	4ε	NUM
ejde-741	440	4	∫	∫	NOUN
ejde-741	440	5	1	1	NUM
ejde-741	440	6	0	0	NUM
ejde-741	440	7	ρ20|g|2	ρ20|g|2	PROPN
ejde-741	440	8	,	,	PUNCT
ejde-741	440	9	−i3	−i3	VERB
ejde-741	440	10	≤	≤	NUM
ejde-741	440	11	∫	∫	PROPN
ejde-741	440	12	1	1	NUM
ejde-741	440	13	0	0	NUM
ejde-741	440	14	|c(t	|c(t	PROPN
ejde-741	440	15	,	,	PUNCT
ejde-741	440	16	x)|ρ2∗|uut|	x)|ρ2∗|uut|	PROPN
ejde-741	440	17	≤	≤	PROPN
ejde-741	441	1	ε	ε	PROPN
ejde-741	441	2	∫	∫	PROPN
ejde-741	441	3	1	1	NUM
ejde-741	441	4	0	0	NUM
ejde-741	441	5	ρ2∗|ut|2	ρ2∗|ut|2	PROPN
ejde-741	442	1	+	+	NUM
ejde-741	442	2	∥c∥∞	∥c∥∞	X
ejde-741	443	1	4ε	4ε	NUM
ejde-741	443	2	∫	∫	NOUN
ejde-741	443	3	1	1	NUM
ejde-741	443	4	0	0	NUM
ejde-741	443	5	ρ2∗|u|2	ρ2∗|u|2	PROPN
ejde-741	443	6	≤	≤	PROPN
ejde-741	443	7	ε	ε	PROPN
ejde-741	443	8	∫	∫	PROPN
ejde-741	443	9	1	1	NUM
ejde-741	443	10	0	0	NUM
ejde-741	443	11	ρ2∗|ut|2	ρ2∗|ut|2	PROPN
ejde-741	444	1	+	+	PUNCT
ejde-741	444	2	c	c	NOUN
ejde-741	444	3	4ε	4ε	NUM
ejde-741	444	4	∫	∫	NOUN
ejde-741	444	5	1	1	NUM
ejde-741	444	6	0	0	NUM
ejde-741	444	7	ρ20|u|2	ρ20|u|2	NOUN
ejde-741	444	8	.	.	PUNCT
ejde-741	445	1	since	since	SCONJ
ejde-741	445	2	ut(t	ut(t	PROPN
ejde-741	445	3	,	,	PUNCT
ejde-741	445	4	0	0	NUM
ejde-741	445	5	)	)	PUNCT
ejde-741	445	6	=	=	SYM
ejde-741	445	7	ut(t	ut(t	NOUN
ejde-741	445	8	,	,	PUNCT
ejde-741	445	9	1	1	X
ejde-741	445	10	)	)	PUNCT
ejde-741	445	11	≡	≡	PROPN
ejde-741	445	12	0	0	NUM
ejde-741	445	13	for	for	ADP
ejde-741	445	14	the	the	DET
ejde-741	445	15	(	(	PUNCT
ejde-741	445	16	wdp	wdp	PROPN
ejde-741	445	17	)	)	PUNCT
ejde-741	445	18	and	and	CCONJ
ejde-741	445	19	aux(t	aux(t	PROPN
ejde-741	445	20	,	,	PUNCT
ejde-741	445	21	0	0	NUM
ejde-741	445	22	)	)	PUNCT
ejde-741	445	23	=	=	SYM
ejde-741	445	24	ut(t	ut(t	NOUN
ejde-741	445	25	,	,	PUNCT
ejde-741	445	26	1	1	X
ejde-741	445	27	)	)	PUNCT
ejde-741	445	28	≡	≡	PROPN
ejde-741	445	29	0	0	NUM
ejde-741	445	30	for	for	ADP
ejde-741	445	31	the	the	DET
ejde-741	445	32	(	(	PUNCT
ejde-741	445	33	sdp	sdp	NOUN
ejde-741	445	34	)	)	PUNCT
ejde-741	445	35	,	,	PUNCT
ejde-741	445	36	we	we	PRON
ejde-741	445	37	integrate	integrate	VERB
ejde-741	445	38	by	by	ADP
ejde-741	445	39	parts	part	NOUN
ejde-741	445	40	to	to	PART
ejde-741	445	41	obtain	obtain	VERB
ejde-741	445	42	i4	i4	PROPN
ejde-741	445	43	=	=	SYM
ejde-741	445	44	ρ2∗auxut	ρ2∗auxut	NOUN
ejde-741	445	45	∣∣x=1	∣∣x=1	NOUN
ejde-741	445	46	x=0	x=0	PUNCT
ejde-741	446	1	−	−	NOUN
ejde-741	447	1	∫	∫	PROPN
ejde-741	447	2	1	1	NUM
ejde-741	447	3	0	0	NUM
ejde-741	447	4	(	(	PUNCT
ejde-741	447	5	ρ2∗utxaux	ρ2∗utxaux	ADJ
ejde-741	447	6	=	=	SYM
ejde-741	447	7	−1	−1	NOUN
ejde-741	447	8	2	2	NUM
ejde-741	447	9	d	d	NOUN
ejde-741	447	10	dt	dt	X
ejde-741	447	11	∫	∫	PROPN
ejde-741	447	12	1	1	NUM
ejde-741	447	13	0	0	NUM
ejde-741	447	14	ρ2∗au	ρ2∗au	NOUN
ejde-741	447	15	2	2	NUM
ejde-741	447	16	x	x	SYM
ejde-741	447	17	+	+	NOUN
ejde-741	447	18	1	1	NUM
ejde-741	447	19	2	2	NUM
ejde-741	447	20	∫	∫	NOUN
ejde-741	447	21	1	1	NUM
ejde-741	447	22	0	0	NUM
ejde-741	447	23	(	(	PUNCT
ejde-741	447	24	ρ2∗)tau	ρ2∗)tau	PROPN
ejde-741	447	25	2	2	NUM
ejde-741	447	26	x	x	X
ejde-741	447	27	=	=	SYM
ejde-741	447	28	−1	−1	NOUN
ejde-741	447	29	2	2	NUM
ejde-741	447	30	d	d	NOUN
ejde-741	447	31	dt	dt	X
ejde-741	447	32	∫	∫	PROPN
ejde-741	447	33	1	1	NUM
ejde-741	447	34	0	0	NUM
ejde-741	447	35	ρ2∗au	ρ2∗au	NOUN
ejde-741	447	36	2	2	NUM
ejde-741	447	37	x	x	SYM
ejde-741	447	38	+	+	NOUN
ejde-741	447	39	1	1	NUM
ejde-741	447	40	2	2	NUM
ejde-741	447	41	i41	i41	NUM
ejde-741	447	42	.	.	PUNCT
ejde-741	448	1	(	(	PUNCT
ejde-741	448	2	3.11	3.11	NUM
ejde-741	448	3	)	)	PUNCT
ejde-741	448	4	hence	hence	ADV
ejde-741	448	5	∫	∫	PROPN
ejde-741	448	6	1	1	NUM
ejde-741	448	7	0	0	NUM
ejde-741	448	8	ρ2∗|ut|2	ρ2∗|ut|2	X
ejde-741	449	1	+	+	CCONJ
ejde-741	449	2	1	1	NUM
ejde-741	449	3	2	2	NUM
ejde-741	449	4	d	d	NOUN
ejde-741	449	5	dt	dt	X
ejde-741	449	6	∫	∫	PROPN
ejde-741	449	7	1	1	NUM
ejde-741	449	8	0	0	NUM
ejde-741	449	9	ρ2∗a|ux|2	ρ2∗a|ux|2	PROPN
ejde-741	449	10	=	=	SYM
ejde-741	449	11	i1	i1	PROPN
ejde-741	449	12	+	+	CCONJ
ejde-741	449	13	i2	i2	PROPN
ejde-741	449	14	−	−	PROPN
ejde-741	449	15	i3	i3	NOUN
ejde-741	449	16	+	+	CCONJ
ejde-741	449	17	1	1	NUM
ejde-741	449	18	2	2	NUM
ejde-741	449	19	i41	i41	NUM
ejde-741	449	20	.	.	PUNCT
ejde-741	450	1	(	(	PUNCT
ejde-741	450	2	3.12	3.12	NUM
ejde-741	450	3	)	)	PUNCT
ejde-741	450	4	at	at	ADP
ejde-741	450	5	this	this	DET
ejde-741	450	6	point	point	NOUN
ejde-741	450	7	,	,	PUNCT
ejde-741	450	8	we	we	PRON
ejde-741	450	9	observe	observe	VERB
ejde-741	450	10	that	that	SCONJ
ejde-741	450	11	(	(	PUNCT
ejde-741	450	12	ρ∗)t	ρ∗)t	PROPN
ejde-741	450	13	=	=	SYM
ejde-741	450	14	−sτtη̄re−sā(ζ∗)−17/6	−sτtη̄re−sā(ζ∗)−17/6	NOUN
ejde-741	450	15	−	−	PROPN
ejde-741	450	16	17	17	NUM
ejde-741	450	17	6	6	NUM
ejde-741	450	18	e−sā(ζ∗)−23/6τtη	e−sā(ζ∗)−23/6τtη	NOUN
ejde-741	450	19	∗	∗	NOUN
ejde-741	450	20	ejde-2025/15	ejde-2025/15	VERB
ejde-741	450	21	null	null	ADJ
ejde-741	450	22	-	-	PUNCT
ejde-741	450	23	controllability	controllability	NOUN
ejde-741	450	24	degenerate	degenerate	ADJ
ejde-741	450	25	quasilinear	quasilinear	NOUN
ejde-741	450	26	equations	equation	NOUN
ejde-741	450	27	17	17	NUM
ejde-741	450	28	and	and	CCONJ
ejde-741	450	29	,	,	PUNCT
ejde-741	450	30	consequently	consequently	ADV
ejde-741	450	31	,	,	PUNCT
ejde-741	451	1	|ρ∗(ρ∗)t|	|ρ∗(ρ∗)t|	ADV
ejde-741	451	2	≤	≤	ADJ
ejde-741	451	3	ce−2sā[|τtη∗|(ζ∗)−17/3	ce−2sā[|τtη∗|(ζ∗)−17/3	PROPN
ejde-741	451	4	+	+	CCONJ
ejde-741	451	5	|τtη∗|(ζ∗)−20/3	|τtη∗|(ζ∗)−20/3	X
ejde-741	451	6	]	]	X
ejde-741	451	7	=	=	X
ejde-741	451	8	ce−2sā(ζ∗)−11/3[(ζ∗)−2	ce−2sā(ζ∗)−11/3[(ζ∗)−2	X
ejde-741	451	9	+	+	CCONJ
ejde-741	451	10	(	(	PUNCT
ejde-741	451	11	ζ∗)−3]|ζ∗t	ζ∗)−3]|ζ∗t	X
ejde-741	451	12	)	)	PUNCT
ejde-741	451	13	|	|	ADV
ejde-741	451	14	≤	≤	PUNCT
ejde-741	452	1	cρ̂2	cρ̂2	PROPN
ejde-741	452	2	.	.	PUNCT
ejde-741	453	1	so	so	SCONJ
ejde-741	453	2	that	that	PRON
ejde-741	453	3	i41	i41	VERB
ejde-741	453	4	≤	≤	NUM
ejde-741	453	5	c	c	NOUN
ejde-741	453	6	∫	∫	PROPN
ejde-741	453	7	1	1	NUM
ejde-741	453	8	0	0	NUM
ejde-741	453	9	ρ̂2au2x	ρ̂2au2x	NUM
ejde-741	453	10	.	.	PUNCT
ejde-741	454	1	as	as	ADP
ejde-741	454	2	a	a	DET
ejde-741	454	3	result	result	NOUN
ejde-741	454	4	,	,	PUNCT
ejde-741	454	5	taking	take	VERB
ejde-741	454	6	a	a	DET
ejde-741	454	7	sufficiently	sufficiently	ADV
ejde-741	454	8	small	small	ADJ
ejde-741	454	9	ε	ε	PROPN
ejde-741	454	10	>	>	X
ejde-741	454	11	0	0	PROPN
ejde-741	454	12	,	,	PUNCT
ejde-741	454	13	we	we	PRON
ejde-741	454	14	obtain∫	obtain∫	VERB
ejde-741	454	15	1	1	NUM
ejde-741	454	16	0	0	NUM
ejde-741	454	17	ρ2∗u	ρ2∗u	SYM
ejde-741	454	18	2	2	NUM
ejde-741	454	19	t	t	NOUN
ejde-741	454	20	+	+	NOUN
ejde-741	454	21	1	1	NUM
ejde-741	454	22	2	2	NUM
ejde-741	454	23	d	d	NOUN
ejde-741	454	24	dt	dt	X
ejde-741	454	25	∫	∫	PROPN
ejde-741	454	26	1	1	NUM
ejde-741	454	27	0	0	NUM
ejde-741	454	28	ρ2∗au	ρ2∗au	NOUN
ejde-741	454	29	2	2	NUM
ejde-741	454	30	x	x	SYM
ejde-741	454	31	≤	≤	NUM
ejde-741	454	32	c	c	X
ejde-741	454	33	(	(	PUNCT
ejde-741	454	34	∫	∫	PROPN
ejde-741	455	1	1	1	NUM
ejde-741	455	2	0	0	NUM
ejde-741	455	3	ρ2∗|hχω|2	ρ2∗|hχω|2	PROPN
ejde-741	455	4	+	+	CCONJ
ejde-741	455	5	∫	∫	PROPN
ejde-741	455	6	1	1	NUM
ejde-741	455	7	0	0	NUM
ejde-741	455	8	ρ20	ρ20	NOUN
ejde-741	455	9	g	g	NOUN
ejde-741	455	10	2	2	NUM
ejde-741	455	11	+	+	CCONJ
ejde-741	455	12	∫	∫	PROPN
ejde-741	455	13	1	1	NUM
ejde-741	455	14	0	0	NUM
ejde-741	455	15	ρ20u	ρ20u	NUM
ejde-741	455	16	2	2	NUM
ejde-741	455	17	+	+	NUM
ejde-741	455	18	∫	∫	PROPN
ejde-741	455	19	1	1	NUM
ejde-741	455	20	0	0	NUM
ejde-741	455	21	ρ̂2au2x	ρ̂2au2x	PROPN
ejde-741	455	22	)	)	PUNCT
ejde-741	455	23	,	,	PUNCT
ejde-741	455	24	which	which	PRON
ejde-741	455	25	implies	imply	VERB
ejde-741	455	26	sup	sup	NOUN
ejde-741	455	27	t∈[0,t	t∈[0,t	X
ejde-741	455	28	]	]	PUNCT
ejde-741	455	29	∥ρ∗	∥ρ∗	PROPN
ejde-741	455	30	√	√	PROPN
ejde-741	455	31	aux(t	aux(t	PROPN
ejde-741	455	32	,	,	PUNCT
ejde-741	455	33	·	·	PUNCT
ejde-741	455	34	)	)	PUNCT
ejde-741	455	35	∥2l2(0,1	∥2l2(0,1	ADV
ejde-741	455	36	)	)	PUNCT
ejde-741	456	1	+	+	CCONJ
ejde-741	456	2	∥ut∥2ρ2∗	∥ut∥2ρ2∗	ADJ
ejde-741	456	3	≤	≤	NUM
ejde-741	456	4	c(∥u∥2ρ20	c(∥u∥2ρ20	X
ejde-741	457	1	+	+	CCONJ
ejde-741	457	2	∥hχω∥2ρ2∗	∥hχω∥2ρ2∗	PROPN
ejde-741	457	3	+	+	CCONJ
ejde-741	457	4	∥g∥2ρ20	∥g∥2ρ20	X
ejde-741	457	5	+	+	CCONJ
ejde-741	457	6	∥u0∥2h1	∥u0∥2h1	NOUN
ejde-741	457	7	a	a	PRON
ejde-741	457	8	)	)	PUNCT
ejde-741	457	9	.	.	PUNCT
ejde-741	458	1	to	to	PART
ejde-741	458	2	estimate	estimate	VERB
ejde-741	458	3	∥(aux)x∥2ρ2∗	∥(aux)x∥2ρ2∗	PRON
ejde-741	458	4	,	,	PUNCT
ejde-741	458	5	we	we	PRON
ejde-741	458	6	proceed	proceed	VERB
ejde-741	458	7	analogously	analogously	ADV
ejde-741	458	8	,	,	PUNCT
ejde-741	458	9	multiplying	multiply	VERB
ejde-741	458	10	the	the	DET
ejde-741	458	11	pde	pde	NOUN
ejde-741	458	12	in	in	ADP
ejde-741	458	13	(	(	PUNCT
ejde-741	458	14	2.2	2.2	NUM
ejde-741	458	15	)	)	PUNCT
ejde-741	458	16	by	by	ADP
ejde-741	458	17	−ρ217(aux)x	−ρ217(aux)x	ADV
ejde-741	458	18	and	and	CCONJ
ejde-741	458	19	integrating	integrate	VERB
ejde-741	458	20	in	in	ADP
ejde-741	458	21	[	[	X
ejde-741	458	22	0	0	NUM
ejde-741	458	23	,	,	PUNCT
ejde-741	458	24	1	1	NUM
ejde-741	458	25	]	]	PUNCT
ejde-741	458	26	.	.	PUNCT
ejde-741	459	1	the	the	DET
ejde-741	459	2	details	detail	NOUN
ejde-741	459	3	can	can	AUX
ejde-741	459	4	be	be	AUX
ejde-741	459	5	seen	see	VERB
ejde-741	459	6	in	in	ADP
ejde-741	459	7	[	[	X
ejde-741	459	8	20	20	NUM
ejde-741	459	9	,	,	PUNCT
ejde-741	459	10	lemma	lemma	PROPN
ejde-741	459	11	4.3	4.3	NUM
ejde-741	459	12	]	]	PUNCT
ejde-741	459	13	.	.	PUNCT
ejde-741	460	1	□	□	PUNCT
ejde-741	460	2	4	4	X
ejde-741	460	3	.	.	X
ejde-741	460	4	main	main	ADJ
ejde-741	460	5	result	result	NOUN
ejde-741	460	6	and	and	CCONJ
ejde-741	460	7	further	further	ADJ
ejde-741	460	8	comments	comment	NOUN
ejde-741	460	9	proof	proof	NOUN
ejde-741	460	10	of	of	ADP
ejde-741	460	11	theorem	theorem	ADJ
ejde-741	460	12	1.6	1.6	NUM
ejde-741	460	13	.	.	PUNCT
ejde-741	461	1	in	in	ADP
ejde-741	461	2	section	section	NOUN
ejde-741	461	3	3	3	NUM
ejde-741	461	4	,	,	PUNCT
ejde-741	461	5	we	we	PRON
ejde-741	461	6	have	have	AUX
ejde-741	461	7	proved	prove	VERB
ejde-741	461	8	that	that	DET
ejde-741	461	9	h	h	NOUN
ejde-741	461	10	:	:	PUNCT
ejde-741	461	11	e	e	X
ejde-741	461	12	→	→	SYM
ejde-741	461	13	f	f	PROPN
ejde-741	461	14	is	be	AUX
ejde-741	461	15	a	a	DET
ejde-741	461	16	continuously	continuously	ADV
ejde-741	461	17	differentiable	differentiable	ADJ
ejde-741	461	18	mapping	mapping	NOUN
ejde-741	461	19	,	,	PUNCT
ejde-741	461	20	whose	whose	DET
ejde-741	461	21	derivative	derivative	ADJ
ejde-741	461	22	h′(0	h′(0	NOUN
ejde-741	461	23	,	,	PUNCT
ejde-741	461	24	0	0	NUM
ejde-741	461	25	)	)	PUNCT
ejde-741	461	26	∈	∈	PROPN
ejde-741	461	27	l(e;f	l(e;f	NOUN
ejde-741	461	28	)	)	PUNCT
ejde-741	461	29	is	be	AUX
ejde-741	461	30	onto	onto	ADP
ejde-741	461	31	(	(	PUNCT
ejde-741	461	32	lemma	lemma	PROPN
ejde-741	461	33	3.2	3.2	NUM
ejde-741	461	34	,	,	PUNCT
ejde-741	461	35	and	and	CCONJ
ejde-741	461	36	propositions	proposition	NOUN
ejde-741	461	37	3.3	3.3	NUM
ejde-741	461	38	and	and	CCONJ
ejde-741	461	39	3.4	3.4	NUM
ejde-741	461	40	)	)	PUNCT
ejde-741	461	41	.	.	PUNCT
ejde-741	462	1	as	as	ADP
ejde-741	462	2	a	a	DET
ejde-741	462	3	result	result	NOUN
ejde-741	462	4	,	,	PUNCT
ejde-741	462	5	theorem	theorem	VERB
ejde-741	462	6	2.1	2.1	NUM
ejde-741	462	7	can	can	AUX
ejde-741	462	8	be	be	AUX
ejde-741	462	9	applied	apply	VERB
ejde-741	462	10	in	in	ADP
ejde-741	462	11	order	order	NOUN
ejde-741	462	12	to	to	PART
ejde-741	462	13	obtain	obtain	VERB
ejde-741	462	14	a	a	DET
ejde-741	462	15	sufficiently	sufficiently	ADV
ejde-741	462	16	small	small	ADJ
ejde-741	462	17	ε	ε	PROPN
ejde-741	462	18	>	>	PUNCT
ejde-741	462	19	0	0	PROPN
ejde-741	462	20	and	and	CCONJ
ejde-741	462	21	a	a	DET
ejde-741	462	22	right	right	ADJ
ejde-741	462	23	inverse	inverse	NOUN
ejde-741	462	24	mapping	mapping	NOUN
ejde-741	462	25	h̃	h̃	PROPN
ejde-741	462	26	:	:	PUNCT
ejde-741	462	27	bε(0	bε(0	X
ejde-741	462	28	)	)	PUNCT
ejde-741	463	1	⊂	⊂	PROPN
ejde-741	463	2	f	f	X
ejde-741	463	3	→	→	SYM
ejde-741	463	4	e	e	PROPN
ejde-741	463	5	of	of	ADP
ejde-741	463	6	h.	h.	PROPN
ejde-741	463	7	hence	hence	ADV
ejde-741	463	8	,	,	PUNCT
ejde-741	463	9	taking	take	VERB
ejde-741	463	10	u0	u0	ADJ
ejde-741	463	11	∈	∈	PROPN
ejde-741	463	12	h1	h1	VERB
ejde-741	463	13	a	a	DET
ejde-741	463	14	satisfying	satisfying	ADJ
ejde-741	463	15	∥u0∥h1	∥u0∥h1	PUNCT
ejde-741	463	16	a	a	DET
ejde-741	463	17	<	<	X
ejde-741	463	18	ε	ε	PROPN
ejde-741	463	19	,	,	PUNCT
ejde-741	463	20	we	we	PRON
ejde-741	463	21	can	can	AUX
ejde-741	463	22	see	see	VERB
ejde-741	463	23	that	that	PRON
ejde-741	463	24	(	(	PUNCT
ejde-741	463	25	u	u	NOUN
ejde-741	463	26	,	,	PUNCT
ejde-741	463	27	h	h	NOUN
ejde-741	463	28	)	)	PUNCT
ejde-741	463	29	:	:	PUNCT
ejde-741	464	1	=	=	SYM
ejde-741	464	2	h̃(0	h̃(0	NOUN
ejde-741	464	3	,	,	PUNCT
ejde-741	464	4	u0	u0	ADJ
ejde-741	464	5	)	)	PUNCT
ejde-741	464	6	solves	solve	NOUN
ejde-741	464	7	ut	ut	PROPN
ejde-741	465	1	−	−	PROPN
ejde-741	465	2	ℓ(au)(a(x)ux)x	ℓ(au)(a(x)ux)x	ADJ
ejde-741	465	3	+	+	CCONJ
ejde-741	465	4	f(t	f(t	NOUN
ejde-741	465	5	,	,	PUNCT
ejde-741	465	6	x	x	NOUN
ejde-741	465	7	,	,	PUNCT
ejde-741	465	8	u	u	NOUN
ejde-741	465	9	)	)	PUNCT
ejde-741	465	10	=	=	SYM
ejde-741	465	11	hχω	hχω	PROPN
ejde-741	465	12	,	,	PUNCT
ejde-741	465	13	(	(	PUNCT
ejde-741	465	14	t	t	PROPN
ejde-741	465	15	,	,	PUNCT
ejde-741	465	16	x	x	X
ejde-741	465	17	)	)	PUNCT
ejde-741	465	18	∈	∈	PROPN
ejde-741	465	19	q	q	NOUN
ejde-741	465	20	,	,	PUNCT
ejde-741	465	21	u(t	u(t	NOUN
ejde-741	465	22	,	,	PUNCT
ejde-741	465	23	1	1	NUM
ejde-741	465	24	)	)	PUNCT
ejde-741	465	25	=	=	SYM
ejde-741	465	26	0	0	NUM
ejde-741	465	27	,	,	PUNCT
ejde-741	465	28	in	in	ADP
ejde-741	465	29	(	(	PUNCT
ejde-741	465	30	0	0	NUM
ejde-741	465	31	,	,	PUNCT
ejde-741	465	32	t	t	NOUN
ejde-741	465	33	)	)	PUNCT
ejde-741	465	34	,	,	PUNCT
ejde-741	465	35			PROPN
ejde-741	465	36	u(t	u(t	NOUN
ejde-741	465	37	,	,	PUNCT
ejde-741	465	38	0	0	NUM
ejde-741	465	39	)	)	PUNCT
ejde-741	465	40	=	=	SYM
ejde-741	465	41	0	0	NUM
ejde-741	465	42	,	,	PUNCT
ejde-741	465	43	(	(	PUNCT
ejde-741	465	44	weak	weak	ADJ
ejde-741	465	45	)	)	PUNCT
ejde-741	465	46	,	,	PUNCT
ejde-741	465	47	t	t	PROPN
ejde-741	465	48	∈	∈	PROPN
ejde-741	465	49	(	(	PUNCT
ejde-741	465	50	0	0	NUM
ejde-741	465	51	,	,	PUNCT
ejde-741	465	52	t	t	NOUN
ejde-741	465	53	)	)	PUNCT
ejde-741	465	54	or	or	CCONJ
ejde-741	465	55	(	(	PUNCT
ejde-741	465	56	aux)(t	aux)(t	NOUN
ejde-741	465	57	,	,	PUNCT
ejde-741	465	58	0	0	NUM
ejde-741	465	59	)	)	PUNCT
ejde-741	465	60	=	=	SYM
ejde-741	465	61	0	0	NUM
ejde-741	465	62	,	,	PUNCT
ejde-741	465	63	(	(	PUNCT
ejde-741	465	64	strong	strong	ADJ
ejde-741	465	65	)	)	PUNCT
ejde-741	465	66	,	,	PUNCT
ejde-741	465	67	t	t	PROPN
ejde-741	465	68	∈	∈	PROPN
ejde-741	465	69	(	(	PUNCT
ejde-741	465	70	0	0	NUM
ejde-741	465	71	,	,	PUNCT
ejde-741	465	72	t	t	NOUN
ejde-741	465	73	)	)	PUNCT
ejde-741	466	1	u(0	u(0	PROPN
ejde-741	466	2	,	,	PUNCT
ejde-741	466	3	x	x	NOUN
ejde-741	466	4	)	)	PUNCT
ejde-741	466	5	=	=	SYM
ejde-741	466	6	u0	u0	ADJ
ejde-741	466	7	,	,	PUNCT
ejde-741	466	8	x	x	SYM
ejde-741	466	9	∈	∈	PROPN
ejde-741	466	10	(	(	PUNCT
ejde-741	466	11	0	0	NUM
ejde-741	466	12	,	,	PUNCT
ejde-741	466	13	1	1	NUM
ejde-741	466	14	)	)	PUNCT
ejde-741	466	15	,	,	PUNCT
ejde-741	466	16	u(t	u(t	NOUN
ejde-741	466	17	,	,	PUNCT
ejde-741	466	18	x	x	NOUN
ejde-741	466	19	)	)	PUNCT
ejde-741	466	20	=	=	SYM
ejde-741	466	21	0	0	NUM
ejde-741	466	22	,	,	PUNCT
ejde-741	466	23	x	x	SYM
ejde-741	466	24	∈	∈	PROPN
ejde-741	466	25	(	(	PUNCT
ejde-741	466	26	0	0	NUM
ejde-741	466	27	,	,	PUNCT
ejde-741	466	28	1	1	NUM
ejde-741	466	29	)	)	PUNCT
ejde-741	466	30	,	,	PUNCT
ejde-741	466	31	(	(	PUNCT
ejde-741	466	32	4.1	4.1	NUM
ejde-741	466	33	)	)	PUNCT
ejde-741	466	34	where	where	SCONJ
ejde-741	466	35	the	the	DET
ejde-741	466	36	last	last	ADJ
ejde-741	466	37	condition	condition	NOUN
ejde-741	466	38	comes	come	VERB
ejde-741	466	39	from	from	ADP
ejde-741	466	40	remark	remark	NOUN
ejde-741	466	41	2.4	2.4	NUM
ejde-741	466	42	.	.	PUNCT
ejde-741	467	1	it	it	PRON
ejde-741	467	2	completes	complete	VERB
ejde-741	467	3	the	the	DET
ejde-741	467	4	proof	proof	NOUN
ejde-741	467	5	.	.	PUNCT
ejde-741	468	1	□	□	PUNCT
ejde-741	468	2	in	in	ADP
ejde-741	468	3	the	the	DET
ejde-741	468	4	context	context	NOUN
ejde-741	468	5	of	of	ADP
ejde-741	468	6	degenerate	degenerate	ADJ
ejde-741	468	7	equations	equation	NOUN
ejde-741	468	8	,	,	PUNCT
ejde-741	468	9	there	there	PRON
ejde-741	468	10	are	be	VERB
ejde-741	468	11	many	many	ADJ
ejde-741	468	12	important	important	ADJ
ejde-741	468	13	questions	question	NOUN
ejde-741	468	14	which	which	PRON
ejde-741	468	15	have	have	AUX
ejde-741	468	16	not	not	PART
ejde-741	468	17	been	be	AUX
ejde-741	468	18	dealt	deal	VERB
ejde-741	468	19	yet	yet	ADV
ejde-741	468	20	,	,	PUNCT
ejde-741	468	21	or	or	CCONJ
ejde-741	468	22	for	for	ADP
ejde-741	468	23	which	which	PRON
ejde-741	468	24	much	much	ADV
ejde-741	468	25	more	more	ADJ
ejde-741	468	26	investigation	investigation	NOUN
ejde-741	468	27	should	should	AUX
ejde-741	468	28	be	be	AUX
ejde-741	468	29	performed	perform	VERB
ejde-741	468	30	.	.	PUNCT
ejde-741	469	1	among	among	ADP
ejde-741	469	2	them	they	PRON
ejde-741	469	3	,	,	PUNCT
ejde-741	469	4	we	we	PRON
ejde-741	469	5	would	would	AUX
ejde-741	469	6	like	like	VERB
ejde-741	469	7	to	to	PART
ejde-741	469	8	emphasize	emphasize	VERB
ejde-741	469	9	the	the	DET
ejde-741	469	10	following	following	ADJ
ejde-741	469	11	ones	one	NOUN
ejde-741	469	12	:	:	PUNCT
ejde-741	469	13	the	the	DET
ejde-741	469	14	controllability	controllability	NOUN
ejde-741	469	15	of	of	ADP
ejde-741	469	16	linear	linear	PROPN
ejde-741	469	17	problems	problem	NOUN
ejde-741	469	18	in	in	ADP
ejde-741	469	19	higher	higher	ADV
ejde-741	469	20	-	-	PUNCT
ejde-741	469	21	dimensional	dimensional	ADJ
ejde-741	469	22	spatial	spatial	ADJ
ejde-741	469	23	domains	domain	NOUN
ejde-741	469	24	,	,	PUNCT
ejde-741	469	25	using	use	VERB
ejde-741	469	26	the	the	DET
ejde-741	469	27	method	method	NOUN
ejde-741	469	28	of	of	ADP
ejde-741	469	29	moments	moment	NOUN
ejde-741	469	30	;	;	PUNCT
ejde-741	469	31	the	the	DET
ejde-741	469	32	boundary	boundary	ADJ
ejde-741	469	33	controllability	controllability	NOUN
ejde-741	469	34	obtained	obtain	VERB
ejde-741	469	35	as	as	ADP
ejde-741	469	36	the	the	DET
ejde-741	469	37	limit	limit	NOUN
ejde-741	469	38	of	of	ADP
ejde-741	469	39	internal	internal	ADJ
ejde-741	469	40	controllability	controllability	NOUN
ejde-741	469	41	,	,	PUNCT
ejde-741	469	42	in	in	ADP
ejde-741	469	43	nonlinear	nonlinear	ADJ
ejde-741	469	44	cases	case	NOUN
ejde-741	469	45	18	18	NUM
ejde-741	469	46	p.	p.	NOUN
ejde-741	469	47	p.	p.	NOUN
ejde-741	469	48	de	de	PROPN
ejde-741	469	49	carvalho	carvalho	PROPN
ejde-741	469	50	,	,	PUNCT
ejde-741	469	51	r.	r.	PROPN
ejde-741	469	52	demarque	demarque	PROPN
ejde-741	469	53	,	,	PUNCT
ejde-741	469	54	j.	j.	PROPN
ejde-741	469	55	límaco	límaco	PROPN
ejde-741	469	56	,	,	PUNCT
ejde-741	469	57	l.	l.	PROPN
ejde-741	469	58	viana	viana	PROPN
ejde-741	469	59	ejde-2025/15	ejde-2025/15	AUX
ejde-741	469	60	specifically	specifically	ADV
ejde-741	469	61	talking	talk	VERB
ejde-741	469	62	about	about	ADP
ejde-741	469	63	some	some	DET
ejde-741	469	64	numerical	numerical	ADJ
ejde-741	469	65	perspective	perspective	NOUN
ejde-741	469	66	regarding	regard	VERB
ejde-741	469	67	this	this	DET
ejde-741	469	68	current	current	ADJ
ejde-741	469	69	paper	paper	NOUN
ejde-741	469	70	,	,	PUNCT
ejde-741	469	71	we	we	PRON
ejde-741	469	72	include	include	VERB
ejde-741	469	73	some	some	DET
ejde-741	469	74	comments	comment	NOUN
ejde-741	469	75	pointing	point	VERB
ejde-741	469	76	out	out	ADP
ejde-741	469	77	possible	possible	ADJ
ejde-741	469	78	future	future	ADJ
ejde-741	469	79	works	work	NOUN
ejde-741	469	80	.	.	PUNCT
ejde-741	470	1	in	in	ADP
ejde-741	470	2	order	order	NOUN
ejde-741	470	3	to	to	PART
ejde-741	470	4	solve	solve	VERB
ejde-741	470	5	the	the	DET
ejde-741	470	6	proposed	propose	VERB
ejde-741	470	7	controllability	controllability	NOUN
ejde-741	470	8	problem	problem	NOUN
ejde-741	470	9	numerically	numerically	ADV
ejde-741	470	10	,	,	PUNCT
ejde-741	470	11	some	some	DET
ejde-741	470	12	different	different	ADJ
ejde-741	470	13	approaches	approach	NOUN
ejde-741	470	14	can	can	AUX
ejde-741	470	15	be	be	AUX
ejde-741	470	16	combined	combine	VERB
ejde-741	470	17	to	to	PART
ejde-741	470	18	perform	perform	VERB
ejde-741	470	19	an	an	DET
ejde-741	470	20	approximate	approximate	ADJ
ejde-741	470	21	and	and	CCONJ
ejde-741	470	22	reliable	reliable	ADJ
ejde-741	470	23	analysis	analysis	NOUN
ejde-741	470	24	of	of	ADP
ejde-741	470	25	the	the	DET
ejde-741	470	26	system	system	NOUN
ejde-741	470	27	.	.	PUNCT
ejde-741	471	1	a	a	DET
ejde-741	471	2	practical	practical	ADJ
ejde-741	471	3	and	and	CCONJ
ejde-741	471	4	well	well	ADV
ejde-741	471	5	-	-	PUNCT
ejde-741	471	6	established	establish	VERB
ejde-741	471	7	strategy	strategy	NOUN
ejde-741	471	8	in	in	ADP
ejde-741	471	9	the	the	DET
ejde-741	471	10	literature	literature	NOUN
ejde-741	471	11	involves	involve	VERB
ejde-741	471	12	the	the	DET
ejde-741	471	13	use	use	NOUN
ejde-741	471	14	of	of	ADP
ejde-741	471	15	the	the	DET
ejde-741	471	16	finite	finite	PROPN
ejde-741	471	17	element	element	NOUN
ejde-741	471	18	method	method	NOUN
ejde-741	471	19	(	(	PUNCT
ejde-741	471	20	fem	fem	PROPN
ejde-741	471	21	)	)	PUNCT
ejde-741	471	22	.	.	PUNCT
ejde-741	472	1	this	this	DET
ejde-741	472	2	methodology	methodology	NOUN
ejde-741	472	3	makes	make	VERB
ejde-741	472	4	possible	possible	ADJ
ejde-741	472	5	a	a	DET
ejde-741	472	6	complete	complete	ADJ
ejde-741	472	7	discretization	discretization	NOUN
ejde-741	472	8	of	of	ADP
ejde-741	472	9	the	the	DET
ejde-741	472	10	problem	problem	NOUN
ejde-741	472	11	into	into	ADP
ejde-741	472	12	a	a	DET
ejde-741	472	13	finite	finite	ADJ
ejde-741	472	14	-	-	ADJ
ejde-741	472	15	dimensional	dimensional	ADJ
ejde-741	472	16	space	space	NOUN
ejde-741	472	17	,	,	PUNCT
ejde-741	472	18	allowing	allow	VERB
ejde-741	472	19	numerical	numerical	ADJ
ejde-741	472	20	approximations	approximation	NOUN
ejde-741	472	21	through	through	ADP
ejde-741	472	22	well	well	ADV
ejde-741	472	23	-	-	PUNCT
ejde-741	472	24	defined	define	VERB
ejde-741	472	25	iterative	iterative	NOUN
ejde-741	472	26	processes	process	NOUN
ejde-741	472	27	.	.	PUNCT
ejde-741	473	1	so	so	ADV
ejde-741	473	2	that	that	SCONJ
ejde-741	473	3	,	,	PUNCT
ejde-741	473	4	a	a	DET
ejde-741	473	5	natural	natural	ADJ
ejde-741	473	6	future	future	ADJ
ejde-741	473	7	study	study	NOUN
ejde-741	473	8	could	could	AUX
ejde-741	473	9	be	be	AUX
ejde-741	473	10	the	the	DET
ejde-741	473	11	comparison	comparison	NOUN
ejde-741	473	12	between	between	ADP
ejde-741	473	13	two	two	NUM
ejde-741	473	14	approaches	approach	NOUN
ejde-741	473	15	:	:	PUNCT
ejde-741	473	16	the	the	DET
ejde-741	473	17	primal	primal	ADJ
ejde-741	473	18	method	method	NOUN
ejde-741	473	19	and	and	CCONJ
ejde-741	473	20	the	the	DET
ejde-741	473	21	dual	dual	ADJ
ejde-741	473	22	one	one	NUM
ejde-741	473	23	.	.	PUNCT
ejde-741	474	1	the	the	DET
ejde-741	474	2	primal	primal	ADJ
ejde-741	474	3	method	method	NOUN
ejde-741	474	4	is	be	AUX
ejde-741	474	5	more	more	ADV
ejde-741	474	6	straightforward	straightforward	ADJ
ejde-741	474	7	,	,	PUNCT
ejde-741	474	8	focusing	focus	VERB
ejde-741	474	9	directly	directly	ADV
ejde-741	474	10	on	on	ADP
ejde-741	474	11	the	the	DET
ejde-741	474	12	finite	finite	ADJ
ejde-741	474	13	element	element	NOUN
ejde-741	474	14	formulation	formulation	NOUN
ejde-741	474	15	by	by	ADP
ejde-741	474	16	discretizing	discretize	VERB
ejde-741	474	17	the	the	DET
ejde-741	474	18	spatial	spatial	ADJ
ejde-741	474	19	and	and	CCONJ
ejde-741	474	20	temporal	temporal	ADJ
ejde-741	474	21	domains	domain	NOUN
ejde-741	474	22	,	,	PUNCT
ejde-741	474	23	updating	update	VERB
ejde-741	474	24	the	the	DET
ejde-741	474	25	solution	solution	NOUN
ejde-741	474	26	iteratively	iteratively	ADV
ejde-741	474	27	in	in	ADP
ejde-741	474	28	an	an	DET
ejde-741	474	29	approximate	approximate	ADJ
ejde-741	474	30	solution	solution	NOUN
ejde-741	474	31	space	space	NOUN
ejde-741	474	32	vh	vh	PROPN
ejde-741	474	33	,	,	PUNCT
ejde-741	474	34	with	with	ADP
ejde-741	474	35	dim(vh	dim(vh	NOUN
ejde-741	474	36	)	)	PUNCT
ejde-741	474	37	<	<	X
ejde-741	474	38	∞.	∞.	PROPN
ejde-741	474	39	on	on	ADP
ejde-741	474	40	the	the	DET
ejde-741	474	41	other	other	ADJ
ejde-741	474	42	hand	hand	NOUN
ejde-741	474	43	,	,	PUNCT
ejde-741	474	44	the	the	DET
ejde-741	474	45	dual	dual	ADJ
ejde-741	474	46	method	method	NOUN
ejde-741	474	47	introduces	introduce	VERB
ejde-741	474	48	some	some	DET
ejde-741	474	49	pre	pre	ADJ
ejde-741	474	50	-	-	ADJ
ejde-741	474	51	programming	programming	ADJ
ejde-741	474	52	complexity	complexity	NOUN
ejde-741	474	53	,	,	PUNCT
ejde-741	474	54	since	since	SCONJ
ejde-741	474	55	it	it	PRON
ejde-741	474	56	is	be	AUX
ejde-741	474	57	incorporated	incorporate	VERB
ejde-741	474	58	dual	dual	ADJ
ejde-741	474	59	variables	variable	NOUN
ejde-741	474	60	into	into	ADP
ejde-741	474	61	the	the	DET
ejde-741	474	62	weak	weak	ADJ
ejde-741	474	63	formulation	formulation	NOUN
ejde-741	474	64	of	of	ADP
ejde-741	474	65	the	the	DET
ejde-741	474	66	problem	problem	NOUN
ejde-741	474	67	.	.	PUNCT
ejde-741	475	1	this	this	DET
ejde-741	475	2	process	process	NOUN
ejde-741	475	3	reformulates	reformulate	VERB
ejde-741	475	4	the	the	DET
ejde-741	475	5	original	original	ADJ
ejde-741	475	6	problem	problem	NOUN
ejde-741	475	7	as	as	ADP
ejde-741	475	8	a	a	DET
ejde-741	475	9	constrained	constrain	VERB
ejde-741	475	10	optimization	optimization	NOUN
ejde-741	475	11	problem	problem	NOUN
ejde-741	475	12	within	within	ADP
ejde-741	475	13	a	a	DET
ejde-741	475	14	variational	variational	ADJ
ejde-741	475	15	framework	framework	NOUN
ejde-741	475	16	,	,	PUNCT
ejde-741	475	17	where	where	SCONJ
ejde-741	475	18	iterative	iterative	NOUN
ejde-741	475	19	algorithms	algorithm	NOUN
ejde-741	475	20	are	be	AUX
ejde-741	475	21	employed	employ	VERB
ejde-741	475	22	to	to	PART
ejde-741	475	23	solve	solve	VERB
ejde-741	475	24	both	both	PRON
ejde-741	475	25	primal	primal	ADJ
ejde-741	475	26	and	and	CCONJ
ejde-741	475	27	dual	dual	ADJ
ejde-741	475	28	variables	variable	NOUN
ejde-741	475	29	,	,	PUNCT
ejde-741	475	30	simultaneously	simultaneously	ADV
ejde-741	475	31	.	.	PUNCT
ejde-741	476	1	the	the	DET
ejde-741	476	2	advantage	advantage	NOUN
ejde-741	476	3	and	and	CCONJ
ejde-741	476	4	disadvantage	disadvantage	NOUN
ejde-741	476	5	of	of	ADP
ejde-741	476	6	each	each	PRON
ejde-741	476	7	on	on	ADP
ejde-741	476	8	of	of	ADP
ejde-741	476	9	this	this	DET
ejde-741	476	10	methods	method	NOUN
ejde-741	476	11	are	be	AUX
ejde-741	476	12	properly	properly	ADV
ejde-741	476	13	discussed	discuss	VERB
ejde-741	476	14	in	in	ADP
ejde-741	476	15	[	[	X
ejde-741	476	16	27	27	NUM
ejde-741	476	17	]	]	PUNCT
ejde-741	476	18	.	.	PUNCT
ejde-741	477	1	at	at	ADP
ejde-741	477	2	this	this	DET
ejde-741	477	3	point	point	NOUN
ejde-741	477	4	,	,	PUNCT
ejde-741	477	5	we	we	PRON
ejde-741	477	6	should	should	AUX
ejde-741	477	7	say	say	VERB
ejde-741	477	8	that	that	SCONJ
ejde-741	477	9	some	some	DET
ejde-741	477	10	initial	initial	ADJ
ejde-741	477	11	numerical	numerical	ADJ
ejde-741	477	12	insights	insight	NOUN
ejde-741	477	13	into	into	ADP
ejde-741	477	14	the	the	DET
ejde-741	477	15	class	class	NOUN
ejde-741	477	16	of	of	ADP
ejde-741	477	17	problems	problem	NOUN
ejde-741	477	18	proposed	propose	VERB
ejde-741	477	19	here	here	ADV
ejde-741	477	20	can	can	AUX
ejde-741	477	21	be	be	AUX
ejde-741	477	22	found	find	VERB
ejde-741	477	23	in	in	ADP
ejde-741	477	24	[	[	X
ejde-741	477	25	19	19	NUM
ejde-741	477	26	]	]	PUNCT
ejde-741	477	27	,	,	PUNCT
ejde-741	477	28	where	where	SCONJ
ejde-741	477	29	iterative	iterative	NOUN
ejde-741	477	30	algorithms	algorithm	NOUN
ejde-741	477	31	,	,	PUNCT
ejde-741	477	32	adapted	adapt	VERB
ejde-741	477	33	for	for	ADP
ejde-741	477	34	nonlinear	nonlinear	ADJ
ejde-741	477	35	parabolic	parabolic	PROPN
ejde-741	477	36	problems	problem	NOUN
ejde-741	477	37	,	,	PUNCT
ejde-741	477	38	are	be	AUX
ejde-741	477	39	presented	present	VERB
ejde-741	477	40	.	.	PUNCT
ejde-741	478	1	besides	besides	SCONJ
ejde-741	478	2	,	,	PUNCT
ejde-741	478	3	in	in	ADP
ejde-741	478	4	[	[	X
ejde-741	478	5	19	19	NUM
ejde-741	478	6	]	]	X
ejde-741	478	7	,	,	PUNCT
ejde-741	478	8	an	an	DET
ejde-741	478	9	effective	effective	ADJ
ejde-741	478	10	approach	approach	NOUN
ejde-741	478	11	for	for	ADP
ejde-741	478	12	numerical	numerical	ADJ
ejde-741	478	13	iterations	iteration	NOUN
ejde-741	478	14	is	be	AUX
ejde-741	478	15	considered	consider	VERB
ejde-741	478	16	,	,	PUNCT
ejde-741	478	17	by	by	ADP
ejde-741	478	18	adjusting	adjust	VERB
ejde-741	478	19	quasi	quasi	PROPN
ejde-741	478	20	-	-	PROPN
ejde-741	478	21	newton	newton	PROPN
ejde-741	478	22	method	method	NOUN
ejde-741	478	23	for	for	ADP
ejde-741	478	24	null	null	ADJ
ejde-741	478	25	controllability	controllability	NOUN
ejde-741	478	26	problems	problem	NOUN
ejde-741	478	27	.	.	PUNCT
ejde-741	479	1	the	the	DET
ejde-741	479	2	whole	whole	ADJ
ejde-741	479	3	numerical	numerical	ADJ
ejde-741	479	4	analysis	analysis	NOUN
ejde-741	479	5	of	of	ADP
ejde-741	479	6	null	null	ADJ
ejde-741	479	7	-	-	PUNCT
ejde-741	479	8	controllability	controllability	NOUN
ejde-741	479	9	problems	problem	NOUN
ejde-741	479	10	is	be	AUX
ejde-741	479	11	completely	completely	ADV
ejde-741	479	12	associated	associate	VERB
ejde-741	479	13	with	with	ADP
ejde-741	479	14	well	well	ADV
ejde-741	479	15	-	-	PUNCT
ejde-741	479	16	chosen	choose	VERB
ejde-741	479	17	weight	weight	NOUN
ejde-741	479	18	functions	function	NOUN
ejde-741	479	19	,	,	PUNCT
ejde-741	479	20	such	such	DET
ejde-741	479	21	those	those	PRON
ejde-741	479	22	defined	define	VERB
ejde-741	479	23	in	in	ADP
ejde-741	479	24	section	section	NOUN
ejde-741	479	25	2	2	NUM
ejde-741	479	26	.	.	PUNCT
ejde-741	480	1	we	we	PRON
ejde-741	480	2	think	think	VERB
ejde-741	480	3	that	that	SCONJ
ejde-741	480	4	numerical	numerical	ADJ
ejde-741	480	5	simulations	simulation	NOUN
ejde-741	480	6	for	for	ADP
ejde-741	480	7	the	the	DET
ejde-741	480	8	null	null	NOUN
ejde-741	480	9	-	-	PUNCT
ejde-741	480	10	controllability	controllability	NOUN
ejde-741	480	11	of	of	ADP
ejde-741	480	12	degenerate	degenerate	ADJ
ejde-741	480	13	quasilinear	quasilinear	NOUN
ejde-741	480	14	equations	equation	NOUN
ejde-741	480	15	could	could	AUX
ejde-741	480	16	be	be	AUX
ejde-741	480	17	based	base	VERB
ejde-741	480	18	on	on	ADP
ejde-741	480	19	[	[	X
ejde-741	480	20	17	17	NUM
ejde-741	480	21	,	,	PUNCT
ejde-741	480	22	18	18	NUM
ejde-741	480	23	,	,	PUNCT
ejde-741	480	24	19	19	NUM
ejde-741	480	25	,	,	PUNCT
ejde-741	480	26	28	28	NUM
ejde-741	480	27	,	,	PUNCT
ejde-741	480	28	30	30	NUM
ejde-741	480	29	]	]	PUNCT
ejde-741	480	30	and	and	CCONJ
ejde-741	480	31	[	[	X
ejde-741	480	32	31	31	NUM
ejde-741	480	33	]	]	SYM
ejde-741	480	34	.	.	PUNCT
ejde-741	481	1	5	5	X
ejde-741	481	2	.	.	X
ejde-741	481	3	appendix	appendix	NOUN
ejde-741	481	4	:	:	PUNCT
ejde-741	481	5	essential	essential	ADJ
ejde-741	481	6	boundedness	boundedness	NOUN
ejde-741	481	7	of	of	ADP
ejde-741	481	8	au	au	ADV
ejde-741	481	9	this	this	DET
ejde-741	481	10	appendix	appendix	NOUN
ejde-741	481	11	shows	show	VERB
ejde-741	481	12	that	that	SCONJ
ejde-741	481	13	,	,	PUNCT
ejde-741	481	14	for	for	ADP
ejde-741	481	15	each	each	DET
ejde-741	481	16	u	u	PROPN
ejde-741	481	17	∈	∈	PROPN
ejde-741	481	18	h1	h1	VERB
ejde-741	481	19	a	a	PRON
ejde-741	481	20	,	,	PUNCT
ejde-741	481	21	we	we	PRON
ejde-741	481	22	have	have	VERB
ejde-741	481	23	au	au	ADP
ejde-741	481	24	∈	∈	PROPN
ejde-741	481	25	l∞(0	l∞(0	PRON
ejde-741	481	26	,	,	PUNCT
ejde-741	481	27	1	1	NUM
ejde-741	481	28	)	)	PUNCT
ejde-741	481	29	.	.	PUNCT
ejde-741	482	1	this	this	DET
ejde-741	482	2	fact	fact	NOUN
ejde-741	482	3	is	be	AUX
ejde-741	482	4	essential	essential	ADJ
ejde-741	482	5	in	in	ADP
ejde-741	482	6	to	to	PART
ejde-741	482	7	prove	prove	VERB
ejde-741	482	8	that	that	SCONJ
ejde-741	482	9	the	the	DET
ejde-741	482	10	mapping	mapping	NOUN
ejde-741	482	11	h	h	NOUN
ejde-741	482	12	:	:	PUNCT
ejde-741	482	13	e	e	X
ejde-741	482	14	→	→	SYM
ejde-741	482	15	f	f	PROPN
ejde-741	482	16	,	,	PUNCT
ejde-741	482	17	set	set	VERB
ejde-741	482	18	in	in	ADP
ejde-741	482	19	(	(	PUNCT
ejde-741	482	20	2.1	2.1	NUM
ejde-741	482	21	)	)	PUNCT
ejde-741	482	22	,	,	PUNCT
ejde-741	482	23	is	be	AUX
ejde-741	482	24	well	well	ADV
ejde-741	482	25	defined	define	VERB
ejde-741	482	26	and	and	CCONJ
ejde-741	482	27	continuously	continuously	ADV
ejde-741	482	28	differentiable	differentiable	ADJ
ejde-741	482	29	.	.	PUNCT
ejde-741	483	1	for	for	ADP
ejde-741	483	2	the	the	DET
ejde-741	483	3	whole	whole	ADJ
ejde-741	483	4	discussion	discussion	NOUN
ejde-741	483	5	,	,	PUNCT
ejde-741	483	6	let	let	VERB
ejde-741	483	7	us	we	PRON
ejde-741	483	8	consider	consider	VERB
ejde-741	483	9	k	k	PROPN
ejde-741	483	10	∈	∈	PROPN
ejde-741	484	1	[	[	X
ejde-741	484	2	0	0	NUM
ejde-741	484	3	,	,	PUNCT
ejde-741	484	4	2	2	NUM
ejde-741	484	5	)	)	PUNCT
ejde-741	484	6	mentioned	mention	VERB
ejde-741	484	7	in	in	ADP
ejde-741	484	8	assumption	assumption	NOUN
ejde-741	484	9	1.1	1.1	NUM
ejde-741	484	10	and	and	CCONJ
ejde-741	484	11	θ	θ	PROPN
ejde-741	484	12	∈	∈	NOUN
ejde-741	484	13	r	r	NOUN
ejde-741	484	14	given	give	VERB
ejde-741	484	15	in	in	ADP
ejde-741	484	16	(	(	PUNCT
ejde-741	484	17	1.3	1.3	NUM
ejde-741	484	18	)	)	PUNCT
ejde-741	484	19	.	.	PUNCT
ejde-741	485	1	proposition	proposition	NOUN
ejde-741	485	2	5.1	5.1	NUM
ejde-741	485	3	.	.	PUNCT
ejde-741	486	1	given	give	VERB
ejde-741	486	2	u	u	PRON
ejde-741	486	3	∈	∈	PROPN
ejde-741	486	4	h1	h1	NOUN
ejde-741	486	5	a	a	PRON
ejde-741	486	6	,	,	PUNCT
ejde-741	486	7	we	we	PRON
ejde-741	486	8	have	have	VERB
ejde-741	486	9	au	au	ADP
ejde-741	486	10	∈	∈	PROPN
ejde-741	486	11	l∞(0	l∞(0	PRON
ejde-741	486	12	,	,	PUNCT
ejde-741	486	13	1	1	NUM
ejde-741	486	14	)	)	PUNCT
ejde-741	486	15	and	and	CCONJ
ejde-741	486	16	ca	can	AUX
ejde-741	486	17	>	>	X
ejde-741	486	18	0	0	NUM
ejde-741	486	19	,	,	PUNCT
ejde-741	486	20	only	only	ADV
ejde-741	486	21	depending	depend	VERB
ejde-741	486	22	on	on	ADP
ejde-741	486	23	the	the	DET
ejde-741	486	24	function	function	NOUN
ejde-741	486	25	a	a	PRON
ejde-741	486	26	,	,	PUNCT
ejde-741	486	27	such	such	ADJ
ejde-741	486	28	that	that	DET
ejde-741	486	29	∥au∥l∞(0,1	∥au∥l∞(0,1	NOUN
ejde-741	486	30	)	)	PUNCT
ejde-741	486	31	≤	≤	PUNCT
ejde-741	486	32	ca∥u∥h1	ca∥u∥h1	NOUN
ejde-741	486	33	a	a	X
ejde-741	486	34	,	,	PUNCT
ejde-741	486	35	provided	provide	VERB
ejde-741	486	36	that	that	SCONJ
ejde-741	486	37	one	one	NUM
ejde-741	486	38	of	of	ADP
ejde-741	486	39	the	the	DET
ejde-741	486	40	following	follow	VERB
ejde-741	486	41	conditions	condition	NOUN
ejde-741	486	42	holds	hold	VERB
ejde-741	486	43	:	:	PUNCT
ejde-741	486	44	(	(	PUNCT
ejde-741	486	45	a	a	X
ejde-741	486	46	)	)	PUNCT
ejde-741	486	47	k	k	PROPN
ejde-741	486	48	̸=	̸=	PROPN
ejde-741	486	49	1	1	NUM
ejde-741	486	50	;	;	PUNCT
ejde-741	486	51	(	(	PUNCT
ejde-741	486	52	b	b	X
ejde-741	486	53	)	)	PUNCT
ejde-741	486	54	k	k	NOUN
ejde-741	487	1	=	=	SYM
ejde-741	487	2	1	1	NUM
ejde-741	487	3	and	and	CCONJ
ejde-741	487	4	θ	θ	PROPN
ejde-741	487	5	≥	≥	NUM
ejde-741	487	6	1/2	1/2	NUM
ejde-741	487	7	.	.	PUNCT
ejde-741	488	1	the	the	DET
ejde-741	488	2	proof	proof	NOUN
ejde-741	488	3	of	of	ADP
ejde-741	488	4	this	this	DET
ejde-741	488	5	proposition	proposition	NOUN
ejde-741	488	6	will	will	AUX
ejde-741	488	7	be	be	AUX
ejde-741	488	8	a	a	DET
ejde-741	488	9	consequence	consequence	NOUN
ejde-741	488	10	of	of	ADP
ejde-741	488	11	the	the	DET
ejde-741	488	12	four	four	NUM
ejde-741	488	13	next	next	ADJ
ejde-741	488	14	lemmas	lemmas	PROPN
ejde-741	488	15	.	.	PUNCT
ejde-741	489	1	lemma	lemma	PROPN
ejde-741	489	2	5.2	5.2	NUM
ejde-741	489	3	.	.	PUNCT
ejde-741	490	1	the	the	DET
ejde-741	490	2	continuous	continuous	ADJ
ejde-741	490	3	embedding	embed	VERB
ejde-741	490	4	h1	h1	PROPN
ejde-741	490	5	a	a	DET
ejde-741	490	6	↪	↪	PROPN
ejde-741	490	7	→	→	SYM
ejde-741	490	8	l∞(0	l∞(0	ADJ
ejde-741	490	9	,	,	PUNCT
ejde-741	490	10	1	1	X
ejde-741	490	11	)	)	PUNCT
ejde-741	490	12	holds	hold	VERB
ejde-741	490	13	for	for	ADP
ejde-741	490	14	the	the	DET
ejde-741	490	15	(	(	PUNCT
ejde-741	490	16	wdc	wdc	PROPN
ejde-741	490	17	)	)	PUNCT
ejde-741	490	18	.	.	PUNCT
ejde-741	491	1	in	in	ADP
ejde-741	491	2	particular	particular	ADJ
ejde-741	491	3	,	,	PUNCT
ejde-741	491	4	au	au	PROPN
ejde-741	491	5	∈	∈	PROPN
ejde-741	491	6	l∞(0	l∞(0	PRON
ejde-741	491	7	,	,	PUNCT
ejde-741	491	8	1	1	NUM
ejde-741	491	9	)	)	PUNCT
ejde-741	491	10	,	,	PUNCT
ejde-741	491	11	for	for	SCONJ
ejde-741	491	12	any	any	DET
ejde-741	491	13	u	u	PROPN
ejde-741	491	14	∈	∈	PROPN
ejde-741	491	15	h1	h1	VERB
ejde-741	491	16	a	a	PRON
ejde-741	491	17	.	.	PUNCT
ejde-741	492	1	proof	proof	NOUN
ejde-741	492	2	.	.	PUNCT
ejde-741	493	1	in	in	ADP
ejde-741	493	2	fact	fact	NOUN
ejde-741	493	3	,	,	PUNCT
ejde-741	493	4	given	give	VERB
ejde-741	493	5	u	u	NOUN
ejde-741	493	6	=	=	SYM
ejde-741	493	7	u(x	u(x	PROPN
ejde-741	493	8	)	)	PUNCT
ejde-741	493	9	∈	∈	PROPN
ejde-741	493	10	h1	h1	VERB
ejde-741	493	11	a	a	PRON
ejde-741	493	12	,	,	PUNCT
ejde-741	493	13	we	we	PRON
ejde-741	493	14	can	can	AUX
ejde-741	493	15	take	take	VERB
ejde-741	493	16	|u(x)|	|u(x)|	PROPN
ejde-741	493	17	≤	≤	X
ejde-741	493	18	∣∣	∣∣	PUNCT
ejde-741	493	19	∫	∫	PROPN
ejde-741	493	20	1	1	NUM
ejde-741	493	21	x	x	SYM
ejde-741	493	22	ux	ux	PROPN
ejde-741	493	23	∣∣	∣∣	X
ejde-741	493	24	≤	≤	PROPN
ejde-741	493	25	(	(	PUNCT
ejde-741	493	26	∫	∫	PROPN
ejde-741	493	27	1	1	NUM
ejde-741	493	28	x	x	SYM
ejde-741	493	29	1	1	NUM
ejde-741	493	30	a	a	PRON
ejde-741	493	31	)	)	PUNCT
ejde-741	493	32	1/2(∫	1/2(∫	NUM
ejde-741	493	33	1	1	NUM
ejde-741	493	34	x	x	NOUN
ejde-741	493	35	au2x	au2x	ADJ
ejde-741	493	36	)	)	PUNCT
ejde-741	493	37	1/2	1/2	NUM
ejde-741	493	38	≤	≤	NOUN
ejde-741	493	39	∥1	∥1	PRON
ejde-741	493	40	a	a	DET
ejde-741	493	41	∥2l1∥u∥h1	∥2l1∥u∥h1	NOUN
ejde-741	493	42	a	a	PRON
ejde-741	493	43	,	,	PUNCT
ejde-741	493	44	ejde-2025/15	ejde-2025/15	PROPN
ejde-741	493	45	null	null	ADJ
ejde-741	493	46	-	-	PUNCT
ejde-741	493	47	controllability	controllability	NOUN
ejde-741	493	48	degenerate	degenerate	ADJ
ejde-741	493	49	quasilinear	quasilinear	NOUN
ejde-741	493	50	equations	equation	NOUN
ejde-741	493	51	19	19	NUM
ejde-741	493	52	for	for	ADP
ejde-741	493	53	each	each	DET
ejde-741	493	54	x	x	SYM
ejde-741	493	55	∈	∈	PROPN
ejde-741	493	56	(	(	PUNCT
ejde-741	493	57	0	0	NUM
ejde-741	493	58	,	,	PUNCT
ejde-741	493	59	1	1	NUM
ejde-741	493	60	]	]	PUNCT
ejde-741	493	61	.	.	PUNCT
ejde-741	494	1	hence	hence	ADV
ejde-741	494	2	,	,	PUNCT
ejde-741	494	3	there	there	PRON
ejde-741	494	4	exists	exist	VERB
ejde-741	494	5	ca	can	AUX
ejde-741	494	6	>	>	X
ejde-741	494	7	0	0	NUM
ejde-741	494	8	,	,	PUNCT
ejde-741	494	9	only	only	ADV
ejde-741	494	10	depending	depend	VERB
ejde-741	494	11	on	on	ADP
ejde-741	494	12	the	the	DET
ejde-741	494	13	function	function	NOUN
ejde-741	494	14	a	a	PRON
ejde-741	494	15	,	,	PUNCT
ejde-741	494	16	such	such	ADJ
ejde-741	494	17	that	that	DET
ejde-741	494	18	∥u∥l∞	∥u∥l∞	NOUN
ejde-741	494	19	≤	≤	ADJ
ejde-741	494	20	ca∥u∥h1	ca∥u∥h1	NOUN
ejde-741	494	21	a	a	PRON
ejde-741	494	22	.	.	PUNCT
ejde-741	495	1	□	□	PUNCT
ejde-741	495	2	lemma	lemma	PROPN
ejde-741	495	3	5.3	5.3	NUM
ejde-741	495	4	.	.	PUNCT
ejde-741	496	1	if	if	SCONJ
ejde-741	496	2	a	a	DET
ejde-741	496	3	∈w	∈w	PROPN
ejde-741	496	4	1,∞(0	1,∞(0	NUM
ejde-741	496	5	,	,	PUNCT
ejde-741	496	6	1	1	NUM
ejde-741	496	7	)	)	PUNCT
ejde-741	496	8	,	,	PUNCT
ejde-741	496	9	then	then	ADV
ejde-741	496	10	au	au	ADP
ejde-741	496	11	∈	∈	PROPN
ejde-741	496	12	l∞(0	l∞(0	PRON
ejde-741	496	13	,	,	PUNCT
ejde-741	496	14	1	1	NUM
ejde-741	496	15	)	)	PUNCT
ejde-741	496	16	,	,	PUNCT
ejde-741	496	17	for	for	SCONJ
ejde-741	496	18	each	each	DET
ejde-741	496	19	u	u	PROPN
ejde-741	496	20	∈	∈	PROPN
ejde-741	496	21	h1	h1	VERB
ejde-741	496	22	a	a	PRON
ejde-741	496	23	.	.	PUNCT
ejde-741	497	1	proof	proof	NOUN
ejde-741	497	2	.	.	PUNCT
ejde-741	498	1	indeed	indeed	ADV
ejde-741	498	2	,	,	PUNCT
ejde-741	498	3	given	give	VERB
ejde-741	498	4	y	y	PROPN
ejde-741	498	5	∈	∈	PROPN
ejde-741	498	6	(	(	PUNCT
ejde-741	498	7	0	0	NUM
ejde-741	498	8	,	,	PUNCT
ejde-741	498	9	1	1	NUM
ejde-741	498	10	]	]	PUNCT
ejde-741	498	11	,	,	PUNCT
ejde-741	498	12	since	since	SCONJ
ejde-741	498	13	a	a	DET
ejde-741	498	14	∈	∈	PROPN
ejde-741	498	15	c1([y	c1([y	NOUN
ejde-741	498	16	,	,	PUNCT
ejde-741	498	17	1	1	NUM
ejde-741	498	18	]	]	PUNCT
ejde-741	498	19	)	)	PUNCT
ejde-741	498	20	and	and	CCONJ
ejde-741	498	21	u	u	NOUN
ejde-741	498	22	is	be	AUX
ejde-741	498	23	absolutely	absolutely	ADV
ejde-741	498	24	continuous	continuous	ADJ
ejde-741	498	25	in	in	ADP
ejde-741	498	26	[	[	X
ejde-741	498	27	y	y	PROPN
ejde-741	498	28	,	,	PUNCT
ejde-741	498	29	1	1	NUM
ejde-741	498	30	]	]	PUNCT
ejde-741	498	31	,	,	PUNCT
ejde-741	498	32	we	we	PRON
ejde-741	498	33	have	have	VERB
ejde-741	498	34	|au(y)|	|au(y)|	ADV
ejde-741	498	35	≤	≤	NUM
ejde-741	498	36	∫	∫	PROPN
ejde-741	498	37	1	1	NUM
ejde-741	498	38	y	y	PROPN
ejde-741	498	39	|a′u|	|a′u|	VERB
ejde-741	499	1	dx+	dx+	ADJ
ejde-741	499	2	∫	∫	PROPN
ejde-741	500	1	1	1	NUM
ejde-741	500	2	y	y	PROPN
ejde-741	500	3	a|ux|	a|ux|	PROPN
ejde-741	500	4	dx	dx	PROPN
ejde-741	500	5	≤	≤	NUM
ejde-741	500	6	∫	∫	PROPN
ejde-741	500	7	1	1	NUM
ejde-741	500	8	0	0	NUM
ejde-741	500	9	|a′u|	|a′u|	NOUN
ejde-741	500	10	dx+	dx+	ADJ
ejde-741	500	11	∫	∫	PROPN
ejde-741	500	12	1	1	NUM
ejde-741	500	13	0	0	NUM
ejde-741	501	1	a|ux|	a|ux|	PUNCT
ejde-741	501	2	dx	dx	PROPN
ejde-741	501	3	≤	≤	PROPN
ejde-741	501	4	∥a′∥l∞(0	∥a′∥l∞(0	PROPN
ejde-741	501	5	,	,	PUNCT
ejde-741	501	6	1)∥u∥l2(0,1	1)∥u∥l2(0,1	NUM
ejde-741	501	7	)	)	PUNCT
ejde-741	502	1	+	+	NUM
ejde-741	502	2	∫	∫	PROPN
ejde-741	502	3	1	1	NUM
ejde-741	502	4	0	0	NUM
ejde-741	502	5	√	√	PROPN
ejde-741	502	6	a	a	DET
ejde-741	502	7	√	√	PROPN
ejde-741	502	8	a|ux|	a|ux|	PROPN
ejde-741	502	9	dx	dx	PROPN
ejde-741	502	10	≤	≤	NOUN
ejde-741	502	11	∥a′∥l∞(0,1)∥u∥l2(0,1	∥a′∥l∞(0,1)∥u∥l2(0,1	PROPN
ejde-741	502	12	)	)	PUNCT
ejde-741	503	1	+	+	CCONJ
ejde-741	503	2	∥a∥l∞(0,1)∥	∥a∥l∞(0,1)∥	VERB
ejde-741	503	3	√	√	NUM
ejde-741	503	4	aux∥l2(0,1	aux∥l2(0,1	NOUN
ejde-741	503	5	)	)	PUNCT
ejde-741	503	6	≤	≤	X
ejde-741	503	7	∥a∥w	∥a∥w	X
ejde-741	503	8	1∞(0,1)∥u∥h1	1∞(0,1)∥u∥h1	PROPN
ejde-741	503	9	a	a	X
ejde-741	503	10	.	.	PUNCT
ejde-741	504	1	since	since	ADV
ejde-741	504	2	,	,	PUNCT
ejde-741	504	3	y	y	PROPN
ejde-741	504	4	∈	∈	PROPN
ejde-741	504	5	(	(	PUNCT
ejde-741	504	6	0	0	NUM
ejde-741	504	7	,	,	PUNCT
ejde-741	504	8	1	1	NUM
ejde-741	504	9	]	]	PUNCT
ejde-741	504	10	is	be	AUX
ejde-741	504	11	arbitrary	arbitrary	ADJ
ejde-741	504	12	,	,	PUNCT
ejde-741	504	13	the	the	DET
ejde-741	504	14	desired	desire	VERB
ejde-741	504	15	result	result	NOUN
ejde-741	504	16	follows	follow	VERB
ejde-741	504	17	.	.	PUNCT
ejde-741	505	1	□	□	PUNCT
ejde-741	505	2	lemma	lemma	PROPN
ejde-741	505	3	5.4	5.4	NUM
ejde-741	505	4	.	.	PUNCT
ejde-741	506	1	if	if	SCONJ
ejde-741	506	2	k	k	PROPN
ejde-741	506	3	>	>	X
ejde-741	506	4	1	1	NUM
ejde-741	506	5	,	,	PUNCT
ejde-741	506	6	then	then	ADV
ejde-741	506	7	a	a	DET
ejde-741	506	8	∈w	∈w	PROPN
ejde-741	506	9	1,∞(0	1,∞(0	NUM
ejde-741	506	10	,	,	PUNCT
ejde-741	506	11	1	1	NUM
ejde-741	506	12	)	)	PUNCT
ejde-741	506	13	.	.	PUNCT
ejde-741	507	1	in	in	ADP
ejde-741	507	2	particular	particular	ADJ
ejde-741	507	3	,	,	PUNCT
ejde-741	507	4	au	au	PROPN
ejde-741	507	5	∈	∈	PROPN
ejde-741	507	6	l∞(0	l∞(0	PRON
ejde-741	507	7	,	,	PUNCT
ejde-741	507	8	1	1	NUM
ejde-741	507	9	)	)	PUNCT
ejde-741	507	10	.	.	PUNCT
ejde-741	508	1	proof	proof	NOUN
ejde-741	508	2	.	.	PUNCT
ejde-741	509	1	we	we	PRON
ejde-741	509	2	only	only	ADV
ejde-741	509	3	need	need	VERB
ejde-741	509	4	to	to	PART
ejde-741	509	5	prove	prove	VERB
ejde-741	509	6	that	that	SCONJ
ejde-741	509	7	a′	a′	PROPN
ejde-741	509	8	∈	∈	PROPN
ejde-741	509	9	l∞(0	l∞(0	PRON
ejde-741	509	10	,	,	PUNCT
ejde-741	509	11	1	1	NUM
ejde-741	509	12	)	)	PUNCT
ejde-741	509	13	.	.	PUNCT
ejde-741	510	1	from	from	ADP
ejde-741	510	2	(	(	PUNCT
ejde-741	510	3	1.3	1.3	NUM
ejde-741	510	4	)	)	PUNCT
ejde-741	510	5	,	,	PUNCT
ejde-741	510	6	there	there	PRON
ejde-741	510	7	exists	exist	VERB
ejde-741	510	8	ε	ε	PROPN
ejde-741	510	9	>	>	X
ejde-741	510	10	0	0	NUM
ejde-741	510	11	such	such	ADJ
ejde-741	510	12	that	that	SCONJ
ejde-741	510	13	θa	θa	NUM
ejde-741	510	14	≤	≤	NUM
ejde-741	510	15	xa′	xa′	NOUN
ejde-741	510	16	,	,	PUNCT
ejde-741	510	17	∀x	∀x	X
ejde-741	510	18	∈	∈	PROPN
ejde-741	510	19	(	(	PUNCT
ejde-741	510	20	0	0	NUM
ejde-741	510	21	,	,	PUNCT
ejde-741	510	22	ε	ε	PROPN
ejde-741	510	23	]	]	PUNCT
ejde-741	510	24	.	.	PUNCT
ejde-741	511	1	in	in	ADP
ejde-741	511	2	particular	particular	ADJ
ejde-741	511	3	a′	a′	PROPN
ejde-741	511	4	>	>	X
ejde-741	511	5	0	0	PUNCT
ejde-741	512	1	and	and	CCONJ
ejde-741	512	2	,	,	PUNCT
ejde-741	512	3	since	since	SCONJ
ejde-741	512	4	θ	θ	PROPN
ejde-741	512	5	>	>	X
ejde-741	512	6	1	1	NUM
ejde-741	512	7	,	,	PUNCT
ejde-741	512	8	the	the	DET
ejde-741	512	9	mapping	mapping	NOUN
ejde-741	512	10	x	x	SYM
ejde-741	512	11	7→	7→	NUM
ejde-741	512	12	a	a	PRON
ejde-741	512	13	x	x	AUX
ejde-741	512	14	is	be	AUX
ejde-741	512	15	increasing	increase	VERB
ejde-741	512	16	in	in	ADP
ejde-741	512	17	(	(	PUNCT
ejde-741	512	18	0	0	NUM
ejde-741	512	19	,	,	PUNCT
ejde-741	512	20	ε	ε	PROPN
ejde-741	512	21	]	]	PUNCT
ejde-741	512	22	.	.	PUNCT
ejde-741	513	1	so	so	ADV
ejde-741	513	2	that	that	SCONJ
ejde-741	513	3	,	,	PUNCT
ejde-741	513	4	using	use	VERB
ejde-741	513	5	(	(	PUNCT
ejde-741	513	6	1.2	1.2	NUM
ejde-741	513	7	)	)	PUNCT
ejde-741	513	8	,	,	PUNCT
ejde-741	513	9	we	we	PRON
ejde-741	513	10	have	have	VERB
ejde-741	513	11	0	0	NUM
ejde-741	513	12	≤	≤	NUM
ejde-741	513	13	a′(x	a′(x	NOUN
ejde-741	513	14	)	)	PUNCT
ejde-741	513	15	≤	≤	NOUN
ejde-741	513	16	ka(x	ka(x	NOUN
ejde-741	513	17	)	)	PUNCT
ejde-741	513	18	x	x	SYM
ejde-741	513	19	≤	≤	NUM
ejde-741	513	20	ka(ε	ka(ε	PUNCT
ejde-741	513	21	)	)	PUNCT
ejde-741	513	22	ε	ε	PROPN
ejde-741	513	23	,	,	PUNCT
ejde-741	513	24	∀x	∀x	X
ejde-741	513	25	∈	∈	PROPN
ejde-741	513	26	(	(	PUNCT
ejde-741	513	27	0	0	NUM
ejde-741	513	28	,	,	PUNCT
ejde-741	513	29	ε	ε	PROPN
ejde-741	513	30	]	]	PUNCT
ejde-741	513	31	.	.	PUNCT
ejde-741	514	1	on	on	ADP
ejde-741	514	2	the	the	DET
ejde-741	514	3	other	other	ADJ
ejde-741	514	4	hand	hand	NOUN
ejde-741	514	5	,	,	PUNCT
ejde-741	514	6	a′	a′	PROPN
ejde-741	514	7	∈	∈	PROPN
ejde-741	514	8	c0([ε	c0([ε	NOUN
ejde-741	514	9	,	,	PUNCT
ejde-741	514	10	1	1	NUM
ejde-741	514	11	]	]	NUM
ejde-741	514	12	)	)	PUNCT
ejde-741	514	13	,	,	PUNCT
ejde-741	514	14	following	follow	VERB
ejde-741	514	15	that	that	SCONJ
ejde-741	514	16	a′	a′	PROPN
ejde-741	514	17	∈	∈	PROPN
ejde-741	514	18	l∞(0	l∞(0	PRON
ejde-741	514	19	,	,	PUNCT
ejde-741	514	20	1	1	NUM
ejde-741	514	21	)	)	PUNCT
ejde-741	514	22	and	and	CCONJ
ejde-741	514	23	∥a′∥l∞(0,1	∥a′∥l∞(0,1	NOUN
ejde-741	514	24	)	)	PUNCT
ejde-741	514	25	≤	≤	NUM
ejde-741	514	26	max	max	PROPN
ejde-741	514	27	{	{	PUNCT
ejde-741	514	28	ka(ε	ka(ε	PROPN
ejde-741	514	29	)	)	PUNCT
ejde-741	514	30	ε	ε	PROPN
ejde-741	514	31	,	,	PUNCT
ejde-741	514	32	max	max	PROPN
ejde-741	514	33	x∈[ε,1	x∈[ε,1	PROPN
ejde-741	514	34	]	]	X
ejde-741	514	35	|a′(x)|	|a′(x)|	NUM
ejde-741	514	36	}	}	PUNCT
ejde-741	514	37	.	.	PUNCT
ejde-741	515	1	□	□	PUNCT
ejde-741	515	2	lemma	lemma	PROPN
ejde-741	515	3	5.5	5.5	NUM
ejde-741	515	4	.	.	PUNCT
ejde-741	516	1	if	if	SCONJ
ejde-741	516	2	k	k	PROPN
ejde-741	516	3	=	=	SYM
ejde-741	516	4	1	1	NUM
ejde-741	516	5	and	and	CCONJ
ejde-741	516	6	θ	θ	PROPN
ejde-741	516	7	≥	≥	NUM
ejde-741	516	8	1/2	1/2	NUM
ejde-741	516	9	,	,	PUNCT
ejde-741	516	10	then	then	ADV
ejde-741	516	11	au	au	ADP
ejde-741	516	12	∈	∈	PROPN
ejde-741	516	13	l∞(0	l∞(0	PRON
ejde-741	516	14	,	,	PUNCT
ejde-741	516	15	1	1	NUM
ejde-741	516	16	)	)	PUNCT
ejde-741	516	17	,	,	PUNCT
ejde-741	516	18	for	for	SCONJ
ejde-741	516	19	any	any	DET
ejde-741	516	20	u	u	PROPN
ejde-741	516	21	∈	∈	PROPN
ejde-741	516	22	h1	h1	VERB
ejde-741	516	23	a	a	PRON
ejde-741	516	24	.	.	PUNCT
ejde-741	517	1	proof	proof	NOUN
ejde-741	517	2	.	.	PUNCT
ejde-741	518	1	from	from	ADP
ejde-741	518	2	(	(	PUNCT
ejde-741	518	3	1.3	1.3	NUM
ejde-741	518	4	)	)	PUNCT
ejde-741	518	5	,	,	PUNCT
ejde-741	518	6	there	there	PRON
ejde-741	518	7	exists	exist	VERB
ejde-741	518	8	ε	ε	PROPN
ejde-741	518	9	>	>	X
ejde-741	518	10	0	0	NUM
ejde-741	518	11	such	such	ADJ
ejde-741	518	12	that	that	SCONJ
ejde-741	518	13	θa	θa	NUM
ejde-741	518	14	≤	≤	NUM
ejde-741	518	15	xa′	xa′	NOUN
ejde-741	519	1	for	for	ADP
ejde-741	519	2	all	all	DET
ejde-741	519	3	x	x	SYM
ejde-741	519	4	∈	∈	PROPN
ejde-741	519	5	(	(	PUNCT
ejde-741	519	6	0	0	NUM
ejde-741	519	7	,	,	PUNCT
ejde-741	519	8	ε	ε	PROPN
ejde-741	519	9	]	]	PUNCT
ejde-741	519	10	.	.	PUNCT
ejde-741	520	1	this	this	PRON
ejde-741	520	2	implies	imply	VERB
ejde-741	520	3	that	that	SCONJ
ejde-741	520	4	a′	a′	PROPN
ejde-741	520	5	>	>	X
ejde-741	520	6	0	0	PUNCT
ejde-741	520	7	and	and	CCONJ
ejde-741	520	8	x	x	SYM
ejde-741	520	9	7→	7→	NOUN
ejde-741	520	10	a	a	DET
ejde-741	520	11	xθ	xθ	PROPN
ejde-741	520	12	is	be	AUX
ejde-741	520	13	nondecreasing	nondecrease	VERB
ejde-741	520	14	in	in	ADP
ejde-741	520	15	(	(	PUNCT
ejde-741	520	16	0	0	NUM
ejde-741	520	17	,	,	PUNCT
ejde-741	520	18	ε	ε	PROPN
ejde-741	520	19	]	]	PUNCT
ejde-741	520	20	.	.	PUNCT
ejde-741	521	1	(	(	PUNCT
ejde-741	521	2	5.1	5.1	NUM
ejde-741	521	3	)	)	PUNCT
ejde-741	521	4	since	since	SCONJ
ejde-741	521	5	a	a	DET
ejde-741	521	6	∈	∈	PROPN
ejde-741	521	7	c0([0	c0([0	NOUN
ejde-741	521	8	,	,	PUNCT
ejde-741	521	9	1	1	NUM
ejde-741	521	10	]	]	PUNCT
ejde-741	521	11	)	)	PUNCT
ejde-741	521	12	and	and	CCONJ
ejde-741	521	13	u	u	PROPN
ejde-741	521	14	∈	∈	PROPN
ejde-741	521	15	h1	h1	VERB
ejde-741	521	16	a	a	DET
ejde-741	521	17	↪	↪	PROPN
ejde-741	521	18	→	→	SYM
ejde-741	521	19	h1(ε	h1(ε	ADJ
ejde-741	521	20	,	,	PUNCT
ejde-741	521	21	1	1	NUM
ejde-741	521	22	)	)	PUNCT
ejde-741	521	23	↪	↪	PROPN
ejde-741	521	24	→	→	SYM
ejde-741	521	25	c0([ε	c0([ε	ADJ
ejde-741	521	26	,	,	PUNCT
ejde-741	521	27	1	1	NUM
ejde-741	521	28	]	]	NUM
ejde-741	521	29	)	)	PUNCT
ejde-741	521	30	,	,	PUNCT
ejde-741	521	31	we	we	PRON
ejde-741	521	32	have	have	VERB
ejde-741	521	33	that	that	DET
ejde-741	521	34	au	au	PROPN
ejde-741	521	35	∈	∈	PROPN
ejde-741	521	36	l∞(ε	l∞(ε	NOUN
ejde-741	521	37	,	,	PUNCT
ejde-741	521	38	1	1	NUM
ejde-741	521	39	)	)	PUNCT
ejde-741	521	40	and	and	CCONJ
ejde-741	521	41	|au(y)|	|au(y)|	PROPN
ejde-741	521	42	≤	≤	X
ejde-741	521	43	∥a∥l∞(0,1)∥u∥h1	∥a∥l∞(0,1)∥u∥h1	PUNCT
ejde-741	521	44	a	a	PRON
ejde-741	521	45	,	,	PUNCT
ejde-741	521	46	for	for	ADP
ejde-741	521	47	all	all	DET
ejde-741	521	48	y	y	PROPN
ejde-741	521	49	∈	∈	PROPN
ejde-741	521	50	[	[	X
ejde-741	521	51	ε	ε	PROPN
ejde-741	521	52	,	,	PUNCT
ejde-741	521	53	1	1	NUM
ejde-741	521	54	]	]	PUNCT
ejde-741	521	55	.	.	PUNCT
ejde-741	522	1	we	we	PRON
ejde-741	522	2	just	just	ADV
ejde-741	522	3	need	need	VERB
ejde-741	522	4	to	to	PART
ejde-741	522	5	prove	prove	VERB
ejde-741	522	6	au	au	ADP
ejde-741	522	7	∈	∈	PROPN
ejde-741	522	8	l∞(0	l∞(0	PRON
ejde-741	522	9	,	,	PUNCT
ejde-741	522	10	ε	ε	PROPN
ejde-741	522	11	)	)	PUNCT
ejde-741	522	12	.	.	PUNCT
ejde-741	523	1	indeed	indeed	ADV
ejde-741	523	2	,	,	PUNCT
ejde-741	523	3	given	give	VERB
ejde-741	523	4	y	y	PROPN
ejde-741	523	5	∈	∈	PROPN
ejde-741	523	6	(	(	PUNCT
ejde-741	523	7	0	0	NUM
ejde-741	523	8	,	,	PUNCT
ejde-741	523	9	ε	ε	PROPN
ejde-741	523	10	]	]	PUNCT
ejde-741	523	11	,	,	PUNCT
ejde-741	523	12	we	we	PRON
ejde-741	523	13	have	have	VERB
ejde-741	523	14	a2u2(y	a2u2(y	NUM
ejde-741	523	15	)	)	PUNCT
ejde-741	524	1	=	=	SYM
ejde-741	524	2	a2u2(ε)−	a2u2(ε)−	NOUN
ejde-741	524	3	∫	∫	PROPN
ejde-741	524	4	ε	ε	PROPN
ejde-741	524	5	y	y	PROPN
ejde-741	524	6	(	(	PUNCT
ejde-741	524	7	a2u2(x))x	a2u2(x))x	PROPN
ejde-741	524	8	dx	dx	PROPN
ejde-741	524	9	,	,	PUNCT
ejde-741	524	10	whence	whence	NOUN
ejde-741	524	11	,	,	PUNCT
ejde-741	524	12	|au(y)|2	|au(y)|2	PROPN
ejde-741	524	13	≤	≤	NUM
ejde-741	524	14	∥a∥2l∞(0,1)∥u∥	∥a∥2l∞(0,1)∥u∥	NOUN
ejde-741	524	15	2	2	NUM
ejde-741	524	16	h1	h1	VERB
ejde-741	524	17	a	a	DET
ejde-741	524	18	+	+	NUM
ejde-741	524	19	2	2	NUM
ejde-741	524	20	∫	∫	NOUN
ejde-741	524	21	ε	ε	PROPN
ejde-741	524	22	y	y	PROPN
ejde-741	524	23	a2|u||ux|	a2|u||ux|	PROPN
ejde-741	524	24	dx+	dx+	NOUN
ejde-741	524	25	2	2	NUM
ejde-741	524	26	∫	∫	NOUN
ejde-741	524	27	ε	ε	PROPN
ejde-741	524	28	y	y	PROPN
ejde-741	524	29	aa′|u|2	aa′|u|2	PROPN
ejde-741	524	30	dx	dx	PROPN
ejde-741	524	31	.	.	PROPN
ejde-741	524	32	20	20	NUM
ejde-741	525	1	p.	p.	NOUN
ejde-741	525	2	p.	p.	NOUN
ejde-741	525	3	de	de	PROPN
ejde-741	525	4	carvalho	carvalho	PROPN
ejde-741	525	5	,	,	PUNCT
ejde-741	525	6	r.	r.	PROPN
ejde-741	525	7	demarque	demarque	PROPN
ejde-741	525	8	,	,	PUNCT
ejde-741	525	9	j.	j.	PROPN
ejde-741	525	10	límaco	límaco	PROPN
ejde-741	525	11	,	,	PUNCT
ejde-741	525	12	l.	l.	PROPN
ejde-741	525	13	viana	viana	PROPN
ejde-741	525	14	ejde-2025/15	ejde-2025/15	PROPN
ejde-741	525	15	let	let	VERB
ejde-741	525	16	us	we	PRON
ejde-741	525	17	estimate	estimate	VERB
ejde-741	525	18	each	each	DET
ejde-741	525	19	one	one	NUM
ejde-741	525	20	of	of	ADP
ejde-741	525	21	the	the	DET
ejde-741	525	22	two	two	NUM
ejde-741	525	23	last	last	ADJ
ejde-741	525	24	integrals	integral	NOUN
ejde-741	525	25	.	.	PUNCT
ejde-741	526	1	the	the	DET
ejde-741	526	2	first	first	ADJ
ejde-741	526	3	of	of	ADP
ejde-741	526	4	them	they	PRON
ejde-741	526	5	is	be	AUX
ejde-741	526	6	easier	easy	ADJ
ejde-741	526	7	,	,	PUNCT
ejde-741	526	8	since	since	SCONJ
ejde-741	526	9	we	we	PRON
ejde-741	526	10	just	just	ADV
ejde-741	526	11	use	use	VERB
ejde-741	526	12	hölder	hölder	NOUN
ejde-741	526	13	inequality	inequality	NOUN
ejde-741	526	14	to	to	PART
ejde-741	526	15	obtain∫	obtain∫	VERB
ejde-741	526	16	ε	ε	PROPN
ejde-741	526	17	y	y	PROPN
ejde-741	526	18	a2|u||ux|	a2|u||ux|	PROPN
ejde-741	526	19	dx	dx	PROPN
ejde-741	526	20	≤	≤	PROPN
ejde-741	526	21	∥a∥3/2l∞(0,1)∥u∥l2(0,1)∥	∥a∥3/2l∞(0,1)∥u∥l2(0,1)∥	PROPN
ejde-741	526	22	√	√	NUM
ejde-741	526	23	aux∥l2(0,1	aux∥l2(0,1	NOUN
ejde-741	526	24	)	)	PUNCT
ejde-741	526	25	≤	≤	NOUN
ejde-741	526	26	∥a∥3/2l∞(0,1)∥u∥	∥a∥3/2l∞(0,1)∥u∥	VERB
ejde-741	526	27	2	2	NUM
ejde-741	526	28	h1	h1	NOUN
ejde-741	526	29	a	a	PRON
ejde-741	526	30	.	.	PUNCT
ejde-741	527	1	to	to	PART
ejde-741	527	2	estimate	estimate	VERB
ejde-741	527	3	the	the	DET
ejde-741	527	4	second	second	ADJ
ejde-741	527	5	integral	integral	ADJ
ejde-741	527	6	,	,	PUNCT
ejde-741	527	7	let	let	VERB
ejde-741	527	8	us	we	PRON
ejde-741	527	9	prove	prove	VERB
ejde-741	527	10	that	that	SCONJ
ejde-741	527	11	aa′	aa′	ADJ
ejde-741	527	12	is	be	AUX
ejde-741	527	13	bounded	bound	VERB
ejde-741	527	14	in	in	ADP
ejde-741	527	15	(	(	PUNCT
ejde-741	527	16	0	0	NUM
ejde-741	527	17	,	,	PUNCT
ejde-741	527	18	ε	ε	PROPN
ejde-741	527	19	]	]	PUNCT
ejde-741	527	20	.	.	PUNCT
ejde-741	528	1	in	in	ADP
ejde-741	528	2	fact	fact	NOUN
ejde-741	528	3	,	,	PUNCT
ejde-741	528	4	using	use	VERB
ejde-741	528	5	(	(	PUNCT
ejde-741	528	6	1.2	1.2	NUM
ejde-741	528	7	)	)	PUNCT
ejde-741	528	8	and	and	CCONJ
ejde-741	528	9	(	(	PUNCT
ejde-741	528	10	5.1	5.1	NUM
ejde-741	528	11	)	)	PUNCT
ejde-741	528	12	,	,	PUNCT
ejde-741	528	13	we	we	PRON
ejde-741	528	14	have	have	VERB
ejde-741	528	15	aa′	aa′	ADJ
ejde-741	528	16	≤	≤	NUM
ejde-741	528	17	a2	a2	PROPN
ejde-741	528	18	x	x	PUNCT
ejde-741	529	1	=	=	PUNCT
ejde-741	529	2	(	(	PUNCT
ejde-741	529	3	a	a	DET
ejde-741	529	4	xθ	xθ	PROPN
ejde-741	529	5	)	)	PUNCT
ejde-741	529	6	2x2θ−1	2x2θ−1	PROPN
ejde-741	529	7	≤	≤	NUM
ejde-741	529	8	a(ε	a(ε	PROPN
ejde-741	529	9	)	)	PUNCT
ejde-741	529	10	ε	ε	PROPN
ejde-741	529	11	,	,	PUNCT
ejde-741	529	12	∀x	∀x	X
ejde-741	529	13	∈	∈	PROPN
ejde-741	529	14	(	(	PUNCT
ejde-741	529	15	0	0	NUM
ejde-741	529	16	,	,	PUNCT
ejde-741	529	17	ε	ε	PROPN
ejde-741	529	18	]	]	PUNCT
ejde-741	529	19	,	,	PUNCT
ejde-741	529	20	since	since	SCONJ
ejde-741	529	21	θ	θ	PROPN
ejde-741	529	22	≥	≥	NUM
ejde-741	529	23	1/2	1/2	NUM
ejde-741	529	24	.	.	PUNCT
ejde-741	529	25	hence,∫	hence,∫	X
ejde-741	530	1	ε	ε	PROPN
ejde-741	530	2	y	y	PROPN
ejde-741	530	3	aa′|u|2	aa′|u|2	PROPN
ejde-741	530	4	dx	dx	PROPN
ejde-741	530	5	≤	≤	PROPN
ejde-741	530	6	a(ε	a(ε	PROPN
ejde-741	530	7	)	)	PUNCT
ejde-741	530	8	ε	ε	PROPN
ejde-741	530	9	∫	∫	PROPN
ejde-741	530	10	1	1	NUM
ejde-741	530	11	0	0	NUM
ejde-741	530	12	|u|2	|u|2	PROPN
ejde-741	530	13	dx	dx	PROPN
ejde-741	530	14	=	=	SYM
ejde-741	530	15	a(ε	a(ε	PROPN
ejde-741	530	16	)	)	PUNCT
ejde-741	530	17	ε	ε	PROPN
ejde-741	530	18	∥u∥2l2(0,1	∥u∥2l2(0,1	NOUN
ejde-741	530	19	)	)	PUNCT
ejde-741	530	20	.	.	PUNCT
ejde-741	531	1	therefore	therefore	ADV
ejde-741	531	2	,	,	PUNCT
ejde-741	531	3	au	au	PROPN
ejde-741	531	4	∈	∈	PROPN
ejde-741	531	5	l∞(0	l∞(0	PRON
ejde-741	531	6	,	,	PUNCT
ejde-741	531	7	1	1	NUM
ejde-741	531	8	)	)	PUNCT
ejde-741	531	9	and	and	CCONJ
ejde-741	531	10	∥au∥l∞(0,1	∥au∥l∞(0,1	NOUN
ejde-741	531	11	)	)	PUNCT
ejde-741	531	12	≤	≤	NUM
ejde-741	531	13	c∥u∥h1	c∥u∥h1	PROPN
ejde-741	531	14	a	a	X
ejde-741	531	15	,	,	PUNCT
ejde-741	531	16	where	where	SCONJ
ejde-741	531	17	c	c	NOUN
ejde-741	531	18	=	=	PRON
ejde-741	531	19	(	(	PUNCT
ejde-741	531	20	max	max	PROPN
ejde-741	531	21	{	{	PUNCT
ejde-741	531	22	∥a∥2l∞(0,1	∥a∥2l∞(0,1	ADV
ejde-741	531	23	)	)	PUNCT
ejde-741	531	24	,	,	PUNCT
ejde-741	531	25	2∥a∥	2∥a∥	NUM
ejde-741	531	26	3/2	3/2	NUM
ejde-741	531	27	l∞(0,1	l∞(0,1	NOUN
ejde-741	531	28	)	)	PUNCT
ejde-741	531	29	,	,	PUNCT
ejde-741	531	30	2a(ε	2a(ε	NUM
ejde-741	531	31	)	)	PUNCT
ejde-741	531	32	ε	ε	PROPN
ejde-741	531	33	}	}	PUNCT
ejde-741	531	34	)	)	PUNCT
ejde-741	531	35	1/2	1/2	NUM
ejde-741	531	36	>	>	X
ejde-741	531	37	0	0	NUM
ejde-741	531	38	.	.	PUNCT
ejde-741	531	39	□	□	PUNCT
ejde-741	531	40	remark	remark	NOUN
ejde-741	531	41	5.6	5.6	NUM
ejde-741	531	42	.	.	PUNCT
ejde-741	532	1	the	the	DET
ejde-741	532	2	prototype	prototype	NOUN
ejde-741	532	3	function	function	NOUN
ejde-741	532	4	ã(x	ã(x	PROPN
ejde-741	532	5	)	)	PUNCT
ejde-741	532	6	=	=	PUNCT
ejde-741	533	1	xα	xα	ADJ
ejde-741	533	2	,	,	PUNCT
ejde-741	533	3	with	with	ADP
ejde-741	533	4	α	α	PRON
ejde-741	533	5	∈	∈	PROPN
ejde-741	534	1	[	[	X
ejde-741	534	2	1	1	NUM
ejde-741	534	3	,	,	PUNCT
ejde-741	534	4	2	2	NUM
ejde-741	534	5	)	)	PUNCT
ejde-741	534	6	,	,	PUNCT
ejde-741	534	7	belongs	belong	VERB
ejde-741	534	8	to	to	ADP
ejde-741	534	9	w	w	PROPN
ejde-741	534	10	1,∞(0	1,∞(0	NOUN
ejde-741	534	11	,	,	PUNCT
ejde-741	534	12	1	1	NUM
ejde-741	534	13	)	)	PUNCT
ejde-741	534	14	,	,	PUNCT
ejde-741	534	15	therefore	therefore	ADV
ejde-741	534	16	ãu	ãu	VERB
ejde-741	534	17	∈	∈	PROPN
ejde-741	534	18	l∞(0	l∞(0	PRON
ejde-741	534	19	,	,	PUNCT
ejde-741	534	20	1	1	NUM
ejde-741	534	21	)	)	PUNCT
ejde-741	534	22	,	,	PUNCT
ejde-741	534	23	for	for	SCONJ
ejde-741	534	24	any	any	DET
ejde-741	534	25	u	u	PROPN
ejde-741	534	26	∈	∈	PROPN
ejde-741	534	27	h1	h1	VERB
ejde-741	534	28	a	a	PRON
ejde-741	534	29	.	.	PUNCT
ejde-741	535	1	nevertheless	nevertheless	ADV
ejde-741	535	2	,	,	PUNCT
ejde-741	535	3	for	for	ADP
ejde-741	535	4	any	any	DET
ejde-741	535	5	p	p	X
ejde-741	535	6	∈	∈	PROPN
ejde-741	535	7	(	(	PUNCT
ejde-741	535	8	0	0	NUM
ejde-741	535	9	,	,	PUNCT
ejde-741	535	10	1	1	NUM
ejde-741	535	11	)	)	PUNCT
ejde-741	535	12	,	,	PUNCT
ejde-741	535	13	consider	consider	VERB
ejde-741	535	14	the	the	DET
ejde-741	535	15	function	function	NOUN
ejde-741	535	16	a(x	a(x	NOUN
ejde-741	535	17	)	)	PUNCT
ejde-741	535	18	=	=	PUNCT
ejde-741	536	1	xp	xp	NOUN
ejde-741	537	1	+	+	CCONJ
ejde-741	537	2	x	x	X
ejde-741	537	3	and	and	CCONJ
ejde-741	537	4	note	note	VERB
ejde-741	537	5	that	that	SCONJ
ejde-741	537	6	•	•	NOUN
ejde-741	537	7	a′	a′	PROPN
ejde-741	537	8	=	=	SYM
ejde-741	537	9	pxp−1	pxp−1	PROPN
ejde-741	537	10	+	+	CCONJ
ejde-741	537	11	1	1	NUM
ejde-741	537	12	⇒	⇒	NOUN
ejde-741	537	13	a	a	DET
ejde-741	537	14	̸∈w	̸∈w	PROPN
ejde-741	537	15	1,∞(0	1,∞(0	NUM
ejde-741	537	16	,	,	PUNCT
ejde-741	537	17	1	1	NUM
ejde-741	537	18	)	)	PUNCT
ejde-741	537	19	;	;	PUNCT
ejde-741	537	20	•	•	X
ejde-741	537	21	xa′	xa′	NOUN
ejde-741	538	1	=	=	SYM
ejde-741	538	2	pxp+x	pxp+x	PROPN
ejde-741	538	3	≤	≤	NOUN
ejde-741	538	4	xp+x	xp+x	PROPN
ejde-741	538	5	=	=	PUNCT
ejde-741	538	6	a	a	PROPN
ejde-741	538	7	,	,	PUNCT
ejde-741	538	8	that	that	ADV
ejde-741	538	9	is	is	ADV
ejde-741	538	10	,	,	PUNCT
ejde-741	538	11	a	a	DET
ejde-741	538	12	satisfies	satisfie	NOUN
ejde-741	538	13	assumption	assumption	NOUN
ejde-741	538	14	1.1	1.1	NUM
ejde-741	538	15	,	,	PUNCT
ejde-741	538	16	with	with	ADP
ejde-741	538	17	k	k	PROPN
ejde-741	538	18	=	=	SYM
ejde-741	538	19	1	1	NUM
ejde-741	538	20	;	;	PUNCT
ejde-741	538	21	•	•	NUM
ejde-741	538	22	aa′	aa′	NOUN
ejde-741	539	1	=	=	X
ejde-741	539	2	px2p−1	px2p−1	X
ejde-741	539	3	+	+	CCONJ
ejde-741	539	4	(	(	PUNCT
ejde-741	539	5	p+	p+	VERB
ejde-741	539	6	1)xp	1)xp	NOUN
ejde-741	539	7	+	+	CCONJ
ejde-741	539	8	x	x	X
ejde-741	539	9	is	be	AUX
ejde-741	539	10	bounded	bound	VERB
ejde-741	539	11	if	if	SCONJ
ejde-741	539	12	,	,	PUNCT
ejde-741	539	13	and	and	CCONJ
ejde-741	539	14	only	only	ADV
ejde-741	539	15	if	if	SCONJ
ejde-741	539	16	,	,	PUNCT
ejde-741	539	17	p	p	PRON
ejde-741	539	18	≥	≥	NUM
ejde-741	539	19	1/2	1/2	NUM
ejde-741	539	20	.	.	PUNCT
ejde-741	540	1	therefore	therefore	ADV
ejde-741	540	2	,	,	PUNCT
ejde-741	540	3	the	the	DET
ejde-741	540	4	proof	proof	NOUN
ejde-741	540	5	given	give	VERB
ejde-741	540	6	in	in	ADP
ejde-741	540	7	lemma	lemma	PROPN
ejde-741	540	8	5.5	5.5	NUM
ejde-741	540	9	does	do	AUX
ejde-741	540	10	not	not	PART
ejde-741	540	11	work	work	VERB
ejde-741	540	12	for	for	ADP
ejde-741	540	13	p	p	NOUN
ejde-741	540	14	<	<	X
ejde-741	540	15	1/2	1/2	NUM
ejde-741	540	16	.	.	PUNCT
ejde-741	541	1	acknowledgments	acknowledgment	NOUN
ejde-741	541	2	.	.	PUNCT
ejde-741	542	1	we	we	PRON
ejde-741	542	2	want	want	VERB
ejde-741	542	3	to	to	PART
ejde-741	542	4	thank	thank	VERB
ejde-741	542	5	the	the	DET
ejde-741	542	6	anonymous	anonymous	ADJ
ejde-741	542	7	referees	referee	NOUN
ejde-741	542	8	for	for	ADP
ejde-741	542	9	their	their	PRON
ejde-741	542	10	thorough	thorough	ADJ
ejde-741	542	11	review	review	NOUN
ejde-741	542	12	and	and	CCONJ
ejde-741	542	13	insightful	insightful	ADJ
ejde-741	542	14	comments	comment	NOUN
ejde-741	542	15	,	,	PUNCT
ejde-741	542	16	which	which	PRON
ejde-741	542	17	significantly	significantly	ADV
ejde-741	542	18	improved	improve	VERB
ejde-741	542	19	the	the	DET
ejde-741	542	20	quality	quality	NOUN
ejde-741	542	21	of	of	ADP
ejde-741	542	22	this	this	DET
ejde-741	542	23	manuscript	manuscript	NOUN
ejde-741	542	24	.	.	PUNCT
ejde-741	543	1	their	their	PRON
ejde-741	543	2	expertise	expertise	NOUN
ejde-741	543	3	and	and	CCONJ
ejde-741	543	4	constructive	constructive	ADJ
ejde-741	543	5	feedback	feedback	NOUN
ejde-741	543	6	greatly	greatly	ADV
ejde-741	543	7	enhanced	enhance	VERB
ejde-741	543	8	the	the	DET
ejde-741	543	9	theoretical	theoretical	ADJ
ejde-741	543	10	and	and	CCONJ
ejde-741	543	11	numerical	numerical	ADJ
ejde-741	543	12	aspects	aspect	NOUN
ejde-741	543	13	,	,	PUNCT
ejde-741	543	14	as	as	ADV
ejde-741	543	15	well	well	ADV
ejde-741	543	16	as	as	ADP
ejde-741	543	17	the	the	DET
ejde-741	543	18	clarity	clarity	NOUN
ejde-741	543	19	and	and	CCONJ
ejde-741	543	20	presentation	presentation	NOUN
ejde-741	543	21	of	of	ADP
ejde-741	543	22	our	our	PRON
ejde-741	543	23	work	work	NOUN
ejde-741	543	24	.	.	PUNCT
ejde-741	544	1	p.	p.	NOUN
ejde-741	544	2	p.	p.	PROPN
ejde-741	545	1	de	de	PROPN
ejde-741	545	2	carvalho	carvalho	PROPN
ejde-741	545	3	was	be	AUX
ejde-741	545	4	partially	partially	ADV
ejde-741	545	5	supported	support	VERB
ejde-741	545	6	by	by	ADP
ejde-741	545	7	cnpq	cnpq	NOUN
ejde-741	545	8	grant	grant	NOUN
ejde-741	545	9	no	no	NOUN
ejde-741	545	10	.	.	PUNCT
ejde-741	546	1	306541/20220	306541/20220	NUM
ejde-741	546	2	(	(	PUNCT
ejde-741	546	3	brazil	brazil	PROPN
ejde-741	546	4	)	)	PUNCT
ejde-741	546	5	.	.	PUNCT
ejde-741	547	1	r.	r.	PROPN
ejde-741	547	2	demarque	demarque	PROPN
ejde-741	547	3	was	be	AUX
ejde-741	547	4	supported	support	VERB
ejde-741	547	5	by	by	ADP
ejde-741	547	6	faperj	faperj	PROPN
ejde-741	547	7	(	(	PUNCT
ejde-741	547	8	fundação	fundação	PROPN
ejde-741	547	9	carlos	carlos	PROPN
ejde-741	547	10	chagas	chagas	PROPN
ejde-741	547	11	filho	filho	PROPN
ejde-741	547	12	de	de	PROPN
ejde-741	547	13	amparo	amparo	PROPN
ejde-741	547	14	à	à	PROPN
ejde-741	547	15	pesquisa	pesquisa	PROPN
ejde-741	547	16	do	do	AUX
ejde-741	547	17	estado	estado	VERB
ejde-741	547	18	do	do	PROPN
ejde-741	547	19	rio	rio	PROPN
ejde-741	547	20	de	de	PROPN
ejde-741	547	21	janeiro	janeiro	PROPN
ejde-741	547	22	)	)	PUNCT
ejde-741	547	23	under	under	ADP
ejde-741	547	24	grant	grant	PROPN
ejde-741	547	25	no	no	PROPN
ejde-741	547	26	.	.	PUNCT
ejde-741	548	1	e26/210.456/2024	e26/210.456/2024	PROPN
ejde-741	548	2	.	.	PUNCT
ejde-741	549	1	j.	j.	PROPN
ejde-741	549	2	ĺımaco	ĺımaco	PRON
ejde-741	549	3	was	be	AUX
ejde-741	549	4	partially	partially	ADV
ejde-741	549	5	supported	support	VERB
ejde-741	549	6	by	by	ADP
ejde-741	549	7	cnpq	cnpq	NOUN
ejde-741	549	8	,	,	PUNCT
ejde-741	549	9	grant	grant	VERB
ejde-741	549	10	no	no	PRON
ejde-741	549	11	.	.	PUNCT
ejde-741	550	1	310860/	310860/	NUM
ejde-741	550	2	2023	2023	NUM
ejde-741	550	3	-	-	SYM
ejde-741	550	4	7	7	NUM
ejde-741	550	5	(	(	PUNCT
ejde-741	550	6	brazil	brazil	PROPN
ejde-741	550	7	)	)	PUNCT
ejde-741	550	8	.	.	PUNCT
ejde-741	551	1	references	reference	NOUN
ejde-741	551	2	[	[	X
ejde-741	551	3	1	1	NUM
ejde-741	551	4	]	]	PUNCT
ejde-741	551	5	fatiha	fatiha	X
ejde-741	551	6	alabau	alabau	PROPN
ejde-741	551	7	-	-	PUNCT
ejde-741	551	8	boussouira	boussouira	PROPN
ejde-741	551	9	,	,	PUNCT
ejde-741	551	10	piermarco	piermarco	NOUN
ejde-741	551	11	cannarsa	cannarsa	PROPN
ejde-741	551	12	,	,	PUNCT
ejde-741	551	13	genni	genni	PROPN
ejde-741	551	14	fragnelli	fragnelli	NOUN
ejde-741	551	15	;	;	PUNCT
ejde-741	551	16	carleman	carleman	ADJ
ejde-741	551	17	estimates	estimate	NOUN
ejde-741	551	18	for	for	ADP
ejde-741	551	19	degenerate	degenerate	ADJ
ejde-741	551	20	parabolic	parabolic	NOUN
ejde-741	551	21	operators	operator	NOUN
ejde-741	551	22	with	with	ADP
ejde-741	551	23	applications	application	NOUN
ejde-741	551	24	to	to	ADP
ejde-741	551	25	null	null	ADJ
ejde-741	551	26	controllability	controllability	NOUN
ejde-741	551	27	,	,	PUNCT
ejde-741	551	28	journal	journal	NOUN
ejde-741	551	29	of	of	ADP
ejde-741	551	30	evolution	evolution	NOUN
ejde-741	551	31	equations	equation	NOUN
ejde-741	551	32	,	,	PUNCT
ejde-741	551	33	6	6	NUM
ejde-741	551	34	(	(	PUNCT
ejde-741	551	35	2006	2006	NUM
ejde-741	551	36	)	)	PUNCT
ejde-741	551	37	,	,	PUNCT
ejde-741	551	38	no	no	INTJ
ejde-741	551	39	.	.	NOUN
ejde-741	551	40	2	2	NUM
ejde-741	551	41	,	,	PUNCT
ejde-741	551	42	161–204	161–204	NUM
ejde-741	551	43	.	.	PUNCT
ejde-741	552	1	[	[	X
ejde-741	552	2	2	2	X
ejde-741	552	3	]	]	PUNCT
ejde-741	552	4	v.	v.	ADP
ejde-741	552	5	m.	m.	NOUN
ejde-741	552	6	alekseev	alekseev	PROPN
ejde-741	552	7	,	,	PUNCT
ejde-741	552	8	v.	v.	ADP
ejde-741	552	9	m.	m.	NOUN
ejde-741	552	10	tikhomirov	tikhomirov	PROPN
ejde-741	552	11	,	,	PUNCT
ejde-741	552	12	s.	s.	PROPN
ejde-741	552	13	v.	v.	PROPN
ejde-741	552	14	fomin	fomin	PROPN
ejde-741	552	15	;	;	PUNCT
ejde-741	552	16	optimal	optimal	ADJ
ejde-741	552	17	control	control	NOUN
ejde-741	552	18	,	,	PUNCT
ejde-741	552	19	consultants	consultants	PROPN
ejde-741	552	20	bureau	bureau	PROPN
ejde-741	552	21	,	,	PUNCT
ejde-741	552	22	new	new	PROPN
ejde-741	552	23	york	york	PROPN
ejde-741	552	24	,	,	PUNCT
ejde-741	552	25	1987	1987	NUM
ejde-741	552	26	.	.	PUNCT
ejde-741	553	1	[	[	X
ejde-741	553	2	3	3	X
ejde-741	553	3	]	]	X
ejde-741	553	4	fágner	fágner	PROPN
ejde-741	553	5	d.	d.	PROPN
ejde-741	553	6	araruna	araruna	PROPN
ejde-741	553	7	,	,	PUNCT
ejde-741	553	8	bruno	bruno	PROPN
ejde-741	554	1	sérgio	sérgio	PROPN
ejde-741	554	2	v.	v.	ADP
ejde-741	554	3	aráujo	aráujo	NUM
ejde-741	554	4	,	,	PUNCT
ejde-741	554	5	enrique	enrique	PROPN
ejde-741	554	6	fernández	fernández	PROPN
ejde-741	554	7	-	-	PUNCT
ejde-741	554	8	cara	cara	NOUN
ejde-741	554	9	;	;	PUNCT
ejde-741	554	10	stackelberg	stackelberg	PROPN
ejde-741	554	11	-	-	PUNCT
ejde-741	554	12	nash	nash	NOUN
ejde-741	554	13	null	null	ADJ
ejde-741	554	14	controllability	controllability	NOUN
ejde-741	554	15	for	for	ADP
ejde-741	554	16	some	some	DET
ejde-741	554	17	linear	linear	ADJ
ejde-741	554	18	and	and	CCONJ
ejde-741	554	19	semilinear	semilinear	ADJ
ejde-741	554	20	degenerate	degenerate	ADJ
ejde-741	554	21	parabolic	parabolic	NOUN
ejde-741	554	22	equations	equation	NOUN
ejde-741	554	23	,	,	PUNCT
ejde-741	554	24	mathematics	mathematic	NOUN
ejde-741	554	25	of	of	ADP
ejde-741	554	26	control	control	NOUN
ejde-741	554	27	signals	signal	NOUN
ejde-741	554	28	and	and	CCONJ
ejde-741	554	29	systems	system	NOUN
ejde-741	554	30	,	,	PUNCT
ejde-741	554	31	30	30	NUM
ejde-741	554	32	(	(	PUNCT
ejde-741	554	33	2018	2018	NUM
ejde-741	554	34	)	)	PUNCT
ejde-741	554	35	,	,	PUNCT
ejde-741	554	36	no	no	INTJ
ejde-741	554	37	.	.	NOUN
ejde-741	555	1	3	3	X
ejde-741	555	2	.	.	X
ejde-741	556	1	ejde-2025/15	ejde-2025/15	VERB
ejde-741	556	2	null	null	ADJ
ejde-741	556	3	-	-	PUNCT
ejde-741	556	4	controllability	controllability	NOUN
ejde-741	556	5	degenerate	degenerate	ADJ
ejde-741	556	6	quasilinear	quasilinear	NOUN
ejde-741	556	7	equations	equation	NOUN
ejde-741	556	8	21	21	NUM
ejde-741	557	1	[	[	X
ejde-741	557	2	4	4	NUM
ejde-741	557	3	]	]	PUNCT
ejde-741	557	4	bruno	bruno	PROPN
ejde-741	558	1	sérgio	sérgio	PROPN
ejde-741	558	2	v.	v.	PROPN
ejde-741	558	3	araújo	araújo	PROPN
ejde-741	558	4	,	,	PUNCT
ejde-741	558	5	reginaldo	reginaldo	PROPN
ejde-741	558	6	demarque	demarque	NOUN
ejde-741	558	7	,	,	PUNCT
ejde-741	558	8	luiz	luiz	NOUN
ejde-741	558	9	viana	viana	PROPN
ejde-741	558	10	;	;	PUNCT
ejde-741	558	11	boundary	boundary	ADJ
ejde-741	558	12	null	null	ADJ
ejde-741	558	13	controllability	controllability	NOUN
ejde-741	558	14	of	of	ADP
ejde-741	558	15	degenerate	degenerate	ADJ
ejde-741	558	16	heat	heat	NOUN
ejde-741	558	17	equation	equation	NOUN
ejde-741	558	18	as	as	ADP
ejde-741	558	19	the	the	DET
ejde-741	558	20	limit	limit	NOUN
ejde-741	558	21	of	of	ADP
ejde-741	558	22	internal	internal	ADJ
ejde-741	558	23	controllability	controllability	NOUN
ejde-741	558	24	,	,	PUNCT
ejde-741	558	25	nonlinear	nonlinear	ADJ
ejde-741	558	26	analysis	analysis	NOUN
ejde-741	558	27	:	:	PUNCT
ejde-741	558	28	real	real	ADJ
ejde-741	558	29	world	world	NOUN
ejde-741	558	30	applications	application	NOUN
ejde-741	558	31	,	,	PUNCT
ejde-741	558	32	66	66	NUM
ejde-741	558	33	(	(	PUNCT
ejde-741	558	34	2022	2022	NUM
ejde-741	558	35	)	)	PUNCT
ejde-741	558	36	,	,	PUNCT
ejde-741	558	37	103519	103519	NUM
ejde-741	558	38	.	.	PUNCT
ejde-741	559	1	[	[	X
ejde-741	559	2	5	5	X
ejde-741	559	3	]	]	PUNCT
ejde-741	559	4	bruno	bruno	PROPN
ejde-741	559	5	sérgio	sérgio	PROPN
ejde-741	559	6	v.	v.	PROPN
ejde-741	559	7	araújo	araújo	PROPN
ejde-741	559	8	,	,	PUNCT
ejde-741	559	9	reginaldo	reginaldo	PROPN
ejde-741	559	10	demarque	demarque	NOUN
ejde-741	559	11	,	,	PUNCT
ejde-741	559	12	luiz	luiz	NOUN
ejde-741	559	13	viana	viana	PROPN
ejde-741	559	14	;	;	PUNCT
ejde-741	559	15	regularity	regularity	NOUN
ejde-741	559	16	results	result	NOUN
ejde-741	559	17	for	for	ADP
ejde-741	559	18	degenerate	degenerate	ADJ
ejde-741	559	19	wave	wave	NOUN
ejde-741	559	20	equations	equation	NOUN
ejde-741	559	21	in	in	ADP
ejde-741	559	22	a	a	DET
ejde-741	559	23	neighborhood	neighborhood	NOUN
ejde-741	559	24	of	of	ADP
ejde-741	559	25	the	the	DET
ejde-741	559	26	boundary	boundary	NOUN
ejde-741	559	27	,	,	PUNCT
ejde-741	559	28	evolution	evolution	NOUN
ejde-741	559	29	equations	equation	NOUN
ejde-741	559	30	&	&	CCONJ
ejde-741	559	31	control	control	PROPN
ejde-741	559	32	theory	theory	NOUN
ejde-741	559	33	,	,	PUNCT
ejde-741	559	34	12	12	NUM
ejde-741	559	35	(	(	PUNCT
ejde-741	559	36	2023	2023	NUM
ejde-741	559	37	)	)	PUNCT
ejde-741	559	38	,	,	PUNCT
ejde-741	559	39	no	no	INTJ
ejde-741	559	40	.	.	NOUN
ejde-741	559	41	5	5	NUM
ejde-741	559	42	.	.	PUNCT
ejde-741	560	1	[	[	X
ejde-741	560	2	6	6	NUM
ejde-741	560	3	]	]	X
ejde-741	560	4	idriss	idriss	NOUN
ejde-741	560	5	boutaayamou	boutaayamou	NOUN
ejde-741	560	6	,	,	PUNCT
ejde-741	560	7	genni	genni	PROPN
ejde-741	560	8	fragnelli	fragnelli	NOUN
ejde-741	560	9	,	,	PUNCT
ejde-741	560	10	lahcen	lahcen	NOUN
ejde-741	560	11	maniar	maniar	NOUN
ejde-741	560	12	;	;	PUNCT
ejde-741	560	13	carleman	carleman	ADJ
ejde-741	560	14	estimates	estimate	NOUN
ejde-741	560	15	for	for	ADP
ejde-741	560	16	parabolic	parabolic	ADJ
ejde-741	560	17	equations	equation	NOUN
ejde-741	560	18	with	with	ADP
ejde-741	560	19	interior	interior	ADJ
ejde-741	560	20	degeneracy	degeneracy	NOUN
ejde-741	560	21	and	and	CCONJ
ejde-741	560	22	neumann	neumann	PROPN
ejde-741	560	23	boundary	boundary	ADJ
ejde-741	560	24	conditions	condition	NOUN
ejde-741	560	25	,	,	PUNCT
ejde-741	560	26	journal	journal	NOUN
ejde-741	560	27	d’analyse	d’analyse	PROPN
ejde-741	560	28	mathématique	mathématique	PROPN
ejde-741	560	29	,	,	PUNCT
ejde-741	560	30	135	135	NUM
ejde-741	560	31	(	(	PUNCT
ejde-741	560	32	2018	2018	NUM
ejde-741	560	33	)	)	PUNCT
ejde-741	560	34	,	,	PUNCT
ejde-741	560	35	no	no	INTJ
ejde-741	560	36	.	.	NOUN
ejde-741	560	37	1	1	NUM
ejde-741	560	38	,	,	PUNCT
ejde-741	560	39	1–35	1–35	NOUN
ejde-741	560	40	.	.	PUNCT
ejde-741	561	1	[	[	X
ejde-741	561	2	7	7	X
ejde-741	561	3	]	]	X
ejde-741	561	4	m.	m.	NOUN
ejde-741	561	5	campiti	campiti	PROPN
ejde-741	561	6	,	,	PUNCT
ejde-741	561	7	g.	g.	PROPN
ejde-741	561	8	metafune	metafune	PROPN
ejde-741	561	9	,	,	PUNCT
ejde-741	561	10	d.	d.	PROPN
ejde-741	561	11	pallara	pallara	PROPN
ejde-741	561	12	;	;	PUNCT
ejde-741	561	13	degenerate	degenerate	ADJ
ejde-741	561	14	self	self	NOUN
ejde-741	561	15	-	-	PUNCT
ejde-741	561	16	adjoint	adjoint	NOUN
ejde-741	561	17	evolution	evolution	NOUN
ejde-741	561	18	equations	equation	NOUN
ejde-741	561	19	on	on	ADP
ejde-741	561	20	the	the	DET
ejde-741	561	21	unit	unit	NOUN
ejde-741	561	22	interval	interval	NOUN
ejde-741	561	23	,	,	PUNCT
ejde-741	561	24	semigroup	semigroup	PROPN
ejde-741	561	25	forum	forum	PROPN
ejde-741	561	26	57	57	NUM
ejde-741	561	27	(	(	PUNCT
ejde-741	561	28	1998	1998	NUM
ejde-741	561	29	)	)	PUNCT
ejde-741	561	30	,	,	PUNCT
ejde-741	561	31	1–36	1–36	NUM
ejde-741	561	32	.	.	PUNCT
ejde-741	562	1	[	[	X
ejde-741	562	2	8	8	NUM
ejde-741	562	3	]	]	PUNCT
ejde-741	562	4	piermarco	piermarco	NOUN
ejde-741	562	5	cannarsa	cannarsa	PROPN
ejde-741	562	6	,	,	PUNCT
ejde-741	562	7	luz	luz	PROPN
ejde-741	562	8	de	de	PROPN
ejde-741	562	9	teresa	teresa	PROPN
ejde-741	562	10	;	;	PUNCT
ejde-741	562	11	controllability	controllability	NOUN
ejde-741	562	12	of	of	ADP
ejde-741	562	13	1	1	NUM
ejde-741	562	14	-	-	PUNCT
ejde-741	562	15	d	d	NOUN
ejde-741	562	16	coupled	couple	VERB
ejde-741	562	17	degenerate	degenerate	ADJ
ejde-741	562	18	parabolic	parabolic	NOUN
ejde-741	562	19	equations	equation	NOUN
ejde-741	562	20	.	.	PUNCT
ejde-741	562	21	,	,	PUNCT
ejde-741	562	22	electronic	electronic	ADJ
ejde-741	562	23	journal	journal	NOUN
ejde-741	562	24	of	of	ADP
ejde-741	562	25	differential	differential	ADJ
ejde-741	562	26	equations	equation	NOUN
ejde-741	562	27	,	,	PUNCT
ejde-741	562	28	2009	2009	NUM
ejde-741	562	29	(	(	PUNCT
ejde-741	562	30	2009	2009	NUM
ejde-741	562	31	)	)	PUNCT
ejde-741	562	32	,	,	PUNCT
ejde-741	562	33	no	no	INTJ
ejde-741	562	34	.	.	NOUN
ejde-741	562	35	73	73	NUM
ejde-741	562	36	,	,	PUNCT
ejde-741	562	37	1–21	1–21	PROPN
ejde-741	562	38	.	.	PUNCT
ejde-741	563	1	[	[	X
ejde-741	563	2	9	9	NUM
ejde-741	563	3	]	]	PUNCT
ejde-741	563	4	piermarco	piermarco	NOUN
ejde-741	563	5	cannarsa	cannarsa	PROPN
ejde-741	563	6	,	,	PUNCT
ejde-741	563	7	genni	genni	PROPN
ejde-741	563	8	fragnelli	fragnelli	NOUN
ejde-741	563	9	;	;	PUNCT
ejde-741	563	10	null	null	ADJ
ejde-741	563	11	controllability	controllability	NOUN
ejde-741	563	12	of	of	ADP
ejde-741	563	13	semilinear	semilinear	PROPN
ejde-741	563	14	degenerate	degenerate	ADJ
ejde-741	563	15	parabolic	parabolic	ADJ
ejde-741	563	16	equations	equation	NOUN
ejde-741	563	17	in	in	ADP
ejde-741	563	18	bounded	bounded	ADJ
ejde-741	563	19	domains	domain	NOUN
ejde-741	563	20	,	,	PUNCT
ejde-741	563	21	electronic	electronic	ADJ
ejde-741	563	22	journal	journal	NOUN
ejde-741	563	23	of	of	ADP
ejde-741	563	24	differential	differential	ADJ
ejde-741	563	25	equations	equation	NOUN
ejde-741	563	26	,	,	PUNCT
ejde-741	563	27	(	(	PUNCT
ejde-741	563	28	2006	2006	NUM
ejde-741	563	29	)	)	PUNCT
ejde-741	563	30	,	,	PUNCT
ejde-741	563	31	no	no	INTJ
ejde-741	563	32	.	.	NOUN
ejde-741	563	33	136	136	NUM
ejde-741	563	34	,	,	PUNCT
ejde-741	563	35	1–20	1–20	NOUN
ejde-741	563	36	.	.	PUNCT
ejde-741	564	1	[	[	X
ejde-741	564	2	10	10	NUM
ejde-741	564	3	]	]	PUNCT
ejde-741	564	4	piermarco	piermarco	NOUN
ejde-741	564	5	cannarsa	cannarsa	PROPN
ejde-741	564	6	,	,	PUNCT
ejde-741	564	7	genni	genni	PROPN
ejde-741	564	8	fragnelli	fragnelli	PROPN
ejde-741	564	9	,	,	PUNCT
ejde-741	564	10	dario	dario	PROPN
ejde-741	564	11	rocchetti	rocchetti	PROPN
ejde-741	564	12	;	;	PUNCT
ejde-741	564	13	null	null	ADJ
ejde-741	564	14	controllability	controllability	NOUN
ejde-741	564	15	of	of	ADP
ejde-741	564	16	degenerate	degenerate	ADJ
ejde-741	564	17	parabolic	parabolic	NOUN
ejde-741	564	18	operators	operator	NOUN
ejde-741	564	19	with	with	ADP
ejde-741	564	20	drift	drift	NOUN
ejde-741	564	21	,	,	PUNCT
ejde-741	564	22	networks	network	NOUN
ejde-741	564	23	&	&	CCONJ
ejde-741	564	24	heterogeneous	heterogeneous	ADJ
ejde-741	564	25	media	medium	NOUN
ejde-741	564	26	,	,	PUNCT
ejde-741	564	27	2	2	NUM
ejde-741	564	28	(	(	PUNCT
ejde-741	564	29	2007	2007	NUM
ejde-741	564	30	)	)	PUNCT
ejde-741	564	31	,	,	PUNCT
ejde-741	564	32	no	no	INTJ
ejde-741	564	33	.	.	NOUN
ejde-741	564	34	4	4	NUM
ejde-741	564	35	,	,	PUNCT
ejde-741	564	36	695	695	NUM
ejde-741	564	37	.	.	PUNCT
ejde-741	565	1	[	[	X
ejde-741	565	2	11	11	NUM
ejde-741	565	3	]	]	PUNCT
ejde-741	565	4	piermarco	piermarco	NOUN
ejde-741	565	5	cannarsa	cannarsa	PROPN
ejde-741	565	6	,	,	PUNCT
ejde-741	565	7	genni	genni	PROPN
ejde-741	565	8	fragnelli	fragnelli	PROPN
ejde-741	565	9	,	,	PUNCT
ejde-741	565	10	dario	dario	PROPN
ejde-741	565	11	rocchetti	rocchetti	PROPN
ejde-741	565	12	;	;	PUNCT
ejde-741	565	13	controllability	controllability	NOUN
ejde-741	565	14	results	result	VERB
ejde-741	565	15	for	for	ADP
ejde-741	565	16	a	a	DET
ejde-741	565	17	class	class	NOUN
ejde-741	565	18	of	of	ADP
ejde-741	565	19	one	one	NUM
ejde-741	565	20	-	-	PUNCT
ejde-741	565	21	dimensional	dimensional	ADJ
ejde-741	565	22	degenerate	degenerate	ADJ
ejde-741	565	23	parabolic	parabolic	ADJ
ejde-741	565	24	problems	problem	NOUN
ejde-741	565	25	in	in	ADP
ejde-741	565	26	nondivergence	nondivergence	NOUN
ejde-741	565	27	form	form	NOUN
ejde-741	565	28	,	,	PUNCT
ejde-741	565	29	journal	journal	NOUN
ejde-741	565	30	of	of	ADP
ejde-741	565	31	evolution	evolution	NOUN
ejde-741	565	32	equations	equation	NOUN
ejde-741	565	33	,	,	PUNCT
ejde-741	565	34	8	8	NUM
ejde-741	565	35	(	(	PUNCT
ejde-741	565	36	2008	2008	NUM
ejde-741	565	37	)	)	PUNCT
ejde-741	565	38	,	,	PUNCT
ejde-741	565	39	no	no	INTJ
ejde-741	565	40	.	.	NOUN
ejde-741	565	41	4	4	NUM
ejde-741	565	42	,	,	PUNCT
ejde-741	565	43	583–616	583–616	NUM
ejde-741	565	44	.	.	PUNCT
ejde-741	566	1	[	[	X
ejde-741	566	2	12	12	NUM
ejde-741	566	3	]	]	PUNCT
ejde-741	566	4	piermarco	piermarco	NOUN
ejde-741	566	5	cannarsa	cannarsa	PROPN
ejde-741	566	6	,	,	PUNCT
ejde-741	566	7	patrick	patrick	PROPN
ejde-741	566	8	martinez	martinez	PROPN
ejde-741	566	9	,	,	PUNCT
ejde-741	566	10	judith	judith	PROPN
ejde-741	566	11	vancostenoble	vancostenoble	ADJ
ejde-741	566	12	;	;	PUNCT
ejde-741	566	13	nulle	nulle	X
ejde-741	566	14	contrôlabilité	contrôlabilité	X
ejde-741	566	15	régionale	régionale	PROPN
ejde-741	566	16	pour	pour	PROPN
ejde-741	566	17	des	des	PROPN
ejde-741	566	18	équations	équations	PROPN
ejde-741	566	19	de	de	X
ejde-741	566	20	la	la	X
ejde-741	566	21	chaleur	chaleur	PROPN
ejde-741	566	22	dégénérées	dégénérées	PROPN
ejde-741	566	23	,	,	PUNCT
ejde-741	566	24	comptes	compte	VERB
ejde-741	566	25	rendus	rendus	PROPN
ejde-741	566	26	-	-	PUNCT
ejde-741	566	27	mécanique	mécanique	PROPN
ejde-741	566	28	6	6	NUM
ejde-741	566	29	(	(	PUNCT
ejde-741	566	30	2002	2002	NUM
ejde-741	566	31	)	)	PUNCT
ejde-741	566	32	,	,	PUNCT
ejde-741	566	33	no	no	INTJ
ejde-741	566	34	.	.	NOUN
ejde-741	566	35	330	330	NUM
ejde-741	566	36	,	,	PUNCT
ejde-741	566	37	397–401	397–401	NUM
ejde-741	566	38	.	.	PUNCT
ejde-741	567	1	[	[	X
ejde-741	567	2	13	13	NUM
ejde-741	567	3	]	]	PUNCT
ejde-741	567	4	piermarco	piermarco	NOUN
ejde-741	567	5	cannarsa	cannarsa	PROPN
ejde-741	567	6	,	,	PUNCT
ejde-741	567	7	patrick	patrick	PROPN
ejde-741	567	8	martinez	martinez	PROPN
ejde-741	567	9	,	,	PUNCT
ejde-741	567	10	judith	judith	PROPN
ejde-741	567	11	vancostenoble	vancostenoble	ADJ
ejde-741	567	12	;	;	PUNCT
ejde-741	567	13	persistent	persistent	ADJ
ejde-741	567	14	regional	regional	ADJ
ejde-741	567	15	null	null	ADJ
ejde-741	567	16	contrillability	contrillability	NOUN
ejde-741	567	17	for	for	ADP
ejde-741	567	18	a	a	DET
ejde-741	567	19	class	class	NOUN
ejde-741	567	20	of	of	ADP
ejde-741	567	21	degenerate	degenerate	ADJ
ejde-741	567	22	parabolic	parabolic	NOUN
ejde-741	567	23	equations	equation	NOUN
ejde-741	567	24	,	,	PUNCT
ejde-741	567	25	communications	communication	NOUN
ejde-741	567	26	on	on	ADP
ejde-741	567	27	pure	pure	ADJ
ejde-741	567	28	&	&	CCONJ
ejde-741	567	29	applied	applied	ADJ
ejde-741	567	30	analysis	analysis	NOUN
ejde-741	567	31	,	,	PUNCT
ejde-741	567	32	3	3	NUM
ejde-741	567	33	(	(	PUNCT
ejde-741	567	34	2004	2004	NUM
ejde-741	567	35	)	)	PUNCT
ejde-741	567	36	,	,	PUNCT
ejde-741	567	37	no	no	INTJ
ejde-741	567	38	.	.	NOUN
ejde-741	567	39	4	4	NUM
ejde-741	567	40	,	,	PUNCT
ejde-741	567	41	607	607	NUM
ejde-741	567	42	.	.	PUNCT
ejde-741	568	1	[	[	X
ejde-741	568	2	14	14	NUM
ejde-741	568	3	]	]	PUNCT
ejde-741	568	4	piermarco	piermarco	NOUN
ejde-741	568	5	cannarsa	cannarsa	PROPN
ejde-741	568	6	,	,	PUNCT
ejde-741	568	7	patrick	patrick	PROPN
ejde-741	568	8	martinez	martinez	PROPN
ejde-741	568	9	,	,	PUNCT
ejde-741	568	10	judith	judith	PROPN
ejde-741	568	11	vancostenoble	vancostenoble	ADJ
ejde-741	568	12	;	;	PUNCT
ejde-741	568	13	null	null	ADJ
ejde-741	568	14	controllability	controllability	NOUN
ejde-741	568	15	of	of	ADP
ejde-741	568	16	degenerate	degenerate	ADJ
ejde-741	568	17	heat	heat	NOUN
ejde-741	568	18	equations	equation	NOUN
ejde-741	568	19	,	,	PUNCT
ejde-741	568	20	advances	advance	NOUN
ejde-741	568	21	in	in	ADP
ejde-741	568	22	differential	differential	ADJ
ejde-741	568	23	equations	equation	NOUN
ejde-741	568	24	,	,	PUNCT
ejde-741	568	25	10	10	NUM
ejde-741	568	26	(	(	PUNCT
ejde-741	568	27	2005	2005	NUM
ejde-741	568	28	)	)	PUNCT
ejde-741	568	29	,	,	PUNCT
ejde-741	568	30	no	no	INTJ
ejde-741	568	31	.	.	NOUN
ejde-741	568	32	2	2	NUM
ejde-741	568	33	,	,	PUNCT
ejde-741	568	34	153–190	153–190	NUM
ejde-741	568	35	.	.	PUNCT
ejde-741	569	1	[	[	X
ejde-741	569	2	15	15	NUM
ejde-741	569	3	]	]	PUNCT
ejde-741	569	4	piermarco	piermarco	NOUN
ejde-741	569	5	cannarsa	cannarsa	PROPN
ejde-741	569	6	,	,	PUNCT
ejde-741	569	7	patrick	patrick	PROPN
ejde-741	569	8	martinez	martinez	PROPN
ejde-741	569	9	,	,	PUNCT
ejde-741	569	10	judith	judith	PROPN
ejde-741	569	11	vancostenoble	vancostenoble	ADJ
ejde-741	569	12	;	;	PUNCT
ejde-741	569	13	carleman	carleman	ADJ
ejde-741	569	14	estimates	estimate	NOUN
ejde-741	569	15	for	for	ADP
ejde-741	569	16	a	a	DET
ejde-741	569	17	class	class	NOUN
ejde-741	569	18	of	of	ADP
ejde-741	569	19	degenerate	degenerate	ADJ
ejde-741	569	20	parabolic	parabolic	NOUN
ejde-741	569	21	operators	operator	NOUN
ejde-741	569	22	,	,	PUNCT
ejde-741	569	23	siam	siam	PROPN
ejde-741	569	24	journal	journal	NOUN
ejde-741	569	25	on	on	ADP
ejde-741	569	26	control	control	NOUN
ejde-741	569	27	and	and	CCONJ
ejde-741	569	28	optimization	optimization	NOUN
ejde-741	569	29	,	,	PUNCT
ejde-741	569	30	47	47	NUM
ejde-741	569	31	(	(	PUNCT
ejde-741	569	32	2008	2008	NUM
ejde-741	569	33	)	)	PUNCT
ejde-741	569	34	,	,	PUNCT
ejde-741	569	35	no	no	INTJ
ejde-741	569	36	.	.	NOUN
ejde-741	569	37	1	1	NUM
ejde-741	569	38	,	,	PUNCT
ejde-741	569	39	1–19	1–19	NOUN
ejde-741	569	40	.	.	PUNCT
ejde-741	570	1	[	[	X
ejde-741	570	2	16	16	NUM
ejde-741	570	3	]	]	X
ejde-741	570	4	felipe	felipe	PROPN
ejde-741	570	5	w.	w.	PROPN
ejde-741	570	6	chaves	chaves	PROPN
ejde-741	570	7	-	-	PUNCT
ejde-741	570	8	silva	silva	PROPN
ejde-741	570	9	,	,	PUNCT
ejde-741	570	10	jean	jean	PROPN
ejde-741	570	11	-	-	PUNCT
ejde-741	570	12	pierre	pierre	PROPN
ejde-741	570	13	puel	puel	NOUN
ejde-741	570	14	,	,	PUNCT
ejde-741	570	15	mauŕıcio	mauŕıcio	PROPN
ejde-741	570	16	c.	c.	PROPN
ejde-741	570	17	santos	santos	PROPN
ejde-741	570	18	;	;	PUNCT
ejde-741	570	19	boundary	boundary	ADJ
ejde-741	570	20	null	null	ADJ
ejde-741	570	21	controllability	controllability	NOUN
ejde-741	570	22	as	as	ADP
ejde-741	570	23	the	the	DET
ejde-741	570	24	limit	limit	NOUN
ejde-741	570	25	of	of	ADP
ejde-741	570	26	internal	internal	ADJ
ejde-741	570	27	controllability	controllability	NOUN
ejde-741	570	28	:	:	PUNCT
ejde-741	570	29	the	the	DET
ejde-741	570	30	heat	heat	NOUN
ejde-741	570	31	case	case	NOUN
ejde-741	570	32	,	,	PUNCT
ejde-741	570	33	esaim	esaim	NOUN
ejde-741	570	34	:	:	PUNCT
ejde-741	570	35	cocv	cocv	NOUN
ejde-741	570	36	26	26	NUM
ejde-741	570	37	(	(	PUNCT
ejde-741	570	38	2020	2020	NUM
ejde-741	570	39	)	)	PUNCT
ejde-741	570	40	,	,	PUNCT
ejde-741	570	41	91	91	NUM
ejde-741	570	42	.	.	PUNCT
ejde-741	571	1	[	[	X
ejde-741	571	2	17	17	NUM
ejde-741	571	3	]	]	X
ejde-741	571	4	pitágoras	pitágoras	PROPN
ejde-741	571	5	p.	p.	PROPN
ejde-741	571	6	de	de	X
ejde-741	571	7	carvalho	carvalho	PROPN
ejde-741	571	8	,	,	PUNCT
ejde-741	571	9	enrique	enrique	PROPN
ejde-741	571	10	fernández	fernández	PROPN
ejde-741	571	11	-	-	PUNCT
ejde-741	571	12	cara	cara	NOUN
ejde-741	571	13	;	;	PUNCT
ejde-741	571	14	numerical	numerical	ADJ
ejde-741	571	15	stackelberg	stackelberg	PROPN
ejde-741	571	16	-	-	PUNCT
ejde-741	571	17	nash	nash	PROPN
ejde-741	571	18	control	control	NOUN
ejde-741	571	19	for	for	ADP
ejde-741	571	20	the	the	DET
ejde-741	571	21	heat	heat	NOUN
ejde-741	571	22	equation	equation	NOUN
ejde-741	571	23	,	,	PUNCT
ejde-741	571	24	siam	siam	ADJ
ejde-741	571	25	journal	journal	NOUN
ejde-741	571	26	on	on	ADP
ejde-741	571	27	scientific	scientific	ADJ
ejde-741	571	28	computing	compute	VERB
ejde-741	571	29	42	42	NUM
ejde-741	571	30	(	(	PUNCT
ejde-741	571	31	2020	2020	NUM
ejde-741	571	32	)	)	PUNCT
ejde-741	571	33	,	,	PUNCT
ejde-741	571	34	no	no	INTJ
ejde-741	571	35	.	.	NOUN
ejde-741	571	36	5	5	NUM
ejde-741	571	37	,	,	PUNCT
ejde-741	571	38	a2678	a2678	NOUN
ejde-741	571	39	–	–	PUNCT
ejde-741	571	40	a2700	a2700	NUM
ejde-741	571	41	.	.	PUNCT
ejde-741	572	1	[	[	X
ejde-741	572	2	18	18	NUM
ejde-741	572	3	]	]	SYM
ejde-741	572	4	pitágoras	pitágoras	PROPN
ejde-741	572	5	p.	p.	PROPN
ejde-741	572	6	de	de	PROPN
ejde-741	572	7	carvalho	carvalho	PROPN
ejde-741	572	8	,	,	PUNCT
ejde-741	572	9	juan	juan	PROPN
ejde-741	572	10	ĺımaco	ĺımaco	PROPN
ejde-741	572	11	,	,	PUNCT
ejde-741	572	12	denilson	denilson	PROPN
ejde-741	572	13	menezes	menezes	PROPN
ejde-741	572	14	,	,	PUNCT
ejde-741	572	15	yuri	yuri	PROPN
ejde-741	572	16	thamsten	thamsten	NOUN
ejde-741	572	17	;	;	PUNCT
ejde-741	572	18	local	local	ADJ
ejde-741	572	19	null	null	ADJ
ejde-741	572	20	controllability	controllability	NOUN
ejde-741	572	21	of	of	ADP
ejde-741	572	22	a	a	DET
ejde-741	572	23	class	class	NOUN
ejde-741	572	24	of	of	ADP
ejde-741	572	25	non	non	ADJ
ejde-741	572	26	-	-	ADJ
ejde-741	572	27	newtonian	newtonian	ADJ
ejde-741	572	28	incompressible	incompressible	ADJ
ejde-741	572	29	viscous	viscous	ADJ
ejde-741	572	30	fluids	fluid	NOUN
ejde-741	572	31	,	,	PUNCT
ejde-741	572	32	evolution	evolution	NOUN
ejde-741	572	33	equations	equation	NOUN
ejde-741	572	34	and	and	CCONJ
ejde-741	572	35	control	control	PROPN
ejde-741	572	36	theory	theory	NOUN
ejde-741	572	37	,	,	PUNCT
ejde-741	572	38	11	11	NUM
ejde-741	572	39	(	(	PUNCT
ejde-741	572	40	2022	2022	NUM
ejde-741	572	41	)	)	PUNCT
ejde-741	572	42	,	,	PUNCT
ejde-741	573	1	no	no	INTJ
ejde-741	573	2	.	.	NOUN
ejde-741	573	3	4	4	NUM
ejde-741	573	4	,	,	PUNCT
ejde-741	573	5	1251–1283	1251–1283	NUM
ejde-741	573	6	.	.	PUNCT
ejde-741	574	1	[	[	X
ejde-741	574	2	19	19	NUM
ejde-741	574	3	]	]	X
ejde-741	574	4	pitágoras	pitágoras	PROPN
ejde-741	574	5	p.	p.	PROPN
ejde-741	574	6	de	de	PROPN
ejde-741	574	7	carvalho	carvalho	PROPN
ejde-741	574	8	,	,	PUNCT
ejde-741	574	9	reginaldo	reginaldo	NOUN
ejde-741	574	10	demarque	demarque	NOUN
ejde-741	574	11	,	,	PUNCT
ejde-741	574	12	juan	juan	PROPN
ejde-741	574	13	ĺımaco	ĺımaco	PROPN
ejde-741	574	14	,	,	PUNCT
ejde-741	574	15	luiz	luiz	NOUN
ejde-741	574	16	viana	viana	PROPN
ejde-741	574	17	;	;	PUNCT
ejde-741	574	18	null	null	ADJ
ejde-741	574	19	controllability	controllability	NOUN
ejde-741	574	20	and	and	CCONJ
ejde-741	574	21	numerical	numerical	ADJ
ejde-741	574	22	simulations	simulation	NOUN
ejde-741	574	23	for	for	ADP
ejde-741	574	24	a	a	DET
ejde-741	574	25	class	class	NOUN
ejde-741	574	26	of	of	ADP
ejde-741	574	27	degenerate	degenerate	ADJ
ejde-741	574	28	parabolic	parabolic	ADJ
ejde-741	574	29	equations	equation	NOUN
ejde-741	574	30	with	with	ADP
ejde-741	574	31	nonlocal	nonlocal	ADJ
ejde-741	574	32	nonlinearities	nonlinearitie	NOUN
ejde-741	574	33	,	,	PUNCT
ejde-741	574	34	nonlinear	nonlinear	ADJ
ejde-741	574	35	differential	differential	ADJ
ejde-741	574	36	equations	equation	NOUN
ejde-741	574	37	and	and	CCONJ
ejde-741	574	38	applications	application	NOUN
ejde-741	574	39	no	no	DET
ejde-741	574	40	dea	dea	NOUN
ejde-741	574	41	,	,	PUNCT
ejde-741	574	42	30	30	NUM
ejde-741	574	43	(	(	PUNCT
ejde-741	574	44	2023	2023	NUM
ejde-741	574	45	)	)	PUNCT
ejde-741	574	46	,	,	PUNCT
ejde-741	574	47	no	no	INTJ
ejde-741	574	48	.	.	NOUN
ejde-741	574	49	3	3	NUM
ejde-741	574	50	,	,	PUNCT
ejde-741	574	51	32	32	NUM
ejde-741	574	52	.	.	PUNCT
ejde-741	575	1	[	[	X
ejde-741	575	2	20	20	NUM
ejde-741	575	3	]	]	PUNCT
ejde-741	575	4	reginaldo	reginaldo	NOUN
ejde-741	575	5	demarque	demarque	NOUN
ejde-741	575	6	,	,	PUNCT
ejde-741	575	7	juan	juan	PROPN
ejde-741	575	8	ĺımaco	ĺımaco	PROPN
ejde-741	575	9	,	,	PUNCT
ejde-741	575	10	luiz	luiz	NOUN
ejde-741	575	11	viana	viana	PROPN
ejde-741	575	12	;	;	PUNCT
ejde-741	575	13	local	local	ADJ
ejde-741	575	14	null	null	ADJ
ejde-741	575	15	controllability	controllability	NOUN
ejde-741	575	16	for	for	ADP
ejde-741	575	17	degenerate	degenerate	ADJ
ejde-741	575	18	parabolic	parabolic	ADJ
ejde-741	575	19	equations	equation	NOUN
ejde-741	575	20	with	with	ADP
ejde-741	575	21	nonlocal	nonlocal	ADJ
ejde-741	575	22	term	term	NOUN
ejde-741	575	23	,	,	PUNCT
ejde-741	575	24	nonlinear	nonlinear	ADJ
ejde-741	575	25	analysis	analysis	NOUN
ejde-741	575	26	:	:	PUNCT
ejde-741	575	27	real	real	ADJ
ejde-741	575	28	world	world	NOUN
ejde-741	575	29	applications	application	NOUN
ejde-741	575	30	,	,	PUNCT
ejde-741	575	31	43	43	NUM
ejde-741	575	32	(	(	PUNCT
ejde-741	575	33	2018	2018	NUM
ejde-741	575	34	)	)	PUNCT
ejde-741	575	35	,	,	PUNCT
ejde-741	575	36	523–547	523–547	NUM
ejde-741	575	37	.	.	PUNCT
ejde-741	576	1	[	[	X
ejde-741	576	2	21	21	NUM
ejde-741	576	3	]	]	PUNCT
ejde-741	576	4	reginaldo	reginaldo	NOUN
ejde-741	576	5	demarque	demarque	NOUN
ejde-741	576	6	,	,	PUNCT
ejde-741	576	7	juan	juan	PROPN
ejde-741	576	8	ĺımaco	ĺımaco	PROPN
ejde-741	576	9	,	,	PUNCT
ejde-741	576	10	luiz	luiz	NOUN
ejde-741	576	11	viana	viana	PROPN
ejde-741	576	12	;	;	PUNCT
ejde-741	576	13	local	local	ADJ
ejde-741	576	14	null	null	ADJ
ejde-741	576	15	controllability	controllability	NOUN
ejde-741	576	16	of	of	ADP
ejde-741	576	17	coupled	couple	VERB
ejde-741	576	18	degenerate	degenerate	ADJ
ejde-741	576	19	systems	system	NOUN
ejde-741	576	20	with	with	ADP
ejde-741	576	21	nonlocal	nonlocal	ADJ
ejde-741	576	22	terms	term	NOUN
ejde-741	576	23	and	and	CCONJ
ejde-741	576	24	one	one	NUM
ejde-741	576	25	control	control	NOUN
ejde-741	576	26	force	force	NOUN
ejde-741	576	27	,	,	PUNCT
ejde-741	576	28	evolution	evolution	NOUN
ejde-741	576	29	equations	equation	NOUN
ejde-741	576	30	&	&	CCONJ
ejde-741	576	31	control	control	PROPN
ejde-741	576	32	theory	theory	NOUN
ejde-741	576	33	,	,	PUNCT
ejde-741	576	34	9	9	NUM
ejde-741	576	35	(	(	PUNCT
ejde-741	576	36	2020	2020	NUM
ejde-741	576	37	)	)	PUNCT
ejde-741	576	38	,	,	PUNCT
ejde-741	576	39	605	605	NUM
ejde-741	576	40	.	.	PUNCT
ejde-741	577	1	[	[	X
ejde-741	577	2	22	22	NUM
ejde-741	577	3	]	]	X
ejde-741	577	4	runmei	runmei	PROPN
ejde-741	577	5	du	du	PROPN
ejde-741	577	6	;	;	PUNCT
ejde-741	577	7	null	null	ADJ
ejde-741	577	8	controllability	controllability	NOUN
ejde-741	577	9	for	for	ADP
ejde-741	577	10	a	a	DET
ejde-741	577	11	class	class	NOUN
ejde-741	577	12	of	of	ADP
ejde-741	577	13	degenerate	degenerate	ADJ
ejde-741	577	14	parabolic	parabolic	ADJ
ejde-741	577	15	equations	equation	NOUN
ejde-741	577	16	with	with	ADP
ejde-741	577	17	the	the	DET
ejde-741	577	18	gradient	gradient	ADJ
ejde-741	577	19	terms	term	NOUN
ejde-741	577	20	,	,	PUNCT
ejde-741	577	21	journal	journal	NOUN
ejde-741	577	22	of	of	ADP
ejde-741	577	23	evolution	evolution	NOUN
ejde-741	577	24	equations	equation	NOUN
ejde-741	577	25	,	,	PUNCT
ejde-741	577	26	19	19	NUM
ejde-741	577	27	(	(	PUNCT
ejde-741	577	28	2019	2019	NUM
ejde-741	577	29	)	)	PUNCT
ejde-741	577	30	,	,	PUNCT
ejde-741	577	31	no	no	INTJ
ejde-741	577	32	.	.	NOUN
ejde-741	577	33	2	2	NUM
ejde-741	577	34	,	,	PUNCT
ejde-741	577	35	585–613	585–613	NUM
ejde-741	577	36	.	.	PUNCT
ejde-741	578	1	[	[	X
ejde-741	578	2	23	23	NUM
ejde-741	578	3	]	]	PUNCT
ejde-741	578	4	ivar	ivar	NOUN
ejde-741	578	5	ekeland	ekeland	NOUN
ejde-741	578	6	,	,	PUNCT
ejde-741	578	7	roger	roger	PROPN
ejde-741	578	8	temam	temam	NOUN
ejde-741	578	9	;	;	PUNCT
ejde-741	578	10	convex	convex	VERB
ejde-741	578	11	analysis	analysis	NOUN
ejde-741	578	12	and	and	CCONJ
ejde-741	578	13	variational	variational	ADJ
ejde-741	578	14	problems	problem	NOUN
ejde-741	578	15	,	,	PUNCT
ejde-741	578	16	vol	vol	NOUN
ejde-741	578	17	.	.	PROPN
ejde-741	578	18	28	28	NUM
ejde-741	578	19	,	,	PUNCT
ejde-741	578	20	siam	siam	NOUN
ejde-741	578	21	,	,	PUNCT
ejde-741	578	22	1999	1999	NUM
ejde-741	578	23	.	.	PUNCT
ejde-741	579	1	[	[	X
ejde-741	579	2	24	24	NUM
ejde-741	579	3	]	]	X
ejde-741	579	4	ait	ait	NOUN
ejde-741	579	5	ben	ben	PROPN
ejde-741	579	6	hassi	hassi	PROPN
ejde-741	579	7	el	el	PROPN
ejde-741	579	8	mustapha	mustapha	PROPN
ejde-741	579	9	,	,	PUNCT
ejde-741	579	10	fadili	fadili	NOUN
ejde-741	579	11	mohamed	mohamed	PROPN
ejde-741	579	12	,	,	PUNCT
ejde-741	579	13	maniar	maniar	NOUN
ejde-741	579	14	lahcen	lahcen	NOUN
ejde-741	579	15	;	;	PUNCT
ejde-741	579	16	on	on	ADP
ejde-741	579	17	algebraic	algebraic	ADJ
ejde-741	579	18	condition	condition	NOUN
ejde-741	579	19	for	for	ADP
ejde-741	579	20	null	null	ADJ
ejde-741	579	21	controllability	controllability	NOUN
ejde-741	579	22	of	of	ADP
ejde-741	579	23	some	some	DET
ejde-741	579	24	coupled	couple	VERB
ejde-741	579	25	degenerate	degenerate	ADJ
ejde-741	579	26	systems	system	NOUN
ejde-741	579	27	,	,	PUNCT
ejde-741	579	28	mathematical	mathematical	ADJ
ejde-741	579	29	control	control	NOUN
ejde-741	579	30	and	and	CCONJ
ejde-741	579	31	related	related	ADJ
ejde-741	579	32	fields	field	NOUN
ejde-741	579	33	9	9	NUM
ejde-741	579	34	(	(	PUNCT
ejde-741	579	35	2019	2019	NUM
ejde-741	579	36	)	)	PUNCT
ejde-741	579	37	,	,	PUNCT
ejde-741	579	38	no	no	INTJ
ejde-741	579	39	.	.	NOUN
ejde-741	579	40	1	1	NUM
ejde-741	579	41	,	,	PUNCT
ejde-741	579	42	77–95	77–95	NUM
ejde-741	579	43	.	.	PUNCT
ejde-741	580	1	[	[	X
ejde-741	580	2	25	25	NUM
ejde-741	580	3	]	]	PUNCT
ejde-741	580	4	caroline	caroline	PROPN
ejde-741	580	5	fabre	fabre	PROPN
ejde-741	580	6	;	;	PUNCT
ejde-741	580	7	exact	exact	ADJ
ejde-741	580	8	boundary	boundary	ADJ
ejde-741	580	9	controllability	controllability	NOUN
ejde-741	580	10	of	of	ADP
ejde-741	580	11	the	the	DET
ejde-741	580	12	wave	wave	NOUN
ejde-741	580	13	equation	equation	NOUN
ejde-741	580	14	as	as	ADP
ejde-741	580	15	the	the	DET
ejde-741	580	16	limit	limit	NOUN
ejde-741	580	17	of	of	ADP
ejde-741	580	18	internal	internal	ADJ
ejde-741	580	19	controllability	controllability	NOUN
ejde-741	580	20	,	,	PUNCT
ejde-741	580	21	siam	siam	ADJ
ejde-741	580	22	journal	journal	NOUN
ejde-741	580	23	on	on	ADP
ejde-741	580	24	control	control	NOUN
ejde-741	580	25	and	and	CCONJ
ejde-741	580	26	optimization	optimization	NOUN
ejde-741	580	27	,	,	PUNCT
ejde-741	580	28	30	30	NUM
ejde-741	580	29	(	(	PUNCT
ejde-741	580	30	1992	1992	NUM
ejde-741	580	31	)	)	PUNCT
ejde-741	580	32	,	,	PUNCT
ejde-741	580	33	no	no	INTJ
ejde-741	580	34	.	.	NOUN
ejde-741	580	35	5	5	NUM
ejde-741	580	36	,	,	PUNCT
ejde-741	580	37	1066–1086	1066–1086	NUM
ejde-741	580	38	.	.	PUNCT
ejde-741	580	39	22	22	NUM
ejde-741	581	1	p.	p.	NOUN
ejde-741	581	2	p.	p.	NOUN
ejde-741	581	3	de	de	PROPN
ejde-741	581	4	carvalho	carvalho	PROPN
ejde-741	581	5	,	,	PUNCT
ejde-741	581	6	r.	r.	PROPN
ejde-741	581	7	demarque	demarque	PROPN
ejde-741	581	8	,	,	PUNCT
ejde-741	581	9	j.	j.	PROPN
ejde-741	581	10	límaco	límaco	PROPN
ejde-741	581	11	,	,	PUNCT
ejde-741	581	12	l.	l.	PROPN
ejde-741	581	13	viana	viana	PROPN
ejde-741	581	14	ejde-2025/15	ejde-2025/15	VERB
ejde-741	582	1	[	[	X
ejde-741	582	2	26	26	NUM
ejde-741	582	3	]	]	X
ejde-741	582	4	josiane	josiane	PROPN
ejde-741	582	5	c.	c.	PROPN
ejde-741	582	6	o.	o.	PROPN
ejde-741	582	7	faria	faria	PROPN
ejde-741	582	8	;	;	PUNCT
ejde-741	582	9	carleman	carleman	ADJ
ejde-741	582	10	estimates	estimate	NOUN
ejde-741	582	11	and	and	CCONJ
ejde-741	582	12	observability	observability	NOUN
ejde-741	582	13	inequalities	inequality	NOUN
ejde-741	582	14	for	for	ADP
ejde-741	582	15	a	a	DET
ejde-741	582	16	class	class	NOUN
ejde-741	582	17	of	of	ADP
ejde-741	582	18	problems	problem	NOUN
ejde-741	582	19	ruled	rule	VERB
ejde-741	582	20	by	by	ADP
ejde-741	582	21	parabolic	parabolic	ADJ
ejde-741	582	22	equations	equation	NOUN
ejde-741	582	23	with	with	ADP
ejde-741	582	24	interior	interior	ADJ
ejde-741	582	25	degenaracy	degenaracy	NOUN
ejde-741	582	26	,	,	PUNCT
ejde-741	582	27	applied	apply	VERB
ejde-741	582	28	mathematics	mathematics	PROPN
ejde-741	582	29	&	&	CCONJ
ejde-741	582	30	optimization	optimization	NOUN
ejde-741	582	31	,	,	PUNCT
ejde-741	582	32	(	(	PUNCT
ejde-741	582	33	2020	2020	NUM
ejde-741	582	34	)	)	PUNCT
ejde-741	582	35	,	,	PUNCT
ejde-741	582	36	1–24	1–24	PROPN
ejde-741	582	37	.	.	PUNCT
ejde-741	583	1	[	[	X
ejde-741	583	2	27	27	NUM
ejde-741	583	3	]	]	X
ejde-741	583	4	enrique	enrique	PROPN
ejde-741	583	5	fernández	fernández	PROPN
ejde-741	583	6	-	-	PUNCT
ejde-741	583	7	cara	cara	PROPN
ejde-741	583	8	,	,	PUNCT
ejde-741	583	9	arnaud	arnaud	PROPN
ejde-741	583	10	munch	munch	PROPN
ejde-741	583	11	;	;	PUNCT
ejde-741	583	12	numerical	numerical	PROPN
ejde-741	583	13	null	null	ADJ
ejde-741	583	14	controllability	controllability	NOUN
ejde-741	583	15	of	of	ADP
ejde-741	583	16	the	the	DET
ejde-741	583	17	1d	1d	NUM
ejde-741	583	18	heat	heat	NOUN
ejde-741	583	19	equation	equation	NOUN
ejde-741	583	20	:	:	PUNCT
ejde-741	583	21	primal	primal	ADJ
ejde-741	583	22	methods	method	NOUN
ejde-741	583	23	,	,	PUNCT
ejde-741	583	24	preprint	preprint	NOUN
ejde-741	583	25	submitted	submit	VERB
ejde-741	583	26	to	to	ADP
ejde-741	583	27	hal-00687884	hal-00687884	NOUN
ejde-741	583	28	(	(	PUNCT
ejde-741	583	29	2011	2011	NUM
ejde-741	583	30	)	)	PUNCT
ejde-741	583	31	.	.	PUNCT
ejde-741	584	1	[	[	X
ejde-741	584	2	28	28	NUM
ejde-741	584	3	]	]	X
ejde-741	584	4	enrique	enrique	PROPN
ejde-741	584	5	fernández	fernández	PROPN
ejde-741	584	6	-	-	PUNCT
ejde-741	584	7	cara	cara	PROPN
ejde-741	584	8	,	,	PUNCT
ejde-741	584	9	dany	dany	PROPN
ejde-741	584	10	nina	nina	PROPN
ejde-741	584	11	-	-	PROPN
ejde-741	584	12	huamán	huamán	PROPN
ejde-741	584	13	,	,	PUNCT
ejde-741	584	14	miguel	miguel	PROPN
ejde-741	584	15	r	r	PROPN
ejde-741	584	16	nuñez	nuñez	PROPN
ejde-741	584	17	-	-	PUNCT
ejde-741	584	18	chávez	chávez	PROPN
ejde-741	584	19	,	,	PUNCT
ejde-741	584	20	franciane	franciane	PROPN
ejde-741	584	21	b.	b.	PROPN
ejde-741	584	22	vieira	vieira	PROPN
ejde-741	584	23	;	;	PUNCT
ejde-741	584	24	on	on	ADP
ejde-741	584	25	the	the	DET
ejde-741	584	26	theoretical	theoretical	ADJ
ejde-741	584	27	and	and	CCONJ
ejde-741	584	28	numerical	numerical	ADJ
ejde-741	584	29	control	control	NOUN
ejde-741	584	30	of	of	ADP
ejde-741	584	31	a	a	DET
ejde-741	584	32	one	one	NUM
ejde-741	584	33	-	-	PUNCT
ejde-741	584	34	dimensional	dimensional	ADJ
ejde-741	584	35	nonlinear	nonlinear	ADJ
ejde-741	584	36	parabolic	parabolic	ADJ
ejde-741	584	37	partial	partial	ADJ
ejde-741	584	38	differential	differential	NOUN
ejde-741	584	39	equation	equation	NOUN
ejde-741	584	40	,	,	PUNCT
ejde-741	584	41	journal	journal	NOUN
ejde-741	584	42	of	of	ADP
ejde-741	584	43	optimization	optimization	NOUN
ejde-741	584	44	theory	theory	NOUN
ejde-741	584	45	and	and	CCONJ
ejde-741	584	46	applications	application	NOUN
ejde-741	584	47	,	,	PUNCT
ejde-741	584	48	175	175	NUM
ejde-741	584	49	(	(	PUNCT
ejde-741	584	50	2017	2017	NUM
ejde-741	584	51	)	)	PUNCT
ejde-741	584	52	,	,	PUNCT
ejde-741	584	53	no	no	INTJ
ejde-741	584	54	.	.	NOUN
ejde-741	584	55	3	3	NUM
ejde-741	584	56	,	,	PUNCT
ejde-741	584	57	652–682	652–682	NUM
ejde-741	584	58	.	.	PUNCT
ejde-741	585	1	[	[	X
ejde-741	585	2	29	29	NUM
ejde-741	585	3	]	]	X
ejde-741	585	4	genni	genni	PROPN
ejde-741	585	5	fragnelli	fragnelli	NOUN
ejde-741	585	6	;	;	PUNCT
ejde-741	585	7	carleman	carleman	ADJ
ejde-741	585	8	estimates	estimate	NOUN
ejde-741	585	9	and	and	CCONJ
ejde-741	585	10	null	null	ADJ
ejde-741	585	11	controllability	controllability	NOUN
ejde-741	585	12	for	for	ADP
ejde-741	585	13	a	a	DET
ejde-741	585	14	degenerate	degenerate	ADJ
ejde-741	585	15	population	population	NOUN
ejde-741	585	16	model	model	NOUN
ejde-741	585	17	,	,	PUNCT
ejde-741	585	18	journal	journal	PROPN
ejde-741	585	19	de	de	PROPN
ejde-741	585	20	mathematiques	mathematiques	PROPN
ejde-741	585	21	pures	pure	NOUN
ejde-741	585	22	et	et	PROPN
ejde-741	585	23	appliquees	appliquee	NOUN
ejde-741	585	24	,	,	PUNCT
ejde-741	585	25	115	115	NUM
ejde-741	585	26	(	(	PUNCT
ejde-741	585	27	2018	2018	NUM
ejde-741	585	28	)	)	PUNCT
ejde-741	585	29	,	,	PUNCT
ejde-741	585	30	74–126	74–126	PROPN
ejde-741	585	31	.	.	PUNCT
ejde-741	586	1	[	[	X
ejde-741	586	2	30	30	NUM
ejde-741	586	3	]	]	X
ejde-741	586	4	andrej	andrej	PROPN
ejde-741	586	5	vladimirovič	vladimirovič	PROPN
ejde-741	586	6	fursikov	fursikov	PROPN
ejde-741	586	7	,	,	PUNCT
ejde-741	586	8	oleg	oleg	PROPN
ejde-741	586	9	yu	yu	PROPN
ejde-741	586	10	imanuvilov	imanuvilov	PROPN
ejde-741	586	11	;	;	PUNCT
ejde-741	586	12	controllability	controllability	NOUN
ejde-741	586	13	of	of	ADP
ejde-741	586	14	evolution	evolution	NOUN
ejde-741	586	15	equations	equation	NOUN
ejde-741	586	16	,	,	PUNCT
ejde-741	586	17	vol	vol	NOUN
ejde-741	586	18	.	.	PUNCT
ejde-741	587	1	lecture	lecture	NOUN
ejde-741	587	2	notes	note	NOUN
ejde-741	587	3	,	,	PUNCT
ejde-741	587	4	n.	n.	NOUN
ejde-741	587	5	34	34	NUM
ejde-741	587	6	,	,	PUNCT
ejde-741	587	7	seoul	seoul	PROPN
ejde-741	587	8	national	national	PROPN
ejde-741	587	9	university	university	PROPN
ejde-741	587	10	,	,	PUNCT
ejde-741	587	11	1996	1996	NUM
ejde-741	587	12	.	.	PUNCT
ejde-741	588	1	[	[	X
ejde-741	588	2	31	31	NUM
ejde-741	588	3	]	]	X
ejde-741	588	4	dany	dany	PROPN
ejde-741	588	5	nina	nina	PROPN
ejde-741	588	6	huaman	huaman	PROPN
ejde-741	588	7	,	,	PUNCT
ejde-741	588	8	miguel	miguel	PROPN
ejde-741	588	9	r.	r.	PROPN
ejde-741	588	10	nuñez	nuñez	PROPN
ejde-741	588	11	-	-	PUNCT
ejde-741	588	12	chávez	chávez	PROPN
ejde-741	588	13	,	,	PUNCT
ejde-741	588	14	juan	juan	PROPN
ejde-741	588	15	ĺımaco	ĺımaco	PROPN
ejde-741	588	16	,	,	PUNCT
ejde-741	588	17	pitágoras	pitágoras	PROPN
ejde-741	588	18	p.	p.	NOUN
ejde-741	588	19	carvalho	carvalho	NOUN
ejde-741	588	20	;	;	PUNCT
ejde-741	588	21	local	local	ADJ
ejde-741	588	22	null	null	ADJ
ejde-741	588	23	controllability	controllability	NOUN
ejde-741	588	24	for	for	ADP
ejde-741	588	25	the	the	DET
ejde-741	588	26	thermistor	thermistor	NOUN
ejde-741	588	27	problem	problem	NOUN
ejde-741	588	28	,	,	PUNCT
ejde-741	588	29	nonlinear	nonlinear	ADJ
ejde-741	588	30	analysis	analysis	NOUN
ejde-741	588	31	236	236	NUM
ejde-741	588	32	(	(	PUNCT
ejde-741	588	33	2023	2023	NUM
ejde-741	588	34	)	)	PUNCT
ejde-741	588	35	,	,	PUNCT
ejde-741	588	36	113330	113330	NUM
ejde-741	588	37	.	.	PUNCT
ejde-741	589	1	[	[	X
ejde-741	589	2	32	32	NUM
ejde-741	589	3	]	]	X
ejde-741	589	4	patrick	patrick	PROPN
ejde-741	589	5	martinez	martinez	PROPN
ejde-741	589	6	,	,	PUNCT
ejde-741	589	7	judith	judith	PROPN
ejde-741	589	8	vancostenoble	vancostenoble	ADJ
ejde-741	589	9	;	;	PUNCT
ejde-741	589	10	carleman	carleman	ADJ
ejde-741	589	11	estimates	estimate	NOUN
ejde-741	589	12	for	for	ADP
ejde-741	589	13	one	one	NUM
ejde-741	589	14	-	-	PUNCT
ejde-741	589	15	dimensional	dimensional	ADJ
ejde-741	589	16	degenerate	degenerate	ADJ
ejde-741	589	17	heat	heat	NOUN
ejde-741	589	18	equations	equation	NOUN
ejde-741	589	19	,	,	PUNCT
ejde-741	589	20	journal	journal	NOUN
ejde-741	589	21	of	of	ADP
ejde-741	589	22	evolution	evolution	NOUN
ejde-741	589	23	equations	equation	NOUN
ejde-741	589	24	,	,	PUNCT
ejde-741	589	25	6	6	NUM
ejde-741	589	26	(	(	PUNCT
ejde-741	589	27	2006	2006	NUM
ejde-741	589	28	)	)	PUNCT
ejde-741	589	29	,	,	PUNCT
ejde-741	589	30	no	no	INTJ
ejde-741	589	31	.	.	NOUN
ejde-741	589	32	2	2	NUM
ejde-741	589	33	,	,	PUNCT
ejde-741	589	34	325–362	325–362	NUM
ejde-741	589	35	.	.	PUNCT
ejde-741	590	1	[	[	X
ejde-741	590	2	33	33	NUM
ejde-741	590	3	]	]	X
ejde-741	590	4	chunpeng	chunpeng	PROPN
ejde-741	590	5	wang	wang	PROPN
ejde-741	590	6	,	,	PUNCT
ejde-741	590	7	yanan	yanan	PROPN
ejde-741	590	8	zhou	zhou	PROPN
ejde-741	590	9	,	,	PUNCT
ejde-741	590	10	runmei	runmei	PROPN
ejde-741	590	11	du	du	PROPN
ejde-741	590	12	,	,	PUNCT
ejde-741	590	13	qiang	qiang	PROPN
ejde-741	590	14	liu	liu	PROPN
ejde-741	590	15	;	;	PUNCT
ejde-741	590	16	carleman	carleman	ADJ
ejde-741	590	17	estimate	estimate	NOUN
ejde-741	590	18	for	for	ADP
ejde-741	590	19	solutions	solution	NOUN
ejde-741	590	20	to	to	ADP
ejde-741	590	21	a	a	DET
ejde-741	590	22	degenerate	degenerate	ADJ
ejde-741	590	23	convection	convection	NOUN
ejde-741	590	24	-	-	PUNCT
ejde-741	590	25	diffusion	diffusion	NOUN
ejde-741	590	26	equation	equation	NOUN
ejde-741	590	27	,	,	PUNCT
ejde-741	590	28	discrete	discrete	ADJ
ejde-741	590	29	and	and	CCONJ
ejde-741	590	30	continuous	continuous	ADJ
ejde-741	590	31	dynamical	dynamical	ADJ
ejde-741	590	32	systems	system	NOUN
ejde-741	590	33	-	-	PUNCT
ejde-741	590	34	series	series	NOUN
ejde-741	590	35	b	b	PROPN
ejde-741	590	36	,	,	PUNCT
ejde-741	590	37	23	23	NUM
ejde-741	590	38	(	(	PUNCT
ejde-741	590	39	2018	2018	NUM
ejde-741	590	40	)	)	PUNCT
ejde-741	590	41	,	,	PUNCT
ejde-741	590	42	no	no	INTJ
ejde-741	590	43	.	.	NOUN
ejde-741	590	44	10	10	NUM
ejde-741	590	45	,	,	PUNCT
ejde-741	590	46	4207–4222	4207–4222	NUM
ejde-741	590	47	.	.	PUNCT
ejde-741	591	1	pitágoras	pitágoras	NUM
ejde-741	591	2	p.	p.	PROPN
ejde-741	591	3	de	de	PROPN
ejde-741	591	4	carvalho	carvalho	PROPN
ejde-741	591	5	coordenação	coordenação	PROPN
ejde-741	591	6	de	de	PROPN
ejde-741	591	7	matemática	matemática	PROPN
ejde-741	591	8	,	,	PUNCT
ejde-741	591	9	universidade	universidade	PROPN
ejde-741	591	10	estadual	estadual	PROPN
ejde-741	591	11	do	do	AUX
ejde-741	591	12	piaúı	piaúı	PROPN
ejde-741	591	13	,	,	PUNCT
ejde-741	591	14	teresina	teresina	NOUN
ejde-741	591	15	,	,	PUNCT
ejde-741	591	16	pi	pi	NOUN
ejde-741	591	17	,	,	PUNCT
ejde-741	591	18	64002	64002	NUM
ejde-741	591	19	-	-	SYM
ejde-741	591	20	150	150	NUM
ejde-741	591	21	,	,	PUNCT
ejde-741	591	22	brazil	brazil	PROPN
ejde-741	591	23	email	email	NOUN
ejde-741	591	24	address	address	NOUN
ejde-741	591	25	:	:	PUNCT
ejde-741	591	26	pitagorascarvalho@gmail.com	pitagorascarvalho@gmail.com	X
ejde-741	591	27	reginaldo	reginaldo	PROPN
ejde-741	591	28	demarque	demarque	ADJ
ejde-741	591	29	(	(	PUNCT
ejde-741	591	30	corresponding	corresponding	ADJ
ejde-741	591	31	author	author	NOUN
ejde-741	591	32	)	)	PUNCT
ejde-741	591	33	departamento	departamento	NOUN
ejde-741	591	34	de	de	PROPN
ejde-741	591	35	ciências	ciências	PROPN
ejde-741	591	36	da	da	PROPN
ejde-741	591	37	natureza	natureza	PROPN
ejde-741	591	38	,	,	PUNCT
ejde-741	591	39	universidade	universidade	PROPN
ejde-741	591	40	federal	federal	PROPN
ejde-741	591	41	fluminense	fluminense	PROPN
ejde-741	591	42	,	,	PUNCT
ejde-741	591	43	rio	rio	PROPN
ejde-741	591	44	das	das	PROPN
ejde-741	591	45	ostras	ostras	PROPN
ejde-741	591	46	,	,	PUNCT
ejde-741	591	47	rj	rj	PROPN
ejde-741	591	48	,	,	PUNCT
ejde-741	591	49	28895	28895	NUM
ejde-741	591	50	-	-	SYM
ejde-741	591	51	532	532	NUM
ejde-741	591	52	,	,	PUNCT
ejde-741	591	53	brazil	brazil	PROPN
ejde-741	591	54	email	email	NOUN
ejde-741	591	55	address	address	NOUN
ejde-741	591	56	:	:	PUNCT
ejde-741	591	57	reginaldodr@id.uff.br	reginaldodr@id.uff.br	PROPN
ejde-741	591	58	juan	juan	PROPN
ejde-741	591	59	ĺımaco	ĺımaco	PROPN
ejde-741	591	60	departamento	departamento	PROPN
ejde-741	591	61	de	de	PROPN
ejde-741	591	62	matemática	matemática	PROPN
ejde-741	591	63	aplicada	aplicada	PROPN
ejde-741	591	64	,	,	PUNCT
ejde-741	591	65	universidade	universidade	PROPN
ejde-741	591	66	federal	federal	PROPN
ejde-741	591	67	fluminense	fluminense	PROPN
ejde-741	591	68	,	,	PUNCT
ejde-741	591	69	niterói	niterói	PROPN
ejde-741	591	70	,	,	PUNCT
ejde-741	591	71	rj	rj	PROPN
ejde-741	591	72	,	,	PUNCT
ejde-741	591	73	24210	24210	NUM
ejde-741	591	74	-	-	SYM
ejde-741	591	75	201	201	NUM
ejde-741	591	76	,	,	PUNCT
ejde-741	591	77	brazil	brazil	PROPN
ejde-741	591	78	email	email	NOUN
ejde-741	591	79	address	address	NOUN
ejde-741	591	80	:	:	PUNCT
ejde-741	591	81	jlimaco@id.uff.br	jlimaco@id.uff.br	ADJ
ejde-741	591	82	luiz	luiz	PROPN
ejde-741	591	83	viana	viana	PROPN
ejde-741	591	84	departamento	departamento	PROPN
ejde-741	591	85	de	de	PROPN
ejde-741	591	86	análise	análise	PROPN
ejde-741	591	87	,	,	PUNCT
ejde-741	591	88	universidade	universidade	PROPN
ejde-741	591	89	federal	federal	PROPN
ejde-741	591	90	fluminense	fluminense	PROPN
ejde-741	591	91	,	,	PUNCT
ejde-741	591	92	niterói	niterói	PROPN
ejde-741	591	93	,	,	PUNCT
ejde-741	591	94	rj	rj	PROPN
ejde-741	591	95	,	,	PUNCT
ejde-741	591	96	24210	24210	NUM
ejde-741	591	97	-	-	SYM
ejde-741	591	98	2010	2010	NUM
ejde-741	591	99	,	,	PUNCT
ejde-741	591	100	brazil	brazil	PROPN
ejde-741	591	101	email	email	NOUN
ejde-741	591	102	address	address	NOUN
ejde-741	591	103	:	:	PUNCT
ejde-741	591	104	luizviana@id.uff.br	luizviana@id.uff.br	ADJ
ejde-741	591	105	1	1	NUM
ejde-741	591	106	.	.	PUNCT
ejde-741	591	107	introduction	introduction	NOUN
ejde-741	591	108	2	2	NUM
ejde-741	591	109	.	.	PUNCT
ejde-741	591	110	preliminary	preliminary	ADJ
ejde-741	591	111	results	result	NOUN
ejde-741	591	112	2.1	2.1	NUM
ejde-741	591	113	.	.	PUNCT
ejde-741	591	114	notation	notation	NOUN
ejde-741	591	115	and	and	CCONJ
ejde-741	591	116	results	result	NOUN
ejde-741	591	117	related	relate	VERB
ejde-741	591	118	to	to	ADP
ejde-741	591	119	the	the	DET
ejde-741	591	120	local	local	ADJ
ejde-741	591	121	inversion	inversion	NOUN
ejde-741	591	122	argument	argument	NOUN
ejde-741	591	123	2.2	2.2	NUM
ejde-741	591	124	.	.	PUNCT
ejde-741	592	1	carleman	carleman	ADJ
ejde-741	592	2	inequality	inequality	NOUN
ejde-741	592	3	3	3	NUM
ejde-741	592	4	.	.	PUNCT
ejde-741	593	1	properties	property	NOUN
ejde-741	593	2	of	of	ADP
ejde-741	593	3	the	the	DET
ejde-741	593	4	mapping	mapping	NOUN
ejde-741	593	5	h	h	NOUN
ejde-741	593	6	3.1	3.1	NUM
ejde-741	593	7	.	.	PUNCT
ejde-741	593	8	surjectiveness	surjectiveness	NOUN
ejde-741	593	9	of	of	ADP
ejde-741	593	10	h'(0,0	h'(0,0	NOUN
ejde-741	593	11	)	)	PUNCT
ejde-741	593	12	3.2	3.2	NUM
ejde-741	593	13	.	.	PUNCT
ejde-741	594	1	h	h	NOUN
ejde-741	594	2	is	be	AUX
ejde-741	594	3	continuously	continuously	ADV
ejde-741	594	4	differentiable	differentiable	ADJ
ejde-741	594	5	3.3	3.3	NUM
ejde-741	594	6	.	.	PUNCT
ejde-741	595	1	proof	proof	NOUN
ejde-741	595	2	of	of	ADP
ejde-741	595	3	proposition	proposition	NOUN
ejde-741	595	4	?	?	PUNCT
ejde-741	595	5	?	?	PUNCT
ejde-741	596	1	4	4	X
ejde-741	596	2	.	.	X
ejde-741	596	3	main	main	ADJ
ejde-741	596	4	result	result	NOUN
ejde-741	596	5	and	and	CCONJ
ejde-741	596	6	further	further	ADJ
ejde-741	596	7	comments	comment	NOUN
ejde-741	596	8	5	5	NUM
ejde-741	596	9	.	.	PUNCT
ejde-741	597	1	appendix	appendix	NOUN
ejde-741	597	2	:	:	PUNCT
ejde-741	597	3	essential	essential	ADJ
ejde-741	597	4	boundedness	boundedness	NOUN
ejde-741	597	5	of	of	ADP
ejde-741	597	6	au	au	PROPN
ejde-741	597	7	acknowledgments	acknowledgment	NOUN
ejde-741	597	8	references	reference	NOUN