id	sid	tid	token	lemma	pos
ejde-744	1	1	electronic	electronic	ADJ
ejde-744	1	2	journal	journal	NOUN
ejde-744	1	3	of	of	ADP
ejde-744	1	4	differential	differential	ADJ
ejde-744	1	5	equations	equation	NOUN
ejde-744	1	6	,	,	PUNCT
ejde-744	1	7	vol	vol	NOUN
ejde-744	1	8	.	.	PUNCT
ejde-744	1	9	2025	2025	NUM
ejde-744	1	10	(	(	PUNCT
ejde-744	1	11	2025	2025	NUM
ejde-744	1	12	)	)	PUNCT
ejde-744	1	13	,	,	PUNCT
ejde-744	1	14	no	no	INTJ
ejde-744	1	15	.	.	NOUN
ejde-744	1	16	04	04	NUM
ejde-744	1	17	,	,	PUNCT
ejde-744	1	18	pp	pp	ADJ
ejde-744	1	19	.	.	PUNCT
ejde-744	2	1	1–17	1–17	NOUN
ejde-744	2	2	.	.	PUNCT
ejde-744	3	1	issn	issn	PROPN
ejde-744	3	2	:	:	PUNCT
ejde-744	3	3	1072	1072	NUM
ejde-744	3	4	-	-	SYM
ejde-744	3	5	6691	6691	NUM
ejde-744	3	6	.	.	PUNCT
ejde-744	4	1	url	url	PROPN
ejde-744	4	2	:	:	PUNCT
ejde-744	4	3	https://ejde.math.txstate.edu	https://ejde.math.txstate.edu	PROPN
ejde-744	4	4	,	,	PUNCT
ejde-744	4	5	https://ejde.math.unt.edu	https://ejde.math.unt.edu	PROPN
ejde-744	4	6	doi	doi	PROPN
ejde-744	4	7	:	:	PUNCT
ejde-744	4	8	10.58997	10.58997	NUM
ejde-744	4	9	/	/	SYM
ejde-744	4	10	ejde.2025.04	ejde.2025.04	NOUN
ejde-744	4	11	exact	exact	ADJ
ejde-744	4	12	controllability	controllability	NOUN
ejde-744	4	13	for	for	ADP
ejde-744	4	14	degenerate	degenerate	ADJ
ejde-744	4	15	and	and	CCONJ
ejde-744	4	16	singular	singular	ADJ
ejde-744	4	17	wave	wave	NOUN
ejde-744	4	18	equations	equation	NOUN
ejde-744	4	19	guang	guang	PROPN
ejde-744	4	20	zhang	zhang	PROPN
ejde-744	4	21	,	,	PUNCT
ejde-744	4	22	shugen	shugen	PROPN
ejde-744	4	23	chai	chai	NOUN
ejde-744	4	24	abstract	abstract	NOUN
ejde-744	4	25	.	.	PUNCT
ejde-744	5	1	in	in	ADP
ejde-744	5	2	this	this	DET
ejde-744	5	3	article	article	NOUN
ejde-744	5	4	,	,	PUNCT
ejde-744	5	5	we	we	PRON
ejde-744	5	6	study	study	VERB
ejde-744	5	7	exact	exact	ADJ
ejde-744	5	8	controllability	controllability	NOUN
ejde-744	5	9	for	for	ADP
ejde-744	5	10	degenerate	degenerate	ADJ
ejde-744	5	11	and	and	CCONJ
ejde-744	5	12	singular	singular	ADJ
ejde-744	5	13	wave	wave	NOUN
ejde-744	5	14	equations	equation	NOUN
ejde-744	5	15	with	with	ADP
ejde-744	5	16	a	a	DET
ejde-744	5	17	general	general	ADJ
ejde-744	5	18	coefficient	coefficient	NOUN
ejde-744	5	19	.	.	PUNCT
ejde-744	6	1	we	we	PRON
ejde-744	6	2	estimate	estimate	VERB
ejde-744	6	3	the	the	DET
ejde-744	6	4	observability	observability	NOUN
ejde-744	6	5	inequality	inequality	NOUN
ejde-744	6	6	by	by	ADP
ejde-744	6	7	the	the	DET
ejde-744	6	8	multiplier	multipli	ADJ
ejde-744	6	9	method	method	NOUN
ejde-744	6	10	and	and	CCONJ
ejde-744	6	11	determine	determine	VERB
ejde-744	6	12	the	the	DET
ejde-744	6	13	observability	observability	NOUN
ejde-744	6	14	time	time	NOUN
ejde-744	6	15	.	.	PUNCT
ejde-744	7	1	we	we	PRON
ejde-744	7	2	also	also	ADV
ejde-744	7	3	deduce	deduce	VERB
ejde-744	7	4	the	the	DET
ejde-744	7	5	exact	exact	ADJ
ejde-744	7	6	controllability	controllability	NOUN
ejde-744	7	7	of	of	ADP
ejde-744	7	8	the	the	DET
ejde-744	7	9	corresponding	corresponding	ADJ
ejde-744	7	10	degenerate	degenerate	ADJ
ejde-744	7	11	and	and	CCONJ
ejde-744	7	12	singular	singular	PROPN
ejde-744	7	13	control	control	NOUN
ejde-744	7	14	problem	problem	NOUN
ejde-744	7	15	at	at	ADP
ejde-744	7	16	a	a	DET
ejde-744	7	17	sufficiently	sufficiently	ADV
ejde-744	7	18	large	large	ADJ
ejde-744	7	19	time	time	NOUN
ejde-744	7	20	,	,	PUNCT
ejde-744	7	21	employing	employ	VERB
ejde-744	7	22	the	the	DET
ejde-744	7	23	hilbert	hilbert	NOUN
ejde-744	7	24	uniqueness	uniqueness	NOUN
ejde-744	7	25	method	method	NOUN
ejde-744	7	26	.	.	PUNCT
ejde-744	8	1	1	1	X
ejde-744	8	2	.	.	X
ejde-744	8	3	introduction	introduction	NOUN
ejde-744	8	4	control	control	NOUN
ejde-744	8	5	issues	issue	NOUN
ejde-744	8	6	for	for	ADP
ejde-744	8	7	non	non	ADJ
ejde-744	8	8	-	-	ADJ
ejde-744	8	9	degenerate	degenerate	ADJ
ejde-744	8	10	parabolic	parabolic	ADJ
ejde-744	8	11	and	and	CCONJ
ejde-744	8	12	hyperbolic	hyperbolic	ADJ
ejde-744	8	13	problems	problem	NOUN
ejde-744	8	14	have	have	AUX
ejde-744	8	15	been	be	AUX
ejde-744	8	16	a	a	DET
ejde-744	8	17	mainstream	mainstream	NOUN
ejde-744	8	18	topic	topic	NOUN
ejde-744	8	19	over	over	ADP
ejde-744	8	20	the	the	DET
ejde-744	8	21	past	past	ADJ
ejde-744	8	22	several	several	ADJ
ejde-744	8	23	years	year	NOUN
ejde-744	8	24	,	,	PUNCT
ejde-744	8	25	and	and	CCONJ
ejde-744	8	26	a	a	DET
ejde-744	8	27	lot	lot	NOUN
ejde-744	8	28	of	of	ADP
ejde-744	8	29	attention	attention	NOUN
ejde-744	8	30	has	have	AUX
ejde-744	8	31	led	lead	VERB
ejde-744	8	32	to	to	ADP
ejde-744	8	33	numerous	numerous	ADJ
ejde-744	8	34	developments	development	NOUN
ejde-744	8	35	being	be	AUX
ejde-744	8	36	pursued	pursue	VERB
ejde-744	8	37	(	(	PUNCT
ejde-744	8	38	see	see	VERB
ejde-744	8	39	[	[	X
ejde-744	8	40	10	10	NUM
ejde-744	8	41	,	,	PUNCT
ejde-744	8	42	20	20	NUM
ejde-744	8	43	,	,	PUNCT
ejde-744	8	44	22	22	NUM
ejde-744	8	45	,	,	PUNCT
ejde-744	8	46	23	23	NUM
ejde-744	8	47	,	,	PUNCT
ejde-744	8	48	25	25	NUM
ejde-744	8	49	,	,	PUNCT
ejde-744	8	50	26	26	NUM
ejde-744	8	51	,	,	PUNCT
ejde-744	8	52	28	28	NUM
ejde-744	8	53	,	,	PUNCT
ejde-744	8	54	32	32	NUM
ejde-744	8	55	]	]	PUNCT
ejde-744	8	56	)	)	PUNCT
ejde-744	8	57	.	.	PUNCT
ejde-744	9	1	let	let	VERB
ejde-744	9	2	us	we	PRON
ejde-744	9	3	recall	recall	VERB
ejde-744	9	4	that	that	DET
ejde-744	9	5	exact	exact	ADJ
ejde-744	9	6	controllability	controllability	NOUN
ejde-744	9	7	for	for	ADP
ejde-744	9	8	the	the	DET
ejde-744	9	9	nondegenerate	nondegenerate	ADJ
ejde-744	9	10	wave	wave	NOUN
ejde-744	9	11	equation	equation	NOUN
ejde-744	9	12	,	,	PUNCT
ejde-744	9	13	which	which	PRON
ejde-744	9	14	is	be	AUX
ejde-744	9	15	characterized	characterize	VERB
ejde-744	9	16	by	by	ADP
ejde-744	9	17	the	the	DET
ejde-744	9	18	system	system	NOUN
ejde-744	9	19	of	of	ADP
ejde-744	9	20	equations	equation	NOUN
ejde-744	9	21	utt	utt	PROPN
ejde-744	10	1	−	−	PROPN
ejde-744	10	2	uxx	uxx	NOUN
ejde-744	10	3	=	=	SYM
ejde-744	10	4	0	0	NUM
ejde-744	10	5	,	,	PUNCT
ejde-744	10	6	(	(	PUNCT
ejde-744	10	7	t	t	PROPN
ejde-744	10	8	,	,	PUNCT
ejde-744	10	9	x	x	NOUN
ejde-744	10	10	)	)	PUNCT
ejde-744	10	11	∈	∈	PROPN
ejde-744	10	12	(	(	PUNCT
ejde-744	10	13	0	0	NUM
ejde-744	10	14	,	,	PUNCT
ejde-744	10	15	t	t	NOUN
ejde-744	10	16	)	)	PUNCT
ejde-744	10	17	×	×	NOUN
ejde-744	10	18	(	(	PUNCT
ejde-744	10	19	0	0	NUM
ejde-744	10	20	,	,	PUNCT
ejde-744	10	21	1	1	NUM
ejde-744	10	22	)	)	PUNCT
ejde-744	10	23	,	,	PUNCT
ejde-744	10	24	u(t	u(t	NOUN
ejde-744	10	25	,	,	PUNCT
ejde-744	10	26	0	0	NUM
ejde-744	10	27	)	)	PUNCT
ejde-744	10	28	=	=	SYM
ejde-744	10	29	0	0	NUM
ejde-744	10	30	,	,	PUNCT
ejde-744	10	31	u(t	u(t	NOUN
ejde-744	10	32	,	,	PUNCT
ejde-744	10	33	1	1	NUM
ejde-744	10	34	)	)	PUNCT
ejde-744	10	35	=	=	SYM
ejde-744	10	36	f(t	f(t	NOUN
ejde-744	10	37	)	)	PUNCT
ejde-744	10	38	,	,	PUNCT
ejde-744	10	39	t	t	PROPN
ejde-744	10	40	∈	∈	PROPN
ejde-744	10	41	(	(	PUNCT
ejde-744	10	42	0	0	NUM
ejde-744	10	43	,	,	PUNCT
ejde-744	10	44	t	t	NOUN
ejde-744	10	45	)	)	PUNCT
ejde-744	10	46	,	,	PUNCT
ejde-744	10	47	u(0	u(0	PROPN
ejde-744	10	48	,	,	PUNCT
ejde-744	10	49	x	x	NOUN
ejde-744	10	50	)	)	PUNCT
ejde-744	10	51	=	=	SYM
ejde-744	10	52	u0(x	u0(x	NUM
ejde-744	10	53	)	)	PUNCT
ejde-744	10	54	,	,	PUNCT
ejde-744	10	55	x	x	PUNCT
ejde-744	10	56	∈	∈	PROPN
ejde-744	10	57	(	(	PUNCT
ejde-744	10	58	0	0	NUM
ejde-744	10	59	,	,	PUNCT
ejde-744	10	60	1	1	NUM
ejde-744	10	61	)	)	PUNCT
ejde-744	10	62	,	,	PUNCT
ejde-744	10	63	ut(0	ut(0	PROPN
ejde-744	10	64	,	,	PUNCT
ejde-744	10	65	x	x	NOUN
ejde-744	10	66	)	)	PUNCT
ejde-744	10	67	=	=	SYM
ejde-744	11	1	u1(x	u1(x	NOUN
ejde-744	11	2	)	)	PUNCT
ejde-744	11	3	,	,	PUNCT
ejde-744	11	4	x	x	PUNCT
ejde-744	11	5	∈	∈	PROPN
ejde-744	11	6	(	(	PUNCT
ejde-744	11	7	0	0	NUM
ejde-744	11	8	,	,	PUNCT
ejde-744	11	9	1	1	NUM
ejde-744	11	10	)	)	PUNCT
ejde-744	11	11	,	,	PUNCT
ejde-744	11	12	(	(	PUNCT
ejde-744	11	13	1.1	1.1	NUM
ejde-744	11	14	)	)	PUNCT
ejde-744	11	15	where	where	SCONJ
ejde-744	11	16	u	u	NOUN
ejde-744	11	17	is	be	AUX
ejde-744	11	18	the	the	DET
ejde-744	11	19	state	state	NOUN
ejde-744	11	20	,	,	PUNCT
ejde-744	11	21	f	f	PROPN
ejde-744	11	22	acts	act	VERB
ejde-744	11	23	as	as	ADP
ejde-744	11	24	a	a	DET
ejde-744	11	25	boundary	boundary	ADJ
ejde-744	11	26	control	control	NOUN
ejde-744	11	27	and	and	CCONJ
ejde-744	11	28	is	be	AUX
ejde-744	11	29	used	use	VERB
ejde-744	11	30	to	to	PART
ejde-744	11	31	drive	drive	VERB
ejde-744	11	32	the	the	DET
ejde-744	11	33	solution	solution	NOUN
ejde-744	11	34	to	to	ADP
ejde-744	11	35	zero	zero	NUM
ejde-744	11	36	at	at	ADP
ejde-744	11	37	a	a	DET
ejde-744	11	38	given	give	VERB
ejde-744	11	39	time	time	NOUN
ejde-744	11	40	t	t	PROPN
ejde-744	11	41	.	.	PUNCT
ejde-744	12	1	to	to	PART
ejde-744	12	2	be	be	AUX
ejde-744	12	3	more	more	ADV
ejde-744	12	4	precise	precise	ADJ
ejde-744	12	5	,	,	PUNCT
ejde-744	12	6	for	for	ADP
ejde-744	12	7	given	give	VERB
ejde-744	12	8	the	the	DET
ejde-744	12	9	initial	initial	ADJ
ejde-744	12	10	data	datum	NOUN
ejde-744	12	11	(	(	PUNCT
ejde-744	12	12	u0	u0	ADJ
ejde-744	12	13	,	,	PUNCT
ejde-744	12	14	u1	u1	NOUN
ejde-744	12	15	)	)	PUNCT
ejde-744	12	16	in	in	ADP
ejde-744	12	17	a	a	DET
ejde-744	12	18	suitable	suitable	ADJ
ejde-744	12	19	space	space	NOUN
ejde-744	12	20	,	,	PUNCT
ejde-744	12	21	we	we	PRON
ejde-744	12	22	look	look	VERB
ejde-744	12	23	for	for	ADP
ejde-744	12	24	a	a	DET
ejde-744	12	25	control	control	NOUN
ejde-744	12	26	f	f	NOUN
ejde-744	12	27	such	such	ADJ
ejde-744	12	28	that	that	SCONJ
ejde-744	12	29	u(t	u(t	NOUN
ejde-744	12	30	,	,	PUNCT
ejde-744	12	31	x	x	NOUN
ejde-744	12	32	)	)	PUNCT
ejde-744	12	33	=	=	SYM
ejde-744	12	34	ut(t	ut(t	NOUN
ejde-744	12	35	,	,	PUNCT
ejde-744	12	36	x	x	X
ejde-744	12	37	)	)	PUNCT
ejde-744	12	38	=	=	SYM
ejde-744	12	39	0	0	NUM
ejde-744	12	40	,	,	PUNCT
ejde-744	12	41	∀x	∀x	X
ejde-744	12	42	∈	∈	PROPN
ejde-744	12	43	(	(	PUNCT
ejde-744	12	44	0	0	NUM
ejde-744	12	45	,	,	PUNCT
ejde-744	12	46	1	1	NUM
ejde-744	12	47	)	)	PUNCT
ejde-744	12	48	.	.	PUNCT
ejde-744	13	1	(	(	PUNCT
ejde-744	13	2	1.2	1.2	NUM
ejde-744	13	3	)	)	PUNCT
ejde-744	13	4	because	because	SCONJ
ejde-744	13	5	of	of	ADP
ejde-744	13	6	the	the	DET
ejde-744	13	7	finite	finite	ADJ
ejde-744	13	8	speed	speed	NOUN
ejde-744	13	9	of	of	ADP
ejde-744	13	10	propagation	propagation	NOUN
ejde-744	13	11	of	of	ADP
ejde-744	13	12	solutions	solution	NOUN
ejde-744	13	13	to	to	ADP
ejde-744	13	14	the	the	DET
ejde-744	13	15	wave	wave	NOUN
ejde-744	13	16	equation	equation	NOUN
ejde-744	13	17	,	,	PUNCT
ejde-744	13	18	exact	exact	ADJ
ejde-744	13	19	controllability	controllability	NOUN
ejde-744	13	20	can	can	AUX
ejde-744	13	21	only	only	ADV
ejde-744	13	22	be	be	AUX
ejde-744	13	23	achieved	achieve	VERB
ejde-744	13	24	at	at	ADP
ejde-744	13	25	a	a	DET
ejde-744	13	26	sufficiently	sufficiently	ADV
ejde-744	13	27	large	large	ADJ
ejde-744	13	28	time	time	NOUN
ejde-744	13	29	t	t	PROPN
ejde-744	13	30	(	(	PUNCT
ejde-744	13	31	in	in	ADP
ejde-744	13	32	the	the	DET
ejde-744	13	33	parabolic	parabolic	NOUN
ejde-744	13	34	case	case	NOUN
ejde-744	13	35	,	,	PUNCT
ejde-744	13	36	we	we	PRON
ejde-744	13	37	have	have	AUX
ejde-744	13	38	null	null	ADJ
ejde-744	13	39	controllability	controllability	NOUN
ejde-744	13	40	at	at	ADP
ejde-744	13	41	any	any	DET
ejde-744	13	42	final	final	ADJ
ejde-744	13	43	time	time	NOUN
ejde-744	13	44	t	t	PROPN
ejde-744	13	45	)	)	PUNCT
ejde-744	13	46	.	.	PUNCT
ejde-744	14	1	as	as	ADP
ejde-744	14	2	a	a	DET
ejde-744	14	3	general	general	ADJ
ejde-744	14	4	conclusion	conclusion	NOUN
ejde-744	14	5	,	,	PUNCT
ejde-744	14	6	we	we	PRON
ejde-744	14	7	consider	consider	VERB
ejde-744	14	8	(	(	PUNCT
ejde-744	14	9	1.1	1.1	NUM
ejde-744	14	10	)	)	PUNCT
ejde-744	14	11	to	to	PART
ejde-744	14	12	represent	represent	VERB
ejde-744	14	13	exact	exact	ADJ
ejde-744	14	14	controllability	controllability	NOUN
ejde-744	14	15	if	if	SCONJ
ejde-744	14	16	t	t	PROPN
ejde-744	14	17	>	>	X
ejde-744	14	18	2	2	X
ejde-744	14	19	.	.	PUNCT
ejde-744	15	1	the	the	DET
ejde-744	15	2	degenerate	degenerate	ADJ
ejde-744	15	3	wave	wave	NOUN
ejde-744	15	4	equations	equation	NOUN
ejde-744	15	5	began	begin	VERB
ejde-744	15	6	to	to	PART
ejde-744	15	7	receive	receive	VERB
ejde-744	15	8	some	some	DET
ejde-744	15	9	attention	attention	NOUN
ejde-744	15	10	within	within	ADP
ejde-744	15	11	the	the	DET
ejde-744	15	12	past	past	ADJ
ejde-744	15	13	decade	decade	NOUN
ejde-744	15	14	,	,	PUNCT
ejde-744	15	15	and	and	CCONJ
ejde-744	15	16	had	have	AUX
ejde-744	15	17	developed	develop	VERB
ejde-744	15	18	rapidly	rapidly	ADV
ejde-744	15	19	[	[	X
ejde-744	15	20	4	4	NUM
ejde-744	15	21	,	,	PUNCT
ejde-744	15	22	5	5	NUM
ejde-744	15	23	,	,	PUNCT
ejde-744	15	24	7	7	NUM
ejde-744	15	25	,	,	PUNCT
ejde-744	15	26	29	29	NUM
ejde-744	15	27	]	]	PUNCT
ejde-744	15	28	.	.	PUNCT
ejde-744	16	1	different	different	ADJ
ejde-744	16	2	from	from	ADP
ejde-744	16	3	the	the	DET
ejde-744	16	4	case	case	NOUN
ejde-744	16	5	nondegenerate	nondegenerate	ADJ
ejde-744	16	6	equations	equation	NOUN
ejde-744	16	7	,	,	PUNCT
ejde-744	16	8	the	the	DET
ejde-744	16	9	main	main	ADJ
ejde-744	16	10	difficulty	difficulty	NOUN
ejde-744	16	11	with	with	ADP
ejde-744	16	12	degenerate	degenerate	ADJ
ejde-744	16	13	wave	wave	NOUN
ejde-744	16	14	equations	equation	NOUN
ejde-744	16	15	is	be	AUX
ejde-744	16	16	introducing	introduce	VERB
ejde-744	16	17	a	a	DET
ejde-744	16	18	suitable	suitable	ADJ
ejde-744	16	19	function	function	NOUN
ejde-744	16	20	space	space	NOUN
ejde-744	16	21	to	to	PART
ejde-744	16	22	deal	deal	VERB
ejde-744	16	23	with	with	ADP
ejde-744	16	24	degenerate	degenerate	ADJ
ejde-744	16	25	terms	term	NOUN
ejde-744	16	26	,	,	PUNCT
ejde-744	16	27	requiring	require	VERB
ejde-744	16	28	the	the	DET
ejde-744	16	29	2020	2020	NUM
ejde-744	16	30	mathematics	mathematic	NOUN
ejde-744	16	31	subject	subject	ADJ
ejde-744	16	32	classification	classification	NOUN
ejde-744	16	33	.	.	PUNCT
ejde-744	17	1	35l80	35l80	NUM
ejde-744	17	2	,	,	PUNCT
ejde-744	17	3	35l81	35l81	NUM
ejde-744	17	4	,	,	PUNCT
ejde-744	17	5	35l05	35l05	NUM
ejde-744	17	6	,	,	PUNCT
ejde-744	17	7	93b05	93b05	NUM
ejde-744	17	8	.	.	PUNCT
ejde-744	18	1	key	key	ADJ
ejde-744	18	2	words	word	NOUN
ejde-744	18	3	and	and	CCONJ
ejde-744	18	4	phrases	phrase	NOUN
ejde-744	18	5	.	.	PUNCT
ejde-744	19	1	degenerate	degenerate	ADJ
ejde-744	19	2	wave	wave	NOUN
ejde-744	19	3	equation	equation	NOUN
ejde-744	19	4	;	;	PUNCT
ejde-744	19	5	singular	singular	PROPN
ejde-744	19	6	wave	wave	NOUN
ejde-744	19	7	equation	equation	NOUN
ejde-744	19	8	;	;	PUNCT
ejde-744	19	9	controllability	controllability	NOUN
ejde-744	19	10	;	;	PUNCT
ejde-744	19	11	boundary	boundary	ADJ
ejde-744	19	12	control	control	NOUN
ejde-744	19	13	.	.	PUNCT
ejde-744	20	1	©	©	PROPN
ejde-744	20	2	2025	2025	NUM
ejde-744	20	3	.	.	PUNCT
ejde-744	21	1	this	this	DET
ejde-744	21	2	work	work	NOUN
ejde-744	21	3	is	be	AUX
ejde-744	21	4	licensed	license	VERB
ejde-744	21	5	under	under	ADP
ejde-744	21	6	a	a	DET
ejde-744	21	7	cc	cc	NOUN
ejde-744	21	8	by	by	ADP
ejde-744	21	9	4.0	4.0	NUM
ejde-744	21	10	license	license	NOUN
ejde-744	21	11	.	.	PUNCT
ejde-744	22	1	submitted	submit	VERB
ejde-744	22	2	september	september	PROPN
ejde-744	22	3	18	18	NUM
ejde-744	22	4	,	,	PUNCT
ejde-744	22	5	2024	2024	NUM
ejde-744	22	6	.	.	PUNCT
ejde-744	23	1	published	publish	VERB
ejde-744	23	2	january	january	PROPN
ejde-744	23	3	9	9	NUM
ejde-744	23	4	,	,	PUNCT
ejde-744	23	5	2025	2025	NUM
ejde-744	23	6	.	.	PUNCT
ejde-744	24	1	1	1	NUM
ejde-744	24	2	2	2	NUM
ejde-744	24	3	g.	g.	PROPN
ejde-744	24	4	zhang	zhang	PROPN
ejde-744	24	5	,	,	PUNCT
ejde-744	24	6	s.	s.	PROPN
ejde-744	24	7	chai	chai	PROPN
ejde-744	24	8	ejde-2025/04	ejde-2025/04	PROPN
ejde-744	24	9	development	development	NOUN
ejde-744	24	10	of	of	ADP
ejde-744	24	11	new	new	ADJ
ejde-744	24	12	rules	rule	NOUN
ejde-744	24	13	for	for	ADP
ejde-744	24	14	analyzing	analyze	VERB
ejde-744	24	15	observability	observability	NOUN
ejde-744	24	16	and	and	CCONJ
ejde-744	24	17	controllability	controllability	NOUN
ejde-744	24	18	.	.	PUNCT
ejde-744	25	1	gueye	gueye	NOUN
ejde-744	25	2	[	[	X
ejde-744	25	3	19	19	NUM
ejde-744	25	4	]	]	PUNCT
ejde-744	25	5	considered	consider	VERB
ejde-744	25	6	the	the	DET
ejde-744	25	7	boundary	boundary	ADJ
ejde-744	25	8	control	control	NOUN
ejde-744	25	9	about	about	ADP
ejde-744	25	10	the	the	DET
ejde-744	25	11	degenerate	degenerate	ADJ
ejde-744	25	12	wave	wave	NOUN
ejde-744	25	13	equation	equation	NOUN
ejde-744	25	14	utt	utt	NOUN
ejde-744	25	15	−	−	PROPN
ejde-744	25	16	(	(	PUNCT
ejde-744	25	17	xαux)x	xαux)x	X
ejde-744	25	18	=	=	SYM
ejde-744	25	19	0	0	NUM
ejde-744	25	20	,	,	PUNCT
ejde-744	25	21	(	(	PUNCT
ejde-744	25	22	t	t	PROPN
ejde-744	25	23	,	,	PUNCT
ejde-744	25	24	x	x	NOUN
ejde-744	25	25	)	)	PUNCT
ejde-744	25	26	∈	∈	PROPN
ejde-744	25	27	(	(	PUNCT
ejde-744	25	28	0	0	NUM
ejde-744	25	29	,	,	PUNCT
ejde-744	25	30	t	t	NOUN
ejde-744	25	31	)	)	PUNCT
ejde-744	25	32	×	×	NOUN
ejde-744	25	33	(	(	PUNCT
ejde-744	25	34	0	0	NUM
ejde-744	25	35	,	,	PUNCT
ejde-744	25	36	1	1	NUM
ejde-744	25	37	)	)	PUNCT
ejde-744	25	38	.	.	PUNCT
ejde-744	26	1	(	(	PUNCT
ejde-744	26	2	1.3	1.3	NUM
ejde-744	26	3	)	)	PUNCT
ejde-744	26	4	we	we	PRON
ejde-744	26	5	also	also	ADV
ejde-744	26	6	refer	refer	VERB
ejde-744	26	7	to	to	ADP
ejde-744	26	8	the	the	DET
ejde-744	26	9	work	work	NOUN
ejde-744	26	10	of	of	ADP
ejde-744	26	11	zhang	zhang	PROPN
ejde-744	26	12	and	and	CCONJ
ejde-744	26	13	gao	gao	PROPN
ejde-744	27	1	[	[	X
ejde-744	27	2	30	30	NUM
ejde-744	27	3	,	,	PUNCT
ejde-744	27	4	31	31	NUM
ejde-744	27	5	]	]	PUNCT
ejde-744	27	6	for	for	ADP
ejde-744	27	7	additional	additional	ADJ
ejde-744	27	8	controllability	controllability	NOUN
ejde-744	27	9	results	result	NOUN
ejde-744	27	10	obtained	obtain	VERB
ejde-744	27	11	through	through	ADP
ejde-744	27	12	the	the	DET
ejde-744	27	13	use	use	NOUN
ejde-744	27	14	of	of	ADP
ejde-744	27	15	a	a	DET
ejde-744	27	16	locally	locally	ADV
ejde-744	27	17	distributed	distribute	VERB
ejde-744	27	18	control	control	NOUN
ejde-744	27	19	.	.	PUNCT
ejde-744	28	1	later	later	ADV
ejde-744	28	2	,	,	PUNCT
ejde-744	28	3	alabauboussouira	alabauboussouira	PROPN
ejde-744	28	4	et	et	PROPN
ejde-744	28	5	al	al	PROPN
ejde-744	28	6	.	.	PUNCT
ejde-744	29	1	[	[	X
ejde-744	29	2	1	1	X
ejde-744	29	3	]	]	PUNCT
ejde-744	29	4	consider	consider	VERB
ejde-744	29	5	the	the	DET
ejde-744	29	6	equation	equation	NOUN
ejde-744	29	7	utt	utt	NOUN
ejde-744	29	8	−	−	PROPN
ejde-744	29	9	(	(	PUNCT
ejde-744	29	10	aux)x	aux)x	PROPN
ejde-744	29	11	=	=	SYM
ejde-744	29	12	0	0	NUM
ejde-744	29	13	,	,	PUNCT
ejde-744	29	14	(	(	PUNCT
ejde-744	29	15	t	t	PROPN
ejde-744	29	16	,	,	PUNCT
ejde-744	29	17	x	x	NOUN
ejde-744	29	18	)	)	PUNCT
ejde-744	29	19	∈	∈	PROPN
ejde-744	29	20	(	(	PUNCT
ejde-744	29	21	0	0	NUM
ejde-744	29	22	,	,	PUNCT
ejde-744	29	23	t	t	NOUN
ejde-744	29	24	)	)	PUNCT
ejde-744	29	25	×	×	NOUN
ejde-744	29	26	(	(	PUNCT
ejde-744	29	27	0	0	NUM
ejde-744	29	28	,	,	PUNCT
ejde-744	29	29	1	1	NUM
ejde-744	29	30	)	)	PUNCT
ejde-744	29	31	,	,	PUNCT
ejde-744	29	32	(	(	PUNCT
ejde-744	29	33	1.4	1.4	NUM
ejde-744	29	34	)	)	PUNCT
ejde-744	29	35	where	where	SCONJ
ejde-744	29	36	a	a	PRON
ejde-744	29	37	is	be	AUX
ejde-744	29	38	positive	positive	ADJ
ejde-744	29	39	on	on	ADP
ejde-744	29	40	(	(	PUNCT
ejde-744	29	41	0	0	NUM
ejde-744	29	42	,	,	PUNCT
ejde-744	29	43	1	1	NUM
ejde-744	29	44	]	]	PUNCT
ejde-744	29	45	and	and	CCONJ
ejde-744	29	46	a(0	a(0	PROPN
ejde-744	29	47	)	)	PUNCT
ejde-744	29	48	=	=	SYM
ejde-744	30	1	0	0	X
ejde-744	30	2	.	.	PUNCT
ejde-744	31	1	the	the	DET
ejde-744	31	2	degeneracy	degeneracy	NOUN
ejde-744	31	3	of	of	ADP
ejde-744	31	4	(	(	PUNCT
ejde-744	31	5	1.4	1.4	NUM
ejde-744	31	6	)	)	PUNCT
ejde-744	31	7	at	at	ADP
ejde-744	31	8	x	x	X
ejde-744	31	9	=	=	SYM
ejde-744	31	10	0	0	NUM
ejde-744	31	11	is	be	AUX
ejde-744	31	12	measured	measure	VERB
ejde-744	31	13	by	by	ADP
ejde-744	31	14	the	the	DET
ejde-744	31	15	parameter	parameter	NOUN
ejde-744	31	16	µa	µa	ADP
ejde-744	31	17	defined	define	VERB
ejde-744	31	18	by	by	ADP
ejde-744	31	19	µa	µa	NOUN
ejde-744	31	20	:	:	PUNCT
ejde-744	31	21	=	=	SYM
ejde-744	31	22	sup	sup	NOUN
ejde-744	31	23	0	0	NUM
ejde-744	31	24	<	<	X
ejde-744	31	25	x≤1	x≤1	PROPN
ejde-744	31	26	x|a′(x)|	x|a′(x)|	PROPN
ejde-744	31	27	a(x	a(x	PROPN
ejde-744	31	28	)	)	PUNCT
ejde-744	31	29	<	<	X
ejde-744	31	30	2	2	NUM
ejde-744	31	31	,	,	PUNCT
ejde-744	31	32	(	(	PUNCT
ejde-744	31	33	1.5	1.5	NUM
ejde-744	31	34	)	)	PUNCT
ejde-744	31	35	and	and	CCONJ
ejde-744	31	36	say	say	VERB
ejde-744	31	37	the	the	DET
ejde-744	31	38	function	function	NOUN
ejde-744	31	39	a	a	PRON
ejde-744	31	40	is	be	AUX
ejde-744	31	41	weakly	weakly	ADV
ejde-744	31	42	degenerate	degenerate	ADJ
ejde-744	31	43	(	(	PUNCT
ejde-744	31	44	wd	wd	PROPN
ejde-744	31	45	)	)	PUNCT
ejde-744	31	46	if	if	SCONJ
ejde-744	31	47	µa	µa	ADP
ejde-744	31	48	∈	∈	PROPN
ejde-744	32	1	[	[	X
ejde-744	32	2	0	0	NUM
ejde-744	32	3	,	,	PUNCT
ejde-744	32	4	1	1	NUM
ejde-744	32	5	)	)	PUNCT
ejde-744	32	6	,	,	PUNCT
ejde-744	32	7	strongly	strongly	ADV
ejde-744	32	8	degenerate	degenerate	ADJ
ejde-744	32	9	(	(	PUNCT
ejde-744	32	10	sd	sd	NOUN
ejde-744	32	11	)	)	PUNCT
ejde-744	32	12	if	if	SCONJ
ejde-744	32	13	µa	µa	ADP
ejde-744	32	14	∈	∈	PROPN
ejde-744	32	15	[	[	X
ejde-744	32	16	1	1	NUM
ejde-744	32	17	,	,	PUNCT
ejde-744	32	18	2	2	NUM
ejde-744	32	19	)	)	PUNCT
ejde-744	32	20	.	.	PUNCT
ejde-744	33	1	the	the	DET
ejde-744	33	2	authors	author	NOUN
ejde-744	33	3	establish	establish	VERB
ejde-744	33	4	observability	observability	NOUN
ejde-744	33	5	inequalities	inequality	NOUN
ejde-744	33	6	for	for	ADP
ejde-744	33	7	weakly	weakly	ADJ
ejde-744	33	8	as	as	ADV
ejde-744	33	9	well	well	ADV
ejde-744	33	10	as	as	ADP
ejde-744	33	11	strongly	strongly	ADV
ejde-744	33	12	degenerate	degenerate	ADJ
ejde-744	33	13	equations	equation	NOUN
ejde-744	33	14	,	,	PUNCT
ejde-744	33	15	and	and	CCONJ
ejde-744	33	16	prove	prove	VERB
ejde-744	33	17	a	a	DET
ejde-744	33	18	negative	negative	ADJ
ejde-744	33	19	result	result	NOUN
ejde-744	33	20	when	when	SCONJ
ejde-744	33	21	the	the	DET
ejde-744	33	22	diffusion	diffusion	NOUN
ejde-744	33	23	coefficient	coefficient	NOUN
ejde-744	33	24	degenerate	degenerate	VERB
ejde-744	33	25	too	too	ADV
ejde-744	33	26	violently	violently	ADV
ejde-744	33	27	(	(	PUNCT
ejde-744	33	28	µa	µa	NOUN
ejde-744	33	29	≥	≥	NOUN
ejde-744	33	30	2	2	NUM
ejde-744	33	31	)	)	PUNCT
ejde-744	33	32	.	.	PUNCT
ejde-744	34	1	moreover	moreover	ADV
ejde-744	34	2	,	,	PUNCT
ejde-744	34	3	the	the	DET
ejde-744	34	4	authors	author	NOUN
ejde-744	34	5	prove	prove	VERB
ejde-744	34	6	observability	observability	NOUN
ejde-744	34	7	(	(	PUNCT
ejde-744	34	8	or	or	CCONJ
ejde-744	34	9	controllability	controllability	NOUN
ejde-744	34	10	)	)	PUNCT
ejde-744	34	11	time	time	NOUN
ejde-744	34	12	blows	blow	VERB
ejde-744	34	13	up	up	ADP
ejde-744	34	14	as	as	ADP
ejde-744	34	15	µa	µa	ADP
ejde-744	34	16	approaches	approach	NOUN
ejde-744	34	17	2	2	NUM
ejde-744	34	18	from	from	ADP
ejde-744	34	19	below	below	ADV
ejde-744	34	20	.	.	PUNCT
ejde-744	35	1	finally	finally	ADV
ejde-744	35	2	,	,	PUNCT
ejde-744	35	3	using	use	VERB
ejde-744	35	4	the	the	DET
ejde-744	35	5	hilbert	hilbert	NOUN
ejde-744	35	6	uniqueness	uniqueness	NOUN
ejde-744	35	7	method	method	NOUN
ejde-744	35	8	(	(	PUNCT
ejde-744	35	9	hum	hum	NOUN
ejde-744	35	10	)	)	PUNCT
ejde-744	35	11	,	,	PUNCT
ejde-744	35	12	they	they	PRON
ejde-744	35	13	deduce	deduce	VERB
ejde-744	35	14	the	the	DET
ejde-744	35	15	exact	exact	ADJ
ejde-744	35	16	controllability	controllability	NOUN
ejde-744	35	17	for	for	ADP
ejde-744	35	18	corresponding	correspond	VERB
ejde-744	35	19	control	control	NOUN
ejde-744	35	20	system	system	NOUN
ejde-744	35	21	when	when	SCONJ
ejde-744	35	22	µa	µa	ADV
ejde-744	35	23	∈	∈	PROPN
ejde-744	36	1	[	[	X
ejde-744	36	2	0	0	NUM
ejde-744	36	3	,	,	PUNCT
ejde-744	36	4	2	2	NUM
ejde-744	36	5	)	)	PUNCT
ejde-744	36	6	.	.	PUNCT
ejde-744	37	1	in	in	ADP
ejde-744	37	2	recent	recent	ADJ
ejde-744	37	3	years	year	NOUN
ejde-744	37	4	,	,	PUNCT
ejde-744	37	5	great	great	ADJ
ejde-744	37	6	attention	attention	NOUN
ejde-744	37	7	has	have	AUX
ejde-744	37	8	been	be	AUX
ejde-744	37	9	given	give	VERB
ejde-744	37	10	to	to	PART
ejde-744	37	11	control	control	VERB
ejde-744	37	12	issues	issue	NOUN
ejde-744	37	13	for	for	ADP
ejde-744	37	14	parabolic	parabolic	ADJ
ejde-744	37	15	equations	equation	NOUN
ejde-744	37	16	with	with	ADP
ejde-744	37	17	both	both	CCONJ
ejde-744	37	18	degenerate	degenerate	ADJ
ejde-744	37	19	and	and	CCONJ
ejde-744	37	20	singular	singular	ADJ
ejde-744	37	21	terms	term	NOUN
ejde-744	37	22	.	.	PUNCT
ejde-744	38	1	however	however	ADV
ejde-744	38	2	,	,	PUNCT
ejde-744	38	3	this	this	DET
ejde-744	38	4	paper	paper	NOUN
ejde-744	38	5	will	will	AUX
ejde-744	38	6	not	not	PART
ejde-744	38	7	delve	delve	VERB
ejde-744	38	8	into	into	ADP
ejde-744	38	9	the	the	DET
ejde-744	38	10	details	detail	NOUN
ejde-744	38	11	here	here	ADV
ejde-744	38	12	;	;	PUNCT
ejde-744	38	13	instead	instead	ADV
ejde-744	38	14	,	,	PUNCT
ejde-744	38	15	we	we	PRON
ejde-744	38	16	note	note	VERB
ejde-744	38	17	that	that	SCONJ
ejde-744	38	18	a	a	DET
ejde-744	38	19	common	common	ADJ
ejde-744	38	20	strategy	strategy	NOUN
ejde-744	38	21	to	to	PART
ejde-744	38	22	demonstrate	demonstrate	VERB
ejde-744	38	23	controllability	controllability	NOUN
ejde-744	38	24	is	be	AUX
ejde-744	38	25	through	through	ADP
ejde-744	38	26	the	the	DET
ejde-744	38	27	proof	proof	NOUN
ejde-744	38	28	of	of	ADP
ejde-744	38	29	global	global	ADJ
ejde-744	38	30	carleman	carleman	ADJ
ejde-744	38	31	estimates	estimate	NOUN
ejde-744	38	32	.	.	PUNCT
ejde-744	39	1	for	for	ADP
ejde-744	39	2	regular	regular	ADJ
ejde-744	39	3	degenerate	degenerate	ADJ
ejde-744	39	4	coefficients	coefficient	NOUN
ejde-744	39	5	,	,	PUNCT
ejde-744	39	6	the	the	DET
ejde-744	39	7	carleman	carleman	ADJ
ejde-744	39	8	estimates	estimate	NOUN
ejde-744	39	9	and	and	CCONJ
ejde-744	39	10	null	null	ADJ
ejde-744	39	11	controllability	controllability	NOUN
ejde-744	39	12	properties	property	NOUN
ejde-744	39	13	are	be	AUX
ejde-744	39	14	discussed	discuss	VERB
ejde-744	39	15	in	in	ADP
ejde-744	39	16	[	[	X
ejde-744	39	17	2	2	NUM
ejde-744	39	18	,	,	PUNCT
ejde-744	39	19	6	6	NUM
ejde-744	39	20	,	,	PUNCT
ejde-744	39	21	8	8	NUM
ejde-744	39	22	,	,	PUNCT
ejde-744	39	23	9	9	NUM
ejde-744	39	24	,	,	PUNCT
ejde-744	39	25	24	24	NUM
ejde-744	39	26	]	]	PUNCT
ejde-744	39	27	,	,	PUNCT
ejde-744	39	28	for	for	ADP
ejde-744	39	29	non	non	ADJ
ejde-744	39	30	-	-	ADJ
ejde-744	39	31	smooth	smooth	ADJ
ejde-744	39	32	degenerate	degenerate	ADJ
ejde-744	39	33	coefficients	coefficient	NOUN
ejde-744	39	34	in	in	ADP
ejde-744	39	35	[	[	X
ejde-744	39	36	14	14	NUM
ejde-744	39	37	,	,	PUNCT
ejde-744	39	38	15	15	NUM
ejde-744	39	39	]	]	PUNCT
ejde-744	39	40	,	,	PUNCT
ejde-744	39	41	and	and	CCONJ
ejde-744	39	42	for	for	ADP
ejde-744	39	43	equations	equation	NOUN
ejde-744	39	44	with	with	ADP
ejde-744	39	45	both	both	CCONJ
ejde-744	39	46	degenerate	degenerate	ADJ
ejde-744	39	47	and	and	CCONJ
ejde-744	39	48	singular	singular	ADJ
ejde-744	39	49	coefficients	coefficient	NOUN
ejde-744	39	50	in	in	ADP
ejde-744	39	51	[	[	X
ejde-744	39	52	13	13	NUM
ejde-744	39	53	,	,	PUNCT
ejde-744	39	54	16	16	NUM
ejde-744	39	55	,	,	PUNCT
ejde-744	39	56	27	27	NUM
ejde-744	39	57	]	]	PUNCT
ejde-744	39	58	.	.	PUNCT
ejde-744	40	1	in	in	ADP
ejde-744	40	2	[	[	X
ejde-744	40	3	3	3	X
ejde-744	40	4	]	]	PUNCT
ejde-744	40	5	the	the	DET
ejde-744	40	6	authors	author	NOUN
ejde-744	40	7	were	be	AUX
ejde-744	40	8	the	the	DET
ejde-744	40	9	first	first	ADJ
ejde-744	40	10	to	to	PART
ejde-744	40	11	study	study	VERB
ejde-744	40	12	the	the	DET
ejde-744	40	13	boundary	boundary	ADJ
ejde-744	40	14	controllability	controllability	NOUN
ejde-744	40	15	of	of	ADP
ejde-744	40	16	the	the	DET
ejde-744	40	17	wave	wave	NOUN
ejde-744	40	18	equation	equation	NOUN
ejde-744	40	19	with	with	ADP
ejde-744	40	20	degenerate	degenerate	ADJ
ejde-744	40	21	and	and	CCONJ
ejde-744	40	22	singular	singular	NOUN
ejde-744	40	23	,	,	PUNCT
ejde-744	40	24	where	where	SCONJ
ejde-744	40	25	the	the	DET
ejde-744	40	26	singularity	singularity	NOUN
ejde-744	40	27	occurs	occur	VERB
ejde-744	40	28	at	at	ADP
ejde-744	40	29	the	the	DET
ejde-744	40	30	same	same	ADJ
ejde-744	40	31	point	point	NOUN
ejde-744	40	32	as	as	ADP
ejde-744	40	33	the	the	DET
ejde-744	40	34	degeneration	degeneration	NOUN
ejde-744	40	35	of	of	ADP
ejde-744	40	36	the	the	DET
ejde-744	40	37	leading	lead	VERB
ejde-744	40	38	coefficient	coefficient	NOUN
ejde-744	40	39	.	.	PUNCT
ejde-744	41	1	to	to	PART
ejde-744	41	2	be	be	AUX
ejde-744	41	3	more	more	ADV
ejde-744	41	4	precise	precise	ADJ
ejde-744	41	5	,	,	PUNCT
ejde-744	41	6	they	they	PRON
ejde-744	41	7	consider	consider	VERB
ejde-744	41	8	the	the	DET
ejde-744	41	9	problem	problem	NOUN
ejde-744	41	10	ytt	ytt	PROPN
ejde-744	42	1	−	−	PROPN
ejde-744	43	1	(	(	PUNCT
ejde-744	43	2	xαyx)x	xαyx)x	PROPN
ejde-744	43	3	−	−	PROPN
ejde-744	43	4	µ	µ	X
ejde-744	43	5	x2−α	x2−α	PROPN
ejde-744	43	6	y	y	PROPN
ejde-744	43	7	=	=	SYM
ejde-744	43	8	0	0	PROPN
ejde-744	43	9	,	,	PUNCT
ejde-744	43	10	(	(	PUNCT
ejde-744	43	11	t	t	PROPN
ejde-744	43	12	,	,	PUNCT
ejde-744	43	13	x	x	X
ejde-744	43	14	)	)	PUNCT
ejde-744	43	15	∈	∈	PROPN
ejde-744	44	1	q	q	NOUN
ejde-744	44	2	:	:	PUNCT
ejde-744	44	3	=	=	SYM
ejde-744	44	4	(	(	PUNCT
ejde-744	44	5	0	0	NUM
ejde-744	44	6	,	,	PUNCT
ejde-744	44	7	t	t	NOUN
ejde-744	44	8	)	)	PUNCT
ejde-744	44	9	×	×	NOUN
ejde-744	44	10	(	(	PUNCT
ejde-744	44	11	0	0	NUM
ejde-744	44	12	,	,	PUNCT
ejde-744	44	13	1	1	NUM
ejde-744	44	14	)	)	PUNCT
ejde-744	44	15	,	,	PUNCT
ejde-744	44	16	y(t	y(t	PROPN
ejde-744	44	17	,	,	PUNCT
ejde-744	44	18	1	1	NUM
ejde-744	44	19	)	)	PUNCT
ejde-744	44	20	=	=	SYM
ejde-744	44	21	f	f	PROPN
ejde-744	44	22	,	,	PUNCT
ejde-744	44	23	t	t	PROPN
ejde-744	44	24	∈	∈	PROPN
ejde-744	44	25	(	(	PUNCT
ejde-744	44	26	0	0	NUM
ejde-744	44	27	,	,	PUNCT
ejde-744	44	28	t	t	NOUN
ejde-744	44	29	)	)	PUNCT
ejde-744	44	30	,	,	PUNCT
ejde-744	44	31	y(t	y(t	PROPN
ejde-744	44	32	,	,	PUNCT
ejde-744	44	33	0	0	NUM
ejde-744	44	34	)	)	PUNCT
ejde-744	44	35	=	=	SYM
ejde-744	44	36	0	0	NUM
ejde-744	44	37	,	,	PUNCT
ejde-744	44	38	α	α	NOUN
ejde-744	44	39	∈	∈	PROPN
ejde-744	45	1	[	[	X
ejde-744	45	2	0	0	NUM
ejde-744	45	3	,	,	PUNCT
ejde-744	45	4	1	1	NUM
ejde-744	45	5	)	)	PUNCT
ejde-744	45	6	,	,	PUNCT
ejde-744	45	7	t	t	PROPN
ejde-744	45	8	∈	∈	PROPN
ejde-744	45	9	(	(	PUNCT
ejde-744	45	10	0	0	NUM
ejde-744	45	11	,	,	PUNCT
ejde-744	45	12	t	t	PROPN
ejde-744	45	13	)	)	PUNCT
ejde-744	45	14	,	,	PUNCT
ejde-744	45	15	xαyx(t	xαyx(t	PROPN
ejde-744	45	16	,	,	PUNCT
ejde-744	45	17	0	0	NUM
ejde-744	45	18	)	)	PUNCT
ejde-744	45	19	=	=	SYM
ejde-744	46	1	0	0	NUM
ejde-744	46	2	,	,	PUNCT
ejde-744	46	3	α	α	NOUN
ejde-744	46	4	∈	∈	PROPN
ejde-744	47	1	[	[	X
ejde-744	47	2	1	1	NUM
ejde-744	47	3	,	,	PUNCT
ejde-744	47	4	2	2	NUM
ejde-744	47	5	)	)	PUNCT
ejde-744	47	6	,	,	PUNCT
ejde-744	47	7	t	t	PROPN
ejde-744	47	8	∈	∈	PROPN
ejde-744	47	9	(	(	PUNCT
ejde-744	47	10	0	0	NUM
ejde-744	47	11	,	,	PUNCT
ejde-744	47	12	t	t	NOUN
ejde-744	47	13	)	)	PUNCT
ejde-744	47	14	,	,	PUNCT
ejde-744	47	15	y(0	y(0	PROPN
ejde-744	47	16	,	,	PUNCT
ejde-744	47	17	x	x	NOUN
ejde-744	47	18	)	)	PUNCT
ejde-744	47	19	=	=	SYM
ejde-744	47	20	y0(x	y0(x	NOUN
ejde-744	47	21	)	)	PUNCT
ejde-744	47	22	,	,	PUNCT
ejde-744	47	23	yt(0	yt(0	PROPN
ejde-744	47	24	,	,	PUNCT
ejde-744	47	25	x	x	NOUN
ejde-744	47	26	)	)	PUNCT
ejde-744	47	27	=	=	SYM
ejde-744	47	28	y1(x	y1(x	NOUN
ejde-744	47	29	)	)	PUNCT
ejde-744	47	30	,	,	PUNCT
ejde-744	47	31	x	x	PUNCT
ejde-744	47	32	∈	∈	PROPN
ejde-744	47	33	(	(	PUNCT
ejde-744	47	34	0	0	NUM
ejde-744	47	35	,	,	PUNCT
ejde-744	47	36	1	1	NUM
ejde-744	47	37	)	)	PUNCT
ejde-744	47	38	,	,	PUNCT
ejde-744	47	39	(	(	PUNCT
ejde-744	47	40	1.6	1.6	NUM
ejde-744	47	41	)	)	PUNCT
ejde-744	47	42	where	where	SCONJ
ejde-744	47	43	α	α	NOUN
ejde-744	47	44	and	and	CCONJ
ejde-744	47	45	µ	µ	NOUN
ejde-744	47	46	are	be	AUX
ejde-744	47	47	two	two	NUM
ejde-744	47	48	parameters	parameter	NOUN
ejde-744	47	49	such	such	ADJ
ejde-744	47	50	that	that	SCONJ
ejde-744	47	51	α	α	PROPN
ejde-744	47	52	∈	∈	PROPN
ejde-744	48	1	[	[	X
ejde-744	48	2	0	0	NUM
ejde-744	48	3	,	,	PUNCT
ejde-744	48	4	2)\{1	2)\{1	NUM
ejde-744	48	5	}	}	PUNCT
ejde-744	48	6	,	,	PUNCT
ejde-744	48	7	µ	µ	X
ejde-744	48	8	≤	≤	X
ejde-744	48	9	(	(	PUNCT
ejde-744	48	10	1	1	NUM
ejde-744	48	11	−	−	PROPN
ejde-744	48	12	α)2/4	α)2/4	NOUN
ejde-744	48	13	.	.	PUNCT
ejde-744	49	1	furthermore	furthermore	ADV
ejde-744	49	2	,	,	PUNCT
ejde-744	49	3	the	the	DET
ejde-744	49	4	problem	problem	NOUN
ejde-744	49	5	is	be	AUX
ejde-744	49	6	weakly	weakly	ADV
ejde-744	49	7	degenerate	degenerate	ADJ
ejde-744	49	8	if	if	SCONJ
ejde-744	49	9	α	α	PRON
ejde-744	49	10	∈	∈	PROPN
ejde-744	50	1	[	[	X
ejde-744	50	2	0	0	NUM
ejde-744	50	3	,	,	PUNCT
ejde-744	50	4	1	1	NUM
ejde-744	50	5	)	)	PUNCT
ejde-744	50	6	,	,	PUNCT
ejde-744	50	7	strongly	strongly	ADV
ejde-744	50	8	degenerate	degenerate	ADJ
ejde-744	50	9	if	if	SCONJ
ejde-744	50	10	α	α	PRON
ejde-744	50	11	∈	∈	PROPN
ejde-744	51	1	[	[	X
ejde-744	51	2	1	1	NUM
ejde-744	51	3	,	,	PUNCT
ejde-744	51	4	2	2	NUM
ejde-744	51	5	)	)	PUNCT
ejde-744	51	6	.	.	PUNCT
ejde-744	52	1	the	the	DET
ejde-744	52	2	authors	author	NOUN
ejde-744	52	3	prove	prove	VERB
ejde-744	52	4	the	the	DET
ejde-744	52	5	observability	observability	NOUN
ejde-744	52	6	estimate	estimate	NOUN
ejde-744	52	7	for	for	ADP
ejde-744	52	8	the	the	DET
ejde-744	52	9	corresponding	corresponding	ADJ
ejde-744	52	10	adjoint	adjoint	NOUN
ejde-744	52	11	system	system	NOUN
ejde-744	52	12	by	by	ADP
ejde-744	52	13	means	mean	NOUN
ejde-744	52	14	of	of	ADP
ejde-744	52	15	the	the	DET
ejde-744	52	16	multiplier	multipli	ADJ
ejde-744	52	17	method	method	NOUN
ejde-744	52	18	and	and	CCONJ
ejde-744	52	19	new	new	ADJ
ejde-744	52	20	hardy	hardy	ADJ
ejde-744	52	21	-	-	PUNCT
ejde-744	52	22	type	type	NOUN
ejde-744	52	23	inequalities	inequality	NOUN
ejde-744	52	24	.	.	PUNCT
ejde-744	53	1	moreover	moreover	ADV
ejde-744	53	2	,	,	PUNCT
ejde-744	53	3	the	the	DET
ejde-744	53	4	null	null	ADJ
ejde-744	53	5	controllability	controllability	NOUN
ejde-744	53	6	is	be	AUX
ejde-744	53	7	proved	prove	VERB
ejde-744	53	8	for	for	ADP
ejde-744	53	9	sufficiently	sufficiently	ADV
ejde-744	53	10	large	large	ADJ
ejde-744	53	11	time	time	NOUN
ejde-744	53	12	by	by	ADP
ejde-744	53	13	hum	hum	NOUN
ejde-744	53	14	.	.	PUNCT
ejde-744	54	1	note	note	VERB
ejde-744	54	2	that	that	SCONJ
ejde-744	54	3	the	the	DET
ejde-744	54	4	degeneracy	degeneracy	NOUN
ejde-744	54	5	and	and	CCONJ
ejde-744	54	6	singularity	singularity	NOUN
ejde-744	54	7	coefficient	coefficient	NOUN
ejde-744	54	8	are	be	AUX
ejde-744	54	9	not	not	PART
ejde-744	54	10	general	general	ADJ
ejde-744	54	11	in	in	ADP
ejde-744	54	12	(	(	PUNCT
ejde-744	54	13	1.6	1.6	NUM
ejde-744	54	14	)	)	PUNCT
ejde-744	54	15	.	.	PUNCT
ejde-744	55	1	therefore	therefore	ADV
ejde-744	55	2	,	,	PUNCT
ejde-744	55	3	the	the	DET
ejde-744	55	4	purpose	purpose	NOUN
ejde-744	55	5	of	of	ADP
ejde-744	55	6	this	this	DET
ejde-744	55	7	paper	paper	NOUN
ejde-744	55	8	is	be	AUX
ejde-744	55	9	to	to	PART
ejde-744	55	10	study	study	VERB
ejde-744	55	11	the	the	DET
ejde-744	55	12	controllability	controllability	NOUN
ejde-744	55	13	and	and	CCONJ
ejde-744	55	14	boundary	boundary	ADJ
ejde-744	55	15	observability	observability	NOUN
ejde-744	55	16	of	of	ADP
ejde-744	55	17	a	a	DET
ejde-744	55	18	degenerate	degenerate	ADJ
ejde-744	55	19	wave	wave	NOUN
ejde-744	55	20	equation	equation	NOUN
ejde-744	55	21	with	with	ADP
ejde-744	55	22	a	a	DET
ejde-744	55	23	singular	singular	ADJ
ejde-744	55	24	term(in	term(in	ADJ
ejde-744	55	25	fact	fact	NOUN
ejde-744	55	26	,	,	PUNCT
ejde-744	55	27	it	it	PRON
ejde-744	55	28	is	be	AUX
ejde-744	55	29	an	an	DET
ejde-744	55	30	open	open	ADJ
ejde-744	55	31	question	question	NOUN
ejde-744	55	32	ejde-2025/04	ejde-2025/04	NOUN
ejde-744	55	33	exact	exact	ADJ
ejde-744	55	34	controllability	controllability	NOUN
ejde-744	55	35	3	3	NUM
ejde-744	55	36	in	in	ADP
ejde-744	55	37	reference	reference	NOUN
ejde-744	55	38	[	[	X
ejde-744	55	39	3	3	NUM
ejde-744	55	40	]	]	NUM
ejde-744	55	41	)	)	PUNCT
ejde-744	55	42	.	.	PUNCT
ejde-744	56	1	to	to	PART
ejde-744	56	2	be	be	AUX
ejde-744	56	3	more	more	ADV
ejde-744	56	4	precise	precise	ADJ
ejde-744	56	5	,	,	PUNCT
ejde-744	56	6	we	we	PRON
ejde-744	56	7	consider	consider	VERB
ejde-744	56	8	the	the	DET
ejde-744	56	9	degenerate	degenerate	ADJ
ejde-744	56	10	/	/	SYM
ejde-744	56	11	singular	singular	ADJ
ejde-744	56	12	wave	wave	NOUN
ejde-744	56	13	equation	equation	NOUN
ejde-744	56	14	ytt	ytt	PROPN
ejde-744	56	15	−	−	PROPN
ejde-744	57	1	(	(	PUNCT
ejde-744	57	2	a(x)yx)x	a(x)yx)x	PROPN
ejde-744	57	3	−	−	X
ejde-744	57	4	λ	λ	NOUN
ejde-744	57	5	b(x	b(x	NOUN
ejde-744	57	6	)	)	PUNCT
ejde-744	57	7	y	y	PROPN
ejde-744	57	8	=	=	SYM
ejde-744	57	9	0	0	PROPN
ejde-744	57	10	,	,	PUNCT
ejde-744	57	11	(	(	PUNCT
ejde-744	57	12	t	t	PROPN
ejde-744	57	13	,	,	PUNCT
ejde-744	57	14	x	x	X
ejde-744	57	15	)	)	PUNCT
ejde-744	57	16	∈	∈	PROPN
ejde-744	57	17	q	q	NOUN
ejde-744	58	1	:	:	PUNCT
ejde-744	58	2	=	=	SYM
ejde-744	58	3	(	(	PUNCT
ejde-744	58	4	0	0	NUM
ejde-744	58	5	,	,	PUNCT
ejde-744	58	6	t	t	NOUN
ejde-744	58	7	)	)	PUNCT
ejde-744	58	8	×	×	NOUN
ejde-744	58	9	(	(	PUNCT
ejde-744	58	10	0	0	NUM
ejde-744	58	11	,	,	PUNCT
ejde-744	58	12	1	1	NUM
ejde-744	58	13	)	)	PUNCT
ejde-744	58	14	,	,	PUNCT
ejde-744	58	15	y(t	y(t	PROPN
ejde-744	58	16	,	,	PUNCT
ejde-744	58	17	1	1	NUM
ejde-744	58	18	)	)	PUNCT
ejde-744	58	19	=	=	SYM
ejde-744	58	20	f	f	PROPN
ejde-744	58	21	,	,	PUNCT
ejde-744	58	22	t	t	PROPN
ejde-744	58	23	∈	∈	PROPN
ejde-744	58	24	(	(	PUNCT
ejde-744	58	25	0	0	NUM
ejde-744	58	26	,	,	PUNCT
ejde-744	58	27	t	t	NOUN
ejde-744	58	28	)	)	PUNCT
ejde-744	58	29	,	,	PUNCT
ejde-744	58	30	y(t	y(t	PROPN
ejde-744	58	31	,	,	PUNCT
ejde-744	58	32	0	0	NUM
ejde-744	58	33	)	)	PUNCT
ejde-744	58	34	=	=	SYM
ejde-744	59	1	0	0	NUM
ejde-744	59	2	,	,	PUNCT
ejde-744	59	3	if	if	SCONJ
ejde-744	59	4	ka	ka	PROPN
ejde-744	59	5	∈	∈	PROPN
ejde-744	60	1	[	[	X
ejde-744	60	2	0	0	NUM
ejde-744	60	3	,	,	PUNCT
ejde-744	60	4	1	1	NUM
ejde-744	60	5	)	)	PUNCT
ejde-744	60	6	,	,	PUNCT
ejde-744	60	7	t	t	PROPN
ejde-744	60	8	∈	∈	PROPN
ejde-744	60	9	(	(	PUNCT
ejde-744	60	10	0	0	NUM
ejde-744	60	11	,	,	PUNCT
ejde-744	60	12	t	t	PROPN
ejde-744	60	13	)	)	PUNCT
ejde-744	60	14	,	,	PUNCT
ejde-744	60	15	lim	lim	PROPN
ejde-744	60	16	x→0	x→0	PROPN
ejde-744	61	1	+	+	PROPN
ejde-744	61	2	ayx(t	ayx(t	PROPN
ejde-744	61	3	,	,	PUNCT
ejde-744	61	4	x	x	NOUN
ejde-744	61	5	)	)	PUNCT
ejde-744	61	6	=	=	SYM
ejde-744	61	7	0	0	NUM
ejde-744	61	8	,	,	PUNCT
ejde-744	61	9	if	if	SCONJ
ejde-744	61	10	ka	ka	PROPN
ejde-744	61	11	∈	∈	PROPN
ejde-744	62	1	[	[	X
ejde-744	62	2	1	1	NUM
ejde-744	62	3	,	,	PUNCT
ejde-744	62	4	2	2	NUM
ejde-744	62	5	)	)	PUNCT
ejde-744	62	6	,	,	PUNCT
ejde-744	62	7	t	t	PROPN
ejde-744	62	8	∈	∈	PROPN
ejde-744	62	9	(	(	PUNCT
ejde-744	62	10	0	0	NUM
ejde-744	62	11	,	,	PUNCT
ejde-744	62	12	t	t	NOUN
ejde-744	62	13	)	)	PUNCT
ejde-744	62	14	,	,	PUNCT
ejde-744	62	15	y(0	y(0	PROPN
ejde-744	62	16	,	,	PUNCT
ejde-744	62	17	x	x	NOUN
ejde-744	62	18	)	)	PUNCT
ejde-744	62	19	=	=	SYM
ejde-744	62	20	y0(x	y0(x	NOUN
ejde-744	62	21	)	)	PUNCT
ejde-744	62	22	,	,	PUNCT
ejde-744	62	23	yt(0	yt(0	PROPN
ejde-744	62	24	,	,	PUNCT
ejde-744	62	25	x	x	NOUN
ejde-744	62	26	)	)	PUNCT
ejde-744	62	27	=	=	SYM
ejde-744	62	28	y1(x	y1(x	NOUN
ejde-744	62	29	)	)	PUNCT
ejde-744	62	30	,	,	PUNCT
ejde-744	62	31	x	x	PUNCT
ejde-744	62	32	∈	∈	PROPN
ejde-744	62	33	(	(	PUNCT
ejde-744	62	34	0	0	NUM
ejde-744	62	35	,	,	PUNCT
ejde-744	62	36	1	1	NUM
ejde-744	62	37	)	)	PUNCT
ejde-744	62	38	,	,	PUNCT
ejde-744	62	39	(	(	PUNCT
ejde-744	62	40	1.7	1.7	NUM
ejde-744	62	41	)	)	PUNCT
ejde-744	62	42	where	where	SCONJ
ejde-744	62	43	a(degenerate	a(degenerate	ADJ
ejde-744	62	44	coefficient	coefficient	NOUN
ejde-744	62	45	)	)	PUNCT
ejde-744	62	46	and	and	CCONJ
ejde-744	62	47	b(singular	b(singular	ADJ
ejde-744	62	48	coefficient	coefficient	NOUN
ejde-744	62	49	)	)	PUNCT
ejde-744	62	50	positive	positive	ADJ
ejde-744	62	51	on	on	ADP
ejde-744	62	52	(	(	PUNCT
ejde-744	62	53	0	0	NUM
ejde-744	62	54	,	,	PUNCT
ejde-744	62	55	1	1	NUM
ejde-744	62	56	]	]	PUNCT
ejde-744	62	57	and	and	CCONJ
ejde-744	62	58	vanish	vanish	VERB
ejde-744	62	59	at	at	ADP
ejde-744	62	60	zero	zero	NUM
ejde-744	62	61	,	,	PUNCT
ejde-744	62	62	λ	λ	PROPN
ejde-744	62	63	∈	∈	NOUN
ejde-744	62	64	r	r	NOUN
ejde-744	62	65	and	and	CCONJ
ejde-744	62	66	u0	u0	ADJ
ejde-744	62	67	,	,	PUNCT
ejde-744	62	68	u1	u1	NOUN
ejde-744	62	69	are	be	AUX
ejde-744	62	70	the	the	DET
ejde-744	62	71	initial	initial	ADJ
ejde-744	62	72	values	value	NOUN
ejde-744	62	73	,	,	PUNCT
ejde-744	62	74	the	the	DET
ejde-744	62	75	parameter	parameter	NOUN
ejde-744	62	76	kg	kg	PROPN
ejde-744	62	77	defined	define	VERB
ejde-744	62	78	by	by	ADP
ejde-744	62	79	(	(	PUNCT
ejde-744	62	80	1.8	1.8	NUM
ejde-744	62	81	)	)	PUNCT
ejde-744	62	82	.	.	PUNCT
ejde-744	63	1	the	the	DET
ejde-744	63	2	control	control	NOUN
ejde-744	63	3	function	function	NOUN
ejde-744	63	4	f	f	PROPN
ejde-744	63	5	acts	act	VERB
ejde-744	63	6	on	on	ADP
ejde-744	63	7	non	non	ADJ
ejde-744	63	8	-	-	ADJ
ejde-744	63	9	degenerate	degenerate	ADJ
ejde-744	63	10	and	and	CCONJ
ejde-744	63	11	non	non	ADJ
ejde-744	63	12	-	-	ADJ
ejde-744	63	13	singular	singular	ADJ
ejde-744	63	14	boundary	boundary	NOUN
ejde-744	63	15	which	which	PRON
ejde-744	63	16	is	be	AUX
ejde-744	63	17	used	use	VERB
ejde-744	63	18	drive	drive	VERB
ejde-744	63	19	the	the	DET
ejde-744	63	20	solution	solution	NOUN
ejde-744	63	21	to	to	ADP
ejde-744	63	22	zero	zero	NUM
ejde-744	63	23	at	at	ADP
ejde-744	63	24	a	a	DET
ejde-744	63	25	sufficiently	sufficiently	ADV
ejde-744	63	26	large	large	ADJ
ejde-744	63	27	time	time	NOUN
ejde-744	63	28	t	t	NOUN
ejde-744	63	29	.	.	PUNCT
ejde-744	64	1	definition	definition	NOUN
ejde-744	64	2	1.1	1.1	NUM
ejde-744	64	3	.	.	PUNCT
ejde-744	65	1	let	let	VERB
ejde-744	65	2	g(x	g(x	NOUN
ejde-744	65	3	)	)	PUNCT
ejde-744	65	4	∈	∈	PROPN
ejde-744	65	5	c1((0	c1((0	NOUN
ejde-744	65	6	,	,	PUNCT
ejde-744	65	7	1	1	NUM
ejde-744	65	8	]	]	PUNCT
ejde-744	65	9	)	)	PUNCT
ejde-744	65	10	∩	∩	NOUN
ejde-744	65	11	c([0	c([0	NOUN
ejde-744	65	12	,	,	PUNCT
ejde-744	65	13	1	1	NUM
ejde-744	65	14	]	]	PUNCT
ejde-744	65	15	)	)	PUNCT
ejde-744	65	16	be	be	AUX
ejde-744	65	17	a	a	DET
ejde-744	65	18	function	function	NOUN
ejde-744	65	19	satisfying	satisfy	VERB
ejde-744	65	20	g(x	g(x	NOUN
ejde-744	65	21	)	)	PUNCT
ejde-744	65	22	>	>	X
ejde-744	65	23	0	0	PUNCT
ejde-744	66	1	on	on	ADP
ejde-744	66	2	(	(	PUNCT
ejde-744	66	3	0	0	NUM
ejde-744	66	4	,	,	PUNCT
ejde-744	66	5	1	1	NUM
ejde-744	66	6	]	]	PUNCT
ejde-744	66	7	,	,	PUNCT
ejde-744	66	8	g(0	g(0	PROPN
ejde-744	66	9	)	)	PUNCT
ejde-744	66	10	=	=	SYM
ejde-744	66	11	0	0	NUM
ejde-744	66	12	and	and	CCONJ
ejde-744	66	13	sup	sup	NOUN
ejde-744	66	14	0	0	NUM
ejde-744	66	15	<	<	X
ejde-744	66	16	x≤1	x≤1	NOUN
ejde-744	66	17	x|g′(x)|	x|g′(x)|	X
ejde-744	66	18	g(x	g(x	NOUN
ejde-744	66	19	)	)	PUNCT
ejde-744	66	20	=	=	SYM
ejde-744	66	21	kg	kg	X
ejde-744	66	22	.	.	PUNCT
ejde-744	66	23	(	(	PUNCT
ejde-744	66	24	1.8	1.8	NUM
ejde-744	66	25	)	)	PUNCT
ejde-744	66	26	when	when	SCONJ
ejde-744	66	27	the	the	DET
ejde-744	66	28	function	function	NOUN
ejde-744	66	29	g	g	NOUN
ejde-744	66	30	as	as	ADP
ejde-744	66	31	a	a	DET
ejde-744	66	32	degenerate	degenerate	ADJ
ejde-744	66	33	coefficient	coefficient	NOUN
ejde-744	66	34	,	,	PUNCT
ejde-744	66	35	we	we	PRON
ejde-744	66	36	say	say	VERB
ejde-744	66	37	that	that	SCONJ
ejde-744	66	38	g	g	PROPN
ejde-744	66	39	is	be	AUX
ejde-744	66	40	weakly	weakly	ADV
ejde-744	66	41	degenerate	degenerate	ADJ
ejde-744	66	42	at	at	ADP
ejde-744	66	43	0	0	NUM
ejde-744	66	44	if	if	SCONJ
ejde-744	66	45	kg	kg	PROPN
ejde-744	66	46	∈	∈	PROPN
ejde-744	67	1	[	[	X
ejde-744	67	2	0	0	NUM
ejde-744	67	3	,	,	PUNCT
ejde-744	67	4	1	1	NUM
ejde-744	67	5	)	)	PUNCT
ejde-744	67	6	,	,	PUNCT
ejde-744	67	7	g	g	PROPN
ejde-744	67	8	is	be	AUX
ejde-744	67	9	considered	consider	VERB
ejde-744	67	10	strongly	strongly	ADV
ejde-744	67	11	degenerate	degenerate	ADJ
ejde-744	67	12	at	at	ADP
ejde-744	67	13	0	0	NUM
ejde-744	67	14	if	if	SCONJ
ejde-744	67	15	kg	kg	PROPN
ejde-744	67	16	∈	∈	PROPN
ejde-744	68	1	[	[	X
ejde-744	68	2	1	1	NUM
ejde-744	68	3	,	,	PUNCT
ejde-744	68	4	2	2	NUM
ejde-744	68	5	)	)	PUNCT
ejde-744	68	6	.	.	PUNCT
ejde-744	69	1	remark	remark	PROPN
ejde-744	69	2	1.2	1.2	NUM
ejde-744	69	3	.	.	PUNCT
ejde-744	70	1	clearly	clearly	ADV
ejde-744	70	2	,	,	PUNCT
ejde-744	70	3	when	when	SCONJ
ejde-744	70	4	g(x	g(x	NOUN
ejde-744	70	5	)	)	PUNCT
ejde-744	70	6	∼	∼	NOUN
ejde-744	70	7	xk	xk	PROPN
ejde-744	70	8	,	,	PUNCT
ejde-744	70	9	it	it	PRON
ejde-744	70	10	is	be	AUX
ejde-744	70	11	considered	consider	VERB
ejde-744	70	12	weakly	weakly	ADV
ejde-744	70	13	degenerate	degenerate	ADJ
ejde-744	70	14	if	if	SCONJ
ejde-744	70	15	k	k	PROPN
ejde-744	70	16	∈	∈	PROPN
ejde-744	71	1	[	[	X
ejde-744	71	2	0	0	NUM
ejde-744	71	3	,	,	PUNCT
ejde-744	71	4	1	1	NUM
ejde-744	71	5	)	)	PUNCT
ejde-744	71	6	,	,	PUNCT
ejde-744	71	7	strongly	strongly	ADV
ejde-744	71	8	degenerate	degenerate	ADJ
ejde-744	71	9	if	if	SCONJ
ejde-744	71	10	k	k	PROPN
ejde-744	71	11	∈	∈	PROPN
ejde-744	71	12	[	[	X
ejde-744	71	13	1	1	NUM
ejde-744	71	14	,	,	PUNCT
ejde-744	71	15	2	2	NUM
ejde-744	71	16	)	)	PUNCT
ejde-744	71	17	.	.	PUNCT
ejde-744	72	1	furthermore	furthermore	ADV
ejde-744	72	2	,	,	PUNCT
ejde-744	72	3	the	the	DET
ejde-744	72	4	case	case	NOUN
ejde-744	72	5	where	where	SCONJ
ejde-744	72	6	kg	kg	PROPN
ejde-744	72	7	≥	≥	PUNCT
ejde-744	72	8	2	2	NUM
ejde-744	72	9	is	be	AUX
ejde-744	72	10	not	not	PART
ejde-744	72	11	considered	consider	VERB
ejde-744	72	12	because	because	SCONJ
ejde-744	72	13	it	it	PRON
ejde-744	72	14	does	do	AUX
ejde-744	72	15	not	not	PART
ejde-744	72	16	achieve	achieve	VERB
ejde-744	72	17	controllability	controllability	NOUN
ejde-744	72	18	,	,	PUNCT
ejde-744	72	19	as	as	SCONJ
ejde-744	72	20	discussed	discuss	VERB
ejde-744	72	21	in	in	ADP
ejde-744	72	22	[	[	X
ejde-744	72	23	1	1	NUM
ejde-744	72	24	]	]	PUNCT
ejde-744	72	25	when	when	SCONJ
ejde-744	72	26	λ	λ	X
ejde-744	72	27	=	=	SYM
ejde-744	72	28	0	0	NUM
ejde-744	72	29	.	.	PUNCT
ejde-744	73	1	finally	finally	ADV
ejde-744	73	2	,	,	PUNCT
ejde-744	73	3	we	we	PRON
ejde-744	73	4	point	point	VERB
ejde-744	73	5	out	out	ADP
ejde-744	73	6	that	that	SCONJ
ejde-744	73	7	studies	study	NOUN
ejde-744	73	8	like	like	ADP
ejde-744	73	9	ours	ours	PRON
ejde-744	73	10	are	be	AUX
ejde-744	73	11	significant	significant	ADJ
ejde-744	73	12	,	,	PUNCT
ejde-744	73	13	particularly	particularly	ADV
ejde-744	73	14	in	in	ADP
ejde-744	73	15	the	the	DET
ejde-744	73	16	fields	field	NOUN
ejde-744	73	17	of	of	ADP
ejde-744	73	18	medical	medical	ADJ
ejde-744	73	19	research	research	NOUN
ejde-744	74	1	[	[	X
ejde-744	74	2	18	18	NUM
ejde-744	74	3	]	]	PUNCT
ejde-744	74	4	,	,	PUNCT
ejde-744	74	5	materials	material	NOUN
ejde-744	74	6	science	science	NOUN
ejde-744	74	7	for	for	ADP
ejde-744	74	8	invisible	invisible	ADJ
ejde-744	74	9	materials	material	NOUN
ejde-744	74	10	[	[	X
ejde-744	74	11	12	12	NUM
ejde-744	74	12	,	,	PUNCT
ejde-744	74	13	21	21	NUM
ejde-744	74	14	]	]	PUNCT
ejde-744	74	15	,	,	PUNCT
ejde-744	74	16	and	and	CCONJ
ejde-744	74	17	climate	climate	NOUN
ejde-744	74	18	science	science	NOUN
ejde-744	75	1	[	[	X
ejde-744	75	2	17	17	NUM
ejde-744	75	3	]	]	PUNCT
ejde-744	75	4	.	.	PUNCT
ejde-744	76	1	this	this	DET
ejde-744	76	2	article	article	NOUN
ejde-744	76	3	is	be	AUX
ejde-744	76	4	organized	organize	VERB
ejde-744	76	5	as	as	SCONJ
ejde-744	76	6	follows	follow	VERB
ejde-744	76	7	.	.	PUNCT
ejde-744	77	1	section	section	NOUN
ejde-744	77	2	2	2	NUM
ejde-744	77	3	presents	present	VERB
ejde-744	77	4	some	some	DET
ejde-744	77	5	function	function	NOUN
ejde-744	77	6	spaces	space	NOUN
ejde-744	77	7	and	and	CCONJ
ejde-744	77	8	preliminary	preliminary	ADJ
ejde-744	77	9	results	result	NOUN
ejde-744	77	10	.	.	PUNCT
ejde-744	78	1	in	in	ADP
ejde-744	78	2	section	section	NOUN
ejde-744	78	3	3	3	NUM
ejde-744	78	4	,	,	PUNCT
ejde-744	78	5	we	we	PRON
ejde-744	78	6	employ	employ	VERB
ejde-744	78	7	the	the	DET
ejde-744	78	8	lax	lax	PROPN
ejde-744	78	9	-	-	PUNCT
ejde-744	78	10	milgram	milgram	NOUN
ejde-744	78	11	theorem	theorem	PROPN
ejde-744	78	12	and	and	CCONJ
ejde-744	78	13	semigroup	semigroup	PROPN
ejde-744	78	14	theory	theory	NOUN
ejde-744	78	15	to	to	PART
ejde-744	78	16	address	address	VERB
ejde-744	78	17	the	the	DET
ejde-744	78	18	dual	dual	ADJ
ejde-744	78	19	problem	problem	NOUN
ejde-744	78	20	and	and	CCONJ
ejde-744	78	21	investigate	investigate	VERB
ejde-744	78	22	the	the	DET
ejde-744	78	23	well	well	NOUN
ejde-744	78	24	-	-	PUNCT
ejde-744	78	25	posedness	posedness	NOUN
ejde-744	78	26	of	of	ADP
ejde-744	78	27	the	the	DET
ejde-744	78	28	associated	associated	ADJ
ejde-744	78	29	problem	problem	NOUN
ejde-744	78	30	under	under	ADP
ejde-744	78	31	dirichlet	dirichlet	PROPN
ejde-744	78	32	and	and	CCONJ
ejde-744	78	33	neumann	neumann	PROPN
ejde-744	78	34	boundary	boundary	ADJ
ejde-744	78	35	conditions	condition	NOUN
ejde-744	78	36	.	.	PUNCT
ejde-744	79	1	in	in	ADP
ejde-744	79	2	section	section	NOUN
ejde-744	79	3	4	4	NUM
ejde-744	79	4	,	,	PUNCT
ejde-744	79	5	an	an	DET
ejde-744	79	6	energy	energy	NOUN
ejde-744	79	7	estimate	estimate	NOUN
ejde-744	79	8	is	be	AUX
ejde-744	79	9	established	establish	VERB
ejde-744	79	10	and	and	CCONJ
ejde-744	79	11	the	the	DET
ejde-744	79	12	direct	direct	ADJ
ejde-744	79	13	inequality	inequality	NOUN
ejde-744	79	14	is	be	AUX
ejde-744	79	15	proven	prove	VERB
ejde-744	79	16	,	,	PUNCT
ejde-744	79	17	which	which	PRON
ejde-744	79	18	will	will	AUX
ejde-744	79	19	eventually	eventually	ADV
ejde-744	79	20	lead	lead	VERB
ejde-744	79	21	to	to	ADP
ejde-744	79	22	the	the	DET
ejde-744	79	23	controllability	controllability	NOUN
ejde-744	79	24	result	result	NOUN
ejde-744	79	25	.	.	PUNCT
ejde-744	80	1	section	section	NOUN
ejde-744	80	2	5	5	NUM
ejde-744	80	3	introduces	introduce	NOUN
ejde-744	80	4	the	the	DET
ejde-744	80	5	observable	observable	ADJ
ejde-744	80	6	inequality	inequality	NOUN
ejde-744	80	7	and	and	CCONJ
ejde-744	80	8	determines	determine	VERB
ejde-744	80	9	the	the	DET
ejde-744	80	10	observable	observable	ADJ
ejde-744	80	11	time	time	NOUN
ejde-744	80	12	.	.	PUNCT
ejde-744	81	1	in	in	ADP
ejde-744	81	2	section	section	NOUN
ejde-744	81	3	6	6	NUM
ejde-744	81	4	,	,	PUNCT
ejde-744	81	5	the	the	DET
ejde-744	81	6	controllable	controllable	ADJ
ejde-744	81	7	result	result	NOUN
ejde-744	81	8	is	be	AUX
ejde-744	81	9	obtained	obtain	VERB
ejde-744	81	10	.	.	PUNCT
ejde-744	82	1	we	we	PRON
ejde-744	82	2	conclude	conclude	VERB
ejde-744	82	3	the	the	DET
ejde-744	82	4	paper	paper	NOUN
ejde-744	82	5	with	with	ADP
ejde-744	82	6	a	a	DET
ejde-744	82	7	summary	summary	NOUN
ejde-744	82	8	in	in	ADP
ejde-744	82	9	the	the	DET
ejde-744	82	10	final	final	ADJ
ejde-744	82	11	section	section	NOUN
ejde-744	82	12	.	.	PUNCT
ejde-744	83	1	2	2	X
ejde-744	83	2	.	.	NUM
ejde-744	83	3	preliminaries	preliminary	NOUN
ejde-744	83	4	assumption	assumption	NOUN
ejde-744	83	5	2.1	2.1	NUM
ejde-744	83	6	.	.	PUNCT
ejde-744	84	1	the	the	DET
ejde-744	84	2	functions	function	NOUN
ejde-744	84	3	a	a	PRON
ejde-744	84	4	,	,	PUNCT
ejde-744	84	5	b	b	NOUN
ejde-744	84	6	belong	belong	VERB
ejde-744	84	7	to	to	ADP
ejde-744	84	8	c1((0	c1((0	PROPN
ejde-744	84	9	,	,	PUNCT
ejde-744	84	10	1	1	NUM
ejde-744	84	11	]	]	PUNCT
ejde-744	84	12	)	)	PUNCT
ejde-744	84	13	∩	∩	NOUN
ejde-744	84	14	c([0	c([0	NOUN
ejde-744	84	15	,	,	PUNCT
ejde-744	84	16	1	1	NUM
ejde-744	84	17	]	]	PUNCT
ejde-744	84	18	)	)	PUNCT
ejde-744	84	19	and	and	CCONJ
ejde-744	84	20	satisfy	satisfy	VERB
ejde-744	84	21	a(x	a(x	NOUN
ejde-744	84	22	)	)	PUNCT
ejde-744	84	23	,	,	PUNCT
ejde-744	84	24	b(x	b(x	NOUN
ejde-744	84	25	)	)	PUNCT
ejde-744	84	26	>	>	SYM
ejde-744	84	27	0	0	PUNCT
ejde-744	84	28	∀x	∀x	X
ejde-744	84	29	∈	∈	PROPN
ejde-744	84	30	(	(	PUNCT
ejde-744	84	31	0	0	NUM
ejde-744	84	32	,	,	PUNCT
ejde-744	84	33	1	1	NUM
ejde-744	84	34	]	]	PUNCT
ejde-744	84	35	,	,	PUNCT
ejde-744	84	36	a(0	a(0	PROPN
ejde-744	84	37	)	)	PUNCT
ejde-744	84	38	=	=	SYM
ejde-744	84	39	b(0	b(0	PROPN
ejde-744	84	40	)	)	PUNCT
ejde-744	84	41	=	=	SYM
ejde-744	84	42	0	0	NUM
ejde-744	84	43	,	,	PUNCT
ejde-744	84	44	ka	ka	PROPN
ejde-744	84	45	∈	∈	PROPN
ejde-744	85	1	[	[	X
ejde-744	85	2	0	0	NUM
ejde-744	85	3	,	,	PUNCT
ejde-744	85	4	2	2	NUM
ejde-744	85	5	)	)	PUNCT
ejde-744	85	6	\	\	NOUN
ejde-744	85	7	{	{	PUNCT
ejde-744	85	8	1},kb	1},kb	NUM
ejde-744	85	9	∈	∈	NOUN
ejde-744	86	1	[	[	X
ejde-744	86	2	0	0	NUM
ejde-744	86	3	,	,	PUNCT
ejde-744	86	4	2	2	NUM
ejde-744	86	5	]	]	PUNCT
ejde-744	86	6	and	and	CCONJ
ejde-744	86	7	ka	ka	X
ejde-744	87	1	+	+	PROPN
ejde-744	87	2	kb	kb	PROPN
ejde-744	87	3	≤	≤	ADJ
ejde-744	87	4	2	2	NUM
ejde-744	87	5	.	.	PUNCT
ejde-744	88	1	(	(	PUNCT
ejde-744	88	2	2.1	2.1	NUM
ejde-744	88	3	)	)	PUNCT
ejde-744	88	4	from	from	ADP
ejde-744	88	5	definition	definition	NOUN
ejde-744	88	6	1.1	1.1	NUM
ejde-744	88	7	and	and	CCONJ
ejde-744	88	8	assumption	assumption	NOUN
ejde-744	88	9	2.1	2.1	NUM
ejde-744	88	10	,	,	PUNCT
ejde-744	88	11	it	it	PRON
ejde-744	88	12	is	be	AUX
ejde-744	88	13	easy	easy	ADJ
ejde-744	88	14	to	to	PART
ejde-744	88	15	draw	draw	VERB
ejde-744	88	16	the	the	DET
ejde-744	88	17	following	follow	VERB
ejde-744	88	18	consequences	consequence	NOUN
ejde-744	88	19	.	.	PUNCT
ejde-744	89	1	remark	remark	PROPN
ejde-744	89	2	2.2	2.2	NUM
ejde-744	89	3	.	.	PUNCT
ejde-744	90	1	by	by	ADP
ejde-744	90	2	integrating	integrate	VERB
ejde-744	90	3	the	the	DET
ejde-744	90	4	inequality	inequality	NOUN
ejde-744	90	5	sg′(s	sg′(	VERB
ejde-744	90	6	)	)	PUNCT
ejde-744	90	7	≤	≤	NUM
ejde-744	90	8	kgg(s	kgg(s	PROPN
ejde-744	90	9	)	)	PUNCT
ejde-744	90	10	,	,	PUNCT
ejde-744	90	11	∀s	∀s	PROPN
ejde-744	90	12	∈	∈	PROPN
ejde-744	90	13	(	(	PUNCT
ejde-744	90	14	0	0	NUM
ejde-744	90	15	,	,	PUNCT
ejde-744	90	16	1	1	NUM
ejde-744	90	17	]	]	SYM
ejde-744	90	18	4	4	NUM
ejde-744	90	19	g.	g.	PROPN
ejde-744	90	20	zhang	zhang	PROPN
ejde-744	90	21	,	,	PUNCT
ejde-744	90	22	s.	s.	PROPN
ejde-744	90	23	chai	chai	PROPN
ejde-744	90	24	ejde-2025/04	ejde-2025/04	NOUN
ejde-744	90	25	over	over	ADP
ejde-744	90	26	[	[	X
ejde-744	90	27	x	x	X
ejde-744	90	28	,	,	PUNCT
ejde-744	90	29	1	1	NUM
ejde-744	90	30	]	]	PUNCT
ejde-744	90	31	,	,	PUNCT
ejde-744	90	32	we	we	PRON
ejde-744	90	33	obtain	obtain	VERB
ejde-744	90	34	g(x	g(x	NOUN
ejde-744	90	35	)	)	PUNCT
ejde-744	90	36	≥	≥	NOUN
ejde-744	90	37	g(1)xkg	g(1)xkg	VERB
ejde-744	90	38	,	,	PUNCT
ejde-744	90	39	∀x	∀x	X
ejde-744	90	40	∈	∈	PROPN
ejde-744	91	1	[	[	X
ejde-744	91	2	0	0	NUM
ejde-744	91	3	,	,	PUNCT
ejde-744	91	4	1	1	NUM
ejde-744	91	5	]	]	PUNCT
ejde-744	91	6	.	.	PUNCT
ejde-744	92	1	hence	hence	ADV
ejde-744	92	2	,	,	PUNCT
ejde-744	92	3	for	for	ADP
ejde-744	92	4	all	all	DET
ejde-744	92	5	x	x	SYM
ejde-744	92	6	∈	∈	PROPN
ejde-744	92	7	[	[	X
ejde-744	92	8	0	0	NUM
ejde-744	92	9	,	,	PUNCT
ejde-744	92	10	1	1	NUM
ejde-744	92	11	]	]	PUNCT
ejde-744	92	12	we	we	PRON
ejde-744	92	13	deduce	deduce	VERB
ejde-744	92	14	that	that	SCONJ
ejde-744	92	15	a(x	a(x	NOUN
ejde-744	92	16	)	)	PUNCT
ejde-744	92	17	≥	≥	NOUN
ejde-744	92	18	a(1)xka	a(1)xka	NOUN
ejde-744	92	19	,	,	PUNCT
ejde-744	92	20	b(x	b(x	NOUN
ejde-744	92	21	)	)	PUNCT
ejde-744	92	22	≥	≥	NOUN
ejde-744	92	23	b(1)xkb	b(1)xkb	NOUN
ejde-744	92	24	.	.	PUNCT
ejde-744	93	1	(	(	PUNCT
ejde-744	93	2	2.2	2.2	NUM
ejde-744	93	3	)	)	PUNCT
ejde-744	93	4	proposition	proposition	NOUN
ejde-744	93	5	2.3	2.3	NUM
ejde-744	93	6	(	(	PUNCT
ejde-744	93	7	hardy	hardy	ADJ
ejde-744	93	8	-	-	PUNCT
ejde-744	93	9	poincare	poincare	NOUN
ejde-744	93	10	inequality	inequality	NOUN
ejde-744	93	11	)	)	PUNCT
ejde-744	93	12	.	.	PUNCT
ejde-744	94	1	under	under	ADP
ejde-744	94	2	assumption	assumption	NOUN
ejde-744	94	3	2.1	2.1	NUM
ejde-744	94	4	,	,	PUNCT
ejde-744	94	5	there	there	PRON
ejde-744	94	6	exists	exist	VERB
ejde-744	94	7	ca	ca	NOUN
ejde-744	94	8	,	,	PUNCT
ejde-744	94	9	b	b	PROPN
ejde-744	94	10	>	>	X
ejde-744	94	11	0	0	NUM
ejde-744	94	12	such	such	ADJ
ejde-744	94	13	that∫	that∫	NOUN
ejde-744	94	14	1	1	NUM
ejde-744	94	15	0	0	NUM
ejde-744	94	16	u2	u2	PROPN
ejde-744	94	17	b	b	PROPN
ejde-744	94	18	dx	dx	PROPN
ejde-744	94	19	≤	≤	PROPN
ejde-744	94	20	ca	can	AUX
ejde-744	94	21	,	,	PUNCT
ejde-744	94	22	b	b	X
ejde-744	94	23	∫	∫	PROPN
ejde-744	94	24	1	1	NUM
ejde-744	94	25	0	0	NUM
ejde-744	94	26	a(u′	a(u′	NUM
ejde-744	94	27	)	)	PUNCT
ejde-744	94	28	2	2	NUM
ejde-744	94	29	dx	dx	NOUN
ejde-744	94	30	,	,	PUNCT
ejde-744	94	31	∀u	∀u	NOUN
ejde-744	94	32	∈	∈	NOUN
ejde-744	94	33	c∞	c∞	PROPN
ejde-744	94	34	c	c	X
ejde-744	94	35	(	(	PUNCT
ejde-744	94	36	0	0	NUM
ejde-744	94	37	,	,	PUNCT
ejde-744	94	38	1	1	NUM
ejde-744	94	39	)	)	PUNCT
ejde-744	94	40	,	,	PUNCT
ejde-744	94	41	(	(	PUNCT
ejde-744	94	42	2.3	2.3	NUM
ejde-744	94	43	)	)	PUNCT
ejde-744	94	44	where	where	SCONJ
ejde-744	94	45	ca	can	AUX
ejde-744	94	46	,	,	PUNCT
ejde-744	94	47	b	b	NOUN
ejde-744	94	48	=	=	SYM
ejde-744	94	49	4	4	NUM
ejde-744	94	50	a(1)b(1)(1−ka)2	a(1)b(1)(1−ka)2	NOUN
ejde-744	94	51	.	.	PUNCT
ejde-744	95	1	proof	proof	NOUN
ejde-744	95	2	.	.	PUNCT
ejde-744	96	1	by	by	ADP
ejde-744	96	2	remark	remark	NOUN
ejde-744	96	3	2.2	2.2	NUM
ejde-744	96	4	and	and	CCONJ
ejde-744	96	5	generalized	generalize	VERB
ejde-744	96	6	hardy	hardy	ADJ
ejde-744	96	7	inequality	inequality	NOUN
ejde-744	96	8	[	[	X
ejde-744	96	9	11	11	NUM
ejde-744	96	10	,	,	PUNCT
ejde-744	96	11	chap	chap	NOUN
ejde-744	96	12	.	.	PUNCT
ejde-744	97	1	5.3	5.3	NUM
ejde-744	97	2	]	]	PUNCT
ejde-744	97	3	,	,	PUNCT
ejde-744	97	4	(	(	PUNCT
ejde-744	97	5	1−ka	1−ka	NOUN
ejde-744	97	6	)	)	PUNCT
ejde-744	97	7	2	2	NUM
ejde-744	97	8	4	4	NUM
ejde-744	97	9	∫	∫	NOUN
ejde-744	97	10	1	1	NUM
ejde-744	97	11	0	0	NUM
ejde-744	97	12	u2	u2	PROPN
ejde-744	97	13	x2−ka	x2−ka	PROPN
ejde-744	98	1	dx	dx	PROPN
ejde-744	98	2	≤	≤	NUM
ejde-744	98	3	∫	∫	PROPN
ejde-744	99	1	1	1	NUM
ejde-744	99	2	0	0	NUM
ejde-744	99	3	xkau2	xkau2	PROPN
ejde-744	100	1	x	x	X
ejde-744	100	2	dx	dx	PROPN
ejde-744	100	3	∀u	∀u	PROPN
ejde-744	100	4	∈	∈	PROPN
ejde-744	100	5	c∞	c∞	PROPN
ejde-744	100	6	c	c	X
ejde-744	100	7	(	(	PUNCT
ejde-744	100	8	0	0	NUM
ejde-744	100	9	,	,	PUNCT
ejde-744	100	10	1	1	NUM
ejde-744	100	11	)	)	PUNCT
ejde-744	100	12	,	,	PUNCT
ejde-744	100	13	we	we	PRON
ejde-744	100	14	have∫	have∫	VERB
ejde-744	100	15	1	1	NUM
ejde-744	100	16	0	0	NUM
ejde-744	100	17	u2	u2	PROPN
ejde-744	100	18	b	b	PROPN
ejde-744	100	19	dx	dx	PROPN
ejde-744	100	20	≤	≤	PROPN
ejde-744	100	21	1	1	NUM
ejde-744	100	22	b(1	b(1	PROPN
ejde-744	100	23	)	)	PUNCT
ejde-744	100	24	∫	∫	NOUN
ejde-744	101	1	1	1	NUM
ejde-744	101	2	0	0	NUM
ejde-744	101	3	u2	u2	PROPN
ejde-744	101	4	xkb	xkb	PROPN
ejde-744	101	5	dx	dx	PROPN
ejde-744	101	6	≤	≤	ADV
ejde-744	101	7	1	1	NUM
ejde-744	101	8	a(1)b(1	a(1)b(1	NUM
ejde-744	101	9	)	)	PUNCT
ejde-744	101	10	∫	∫	PROPN
ejde-744	101	11	1	1	NUM
ejde-744	101	12	0	0	NUM
ejde-744	101	13	a	a	DET
ejde-744	101	14	u2	u2	PROPN
ejde-744	101	15	xka+kb	xka+kb	PROPN
ejde-744	101	16	dx	dx	PROPN
ejde-744	101	17	≤	≤	NUM
ejde-744	101	18	1	1	NUM
ejde-744	101	19	a(1)b(1	a(1)b(1	NUM
ejde-744	101	20	)	)	PUNCT
ejde-744	101	21	∫	∫	PROPN
ejde-744	102	1	1	1	NUM
ejde-744	102	2	0	0	NUM
ejde-744	102	3	a	a	DET
ejde-744	102	4	u2	u2	PROPN
ejde-744	102	5	x2	x2	PROPN
ejde-744	102	6	dx	dx	PROPN
ejde-744	102	7	≤	≤	NUM
ejde-744	102	8	4	4	NUM
ejde-744	102	9	a(1)b(1)(1−ka)2	a(1)b(1)(1−ka)2	ADP
ejde-744	102	10	∫	∫	PROPN
ejde-744	102	11	1	1	NUM
ejde-744	102	12	0	0	NUM
ejde-744	102	13	a(u′	a(u′	NUM
ejde-744	102	14	)	)	PUNCT
ejde-744	102	15	2	2	NUM
ejde-744	102	16	dx	dx	PROPN
ejde-744	102	17	.	.	PUNCT
ejde-744	103	1	(	(	PUNCT
ejde-744	103	2	2.4	2.4	NUM
ejde-744	103	3	)	)	PUNCT
ejde-744	103	4	□	□	PUNCT
ejde-744	103	5	assumption	assumption	NOUN
ejde-744	103	6	2.4	2.4	NUM
ejde-744	103	7	.	.	PUNCT
ejde-744	104	1	the	the	DET
ejde-744	104	2	constant	constant	ADJ
ejde-744	104	3	λ	λ	X
ejde-744	104	4	∈	∈	NOUN
ejde-744	104	5	r	r	NOUN
ejde-744	104	6	satisfies	satisfie	NOUN
ejde-744	104	7	λ	λ	X
ejde-744	104	8	<	<	X
ejde-744	104	9	1	1	NUM
ejde-744	104	10	ca	ca	NOUN
ejde-744	104	11	,	,	PUNCT
ejde-744	104	12	b	b	PROPN
ejde-744	104	13	.	.	PUNCT
ejde-744	105	1	(	(	PUNCT
ejde-744	105	2	2.5	2.5	NUM
ejde-744	105	3	)	)	PUNCT
ejde-744	105	4	under	under	ADP
ejde-744	105	5	assumptions	assumption	NOUN
ejde-744	105	6	2.1	2.1	NUM
ejde-744	105	7	and	and	CCONJ
ejde-744	105	8	2.4	2.4	NUM
ejde-744	105	9	,	,	PUNCT
ejde-744	105	10	as	as	ADP
ejde-744	105	11	in	in	ADP
ejde-744	105	12	[	[	X
ejde-744	105	13	1	1	NUM
ejde-744	105	14	]	]	PUNCT
ejde-744	105	15	and	and	CCONJ
ejde-744	105	16	[	[	X
ejde-744	105	17	3	3	NUM
ejde-744	105	18	]	]	PUNCT
ejde-744	105	19	,	,	PUNCT
ejde-744	105	20	we	we	PRON
ejde-744	105	21	introduce	introduce	VERB
ejde-744	105	22	the	the	DET
ejde-744	105	23	following	follow	VERB
ejde-744	105	24	spaces	space	NOUN
ejde-744	105	25	with	with	ADP
ejde-744	105	26	related	related	ADJ
ejde-744	105	27	inner	inner	ADJ
ejde-744	105	28	product	product	NOUN
ejde-744	105	29	v	v	ADP
ejde-744	105	30	1	1	NUM
ejde-744	105	31	a	a	DET
ejde-744	105	32	(	(	PUNCT
ejde-744	105	33	0	0	NUM
ejde-744	105	34	,	,	PUNCT
ejde-744	105	35	1	1	NUM
ejde-744	105	36	)	)	PUNCT
ejde-744	105	37	=	=	PRON
ejde-744	105	38	{	{	PUNCT
ejde-744	105	39	u	u	NOUN
ejde-744	105	40	∈	∈	PROPN
ejde-744	105	41	l2(0	l2(0	NOUN
ejde-744	105	42	,	,	PUNCT
ejde-744	105	43	1	1	NUM
ejde-744	105	44	)	)	PUNCT
ejde-744	105	45	∩h1	∩h1	NOUN
ejde-744	105	46	loc(0	loc(0	ADJ
ejde-744	105	47	,	,	PUNCT
ejde-744	105	48	1	1	X
ejde-744	105	49	]	]	PUNCT
ejde-744	105	50	:	:	PUNCT
ejde-744	105	51	√	√	PROPN
ejde-744	105	52	au′	au′	PROPN
ejde-744	105	53	∈	∈	PROPN
ejde-744	105	54	l2(0	l2(0	NOUN
ejde-744	105	55	,	,	PUNCT
ejde-744	105	56	1	1	NUM
ejde-744	105	57	)	)	PUNCT
ejde-744	105	58	}	}	PUNCT
ejde-744	105	59	,	,	PUNCT
ejde-744	105	60	∥u∥2v	∥u∥2v	PROPN
ejde-744	105	61	1	1	NUM
ejde-744	105	62	a	a	DET
ejde-744	105	63	(	(	PUNCT
ejde-744	105	64	0,1	0,1	NUM
ejde-744	105	65	)	)	PUNCT
ejde-744	105	66	=	=	SYM
ejde-744	106	1	∫	∫	PROPN
ejde-744	106	2	1	1	NUM
ejde-744	106	3	0	0	NUM
ejde-744	106	4	u2	u2	PROPN
ejde-744	106	5	+	+	CCONJ
ejde-744	106	6	a(u′	a(u′	PROPN
ejde-744	106	7	)	)	PUNCT
ejde-744	106	8	2	2	NUM
ejde-744	106	9	dx	dx	NOUN
ejde-744	106	10	,	,	PUNCT
ejde-744	106	11	∀u	∀u	NOUN
ejde-744	106	12	∈	∈	NOUN
ejde-744	106	13	v	v	ADP
ejde-744	106	14	1	1	NUM
ejde-744	106	15	a	a	PRON
ejde-744	106	16	(	(	PUNCT
ejde-744	106	17	0	0	NUM
ejde-744	106	18	,	,	PUNCT
ejde-744	106	19	1	1	NUM
ejde-744	106	20	)	)	PUNCT
ejde-744	106	21	,	,	PUNCT
ejde-744	106	22	⟨u	⟨u	NOUN
ejde-744	106	23	,	,	PUNCT
ejde-744	106	24	v⟩v	v⟩v	NUM
ejde-744	106	25	1	1	NUM
ejde-744	106	26	a	a	DET
ejde-744	106	27	(	(	PUNCT
ejde-744	106	28	0,1	0,1	NUM
ejde-744	106	29	)	)	PUNCT
ejde-744	106	30	=	=	SYM
ejde-744	107	1	∫	∫	PROPN
ejde-744	108	1	1	1	NUM
ejde-744	108	2	0	0	NUM
ejde-744	108	3	uv	uv	NOUN
ejde-744	108	4	+	+	PROPN
ejde-744	108	5	au′v′	au′v′	PROPN
ejde-744	108	6	dx	dx	PROPN
ejde-744	108	7	,	,	PUNCT
ejde-744	108	8	∀u	∀u	NOUN
ejde-744	108	9	,	,	PUNCT
ejde-744	108	10	v	v	ADP
ejde-744	108	11	∈	∈	PROPN
ejde-744	108	12	v	v	ADP
ejde-744	108	13	1	1	NUM
ejde-744	108	14	a	a	DET
ejde-744	108	15	(	(	PUNCT
ejde-744	108	16	0	0	NUM
ejde-744	108	17	,	,	PUNCT
ejde-744	108	18	1	1	NUM
ejde-744	108	19	)	)	PUNCT
ejde-744	108	20	,	,	PUNCT
ejde-744	108	21	v	v	ADP
ejde-744	108	22	2	2	NUM
ejde-744	108	23	a	a	DET
ejde-744	108	24	(	(	PUNCT
ejde-744	108	25	0	0	NUM
ejde-744	108	26	,	,	PUNCT
ejde-744	108	27	1	1	NUM
ejde-744	108	28	)	)	PUNCT
ejde-744	108	29	=	=	PRON
ejde-744	108	30	{	{	PUNCT
ejde-744	108	31	u	u	NOUN
ejde-744	108	32	∈	∈	PROPN
ejde-744	108	33	v	v	ADP
ejde-744	108	34	1	1	NUM
ejde-744	108	35	a	a	DET
ejde-744	108	36	(	(	PUNCT
ejde-744	108	37	0	0	NUM
ejde-744	108	38	,	,	PUNCT
ejde-744	108	39	1)|au′	1)|au′	PROPN
ejde-744	108	40	∈	∈	PROPN
ejde-744	108	41	h1(0	h1(0	NOUN
ejde-744	108	42	,	,	PUNCT
ejde-744	108	43	1	1	NUM
ejde-744	108	44	)	)	PUNCT
ejde-744	108	45	}	}	PUNCT
ejde-744	108	46	;	;	PUNCT
ejde-744	108	47	and	and	CCONJ
ejde-744	108	48	v	v	X
ejde-744	108	49	1	1	NUM
ejde-744	108	50	a	a	DET
ejde-744	108	51	,	,	PUNCT
ejde-744	108	52	b(0	b(0	NOUN
ejde-744	108	53	,	,	PUNCT
ejde-744	108	54	1	1	NUM
ejde-744	108	55	)	)	PUNCT
ejde-744	108	56	=	=	PRON
ejde-744	108	57	{	{	PUNCT
ejde-744	108	58	u	u	NOUN
ejde-744	108	59	∈	∈	PROPN
ejde-744	108	60	l2(0	l2(0	NOUN
ejde-744	108	61	,	,	PUNCT
ejde-744	108	62	1	1	NUM
ejde-744	108	63	)	)	PUNCT
ejde-744	108	64	∩h1	∩h1	NOUN
ejde-744	108	65	loc(0	loc(0	NOUN
ejde-744	108	66	,	,	PUNCT
ejde-744	108	67	1]|a(u′	1]|a(u′	NUM
ejde-744	108	68	)	)	PUNCT
ejde-744	109	1	2	2	NUM
ejde-744	109	2	−	−	PROPN
ejde-744	109	3	λ	λ	X
ejde-744	109	4	b	b	PROPN
ejde-744	109	5	u2	u2	PROPN
ejde-744	109	6	∈	∈	PROPN
ejde-744	109	7	l1(0	l1(0	PROPN
ejde-744	109	8	,	,	PUNCT
ejde-744	109	9	1	1	NUM
ejde-744	109	10	)	)	PUNCT
ejde-744	109	11	}	}	PUNCT
ejde-744	109	12	,	,	PUNCT
ejde-744	109	13	∥u∥2v	∥u∥2v	PROPN
ejde-744	109	14	1	1	NUM
ejde-744	109	15	a	a	DET
ejde-744	109	16	,	,	PUNCT
ejde-744	109	17	b(0,1	b(0,1	NOUN
ejde-744	109	18	)	)	PUNCT
ejde-744	109	19	=	=	SYM
ejde-744	110	1	∫	∫	PROPN
ejde-744	110	2	1	1	NUM
ejde-744	110	3	0	0	NUM
ejde-744	110	4	u2	u2	PROPN
ejde-744	110	5	+	+	CCONJ
ejde-744	110	6	a(u′	a(u′	NUM
ejde-744	110	7	)	)	PUNCT
ejde-744	111	1	2	2	NUM
ejde-744	111	2	−	−	PROPN
ejde-744	111	3	λ	λ	PROPN
ejde-744	111	4	b	b	PROPN
ejde-744	111	5	u2	u2	PROPN
ejde-744	111	6	dx	dx	PROPN
ejde-744	111	7	,	,	PUNCT
ejde-744	111	8	∀u	∀u	NOUN
ejde-744	111	9	∈	∈	NOUN
ejde-744	111	10	v	v	ADP
ejde-744	111	11	1	1	NUM
ejde-744	111	12	a	a	PRON
ejde-744	111	13	,	,	PUNCT
ejde-744	111	14	b(0	b(0	NOUN
ejde-744	111	15	,	,	PUNCT
ejde-744	111	16	1	1	NUM
ejde-744	111	17	)	)	PUNCT
ejde-744	111	18	,	,	PUNCT
ejde-744	111	19	⟨u	⟨u	NOUN
ejde-744	111	20	,	,	PUNCT
ejde-744	111	21	v⟩v	v⟩v	NUM
ejde-744	111	22	1	1	NUM
ejde-744	111	23	a	a	DET
ejde-744	111	24	,	,	PUNCT
ejde-744	111	25	b(0,1	b(0,1	NOUN
ejde-744	111	26	)	)	PUNCT
ejde-744	111	27	=	=	SYM
ejde-744	112	1	∫	∫	PROPN
ejde-744	113	1	1	1	NUM
ejde-744	113	2	0	0	NUM
ejde-744	113	3	uv	uv	NOUN
ejde-744	113	4	+	+	PROPN
ejde-744	113	5	au′v′	au′v′	PROPN
ejde-744	113	6	−	−	PROPN
ejde-744	114	1	λ	λ	X
ejde-744	114	2	b	b	PROPN
ejde-744	114	3	uv	uv	PROPN
ejde-744	114	4	dx	dx	PROPN
ejde-744	114	5	,	,	PUNCT
ejde-744	114	6	∀u	∀u	NOUN
ejde-744	114	7	,	,	PUNCT
ejde-744	114	8	v	v	ADP
ejde-744	114	9	∈	∈	PROPN
ejde-744	114	10	v	v	ADP
ejde-744	114	11	1	1	NUM
ejde-744	114	12	a	a	PRON
ejde-744	114	13	,	,	PUNCT
ejde-744	114	14	b(0	b(0	NOUN
ejde-744	114	15	,	,	PUNCT
ejde-744	114	16	1	1	NUM
ejde-744	114	17	)	)	PUNCT
ejde-744	114	18	,	,	PUNCT
ejde-744	114	19	v	v	ADP
ejde-744	114	20	2	2	NUM
ejde-744	114	21	a	a	PRON
ejde-744	114	22	,	,	PUNCT
ejde-744	114	23	b(0	b(0	NOUN
ejde-744	114	24	,	,	PUNCT
ejde-744	114	25	1	1	NUM
ejde-744	114	26	)	)	PUNCT
ejde-744	114	27	=	=	PRON
ejde-744	115	1	{	{	PUNCT
ejde-744	115	2	u	u	NOUN
ejde-744	115	3	∈	∈	PROPN
ejde-744	115	4	v	v	ADP
ejde-744	115	5	1	1	NUM
ejde-744	115	6	a	a	DET
ejde-744	115	7	,	,	PUNCT
ejde-744	115	8	b(0	b(0	NOUN
ejde-744	115	9	,	,	PUNCT
ejde-744	115	10	1)|(au′)′	1)|(au′)′	NUM
ejde-744	115	11	−	−	PROPN
ejde-744	116	1	λ	λ	X
ejde-744	116	2	b	b	X
ejde-744	116	3	u	u	PROPN
ejde-744	116	4	∈	∈	PROPN
ejde-744	116	5	l2(0	l2(0	NOUN
ejde-744	116	6	,	,	PUNCT
ejde-744	116	7	1	1	NUM
ejde-744	116	8	)	)	PUNCT
ejde-744	116	9	}	}	PUNCT
ejde-744	116	10	.	.	PUNCT
ejde-744	117	1	under	under	ADP
ejde-744	117	2	the	the	DET
ejde-744	117	3	boundary	boundary	ADJ
ejde-744	117	4	conditions	condition	NOUN
ejde-744	117	5	of	of	ADP
ejde-744	117	6	(	(	PUNCT
ejde-744	117	7	1.7	1.7	NUM
ejde-744	117	8	)	)	PUNCT
ejde-744	117	9	,	,	PUNCT
ejde-744	117	10	we	we	PRON
ejde-744	117	11	introduce	introduce	VERB
ejde-744	117	12	the	the	DET
ejde-744	117	13	space	space	NOUN
ejde-744	117	14	h1	h1	NOUN
ejde-744	117	15	a	a	DET
ejde-744	117	16	,	,	PUNCT
ejde-744	117	17	b(0	b(0	NOUN
ejde-744	117	18	,	,	PUNCT
ejde-744	117	19	1	1	NUM
ejde-744	117	20	)	)	PUNCT
ejde-744	117	21	.	.	PUNCT
ejde-744	118	1	ejde-2025/04	ejde-2025/04	X
ejde-744	118	2	exact	exact	ADJ
ejde-744	118	3	controllability	controllability	NOUN
ejde-744	118	4	5	5	NUM
ejde-744	118	5	(	(	PUNCT
ejde-744	118	6	i	i	NOUN
ejde-744	118	7	)	)	PUNCT
ejde-744	118	8	if	if	SCONJ
ejde-744	118	9	ka	ka	PROPN
ejde-744	118	10	∈	∈	PROPN
ejde-744	119	1	[	[	X
ejde-744	119	2	0	0	NUM
ejde-744	119	3	,	,	PUNCT
ejde-744	119	4	1	1	NUM
ejde-744	119	5	)	)	PUNCT
ejde-744	119	6	,	,	PUNCT
ejde-744	119	7	h1	h1	VERB
ejde-744	119	8	a	a	DET
ejde-744	119	9	,	,	PUNCT
ejde-744	119	10	b(0	b(0	NOUN
ejde-744	119	11	,	,	PUNCT
ejde-744	119	12	1	1	NUM
ejde-744	119	13	)	)	PUNCT
ejde-744	119	14	=	=	PRON
ejde-744	119	15	{	{	PUNCT
ejde-744	119	16	u	u	NOUN
ejde-744	119	17	∈	∈	PROPN
ejde-744	119	18	v	v	ADP
ejde-744	119	19	1	1	NUM
ejde-744	119	20	a	a	PRON
ejde-744	119	21	,	,	PUNCT
ejde-744	119	22	b(0	b(0	NOUN
ejde-744	119	23	,	,	PUNCT
ejde-744	119	24	1)|u(0	1)|u(0	NUM
ejde-744	119	25	)	)	PUNCT
ejde-744	119	26	=	=	SYM
ejde-744	119	27	u(1	u(1	NOUN
ejde-744	119	28	)	)	PUNCT
ejde-744	119	29	=	=	PUNCT
ejde-744	119	30	0	0	NUM
ejde-744	119	31	}	}	PUNCT
ejde-744	119	32	;	;	PUNCT
ejde-744	119	33	(	(	PUNCT
ejde-744	119	34	ii	ii	NOUN
ejde-744	119	35	)	)	PUNCT
ejde-744	119	36	if	if	SCONJ
ejde-744	119	37	ka	ka	PROPN
ejde-744	119	38	∈	∈	PROPN
ejde-744	119	39	(	(	PUNCT
ejde-744	119	40	1	1	NUM
ejde-744	119	41	,	,	PUNCT
ejde-744	119	42	2	2	NUM
ejde-744	119	43	)	)	PUNCT
ejde-744	119	44	,	,	PUNCT
ejde-744	119	45	h1	h1	VERB
ejde-744	119	46	a	a	DET
ejde-744	119	47	,	,	PUNCT
ejde-744	119	48	b(0	b(0	NOUN
ejde-744	119	49	,	,	PUNCT
ejde-744	119	50	1	1	NUM
ejde-744	119	51	)	)	PUNCT
ejde-744	119	52	=	=	PRON
ejde-744	119	53	{	{	PUNCT
ejde-744	119	54	u	u	NOUN
ejde-744	119	55	∈	∈	PROPN
ejde-744	119	56	v	v	ADP
ejde-744	119	57	1	1	NUM
ejde-744	119	58	a	a	PRON
ejde-744	119	59	,	,	PUNCT
ejde-744	119	60	b(0	b(0	NOUN
ejde-744	119	61	,	,	PUNCT
ejde-744	119	62	1)|u(1	1)|u(1	NUM
ejde-744	119	63	)	)	PUNCT
ejde-744	119	64	=	=	SYM
ejde-744	119	65	0	0	NUM
ejde-744	119	66	}	}	PUNCT
ejde-744	119	67	.	.	PUNCT
ejde-744	120	1	also	also	ADV
ejde-744	120	2	,	,	PUNCT
ejde-744	120	3	h−1	h−1	PROPN
ejde-744	120	4	a	a	DET
ejde-744	120	5	,	,	PUNCT
ejde-744	120	6	b	b	PROPN
ejde-744	120	7	(	(	PUNCT
ejde-744	120	8	0	0	NUM
ejde-744	120	9	,	,	PUNCT
ejde-744	120	10	1	1	X
ejde-744	120	11	)	)	PUNCT
ejde-744	120	12	denotes	denote	VERB
ejde-744	120	13	the	the	DET
ejde-744	120	14	conjugate	conjugate	ADJ
ejde-744	120	15	space	space	NOUN
ejde-744	120	16	of	of	ADP
ejde-744	120	17	h1	h1	PROPN
ejde-744	120	18	a	a	DET
ejde-744	120	19	,	,	PUNCT
ejde-744	120	20	b(0	b(0	NOUN
ejde-744	120	21	,	,	PUNCT
ejde-744	120	22	1	1	NUM
ejde-744	120	23	)	)	PUNCT
ejde-744	120	24	.	.	PUNCT
ejde-744	121	1	we	we	PRON
ejde-744	121	2	set	set	VERB
ejde-744	121	3	h2	h2	PROPN
ejde-744	121	4	a	a	PRON
ejde-744	121	5	,	,	PUNCT
ejde-744	121	6	b(0	b(0	NOUN
ejde-744	121	7	,	,	PUNCT
ejde-744	121	8	1	1	NUM
ejde-744	121	9	)	)	PUNCT
ejde-744	121	10	=	=	SYM
ejde-744	121	11	v	v	ADP
ejde-744	121	12	2	2	NUM
ejde-744	121	13	a	a	PRON
ejde-744	121	14	,	,	PUNCT
ejde-744	121	15	b(0	b(0	NOUN
ejde-744	121	16	,	,	PUNCT
ejde-744	121	17	1	1	NUM
ejde-744	121	18	)	)	PUNCT
ejde-744	121	19	∩h1	∩h1	PRON
ejde-744	121	20	a	a	DET
ejde-744	121	21	,	,	PUNCT
ejde-744	121	22	b(0	b(0	NOUN
ejde-744	121	23	,	,	PUNCT
ejde-744	121	24	1	1	NUM
ejde-744	121	25	)	)	PUNCT
ejde-744	121	26	.	.	PUNCT
ejde-744	122	1	from	from	ADP
ejde-744	122	2	assumption	assumption	NOUN
ejde-744	122	3	2.4	2.4	NUM
ejde-744	122	4	,	,	PUNCT
ejde-744	122	5	when	when	SCONJ
ejde-744	122	6	λ	λ	X
ejde-744	122	7	>	>	X
ejde-744	122	8	0	0	PUNCT
ejde-744	122	9	there	there	PRON
ejde-744	122	10	exists	exist	VERB
ejde-744	122	11	θ	θ	PROPN
ejde-744	122	12	∈	∈	PROPN
ejde-744	122	13	(	(	PUNCT
ejde-744	122	14	0	0	NUM
ejde-744	122	15	,	,	PUNCT
ejde-744	122	16	1	1	NUM
ejde-744	122	17	)	)	PUNCT
ejde-744	123	1	such	such	ADJ
ejde-744	123	2	that	that	SCONJ
ejde-744	123	3	λ	λ	NOUN
ejde-744	123	4	=	=	SYM
ejde-744	123	5	1	1	NUM
ejde-744	123	6	ca	ca	NOUN
ejde-744	123	7	,	,	PUNCT
ejde-744	123	8	b	b	PROPN
ejde-744	123	9	−	−	PROPN
ejde-744	123	10	θ	θ	PROPN
ejde-744	123	11	ca	can	AUX
ejde-744	123	12	,	,	PUNCT
ejde-744	123	13	b	b	PROPN
ejde-744	123	14	.	.	PUNCT
ejde-744	124	1	(	(	PUNCT
ejde-744	124	2	2.6	2.6	NUM
ejde-744	124	3	)	)	PUNCT
ejde-744	124	4	further	far	ADV
ejde-744	124	5	,	,	PUNCT
ejde-744	124	6	one	one	PRON
ejde-744	124	7	can	can	AUX
ejde-744	124	8	prove	prove	VERB
ejde-744	124	9	the	the	DET
ejde-744	124	10	next	next	ADJ
ejde-744	124	11	result	result	NOUN
ejde-744	124	12	.	.	PUNCT
ejde-744	125	1	lemma	lemma	PROPN
ejde-744	125	2	2.5	2.5	NUM
ejde-744	125	3	.	.	PUNCT
ejde-744	126	1	under	under	ADP
ejde-744	126	2	assumptions	assumption	NOUN
ejde-744	126	3	2.1	2.1	NUM
ejde-744	126	4	and	and	CCONJ
ejde-744	126	5	2.4	2.4	NUM
ejde-744	126	6	,	,	PUNCT
ejde-744	126	7	we	we	PRON
ejde-744	126	8	have∫	have∫	VERB
ejde-744	126	9	1	1	NUM
ejde-744	126	10	0	0	NUM
ejde-744	126	11	a(u′	a(u′	NUM
ejde-744	126	12	)	)	PUNCT
ejde-744	126	13	2	2	NUM
ejde-744	126	14	dx	dx	PROPN
ejde-744	126	15	≤	≤	NUM
ejde-744	126	16	1	1	NUM
ejde-744	126	17	cθ	cθ	ADP
ejde-744	126	18	∫	∫	PROPN
ejde-744	126	19	1	1	NUM
ejde-744	126	20	0	0	NUM
ejde-744	126	21	a(u′	a(u′	NUM
ejde-744	126	22	)	)	PUNCT
ejde-744	126	23	2	2	NUM
ejde-744	126	24	−	−	PROPN
ejde-744	127	1	λ	λ	PROPN
ejde-744	127	2	b	b	PROPN
ejde-744	127	3	u2	u2	PROPN
ejde-744	127	4	dx	dx	PROPN
ejde-744	127	5	,	,	PUNCT
ejde-744	127	6	(	(	PUNCT
ejde-744	127	7	2.7	2.7	NUM
ejde-744	127	8	)	)	PUNCT
ejde-744	127	9	where	where	SCONJ
ejde-744	127	10	cθ	cθ	ADP
ejde-744	127	11	=	=	SYM
ejde-744	127	12	θ	θ	PROPN
ejde-744	127	13	if	if	SCONJ
ejde-744	127	14	λ	λ	X
ejde-744	127	15	∈	∈	PROPN
ejde-744	127	16	(	(	PUNCT
ejde-744	127	17	0	0	NUM
ejde-744	127	18	,	,	PUNCT
ejde-744	127	19	1	1	NUM
ejde-744	127	20	ca	ca	NOUN
ejde-744	127	21	,	,	PUNCT
ejde-744	127	22	b	b	PROPN
ejde-744	127	23	)	)	PUNCT
ejde-744	127	24	,	,	PUNCT
ejde-744	127	25	cθ	cθ	ADP
ejde-744	127	26	=	=	NOUN
ejde-744	127	27	1	1	NUM
ejde-744	127	28	if	if	SCONJ
ejde-744	127	29	λ	λ	PROPN
ejde-744	127	30	≤	≤	NOUN
ejde-744	127	31	0	0	NUM
ejde-744	127	32	.	.	PUNCT
ejde-744	128	1	proof	proof	NOUN
ejde-744	128	2	.	.	PUNCT
ejde-744	129	1	(	(	PUNCT
ejde-744	129	2	i	i	NOUN
ejde-744	129	3	)	)	PUNCT
ejde-744	129	4	if	if	SCONJ
ejde-744	129	5	λ	λ	X
ejde-744	129	6	∈	∈	PROPN
ejde-744	129	7	(	(	PUNCT
ejde-744	129	8	0	0	NUM
ejde-744	129	9	,	,	PUNCT
ejde-744	129	10	1	1	NUM
ejde-744	129	11	ca	ca	NOUN
ejde-744	129	12	,	,	PUNCT
ejde-744	129	13	b	b	PROPN
ejde-744	129	14	)	)	PUNCT
ejde-744	129	15	,	,	PUNCT
ejde-744	129	16	then	then	ADV
ejde-744	129	17	by	by	ADP
ejde-744	129	18	(	(	PUNCT
ejde-744	129	19	2.3	2.3	NUM
ejde-744	129	20	)	)	PUNCT
ejde-744	129	21	(	(	PUNCT
ejde-744	129	22	2.6	2.6	NUM
ejde-744	129	23	)	)	PUNCT
ejde-744	129	24	,	,	PUNCT
ejde-744	129	25	we	we	PRON
ejde-744	129	26	deduce	deduce	VERB
ejde-744	129	27	that∫	that∫	NOUN
ejde-744	129	28	1	1	NUM
ejde-744	129	29	0	0	NUM
ejde-744	129	30	a(u′	a(u′	NUM
ejde-744	129	31	)	)	PUNCT
ejde-744	130	1	2	2	NUM
ejde-744	130	2	−	−	PROPN
ejde-744	130	3	λ	λ	PROPN
ejde-744	130	4	b	b	PROPN
ejde-744	130	5	u2	u2	PROPN
ejde-744	130	6	dx	dx	PROPN
ejde-744	130	7	≥	≥	PROPN
ejde-744	130	8	∫	∫	PROPN
ejde-744	130	9	1	1	NUM
ejde-744	130	10	0	0	NUM
ejde-744	130	11	a(u′	a(u′	NUM
ejde-744	130	12	)	)	PUNCT
ejde-744	130	13	2	2	NUM
ejde-744	130	14	dx−	dx−	NUM
ejde-744	130	15	(	(	PUNCT
ejde-744	130	16	1−	1−	NUM
ejde-744	130	17	θ	θ	NOUN
ejde-744	130	18	)	)	PUNCT
ejde-744	130	19	∫	∫	PROPN
ejde-744	130	20	1	1	NUM
ejde-744	130	21	0	0	NUM
ejde-744	130	22	a(u′	a(u′	NUM
ejde-744	130	23	)	)	PUNCT
ejde-744	130	24	2	2	NUM
ejde-744	130	25	dx	dx	NOUN
ejde-744	130	26	=	=	SYM
ejde-744	130	27	θ	θ	PROPN
ejde-744	130	28	∫	∫	PROPN
ejde-744	130	29	1	1	NUM
ejde-744	130	30	0	0	NUM
ejde-744	130	31	a(u′	a(u′	NUM
ejde-744	130	32	)	)	PUNCT
ejde-744	130	33	2	2	NUM
ejde-744	130	34	dx	dx	X
ejde-744	130	35	.	.	PUNCT
ejde-744	130	36	(	(	PUNCT
ejde-744	130	37	ii	ii	NOUN
ejde-744	130	38	)	)	PUNCT
ejde-744	130	39	if	if	SCONJ
ejde-744	130	40	λ	λ	PROPN
ejde-744	130	41	≤	≤	NOUN
ejde-744	130	42	0	0	NUM
ejde-744	130	43	,	,	PUNCT
ejde-744	130	44	obviously	obviously	ADV
ejde-744	130	45	,	,	PUNCT
ejde-744	130	46	∫	∫	PROPN
ejde-744	130	47	1	1	NUM
ejde-744	130	48	0	0	NUM
ejde-744	130	49	a(u′	a(u′	NUM
ejde-744	130	50	)	)	PUNCT
ejde-744	130	51	2	2	NUM
ejde-744	130	52	−	−	PROPN
ejde-744	131	1	λ	λ	PROPN
ejde-744	131	2	b	b	PROPN
ejde-744	131	3	u2	u2	PROPN
ejde-744	131	4	dx	dx	PROPN
ejde-744	131	5	≥	≥	PROPN
ejde-744	131	6	∫	∫	PROPN
ejde-744	131	7	1	1	NUM
ejde-744	131	8	0	0	NUM
ejde-744	131	9	a(u′	a(u′	NUM
ejde-744	131	10	)	)	PUNCT
ejde-744	131	11	2	2	NUM
ejde-744	131	12	dx	dx	X
ejde-744	131	13	.	.	PUNCT
ejde-744	132	1	□	□	PUNCT
ejde-744	132	2	assumption	assumption	NOUN
ejde-744	132	3	2.6	2.6	NUM
ejde-744	132	4	.	.	PUNCT
ejde-744	133	1	under	under	ADP
ejde-744	133	2	assumptions	assumption	NOUN
ejde-744	133	3	2.1	2.1	NUM
ejde-744	133	4	and	and	CCONJ
ejde-744	133	5	2.4	2.4	NUM
ejde-744	133	6	,	,	PUNCT
ejde-744	133	7	the	the	DET
ejde-744	133	8	function	function	NOUN
ejde-744	133	9	x	x	INTJ
ejde-744	133	10	→	→	SYM
ejde-744	133	11	xkb	xkb	NOUN
ejde-744	133	12	b(x	b(x	NOUN
ejde-744	133	13	)	)	PUNCT
ejde-744	133	14	is	be	AUX
ejde-744	133	15	nondecreasing	nondecrease	VERB
ejde-744	133	16	in	in	ADP
ejde-744	133	17	a	a	DET
ejde-744	133	18	right	right	ADJ
ejde-744	133	19	neighborhood	neighborhood	NOUN
ejde-744	133	20	of	of	ADP
ejde-744	133	21	x	x	X
ejde-744	133	22	=	=	SYM
ejde-744	133	23	0	0	X
ejde-744	133	24	.	.	PUNCT
ejde-744	133	25	remark	remark	PROPN
ejde-744	133	26	2.7	2.7	NUM
ejde-744	133	27	.	.	PUNCT
ejde-744	134	1	it	it	PRON
ejde-744	134	2	is	be	AUX
ejde-744	134	3	clear	clear	ADJ
ejde-744	134	4	that	that	SCONJ
ejde-744	134	5	,	,	PUNCT
ejde-744	134	6	if	if	SCONJ
ejde-744	134	7	assumption	assumption	NOUN
ejde-744	134	8	2.6	2.6	NUM
ejde-744	134	9	holds	hold	NOUN
ejde-744	134	10	,	,	PUNCT
ejde-744	134	11	then	then	ADV
ejde-744	134	12	lim	lim	PROPN
ejde-744	134	13	x→0	x→0	PROPN
ejde-744	135	1	+	+	CCONJ
ejde-744	135	2	xγ	xγ	NOUN
ejde-744	135	3	b(x	b(x	NOUN
ejde-744	135	4	)	)	PUNCT
ejde-744	136	1	=	=	SYM
ejde-744	136	2	0	0	NUM
ejde-744	136	3	,	,	PUNCT
ejde-744	136	4	γ	γ	X
ejde-744	136	5	>	>	X
ejde-744	136	6	kb	kb	PROPN
ejde-744	136	7	.	.	PUNCT
ejde-744	137	1	(	(	PUNCT
ejde-744	137	2	2.8	2.8	NUM
ejde-744	137	3	)	)	PUNCT
ejde-744	137	4	in	in	ADP
ejde-744	137	5	particular	particular	ADJ
ejde-744	137	6	,	,	PUNCT
ejde-744	137	7	lim	lim	PROPN
ejde-744	137	8	x→0	x→0	PROPN
ejde-744	137	9	+	+	PROPN
ejde-744	137	10	x2	x2	ADJ
ejde-744	137	11	b(x	b(x	NOUN
ejde-744	137	12	)	)	PUNCT
ejde-744	137	13	=	=	SYM
ejde-744	138	1	0	0	X
ejde-744	138	2	.	.	PUNCT
ejde-744	138	3	(	(	PUNCT
ejde-744	138	4	2.9	2.9	NUM
ejde-744	138	5	)	)	PUNCT
ejde-744	138	6	lemma	lemma	PROPN
ejde-744	138	7	2.8	2.8	NUM
ejde-744	138	8	.	.	PUNCT
ejde-744	139	1	under	under	ADP
ejde-744	139	2	assumption	assumption	NOUN
ejde-744	139	3	2.6	2.6	NUM
ejde-744	139	4	,	,	PUNCT
ejde-744	139	5	for	for	ADP
ejde-744	139	6	all	all	DET
ejde-744	139	7	u	u	PROPN
ejde-744	139	8	∈	∈	PROPN
ejde-744	139	9	h1	h1	VERB
ejde-744	139	10	a	a	DET
ejde-744	139	11	,	,	PUNCT
ejde-744	139	12	b(0	b(0	NOUN
ejde-744	139	13	,	,	PUNCT
ejde-744	139	14	1	1	NUM
ejde-744	139	15	)	)	PUNCT
ejde-744	139	16	,	,	PUNCT
ejde-744	139	17	we	we	PRON
ejde-744	139	18	have	have	VERB
ejde-744	139	19	lim	lim	PROPN
ejde-744	139	20	x→0	x→0	PROPN
ejde-744	139	21	+	+	CCONJ
ejde-744	139	22	x	x	PUNCT
ejde-744	139	23	b(x	b(x	ADJ
ejde-744	139	24	)	)	PUNCT
ejde-744	139	25	u2	u2	NOUN
ejde-744	139	26	=	=	NOUN
ejde-744	139	27	0	0	PROPN
ejde-744	139	28	.	.	PUNCT
ejde-744	140	1	(	(	PUNCT
ejde-744	140	2	2.10	2.10	NUM
ejde-744	140	3	)	)	PUNCT
ejde-744	140	4	6	6	NUM
ejde-744	140	5	g.	g.	PROPN
ejde-744	140	6	zhang	zhang	PROPN
ejde-744	140	7	,	,	PUNCT
ejde-744	140	8	s.	s.	PROPN
ejde-744	140	9	chai	chai	NOUN
ejde-744	140	10	ejde-2025/04	ejde-2025/04	NOUN
ejde-744	140	11	proof	proof	NOUN
ejde-744	140	12	.	.	PUNCT
ejde-744	141	1	if	if	SCONJ
ejde-744	141	2	0	0	NUM
ejde-744	141	3	≤	≤	NUM
ejde-744	141	4	ka	ka	X
ejde-744	141	5	<	<	X
ejde-744	141	6	1	1	NUM
ejde-744	141	7	,	,	PUNCT
ejde-744	141	8	using	use	VERB
ejde-744	141	9	that	that	DET
ejde-744	141	10	u(0	u(0	NOUN
ejde-744	141	11	)	)	PUNCT
ejde-744	141	12	=	=	SYM
ejde-744	141	13	0	0	NUM
ejde-744	142	1	,	,	PUNCT
ejde-744	142	2	we	we	PRON
ejde-744	142	3	have	have	VERB
ejde-744	142	4	|u(x)|	|u(x)|	NOUN
ejde-744	142	5	≤	≤	ADJ
ejde-744	142	6	∫	∫	PROPN
ejde-744	142	7	x	x	SYM
ejde-744	142	8	0	0	PUNCT
ejde-744	142	9	|u(ξ)|dξ	|u(ξ)|dξ	NUM
ejde-744	142	10	≤	≤	NUM
ejde-744	142	11	√	√	NUM
ejde-744	142	12	x∥u′∥l2(0,1	x∥u′∥l2(0,1	NOUN
ejde-744	142	13	)	)	PUNCT
ejde-744	142	14	.	.	PUNCT
ejde-744	143	1	then	then	ADV
ejde-744	143	2	x	x	PUNCT
ejde-744	143	3	b(x	b(x	NOUN
ejde-744	143	4	)	)	PUNCT
ejde-744	143	5	u2(x	u2(x	NOUN
ejde-744	143	6	)	)	PUNCT
ejde-744	143	7	≤	≤	NUM
ejde-744	143	8	x2	x2	PUNCT
ejde-744	143	9	b(x	b(x	NOUN
ejde-744	143	10	)	)	PUNCT
ejde-744	143	11	∥u′∥l2(0,1	∥u′∥l2(0,1	NOUN
ejde-744	143	12	)	)	PUNCT
ejde-744	143	13	.	.	PUNCT
ejde-744	144	1	by	by	ADP
ejde-744	144	2	equation	equation	NOUN
ejde-744	144	3	(	(	PUNCT
ejde-744	144	4	2.9	2.9	NUM
ejde-744	144	5	)	)	PUNCT
ejde-744	144	6	,	,	PUNCT
ejde-744	144	7	the	the	DET
ejde-744	144	8	lemma	lemma	PROPN
ejde-744	144	9	follows	follow	VERB
ejde-744	144	10	.	.	PUNCT
ejde-744	145	1	if	if	SCONJ
ejde-744	145	2	1	1	NUM
ejde-744	145	3	<	<	X
ejde-744	145	4	ka	ka	X
ejde-744	145	5	<	<	X
ejde-744	145	6	2	2	NUM
ejde-744	145	7	,	,	PUNCT
ejde-744	145	8	then	then	ADV
ejde-744	145	9	0	0	NUM
ejde-744	145	10	≤	≤	NUM
ejde-744	145	11	kb	kb	PROPN
ejde-744	145	12	<	<	X
ejde-744	145	13	1	1	NUM
ejde-744	145	14	,	,	PUNCT
ejde-744	145	15	the	the	DET
ejde-744	145	16	conclusion	conclusion	NOUN
ejde-744	145	17	follows	follow	VERB
ejde-744	145	18	directly	directly	ADV
ejde-744	145	19	from	from	ADP
ejde-744	145	20	remark	remark	NOUN
ejde-744	145	21	2.7	2.7	NUM
ejde-744	145	22	.	.	PUNCT
ejde-744	146	1	□	□	PUNCT
ejde-744	146	2	3	3	X
ejde-744	146	3	.	.	X
ejde-744	146	4	well	well	ADJ
ejde-744	146	5	-	-	PUNCT
ejde-744	146	6	posedness	posedness	NOUN
ejde-744	146	7	first	first	ADV
ejde-744	146	8	,	,	PUNCT
ejde-744	146	9	we	we	PRON
ejde-744	146	10	consider	consider	VERB
ejde-744	146	11	the	the	DET
ejde-744	146	12	degenerate	degenerate	ADJ
ejde-744	146	13	/	/	SYM
ejde-744	146	14	singular	singular	ADJ
ejde-744	146	15	wave	wave	NOUN
ejde-744	146	16	problem	problem	NOUN
ejde-744	146	17	with	with	ADP
ejde-744	146	18	dirichlet	dirichlet	PROPN
ejde-744	146	19	/	/	SYM
ejde-744	146	20	neumann	neumann	PROPN
ejde-744	146	21	boundary	boundary	ADJ
ejde-744	146	22	conditions	condition	NOUN
ejde-744	146	23	:	:	PUNCT
ejde-744	146	24	utt	utt	ADJ
ejde-744	146	25	−	−	PROPN
ejde-744	146	26	(	(	PUNCT
ejde-744	146	27	a(x)ux)x	a(x)ux)x	VERB
ejde-744	146	28	−	−	PROPN
ejde-744	147	1	λ	λ	NOUN
ejde-744	147	2	b(x	b(x	NOUN
ejde-744	147	3	)	)	PUNCT
ejde-744	147	4	u	u	NOUN
ejde-744	147	5	=	=	PROPN
ejde-744	147	6	0	0	NUM
ejde-744	147	7	,	,	PUNCT
ejde-744	147	8	(	(	PUNCT
ejde-744	147	9	t	t	PROPN
ejde-744	147	10	,	,	PUNCT
ejde-744	147	11	x	x	NOUN
ejde-744	147	12	)	)	PUNCT
ejde-744	147	13	∈	∈	PROPN
ejde-744	147	14	(	(	PUNCT
ejde-744	147	15	0	0	NUM
ejde-744	147	16	,	,	PUNCT
ejde-744	147	17	t	t	NOUN
ejde-744	147	18	)	)	PUNCT
ejde-744	147	19	×	×	NOUN
ejde-744	147	20	(	(	PUNCT
ejde-744	147	21	0	0	NUM
ejde-744	147	22	,	,	PUNCT
ejde-744	147	23	1	1	NUM
ejde-744	147	24	)	)	PUNCT
ejde-744	147	25	,	,	PUNCT
ejde-744	147	26	u(t	u(t	NOUN
ejde-744	147	27	,	,	PUNCT
ejde-744	147	28	1	1	NUM
ejde-744	147	29	)	)	PUNCT
ejde-744	147	30	=	=	SYM
ejde-744	147	31	0	0	NUM
ejde-744	147	32	,	,	PUNCT
ejde-744	147	33	t	t	PROPN
ejde-744	147	34	∈	∈	PROPN
ejde-744	147	35	(	(	PUNCT
ejde-744	147	36	0	0	NUM
ejde-744	147	37	,	,	PUNCT
ejde-744	147	38	t	t	NOUN
ejde-744	147	39	)	)	PUNCT
ejde-744	147	40	,	,	PUNCT
ejde-744	147	41	u(t	u(t	NOUN
ejde-744	147	42	,	,	PUNCT
ejde-744	147	43	0	0	NUM
ejde-744	147	44	)	)	PUNCT
ejde-744	147	45	=	=	SYM
ejde-744	147	46	0	0	NUM
ejde-744	147	47	,	,	PUNCT
ejde-744	147	48	ka	ka	PROPN
ejde-744	147	49	∈	∈	PROPN
ejde-744	148	1	[	[	X
ejde-744	148	2	0	0	NUM
ejde-744	148	3	,	,	PUNCT
ejde-744	148	4	1	1	NUM
ejde-744	148	5	)	)	PUNCT
ejde-744	148	6	,	,	PUNCT
ejde-744	148	7	t	t	PROPN
ejde-744	148	8	∈	∈	PROPN
ejde-744	148	9	(	(	PUNCT
ejde-744	148	10	0	0	NUM
ejde-744	148	11	,	,	PUNCT
ejde-744	148	12	t	t	PROPN
ejde-744	148	13	)	)	PUNCT
ejde-744	148	14	,	,	PUNCT
ejde-744	148	15	lim	lim	PROPN
ejde-744	148	16	x→0	x→0	PROPN
ejde-744	149	1	+	+	PROPN
ejde-744	149	2	aux(t	aux(t	PROPN
ejde-744	149	3	,	,	PUNCT
ejde-744	149	4	x	x	NOUN
ejde-744	149	5	)	)	PUNCT
ejde-744	149	6	=	=	SYM
ejde-744	149	7	0	0	NUM
ejde-744	149	8	,	,	PUNCT
ejde-744	149	9	ka	ka	PROPN
ejde-744	149	10	∈	∈	PROPN
ejde-744	149	11	(	(	PUNCT
ejde-744	149	12	1	1	NUM
ejde-744	149	13	,	,	PUNCT
ejde-744	149	14	2	2	NUM
ejde-744	149	15	)	)	PUNCT
ejde-744	149	16	,	,	PUNCT
ejde-744	149	17	t	t	PROPN
ejde-744	149	18	∈	∈	PROPN
ejde-744	149	19	(	(	PUNCT
ejde-744	149	20	0	0	NUM
ejde-744	149	21	,	,	PUNCT
ejde-744	149	22	t	t	NOUN
ejde-744	149	23	)	)	PUNCT
ejde-744	149	24	,	,	PUNCT
ejde-744	149	25	u(0	u(0	PROPN
ejde-744	149	26	,	,	PUNCT
ejde-744	149	27	x	x	NOUN
ejde-744	149	28	)	)	PUNCT
ejde-744	149	29	=	=	SYM
ejde-744	149	30	u0(x	u0(x	NOUN
ejde-744	149	31	)	)	PUNCT
ejde-744	149	32	,	,	PUNCT
ejde-744	149	33	ut(0	ut(0	PROPN
ejde-744	149	34	,	,	PUNCT
ejde-744	149	35	x	x	NOUN
ejde-744	149	36	)	)	PUNCT
ejde-744	149	37	=	=	SYM
ejde-744	149	38	u1(x	u1(x	NOUN
ejde-744	149	39	)	)	PUNCT
ejde-744	149	40	,	,	PUNCT
ejde-744	149	41	x	x	PUNCT
ejde-744	149	42	∈	∈	PROPN
ejde-744	149	43	(	(	PUNCT
ejde-744	149	44	0	0	NUM
ejde-744	149	45	,	,	PUNCT
ejde-744	149	46	1	1	NUM
ejde-744	149	47	)	)	PUNCT
ejde-744	149	48	.	.	PUNCT
ejde-744	150	1	(	(	PUNCT
ejde-744	150	2	3.1	3.1	NUM
ejde-744	150	3	)	)	PUNCT
ejde-744	150	4	let	let	VERB
ejde-744	150	5	us	we	PRON
ejde-744	150	6	recall	recall	VERB
ejde-744	150	7	the	the	DET
ejde-744	150	8	typical	typical	ADJ
ejde-744	150	9	abstract	abstract	ADJ
ejde-744	150	10	setup	setup	NOUN
ejde-744	150	11	of	of	ADP
ejde-744	150	12	semigroup	semigroup	PROPN
ejde-744	150	13	theory	theory	NOUN
ejde-744	150	14	,	,	PUNCT
ejde-744	150	15	which	which	PRON
ejde-744	150	16	provides	provide	VERB
ejde-744	150	17	weak	weak	ADJ
ejde-744	150	18	and	and	CCONJ
ejde-744	150	19	classical	classical	ADJ
ejde-744	150	20	solutions	solution	NOUN
ejde-744	150	21	for	for	ADP
ejde-744	150	22	the	the	DET
ejde-744	150	23	above	above	ADJ
ejde-744	150	24	system	system	NOUN
ejde-744	150	25	.	.	PUNCT
ejde-744	151	1	consider	consider	VERB
ejde-744	151	2	the	the	DET
ejde-744	151	3	hilbert	hilbert	NOUN
ejde-744	151	4	space	space	NOUN
ejde-744	151	5	h	h	NOUN
ejde-744	151	6	=	=	PRON
ejde-744	151	7	h1	h1	VERB
ejde-744	151	8	a	a	PRON
ejde-744	151	9	,	,	PUNCT
ejde-744	151	10	b(0	b(0	NOUN
ejde-744	151	11	,	,	PUNCT
ejde-744	151	12	1)×	1)×	NUM
ejde-744	151	13	l2(0	l2(0	NOUN
ejde-744	151	14	,	,	PUNCT
ejde-744	151	15	1	1	NUM
ejde-744	151	16	)	)	PUNCT
ejde-744	151	17	endowed	endow	VERB
ejde-744	151	18	with	with	ADP
ejde-744	151	19	the	the	DET
ejde-744	151	20	inner	inner	ADJ
ejde-744	151	21	product	product	NOUN
ejde-744	151	22	⟨(u	⟨(u	PROPN
ejde-744	151	23	,	,	PUNCT
ejde-744	151	24	v	v	NOUN
ejde-744	151	25	)	)	PUNCT
ejde-744	151	26	,	,	PUNCT
ejde-744	151	27	(	(	PUNCT
ejde-744	151	28	ũ	ũ	PROPN
ejde-744	151	29	,	,	PUNCT
ejde-744	151	30	ṽ)⟩h	ṽ)⟩h	PROPN
ejde-744	151	31	=	=	SYM
ejde-744	151	32	∫	∫	PROPN
ejde-744	151	33	1	1	NUM
ejde-744	151	34	0	0	NUM
ejde-744	151	35	vṽ	vṽ	PUNCT
ejde-744	151	36	+	+	PUNCT
ejde-744	151	37	au′ũ′	au′ũ′	PROPN
ejde-744	151	38	−	−	NOUN
ejde-744	151	39	λ	λ	PROPN
ejde-744	151	40	b	b	PROPN
ejde-744	151	41	uũ	uũ	PROPN
ejde-744	151	42	dx	dx	PROPN
ejde-744	151	43	,	,	PUNCT
ejde-744	151	44	∀(u	∀(u	PROPN
ejde-744	151	45	,	,	PUNCT
ejde-744	151	46	v	v	NOUN
ejde-744	151	47	)	)	PUNCT
ejde-744	151	48	,	,	PUNCT
ejde-744	151	49	(	(	PUNCT
ejde-744	151	50	ũ	ũ	PROPN
ejde-744	151	51	,	,	PUNCT
ejde-744	151	52	ṽ	ṽ	PROPN
ejde-744	151	53	)	)	PUNCT
ejde-744	151	54	∈	∈	PROPN
ejde-744	151	55	h.	h.	PROPN
ejde-744	151	56	by	by	ADP
ejde-744	151	57	assumption	assumption	NOUN
ejde-744	151	58	2.4	2.4	NUM
ejde-744	151	59	and	and	CCONJ
ejde-744	151	60	the	the	DET
ejde-744	151	61	hardy	hardy	ADJ
ejde-744	151	62	-	-	PUNCT
ejde-744	151	63	poincare	poincare	NOUN
ejde-744	151	64	inequality	inequality	NOUN
ejde-744	151	65	(	(	PUNCT
ejde-744	151	66	2.3	2.3	NUM
ejde-744	151	67	)	)	PUNCT
ejde-744	151	68	,	,	PUNCT
ejde-744	151	69	we	we	PRON
ejde-744	151	70	have	have	VERB
ejde-744	151	71	⟨(u	⟨(u	PROPN
ejde-744	151	72	,	,	PUNCT
ejde-744	151	73	v	v	NOUN
ejde-744	151	74	)	)	PUNCT
ejde-744	151	75	,	,	PUNCT
ejde-744	151	76	(	(	PUNCT
ejde-744	151	77	u	u	NOUN
ejde-744	151	78	,	,	PUNCT
ejde-744	152	1	v)⟩h	v)⟩h	NOUN
ejde-744	152	2	=	=	SYM
ejde-744	152	3	∫	∫	PROPN
ejde-744	152	4	1	1	NUM
ejde-744	152	5	0	0	NUM
ejde-744	152	6	v2	v2	PROPN
ejde-744	152	7	+	+	NUM
ejde-744	152	8	a(u′	a(u′	NUM
ejde-744	152	9	)	)	PUNCT
ejde-744	153	1	2	2	NUM
ejde-744	153	2	−	−	PROPN
ejde-744	153	3	λ	λ	PROPN
ejde-744	153	4	b	b	PROPN
ejde-744	153	5	u2	u2	PROPN
ejde-744	153	6	dx	dx	PROPN
ejde-744	153	7	≥	≥	PROPN
ejde-744	153	8	0	0	NUM
ejde-744	153	9	,	,	PUNCT
ejde-744	153	10	∀(u	∀(u	PROPN
ejde-744	153	11	,	,	PUNCT
ejde-744	153	12	v	v	NOUN
ejde-744	153	13	)	)	PUNCT
ejde-744	153	14	∈	∈	PROPN
ejde-744	153	15	h.	h.	PROPN
ejde-744	153	16	thus	thus	ADV
ejde-744	153	17	,	,	PUNCT
ejde-744	153	18	⟨	⟨	NOUN
ejde-744	153	19	·	·	SYM
ejde-744	153	20	,	,	PUNCT
ejde-744	153	21	·	·	PUNCT
ejde-744	153	22	⟩h	⟩h	NOUN
ejde-744	153	23	forms	form	VERB
ejde-744	153	24	the	the	DET
ejde-744	153	25	scalar	scalar	ADJ
ejde-744	153	26	product	product	NOUN
ejde-744	153	27	.	.	PUNCT
ejde-744	154	1	arguing	argue	VERB
ejde-744	154	2	as	as	ADP
ejde-744	154	3	for	for	ADP
ejde-744	154	4	the	the	DET
ejde-744	154	5	classical	classical	ADJ
ejde-744	154	6	wave	wave	NOUN
ejde-744	154	7	equation	equation	NOUN
ejde-744	154	8	,	,	PUNCT
ejde-744	154	9	the	the	DET
ejde-744	154	10	unbounded	unbounded	ADJ
ejde-744	154	11	operator	operator	NOUN
ejde-744	154	12	a	a	PRON
ejde-744	154	13	:	:	PUNCT
ejde-744	154	14	d(a	d(a	PROPN
ejde-744	154	15	)	)	PUNCT
ejde-744	155	1	⊂	⊂	PROPN
ejde-744	155	2	h	h	PROPN
ejde-744	155	3	→	→	PUNCT
ejde-744	155	4	h	h	NOUN
ejde-744	155	5	is	be	AUX
ejde-744	155	6	defined	define	VERB
ejde-744	155	7	by	by	ADP
ejde-744	155	8	a(u	a(u	PROPN
ejde-744	155	9	,	,	PUNCT
ejde-744	155	10	v	v	NOUN
ejde-744	155	11	)	)	PUNCT
ejde-744	155	12	=	=	SYM
ejde-744	155	13	(	(	PUNCT
ejde-744	155	14	v	v	NOUN
ejde-744	155	15	,	,	PUNCT
ejde-744	155	16	a(u′)′	a(u′)′	PROPN
ejde-744	155	17	+	+	PROPN
ejde-744	155	18	λ	λ	PROPN
ejde-744	155	19	b	b	X
ejde-744	155	20	u	u	PROPN
ejde-744	155	21	)	)	PUNCT
ejde-744	155	22	,	,	PUNCT
ejde-744	155	23	∀(u	∀(u	PROPN
ejde-744	155	24	,	,	PUNCT
ejde-744	155	25	v	v	NOUN
ejde-744	155	26	)	)	PUNCT
ejde-744	155	27	∈	∈	PROPN
ejde-744	155	28	d(a	d(a	PROPN
ejde-744	155	29	)	)	PUNCT
ejde-744	155	30	(	(	PUNCT
ejde-744	155	31	3.2	3.2	NUM
ejde-744	155	32	)	)	PUNCT
ejde-744	155	33	with	with	ADP
ejde-744	155	34	d(a	d(a	PROPN
ejde-744	155	35	)	)	PUNCT
ejde-744	155	36	=	=	SYM
ejde-744	155	37	h2	h2	PROPN
ejde-744	155	38	a	a	PRON
ejde-744	155	39	,	,	PUNCT
ejde-744	155	40	b(0	b(0	NOUN
ejde-744	155	41	,	,	PUNCT
ejde-744	155	42	1)×h1	1)×h1	NUM
ejde-744	155	43	a	a	PRON
ejde-744	155	44	,	,	PUNCT
ejde-744	155	45	b(0	b(0	NOUN
ejde-744	155	46	,	,	PUNCT
ejde-744	155	47	1	1	NUM
ejde-744	155	48	)	)	PUNCT
ejde-744	155	49	,	,	PUNCT
ejde-744	155	50	if	if	SCONJ
ejde-744	155	51	ka	ka	PROPN
ejde-744	155	52	∈	∈	PROPN
ejde-744	156	1	[	[	X
ejde-744	156	2	0	0	NUM
ejde-744	156	3	,	,	PUNCT
ejde-744	156	4	1	1	NUM
ejde-744	156	5	)	)	PUNCT
ejde-744	156	6	,	,	PUNCT
ejde-744	156	7	or	or	CCONJ
ejde-744	156	8	d(a	d(a	PROPN
ejde-744	156	9	)	)	PUNCT
ejde-744	156	10	=	=	PRON
ejde-744	156	11	{	{	PUNCT
ejde-744	156	12	(	(	PUNCT
ejde-744	156	13	u	u	NOUN
ejde-744	156	14	,	,	PUNCT
ejde-744	156	15	v	v	NOUN
ejde-744	156	16	)	)	PUNCT
ejde-744	156	17	∈	∈	PROPN
ejde-744	156	18	h2	h2	PROPN
ejde-744	156	19	a	a	PRON
ejde-744	156	20	,	,	PUNCT
ejde-744	156	21	b(0	b(0	NOUN
ejde-744	156	22	,	,	PUNCT
ejde-744	156	23	1)×h1	1)×h1	NUM
ejde-744	156	24	a	a	DET
ejde-744	156	25	,	,	PUNCT
ejde-744	156	26	b(0	b(0	NOUN
ejde-744	156	27	,	,	PUNCT
ejde-744	156	28	1	1	NUM
ejde-744	156	29	)	)	PUNCT
ejde-744	156	30	:	:	PUNCT
ejde-744	156	31	aux(0	aux(0	ADJ
ejde-744	156	32	)	)	PUNCT
ejde-744	157	1	=	=	PUNCT
ejde-744	157	2	0	0	NUM
ejde-744	157	3	}	}	PUNCT
ejde-744	157	4	,	,	PUNCT
ejde-744	157	5	provide	provide	VERB
ejde-744	157	6	ka	ka	PROPN
ejde-744	157	7	∈	∈	PROPN
ejde-744	157	8	(	(	PUNCT
ejde-744	157	9	1	1	NUM
ejde-744	157	10	,	,	PUNCT
ejde-744	157	11	2	2	NUM
ejde-744	157	12	)	)	PUNCT
ejde-744	157	13	.	.	PUNCT
ejde-744	158	1	proposition	proposition	NOUN
ejde-744	158	2	3.1	3.1	NUM
ejde-744	158	3	.	.	PUNCT
ejde-744	159	1	under	under	ADP
ejde-744	159	2	assumption	assumption	NOUN
ejde-744	159	3	2.6	2.6	NUM
ejde-744	159	4	,	,	PUNCT
ejde-744	159	5	the	the	DET
ejde-744	159	6	operator	operator	NOUN
ejde-744	159	7	a	a	PRON
ejde-744	159	8	is	be	AUX
ejde-744	159	9	maximally	maximally	ADV
ejde-744	159	10	dissipative	dissipative	ADJ
ejde-744	159	11	on	on	ADP
ejde-744	159	12	h.	h.	PROPN
ejde-744	159	13	ejde-2025/04	ejde-2025/04	PROPN
ejde-744	159	14	exact	exact	ADJ
ejde-744	159	15	controllability	controllability	NOUN
ejde-744	159	16	7	7	NUM
ejde-744	159	17	proof	proof	NOUN
ejde-744	159	18	.	.	PUNCT
ejde-744	160	1	let	let	VERB
ejde-744	160	2	(	(	PUNCT
ejde-744	160	3	u	u	NOUN
ejde-744	160	4	,	,	PUNCT
ejde-744	160	5	v	v	NOUN
ejde-744	160	6	)	)	PUNCT
ejde-744	160	7	∈	∈	PROPN
ejde-744	160	8	d(a	d(a	PROPN
ejde-744	160	9	)	)	PUNCT
ejde-744	160	10	.	.	PUNCT
ejde-744	161	1	then	then	ADV
ejde-744	161	2	⟨a(u	⟨a(u	NUM
ejde-744	161	3	,	,	PUNCT
ejde-744	161	4	v	v	NOUN
ejde-744	161	5	)	)	PUNCT
ejde-744	161	6	,	,	PUNCT
ejde-744	161	7	(	(	PUNCT
ejde-744	161	8	u	u	NOUN
ejde-744	161	9	,	,	PUNCT
ejde-744	161	10	v)⟩h	v)⟩h	NOUN
ejde-744	161	11	=	=	SYM
ejde-744	161	12	∫	∫	PROPN
ejde-744	161	13	1	1	NUM
ejde-744	161	14	0	0	NUM
ejde-744	161	15	(	(	PUNCT
ejde-744	161	16	a(u′)′v	a(u′)′v	PROPN
ejde-744	162	1	+	+	PUNCT
ejde-744	162	2	λ	λ	PROPN
ejde-744	162	3	b	b	PROPN
ejde-744	162	4	uv	uv	NOUN
ejde-744	162	5	+	+	PROPN
ejde-744	162	6	au′v′	au′v′	PROPN
ejde-744	162	7	−	−	PROPN
ejde-744	163	1	λ	λ	PROPN
ejde-744	163	2	b	b	PROPN
ejde-744	163	3	uv	uv	NOUN
ejde-744	163	4	)	)	PUNCT
ejde-744	163	5	dx	dx	PROPN
ejde-744	164	1	=	=	SYM
ejde-744	164	2	0	0	PROPN
ejde-744	164	3	.	.	PUNCT
ejde-744	165	1	therefore	therefore	ADV
ejde-744	165	2	,	,	PUNCT
ejde-744	165	3	a	a	PRON
ejde-744	165	4	is	be	AUX
ejde-744	165	5	dissipative	dissipative	ADJ
ejde-744	165	6	.	.	PUNCT
ejde-744	166	1	it	it	PRON
ejde-744	166	2	remains	remain	VERB
ejde-744	166	3	to	to	PART
ejde-744	166	4	be	be	AUX
ejde-744	166	5	proved	prove	VERB
ejde-744	166	6	that	that	SCONJ
ejde-744	166	7	the	the	DET
ejde-744	166	8	operator	operator	NOUN
ejde-744	166	9	is	be	AUX
ejde-744	166	10	maximally	maximally	ADV
ejde-744	166	11	dissipative	dissipative	ADJ
ejde-744	166	12	,	,	PUNCT
ejde-744	166	13	which	which	PRON
ejde-744	166	14	is	be	AUX
ejde-744	166	15	equivalent	equivalent	ADJ
ejde-744	166	16	to	to	ADP
ejde-744	166	17	showing	show	VERB
ejde-744	166	18	that	that	SCONJ
ejde-744	166	19	i	i	PRON
ejde-744	166	20	−a	−a	VERB
ejde-744	166	21	is	be	AUX
ejde-744	166	22	surjective	surjective	ADJ
ejde-744	166	23	.	.	PUNCT
ejde-744	167	1	specifically	specifically	ADV
ejde-744	167	2	,	,	PUNCT
ejde-744	167	3	for	for	ADP
ejde-744	167	4	any	any	DET
ejde-744	167	5	(	(	PUNCT
ejde-744	167	6	g1	g1	PROPN
ejde-744	167	7	,	,	PUNCT
ejde-744	167	8	g2	g2	PROPN
ejde-744	167	9	)	)	PUNCT
ejde-744	167	10	∈	∈	PROPN
ejde-744	167	11	h	h	NOUN
ejde-744	167	12	,	,	PUNCT
ejde-744	167	13	we	we	PRON
ejde-744	167	14	need	need	VERB
ejde-744	167	15	to	to	PART
ejde-744	167	16	find	find	VERB
ejde-744	167	17	(	(	PUNCT
ejde-744	167	18	u	u	NOUN
ejde-744	167	19	,	,	PUNCT
ejde-744	167	20	v	v	NOUN
ejde-744	167	21	)	)	PUNCT
ejde-744	167	22	∈	∈	PROPN
ejde-744	167	23	d(a	d(a	PROPN
ejde-744	167	24	)	)	PUNCT
ejde-744	167	25	such	such	ADJ
ejde-744	167	26	that	that	SCONJ
ejde-744	167	27	the	the	DET
ejde-744	167	28	problem	problem	NOUN
ejde-744	167	29	v	v	ADP
ejde-744	167	30	=	=	SYM
ejde-744	167	31	u−	u−	PROPN
ejde-744	167	32	g1	g1	NOUN
ejde-744	167	33	,	,	PUNCT
ejde-744	167	34	u−	u−	PROPN
ejde-744	167	35	a(u′)′	a(u′)′	PROPN
ejde-744	167	36	−	−	PROPN
ejde-744	168	1	λ	λ	X
ejde-744	168	2	b	b	X
ejde-744	168	3	u	u	X
ejde-744	168	4	=	=	PROPN
ejde-744	168	5	g1	g1	PROPN
ejde-744	168	6	+	+	CCONJ
ejde-744	168	7	g2	g2	PROPN
ejde-744	168	8	.	.	PUNCT
ejde-744	169	1	(	(	PUNCT
ejde-744	169	2	3.3	3.3	NUM
ejde-744	169	3	)	)	PUNCT
ejde-744	169	4	so	so	ADV
ejde-744	169	5	far	far	ADV
ejde-744	169	6	that	that	SCONJ
ejde-744	169	7	,	,	PUNCT
ejde-744	169	8	we	we	PRON
ejde-744	169	9	consider	consider	VERB
ejde-744	169	10	the	the	DET
ejde-744	169	11	bilinear	bilinear	NOUN
ejde-744	169	12	form	form	NOUN
ejde-744	169	13	β	β	X
ejde-744	169	14	:	:	PUNCT
ejde-744	169	15	h1	h1	VERB
ejde-744	169	16	a	a	PRON
ejde-744	169	17	,	,	PUNCT
ejde-744	169	18	b(0	b(0	NOUN
ejde-744	169	19	,	,	PUNCT
ejde-744	169	20	1	1	NUM
ejde-744	169	21	)	)	PUNCT
ejde-744	169	22	×h1	×h1	PROPN
ejde-744	169	23	a	a	DET
ejde-744	169	24	,	,	PUNCT
ejde-744	169	25	b(0	b(0	NOUN
ejde-744	169	26	,	,	PUNCT
ejde-744	169	27	1	1	NUM
ejde-744	169	28	)	)	PUNCT
ejde-744	169	29	→	→	SYM
ejde-744	170	1	r	r	NOUN
ejde-744	170	2	given	give	VERB
ejde-744	170	3	by	by	ADP
ejde-744	170	4	β(u	β(u	PROPN
ejde-744	170	5	,	,	PUNCT
ejde-744	170	6	φ	φ	NOUN
ejde-744	170	7	)	)	PUNCT
ejde-744	170	8	=	=	SYM
ejde-744	171	1	∫	∫	PROPN
ejde-744	171	2	1	1	NUM
ejde-744	171	3	0	0	NUM
ejde-744	171	4	(	(	PUNCT
ejde-744	171	5	uφ+	uφ+	PROPN
ejde-744	171	6	au′φ′	au′φ′	VERB
ejde-744	171	7	−	−	PROPN
ejde-744	171	8	λ	λ	PROPN
ejde-744	171	9	b	b	X
ejde-744	171	10	uφ	uφ	ADJ
ejde-744	171	11	)	)	PUNCT
ejde-744	171	12	dx	dx	PROPN
ejde-744	171	13	,	,	PUNCT
ejde-744	171	14	and	and	CCONJ
ejde-744	171	15	the	the	DET
ejde-744	171	16	linear	linear	ADJ
ejde-744	171	17	functional	functional	ADJ
ejde-744	171	18	l	l	NOUN
ejde-744	171	19	:	:	PUNCT
ejde-744	171	20	h1	h1	VERB
ejde-744	171	21	a	a	DET
ejde-744	171	22	,	,	PUNCT
ejde-744	171	23	b(0	b(0	NOUN
ejde-744	171	24	,	,	PUNCT
ejde-744	171	25	1	1	NUM
ejde-744	171	26	)	)	PUNCT
ejde-744	171	27	→	→	SYM
ejde-744	172	1	r	r	NOUN
ejde-744	172	2	given	give	VERB
ejde-744	172	3	by	by	ADP
ejde-744	172	4	lφ	lφ	PROPN
ejde-744	172	5	=	=	SYM
ejde-744	172	6	∫	∫	PROPN
ejde-744	172	7	1	1	NUM
ejde-744	172	8	0	0	NUM
ejde-744	172	9	(	(	PUNCT
ejde-744	172	10	g1	g1	PROPN
ejde-744	172	11	+	+	PROPN
ejde-744	172	12	g2)φdx	g2)φdx	PROPN
ejde-744	172	13	.	.	PUNCT
ejde-744	173	1	one	one	PRON
ejde-744	173	2	can	can	AUX
ejde-744	173	3	verify	verify	VERB
ejde-744	173	4	that	that	SCONJ
ejde-744	173	5	β	β	NOUN
ejde-744	173	6	is	be	AUX
ejde-744	173	7	a	a	DET
ejde-744	173	8	continuous	continuous	ADJ
ejde-744	173	9	and	and	CCONJ
ejde-744	173	10	coercive	coercive	ADJ
ejde-744	173	11	bilinear	bilinear	ADJ
ejde-744	173	12	functional	functional	ADJ
ejde-744	173	13	on	on	ADP
ejde-744	173	14	h.	h.	PROPN
ejde-744	173	15	also	also	ADV
ejde-744	173	16	,	,	PUNCT
ejde-744	173	17	l	l	NOUN
ejde-744	173	18	is	be	AUX
ejde-744	173	19	a	a	DET
ejde-744	173	20	continuous	continuous	ADJ
ejde-744	173	21	linear	linear	ADJ
ejde-744	173	22	functional	functional	NOUN
ejde-744	173	23	.	.	PUNCT
ejde-744	174	1	consequently	consequently	ADV
ejde-744	174	2	,	,	PUNCT
ejde-744	174	3	by	by	ADP
ejde-744	174	4	the	the	DET
ejde-744	174	5	lax	lax	PROPN
ejde-744	174	6	-	-	PUNCT
ejde-744	174	7	milgram	milgram	NOUN
ejde-744	174	8	theorem	theorem	NOUN
ejde-744	174	9	,	,	PUNCT
ejde-744	174	10	there	there	PRON
ejde-744	174	11	exist	exist	VERB
ejde-744	174	12	a	a	DET
ejde-744	174	13	unique	unique	ADJ
ejde-744	174	14	solution	solution	NOUN
ejde-744	174	15	u	u	NOUN
ejde-744	174	16	∈	∈	PROPN
ejde-744	174	17	h1	h1	VERB
ejde-744	174	18	a	a	DET
ejde-744	174	19	,	,	PUNCT
ejde-744	174	20	b(0	b(0	NOUN
ejde-744	174	21	,	,	PUNCT
ejde-744	174	22	1	1	NUM
ejde-744	174	23	)	)	PUNCT
ejde-744	174	24	to	to	ADP
ejde-744	174	25	the	the	DET
ejde-744	174	26	variational	variational	ADJ
ejde-744	174	27	problem	problem	NOUN
ejde-744	174	28	β(u	β(u	PROPN
ejde-744	174	29	,	,	PUNCT
ejde-744	174	30	φ	φ	NOUN
ejde-744	174	31	)	)	PUNCT
ejde-744	174	32	=	=	SYM
ejde-744	174	33	lφ	lφ	PROPN
ejde-744	174	34	,	,	PUNCT
ejde-744	174	35	∀φ	∀φ	PROPN
ejde-744	174	36	∈	∈	PROPN
ejde-744	174	37	h1	h1	PROPN
ejde-744	174	38	a	a	DET
ejde-744	174	39	,	,	PUNCT
ejde-744	174	40	b(0	b(0	NOUN
ejde-744	174	41	,	,	PUNCT
ejde-744	174	42	1	1	NUM
ejde-744	174	43	)	)	PUNCT
ejde-744	174	44	.	.	PUNCT
ejde-744	175	1	(	(	PUNCT
ejde-744	175	2	3.4	3.4	NUM
ejde-744	175	3	)	)	PUNCT
ejde-744	175	4	since	since	SCONJ
ejde-744	175	5	c∞	c∞	PROPN
ejde-744	175	6	c	c	X
ejde-744	175	7	(	(	PUNCT
ejde-744	175	8	0	0	NUM
ejde-744	175	9	,	,	PUNCT
ejde-744	175	10	1	1	NUM
ejde-744	175	11	)	)	PUNCT
ejde-744	175	12	⊂	⊂	PRON
ejde-744	175	13	h1	h1	VERB
ejde-744	175	14	a	a	DET
ejde-744	175	15	,	,	PUNCT
ejde-744	175	16	b(0	b(0	NOUN
ejde-744	175	17	,	,	PUNCT
ejde-744	175	18	1	1	NUM
ejde-744	175	19	)	)	PUNCT
ejde-744	175	20	,	,	PUNCT
ejde-744	175	21	we	we	PRON
ejde-744	175	22	have∫	have∫	VERB
ejde-744	175	23	1	1	NUM
ejde-744	175	24	0	0	NUM
ejde-744	175	25	(	(	PUNCT
ejde-744	175	26	uφ+	uφ+	PROPN
ejde-744	175	27	au′φ′	au′φ′	VERB
ejde-744	175	28	−	−	PROPN
ejde-744	176	1	λ	λ	PROPN
ejde-744	176	2	b	b	X
ejde-744	176	3	uφ	uφ	PROPN
ejde-744	176	4	)	)	PUNCT
ejde-744	176	5	dx	dx	PROPN
ejde-744	177	1	=	=	SYM
ejde-744	177	2	∫	∫	PROPN
ejde-744	177	3	1	1	NUM
ejde-744	177	4	0	0	NUM
ejde-744	177	5	(	(	PUNCT
ejde-744	177	6	g1	g1	PROPN
ejde-744	177	7	+	+	CCONJ
ejde-744	177	8	g2)φdx	g2)φdx	PROPN
ejde-744	177	9	,	,	PUNCT
ejde-744	177	10	∀φ	∀φ	PROPN
ejde-744	177	11	∈	∈	PROPN
ejde-744	177	12	c∞	c∞	PROPN
ejde-744	177	13	c	c	X
ejde-744	177	14	(	(	PUNCT
ejde-744	177	15	0	0	NUM
ejde-744	177	16	,	,	PUNCT
ejde-744	177	17	1	1	NUM
ejde-744	177	18	)	)	PUNCT
ejde-744	177	19	.	.	PUNCT
ejde-744	178	1	(	(	PUNCT
ejde-744	178	2	3.5	3.5	NUM
ejde-744	178	3	)	)	PUNCT
ejde-744	178	4	by	by	ADP
ejde-744	178	5	duality	duality	NOUN
ejde-744	178	6	,	,	PUNCT
ejde-744	178	7	this	this	PRON
ejde-744	178	8	implies	imply	VERB
ejde-744	178	9	that	that	SCONJ
ejde-744	178	10	u−	u−	PROPN
ejde-744	178	11	a(u′)′	a(u′)′	PROPN
ejde-744	178	12	−	−	PROPN
ejde-744	179	1	λ	λ	X
ejde-744	179	2	b	b	X
ejde-744	179	3	u	u	X
ejde-744	179	4	=	=	PROPN
ejde-744	179	5	g1	g1	PROPN
ejde-744	179	6	+	+	CCONJ
ejde-744	179	7	g2	g2	PROPN
ejde-744	179	8	in	in	ADP
ejde-744	179	9	the	the	DET
ejde-744	179	10	sense	sense	NOUN
ejde-744	179	11	of	of	ADP
ejde-744	179	12	distributions	distribution	NOUN
ejde-744	179	13	.	.	PUNCT
ejde-744	180	1	thus	thus	ADV
ejde-744	180	2	,	,	PUNCT
ejde-744	180	3	u	u	PROPN
ejde-744	180	4	∈	∈	PROPN
ejde-744	180	5	h2	h2	PROPN
ejde-744	180	6	a	a	PRON
ejde-744	180	7	,	,	PUNCT
ejde-744	180	8	b(0	b(0	NOUN
ejde-744	180	9	,	,	PUNCT
ejde-744	180	10	1	1	NUM
ejde-744	180	11	)	)	PUNCT
ejde-744	180	12	and	and	CCONJ
ejde-744	180	13	u	u	NOUN
ejde-744	180	14	−	−	PROPN
ejde-744	180	15	a(u′)′	a(u′)′	PROPN
ejde-744	180	16	−	−	PROPN
ejde-744	180	17	λ	λ	X
ejde-744	180	18	b	b	X
ejde-744	180	19	u	u	X
ejde-744	180	20	=	=	PROPN
ejde-744	180	21	g1	g1	PROPN
ejde-744	180	22	+	+	CCONJ
ejde-744	180	23	g2	g2	PROPN
ejde-744	180	24	almost	almost	ADV
ejde-744	180	25	everywhere	everywhere	ADV
ejde-744	180	26	in	in	ADP
ejde-744	180	27	(	(	PUNCT
ejde-744	180	28	0	0	NUM
ejde-744	180	29	,	,	PUNCT
ejde-744	180	30	1	1	NUM
ejde-744	180	31	)	)	PUNCT
ejde-744	180	32	.	.	PUNCT
ejde-744	181	1	setting	set	VERB
ejde-744	181	2	v	v	ADP
ejde-744	181	3	=	=	SYM
ejde-744	181	4	u	u	NOUN
ejde-744	181	5	−	−	PROPN
ejde-744	181	6	g1	g1	NOUN
ejde-744	181	7	,	,	PUNCT
ejde-744	181	8	we	we	PRON
ejde-744	181	9	conclude	conclude	VERB
ejde-744	181	10	that	that	SCONJ
ejde-744	181	11	(	(	PUNCT
ejde-744	181	12	u	u	NOUN
ejde-744	181	13	,	,	PUNCT
ejde-744	181	14	v	v	NOUN
ejde-744	181	15	)	)	PUNCT
ejde-744	181	16	∈	∈	PROPN
ejde-744	181	17	d(a	d(a	PROPN
ejde-744	181	18	)	)	PUNCT
ejde-744	181	19	and	and	CCONJ
ejde-744	181	20	the	the	DET
ejde-744	181	21	problem	problem	NOUN
ejde-744	181	22	(	(	PUNCT
ejde-744	181	23	3.3	3.3	NUM
ejde-744	181	24	)	)	PUNCT
ejde-744	181	25	is	be	AUX
ejde-744	181	26	solved	solve	VERB
ejde-744	181	27	.	.	PUNCT
ejde-744	182	1	□	□	PUNCT
ejde-744	182	2	therefore	therefore	ADV
ejde-744	182	3	a	a	PRON
ejde-744	182	4	is	be	AUX
ejde-744	182	5	the	the	DET
ejde-744	182	6	generator	generator	NOUN
ejde-744	182	7	of	of	ADP
ejde-744	182	8	a	a	DET
ejde-744	182	9	contraction	contraction	NOUN
ejde-744	182	10	semigroup	semigroup	NOUN
ejde-744	182	11	in	in	ADP
ejde-744	182	12	h	h	NOUN
ejde-744	182	13	,	,	PUNCT
ejde-744	182	14	denoted	denote	VERB
ejde-744	182	15	by	by	ADP
ejde-744	182	16	eta	eta	PROPN
ejde-744	182	17	.	.	PUNCT
ejde-744	183	1	for	for	ADP
ejde-744	183	2	any	any	DET
ejde-744	183	3	u0	u0	ADJ
ejde-744	183	4	=	=	PUNCT
ejde-744	183	5	(	(	PUNCT
ejde-744	183	6	u0	u0	ADJ
ejde-744	183	7	,	,	PUNCT
ejde-744	183	8	u1	u1	NOUN
ejde-744	183	9	)	)	PUNCT
ejde-744	183	10	∈	∈	PROPN
ejde-744	183	11	h	h	NOUN
ejde-744	183	12	,	,	PUNCT
ejde-744	183	13	u(t	u(t	NOUN
ejde-744	183	14	)	)	PUNCT
ejde-744	183	15	=	=	SYM
ejde-744	183	16	etau0	etau0	NOUN
ejde-744	183	17	can	can	AUX
ejde-744	183	18	be	be	AUX
ejde-744	183	19	seen	see	VERB
ejde-744	183	20	as	as	ADP
ejde-744	183	21	the	the	DET
ejde-744	183	22	solution	solution	NOUN
ejde-744	183	23	of	of	ADP
ejde-744	183	24	the	the	DET
ejde-744	183	25	cauchy	cauchy	PROPN
ejde-744	183	26	problem	problem	NOUN
ejde-744	183	27	u	u	PROPN
ejde-744	183	28	′(t	′(t	VERB
ejde-744	183	29	)	)	PUNCT
ejde-744	183	30	=	=	SYM
ejde-744	183	31	au(t	au(t	NUM
ejde-744	183	32	)	)	PUNCT
ejde-744	183	33	,	,	PUNCT
ejde-744	183	34	u(0	u(0	PROPN
ejde-744	183	35	)	)	PUNCT
ejde-744	183	36	=	=	PUNCT
ejde-744	184	1	u0	u0	ADJ
ejde-744	184	2	.	.	PUNCT
ejde-744	185	1	(	(	PUNCT
ejde-744	185	2	3.6	3.6	NUM
ejde-744	185	3	)	)	PUNCT
ejde-744	185	4	hence	hence	ADV
ejde-744	185	5	,	,	PUNCT
ejde-744	185	6	as	as	ADP
ejde-744	185	7	in	in	ADP
ejde-744	185	8	[	[	X
ejde-744	185	9	1	1	NUM
ejde-744	185	10	]	]	PUNCT
ejde-744	185	11	or	or	CCONJ
ejde-744	185	12	[	[	X
ejde-744	185	13	3	3	NUM
ejde-744	185	14	]	]	PUNCT
ejde-744	185	15	,	,	PUNCT
ejde-744	185	16	we	we	PRON
ejde-744	185	17	have	have	VERB
ejde-744	185	18	the	the	DET
ejde-744	185	19	following	follow	VERB
ejde-744	185	20	conclusions	conclusion	NOUN
ejde-744	185	21	.	.	PUNCT
ejde-744	186	1	corollary	corollary	ADJ
ejde-744	186	2	3.2	3.2	NUM
ejde-744	186	3	.	.	PUNCT
ejde-744	187	1	assume	assume	VERB
ejde-744	187	2	assumption	assumption	NOUN
ejde-744	187	3	2.6	2.6	NUM
ejde-744	187	4	,	,	PUNCT
ejde-744	187	5	for	for	SCONJ
ejde-744	187	6	given	give	VERB
ejde-744	187	7	(	(	PUNCT
ejde-744	187	8	u0	u0	ADJ
ejde-744	187	9	,	,	PUNCT
ejde-744	187	10	u1	u1	NOUN
ejde-744	187	11	)	)	PUNCT
ejde-744	187	12	∈	∈	PROPN
ejde-744	187	13	h1	h1	VERB
ejde-744	187	14	a	a	DET
ejde-744	187	15	,	,	PUNCT
ejde-744	187	16	b(0	b(0	NOUN
ejde-744	187	17	,	,	PUNCT
ejde-744	187	18	1)×l2(0	1)×l2(0	NUM
ejde-744	187	19	,	,	PUNCT
ejde-744	187	20	1	1	NUM
ejde-744	187	21	)	)	PUNCT
ejde-744	187	22	,	,	PUNCT
ejde-744	187	23	there	there	PRON
ejde-744	187	24	exist	exist	VERB
ejde-744	187	25	a	a	DET
ejde-744	187	26	unique	unique	ADJ
ejde-744	187	27	mild	mild	ADJ
ejde-744	187	28	solution	solution	NOUN
ejde-744	187	29	u	u	NOUN
ejde-744	187	30	to	to	ADP
ejde-744	187	31	the	the	DET
ejde-744	187	32	problem	problem	NOUN
ejde-744	187	33	(	(	PUNCT
ejde-744	187	34	3.1	3.1	NUM
ejde-744	187	35	)	)	PUNCT
ejde-744	187	36	satisfying	satisfy	VERB
ejde-744	187	37	u	u	NOUN
ejde-744	187	38	∈	∈	PROPN
ejde-744	187	39	c1([0	c1([0	PROPN
ejde-744	187	40	,	,	PUNCT
ejde-744	187	41	t	t	X
ejde-744	187	42	]	]	PUNCT
ejde-744	187	43	;	;	PUNCT
ejde-744	187	44	l2(0	l2(0	NOUN
ejde-744	187	45	,	,	PUNCT
ejde-744	187	46	1	1	NUM
ejde-744	187	47	)	)	PUNCT
ejde-744	187	48	)	)	PUNCT
ejde-744	188	1	∩	∩	NOUN
ejde-744	188	2	c([0	c([0	NOUN
ejde-744	188	3	,	,	PUNCT
ejde-744	188	4	t	t	X
ejde-744	188	5	]	]	PUNCT
ejde-744	188	6	;	;	PUNCT
ejde-744	188	7	h1	h1	VERB
ejde-744	188	8	a	a	DET
ejde-744	188	9	,	,	PUNCT
ejde-744	188	10	b(0	b(0	NOUN
ejde-744	188	11	,	,	PUNCT
ejde-744	188	12	1	1	NUM
ejde-744	188	13	)	)	PUNCT
ejde-744	188	14	)	)	PUNCT
ejde-744	188	15	;	;	PUNCT
ejde-744	188	16	if	if	SCONJ
ejde-744	188	17	(	(	PUNCT
ejde-744	188	18	u0	u0	ADJ
ejde-744	188	19	,	,	PUNCT
ejde-744	188	20	u1	u1	NOUN
ejde-744	188	21	)	)	PUNCT
ejde-744	188	22	∈	∈	PROPN
ejde-744	188	23	d(a	d(a	PROPN
ejde-744	188	24	)	)	PUNCT
ejde-744	188	25	,	,	PUNCT
ejde-744	188	26	then	then	ADV
ejde-744	188	27	the	the	DET
ejde-744	188	28	solution	solution	NOUN
ejde-744	188	29	u	u	NOUN
ejde-744	188	30	is	be	AUX
ejde-744	188	31	classical	classical	ADJ
ejde-744	188	32	,	,	PUNCT
ejde-744	188	33	in	in	ADP
ejde-744	188	34	the	the	DET
ejde-744	188	35	sense	sense	NOUN
ejde-744	188	36	that	that	SCONJ
ejde-744	188	37	u	u	PROPN
ejde-744	188	38	∈	∈	PROPN
ejde-744	188	39	c2([0	c2([0	PROPN
ejde-744	188	40	,	,	PUNCT
ejde-744	188	41	t	t	X
ejde-744	188	42	]	]	PUNCT
ejde-744	188	43	;	;	PUNCT
ejde-744	188	44	l2(0	l2(0	NOUN
ejde-744	188	45	,	,	PUNCT
ejde-744	188	46	1	1	NUM
ejde-744	188	47	)	)	PUNCT
ejde-744	188	48	)	)	PUNCT
ejde-744	188	49	∩	∩	NOUN
ejde-744	188	50	c1([0	c1([0	PROPN
ejde-744	188	51	,	,	PUNCT
ejde-744	188	52	t	t	X
ejde-744	188	53	]	]	PUNCT
ejde-744	188	54	;	;	PUNCT
ejde-744	188	55	h1	h1	VERB
ejde-744	188	56	a	a	DET
ejde-744	188	57	,	,	PUNCT
ejde-744	188	58	b(0	b(0	NOUN
ejde-744	188	59	,	,	PUNCT
ejde-744	188	60	1	1	NUM
ejde-744	188	61	)	)	PUNCT
ejde-744	188	62	∩	∩	NOUN
ejde-744	188	63	c([0	c([0	NOUN
ejde-744	188	64	,	,	PUNCT
ejde-744	188	65	t	t	X
ejde-744	188	66	]	]	PUNCT
ejde-744	188	67	;	;	PUNCT
ejde-744	188	68	h2	h2	PROPN
ejde-744	188	69	a	a	PRON
ejde-744	188	70	,	,	PUNCT
ejde-744	188	71	b(0	b(0	NOUN
ejde-744	188	72	,	,	PUNCT
ejde-744	188	73	1	1	NUM
ejde-744	188	74	)	)	PUNCT
ejde-744	188	75	)	)	PUNCT
ejde-744	188	76	.	.	PUNCT
ejde-744	189	1	8	8	NUM
ejde-744	189	2	g.	g.	PROPN
ejde-744	189	3	zhang	zhang	PROPN
ejde-744	189	4	,	,	PUNCT
ejde-744	189	5	s.	s.	PROPN
ejde-744	189	6	chai	chai	PROPN
ejde-744	189	7	ejde-2025/04	ejde-2025/04	NOUN
ejde-744	189	8	4	4	NUM
ejde-744	189	9	.	.	PUNCT
ejde-744	189	10	energy	energy	NOUN
ejde-744	189	11	estimate	estimate	NOUN
ejde-744	189	12	in	in	ADP
ejde-744	189	13	this	this	DET
ejde-744	189	14	section	section	NOUN
ejde-744	189	15	,	,	PUNCT
ejde-744	189	16	we	we	PRON
ejde-744	189	17	establish	establish	VERB
ejde-744	189	18	an	an	DET
ejde-744	189	19	estimate	estimate	NOUN
ejde-744	189	20	of	of	ADP
ejde-744	189	21	the	the	DET
ejde-744	189	22	energy	energy	NOUN
ejde-744	189	23	and	and	CCONJ
ejde-744	189	24	a	a	DET
ejde-744	189	25	direct	direct	ADJ
ejde-744	189	26	inequality	inequality	NOUN
ejde-744	189	27	associated	associate	VERB
ejde-744	189	28	to	to	ADP
ejde-744	189	29	the	the	DET
ejde-744	189	30	solution	solution	NOUN
ejde-744	189	31	of	of	ADP
ejde-744	189	32	the	the	DET
ejde-744	189	33	initial	initial	ADJ
ejde-744	189	34	value	value	NOUN
ejde-744	189	35	problem	problem	NOUN
ejde-744	189	36	,	,	PUNCT
ejde-744	189	37	which	which	PRON
ejde-744	189	38	will	will	AUX
ejde-744	189	39	be	be	AUX
ejde-744	189	40	used	use	VERB
ejde-744	189	41	to	to	PART
ejde-744	189	42	prove	prove	VERB
ejde-744	189	43	a	a	DET
ejde-744	189	44	controllability	controllability	NOUN
ejde-744	189	45	in	in	ADP
ejde-744	189	46	section	section	NOUN
ejde-744	189	47	6	6	NUM
ejde-744	189	48	.	.	PUNCT
ejde-744	189	49	definition	definition	NOUN
ejde-744	189	50	4.1	4.1	NUM
ejde-744	189	51	.	.	PUNCT
ejde-744	189	52	using	use	VERB
ejde-744	189	53	assumption	assumption	NOUN
ejde-744	189	54	2.6	2.6	NUM
ejde-744	189	55	,	,	PUNCT
ejde-744	189	56	we	we	PRON
ejde-744	189	57	define	define	VERB
ejde-744	189	58	the	the	DET
ejde-744	189	59	generalized	generalized	ADJ
ejde-744	189	60	energy	energy	NOUN
ejde-744	189	61	of	of	ADP
ejde-744	189	62	a	a	DET
ejde-744	189	63	mild	mild	ADJ
ejde-744	189	64	solution	solution	NOUN
ejde-744	189	65	u	u	NOUN
ejde-744	189	66	of	of	ADP
ejde-744	189	67	(	(	PUNCT
ejde-744	189	68	3.1	3.1	NUM
ejde-744	189	69	)	)	PUNCT
ejde-744	189	70	as	as	SCONJ
ejde-744	189	71	follows	follow	VERB
ejde-744	189	72	,	,	PUNCT
ejde-744	189	73	eu(t	eu(t	PUNCT
ejde-744	189	74	)	)	PUNCT
ejde-744	189	75	=	=	SYM
ejde-744	189	76	1	1	NUM
ejde-744	189	77	2	2	NUM
ejde-744	189	78	∫	∫	NOUN
ejde-744	189	79	1	1	NUM
ejde-744	189	80	0	0	NUM
ejde-744	189	81	(	(	PUNCT
ejde-744	189	82	u2	u2	PROPN
ejde-744	189	83	t	t	PROPN
ejde-744	189	84	+	+	CCONJ
ejde-744	189	85	au2	au2	NOUN
ejde-744	189	86	x	x	X
ejde-744	189	87	−	−	PUNCT
ejde-744	189	88	λ	λ	X
ejde-744	189	89	b	b	PROPN
ejde-744	189	90	u2	u2	PROPN
ejde-744	189	91	)	)	PUNCT
ejde-744	189	92	dx	dx	PROPN
ejde-744	189	93	,	,	PUNCT
ejde-744	189	94	∀t	∀t	PROPN
ejde-744	189	95	≥	≥	NOUN
ejde-744	189	96	0	0	NUM
ejde-744	189	97	.	.	PUNCT
ejde-744	190	1	(	(	PUNCT
ejde-744	190	2	4.1	4.1	NUM
ejde-744	190	3	)	)	PUNCT
ejde-744	190	4	computations	computation	NOUN
ejde-744	190	5	show	show	VERB
ejde-744	190	6	that	that	SCONJ
ejde-744	190	7	the	the	DET
ejde-744	190	8	conservation	conservation	NOUN
ejde-744	190	9	of	of	ADP
ejde-744	190	10	the	the	DET
ejde-744	190	11	energy	energy	NOUN
ejde-744	190	12	eu	eu	PROPN
ejde-744	190	13	remains	remain	VERB
ejde-744	190	14	valid	valid	ADJ
ejde-744	190	15	in	in	ADP
ejde-744	190	16	the	the	DET
ejde-744	190	17	degenerate	degenerate	ADJ
ejde-744	190	18	and	and	CCONJ
ejde-744	190	19	singular	singular	ADJ
ejde-744	190	20	situation	situation	NOUN
ejde-744	190	21	.	.	PUNCT
ejde-744	191	1	proposition	proposition	NOUN
ejde-744	191	2	4.2	4.2	NUM
ejde-744	191	3	.	.	PUNCT
ejde-744	192	1	under	under	ADP
ejde-744	192	2	assumption	assumption	NOUN
ejde-744	192	3	2.6	2.6	NUM
ejde-744	192	4	and	and	CCONJ
ejde-744	192	5	considering	consider	VERB
ejde-744	192	6	(	(	PUNCT
ejde-744	192	7	u0	u0	ADJ
ejde-744	192	8	,	,	PUNCT
ejde-744	192	9	u1	u1	NOUN
ejde-744	192	10	)	)	PUNCT
ejde-744	192	11	∈	∈	PROPN
ejde-744	192	12	h1	h1	VERB
ejde-744	192	13	a	a	PRON
ejde-744	192	14	,	,	PUNCT
ejde-744	192	15	b(0	b(0	NOUN
ejde-744	192	16	,	,	PUNCT
ejde-744	192	17	1	1	NUM
ejde-744	192	18	)	)	PUNCT
ejde-744	192	19	×	×	NOUN
ejde-744	192	20	l2(0	l2(0	NOUN
ejde-744	192	21	,	,	PUNCT
ejde-744	192	22	1	1	NUM
ejde-744	192	23	)	)	PUNCT
ejde-744	192	24	,	,	PUNCT
ejde-744	192	25	the	the	DET
ejde-744	192	26	energy	energy	NOUN
ejde-744	192	27	eu(t	eu(t	PUNCT
ejde-744	192	28	)	)	PUNCT
ejde-744	192	29	of	of	ADP
ejde-744	192	30	the	the	DET
ejde-744	192	31	mild	mild	ADJ
ejde-744	192	32	solution	solution	NOUN
ejde-744	192	33	u	u	NOUN
ejde-744	192	34	of	of	ADP
ejde-744	192	35	(	(	PUNCT
ejde-744	192	36	3.1	3.1	NUM
ejde-744	192	37	)	)	PUNCT
ejde-744	192	38	is	be	AUX
ejde-744	192	39	constant	constant	ADJ
ejde-744	192	40	in	in	ADP
ejde-744	192	41	time	time	NOUN
ejde-744	192	42	,	,	PUNCT
ejde-744	192	43	that	that	ADV
ejde-744	192	44	is	is	ADV
ejde-744	192	45	,	,	PUNCT
ejde-744	192	46	eu(t	eu(t	PUNCT
ejde-744	192	47	)	)	PUNCT
ejde-744	192	48	=	=	SYM
ejde-744	192	49	eu(0	eu(0	PROPN
ejde-744	192	50	)	)	PUNCT
ejde-744	192	51	,	,	PUNCT
ejde-744	192	52	∀t	∀t	PROPN
ejde-744	192	53	≥	≥	NOUN
ejde-744	192	54	0	0	NUM
ejde-744	192	55	.	.	PUNCT
ejde-744	193	1	(	(	PUNCT
ejde-744	193	2	4.2	4.2	NUM
ejde-744	193	3	)	)	PUNCT
ejde-744	193	4	proof	proof	NOUN
ejde-744	193	5	.	.	PUNCT
ejde-744	194	1	first	first	ADV
ejde-744	194	2	,	,	PUNCT
ejde-744	194	3	suppose	suppose	VERB
ejde-744	194	4	that	that	SCONJ
ejde-744	194	5	u	u	PROPN
ejde-744	194	6	is	be	AUX
ejde-744	194	7	a	a	DET
ejde-744	194	8	classical	classical	ADJ
ejde-744	194	9	solution	solution	NOUN
ejde-744	194	10	.	.	PUNCT
ejde-744	195	1	then	then	ADV
ejde-744	195	2	,	,	PUNCT
ejde-744	195	3	multiplying	multiply	VERB
ejde-744	195	4	the	the	DET
ejde-744	195	5	equation	equation	NOUN
ejde-744	195	6	by	by	ADP
ejde-744	195	7	ut	ut	PROPN
ejde-744	195	8	and	and	CCONJ
ejde-744	195	9	integrating	integrate	VERB
ejde-744	195	10	over	over	ADP
ejde-744	195	11	(	(	PUNCT
ejde-744	195	12	0	0	NUM
ejde-744	195	13	,	,	PUNCT
ejde-744	195	14	1	1	NUM
ejde-744	195	15	)	)	PUNCT
ejde-744	195	16	,	,	PUNCT
ejde-744	195	17	we	we	PRON
ejde-744	195	18	obtain	obtain	VERB
ejde-744	195	19	0	0	NUM
ejde-744	196	1	=	=	SYM
ejde-744	196	2	∫	∫	PROPN
ejde-744	196	3	1	1	NUM
ejde-744	196	4	0	0	NUM
ejde-744	196	5	ut(t	ut(t	NOUN
ejde-744	196	6	,	,	PUNCT
ejde-744	196	7	x	x	X
ejde-744	196	8	)	)	PUNCT
ejde-744	196	9	(	(	PUNCT
ejde-744	196	10	utt(t	utt(t	PROPN
ejde-744	196	11	,	,	PUNCT
ejde-744	196	12	x)−	x)−	PROPN
ejde-744	196	13	(	(	PUNCT
ejde-744	196	14	a(x)ux(t	a(x)ux(t	PROPN
ejde-744	196	15	,	,	PUNCT
ejde-744	196	16	x))x	x))x	ADJ
ejde-744	196	17	−	−	PROPN
ejde-744	197	1	λ	λ	NOUN
ejde-744	197	2	b(x	b(x	NOUN
ejde-744	197	3	)	)	PUNCT
ejde-744	197	4	u(t	u(t	NOUN
ejde-744	197	5	,	,	PUNCT
ejde-744	197	6	x	x	NOUN
ejde-744	197	7	)	)	PUNCT
ejde-744	197	8	)	)	PUNCT
ejde-744	198	1	dx	dx	PROPN
ejde-744	199	1	=	=	SYM
ejde-744	199	2	∫	∫	PROPN
ejde-744	199	3	1	1	NUM
ejde-744	199	4	0	0	NUM
ejde-744	199	5	(	(	PUNCT
ejde-744	199	6	ut(t	ut(t	PROPN
ejde-744	199	7	,	,	PUNCT
ejde-744	199	8	x)utt(t	x)utt(t	PROPN
ejde-744	199	9	,	,	PUNCT
ejde-744	199	10	x)−	x)−	PROPN
ejde-744	199	11	a(x)ux(t	a(x)ux(t	PROPN
ejde-744	199	12	,	,	PUNCT
ejde-744	199	13	x)utx(t	x)utx(t	PROPN
ejde-744	199	14	,	,	PUNCT
ejde-744	199	15	x)−	x)−	PROPN
ejde-744	199	16	λ	λ	NOUN
ejde-744	199	17	b(x	b(x	NOUN
ejde-744	199	18	)	)	PUNCT
ejde-744	199	19	u(t	u(t	NOUN
ejde-744	199	20	,	,	PUNCT
ejde-744	199	21	x)ut(t	x)ut(t	PROPN
ejde-744	199	22	,	,	PUNCT
ejde-744	199	23	x	x	NOUN
ejde-744	199	24	)	)	PUNCT
ejde-744	199	25	)	)	PUNCT
ejde-744	200	1	dx︸	dx︸	PROPN
ejde-744	200	2	︷︷	︷︷	PROPN
ejde-744	200	3	︸	︸	X
ejde-744	200	4	=	=	SYM
ejde-744	200	5	d	d	NUM
ejde-744	200	6	dteu(t	dteu(t	PROPN
ejde-744	200	7	)	)	PUNCT
ejde-744	200	8	−	−	PROPN
ejde-744	201	1	[	[	X
ejde-744	201	2	a(x)ux(t	a(x)ux(t	PROPN
ejde-744	201	3	,	,	PUNCT
ejde-744	201	4	x)ut(t	x)ut(t	PROPN
ejde-744	201	5	,	,	PUNCT
ejde-744	201	6	x	x	NOUN
ejde-744	201	7	)	)	PUNCT
ejde-744	201	8	]	]	PUNCT
ejde-744	201	9	1	1	NUM
ejde-744	201	10	0	0	NUM
ejde-744	201	11	.	.	PUNCT
ejde-744	202	1	(	(	PUNCT
ejde-744	202	2	4.3	4.3	NUM
ejde-744	202	3	)	)	PUNCT
ejde-744	202	4	using	use	VERB
ejde-744	202	5	the	the	DET
ejde-744	202	6	boundary	boundary	ADJ
ejde-744	202	7	conditions	condition	NOUN
ejde-744	202	8	and	and	CCONJ
ejde-744	202	9	alabau	alabau	NOUN
ejde-744	202	10	-	-	PUNCT
ejde-744	202	11	boussouira	boussouira	PROPN
ejde-744	202	12	et	et	PROPN
ejde-744	202	13	al	al	PROPN
ejde-744	202	14	.	.	PUNCT
ejde-744	203	1	[	[	X
ejde-744	203	2	1	1	NUM
ejde-744	203	3	,	,	PUNCT
ejde-744	203	4	proposition	proposition	NOUN
ejde-744	203	5	2.5	2.5	NUM
ejde-744	203	6	]	]	PUNCT
ejde-744	203	7	,	,	PUNCT
ejde-744	203	8	we	we	PRON
ejde-744	203	9	have	have	VERB
ejde-744	203	10	that	that	SCONJ
ejde-744	203	11	the	the	DET
ejde-744	203	12	boundary	boundary	ADJ
ejde-744	203	13	a(x)ux(t	a(x)ux(t	NOUN
ejde-744	203	14	,	,	PUNCT
ejde-744	203	15	x)ut(t	x)ut(t	PROPN
ejde-744	203	16	,	,	PUNCT
ejde-744	203	17	x	x	X
ejde-744	203	18	)	)	PUNCT
ejde-744	203	19	vanishes	vanish	VERB
ejde-744	203	20	at	at	ADP
ejde-744	203	21	x	x	X
ejde-744	203	22	=	=	SYM
ejde-744	203	23	1	1	NUM
ejde-744	203	24	and	and	CCONJ
ejde-744	203	25	x	x	SYM
ejde-744	203	26	=	=	NOUN
ejde-744	203	27	0	0	X
ejde-744	203	28	.	.	PUNCT
ejde-744	204	1	now	now	ADV
ejde-744	204	2	,	,	PUNCT
ejde-744	204	3	let	let	VERB
ejde-744	204	4	u	u	PRON
ejde-744	204	5	be	be	AUX
ejde-744	204	6	the	the	DET
ejde-744	204	7	mild	mild	ADJ
ejde-744	204	8	solution	solution	NOUN
ejde-744	204	9	associated	associate	VERB
ejde-744	204	10	with	with	ADP
ejde-744	204	11	the	the	DET
ejde-744	204	12	initial	initial	ADJ
ejde-744	204	13	data	datum	NOUN
ejde-744	204	14	(	(	PUNCT
ejde-744	204	15	u0	u0	ADJ
ejde-744	204	16	,	,	PUNCT
ejde-744	204	17	u1	u1	NOUN
ejde-744	204	18	)	)	PUNCT
ejde-744	204	19	∈	∈	PROPN
ejde-744	204	20	h1	h1	VERB
ejde-744	204	21	a	a	PRON
ejde-744	204	22	,	,	PUNCT
ejde-744	204	23	b(0	b(0	NOUN
ejde-744	204	24	,	,	PUNCT
ejde-744	204	25	1	1	NUM
ejde-744	204	26	)	)	PUNCT
ejde-744	204	27	×	×	NOUN
ejde-744	204	28	l2(0	l2(0	NOUN
ejde-744	204	29	,	,	PUNCT
ejde-744	204	30	1	1	NUM
ejde-744	204	31	)	)	PUNCT
ejde-744	204	32	.	.	PUNCT
ejde-744	205	1	consider	consider	VERB
ejde-744	205	2	a	a	DET
ejde-744	205	3	sequence	sequence	NOUN
ejde-744	205	4	{	{	PUNCT
ejde-744	205	5	un	un	PROPN
ejde-744	205	6	0	0	PROPN
ejde-744	205	7	,	,	PUNCT
ejde-744	205	8	u	u	PROPN
ejde-744	205	9	n	n	PROPN
ejde-744	205	10	1}n∈n	1}n∈n	NUM
ejde-744	205	11	⊂	⊂	PUNCT
ejde-744	206	1	d(a	d(a	PROPN
ejde-744	206	2	)	)	PUNCT
ejde-744	206	3	=	=	SYM
ejde-744	206	4	h2	h2	PROPN
ejde-744	206	5	a	a	PRON
ejde-744	206	6	,	,	PUNCT
ejde-744	206	7	b(0	b(0	NOUN
ejde-744	206	8	,	,	PUNCT
ejde-744	206	9	1	1	NUM
ejde-744	206	10	)	)	PUNCT
ejde-744	206	11	×	×	NOUN
ejde-744	206	12	h1	h1	NOUN
ejde-744	206	13	a	a	DET
ejde-744	206	14	,	,	PUNCT
ejde-744	206	15	b(0	b(0	NOUN
ejde-744	206	16	,	,	PUNCT
ejde-744	206	17	1	1	NUM
ejde-744	206	18	)	)	PUNCT
ejde-744	206	19	that	that	PRON
ejde-744	206	20	approximates	approximate	VERB
ejde-744	206	21	(	(	PUNCT
ejde-744	206	22	u0	u0	ADJ
ejde-744	206	23	,	,	PUNCT
ejde-744	206	24	u1	u1	NOUN
ejde-744	206	25	)	)	PUNCT
ejde-744	206	26	,	,	PUNCT
ejde-744	206	27	and	and	CCONJ
ejde-744	206	28	let	let	VERB
ejde-744	206	29	un	un	PROPN
ejde-744	206	30	be	be	AUX
ejde-744	206	31	the	the	DET
ejde-744	206	32	classical	classical	ADJ
ejde-744	206	33	solution	solution	NOUN
ejde-744	206	34	of	of	ADP
ejde-744	206	35	(	(	PUNCT
ejde-744	206	36	3.1	3.1	NUM
ejde-744	206	37	)	)	PUNCT
ejde-744	206	38	associated	associate	VERB
ejde-744	206	39	to	to	ADP
ejde-744	206	40	(	(	PUNCT
ejde-744	206	41	un	un	PROPN
ejde-744	206	42	0	0	PROPN
ejde-744	206	43	,	,	PUNCT
ejde-744	206	44	u	u	NOUN
ejde-744	206	45	n	n	ADV
ejde-744	206	46	1	1	NUM
ejde-744	206	47	)	)	PUNCT
ejde-744	206	48	.	.	PUNCT
ejde-744	207	1	u	u	PRON
ejde-744	207	2	n	n	PRON
ejde-744	207	3	satisfies	satisfie	NOUN
ejde-744	207	4	(	(	PUNCT
ejde-744	207	5	4.2	4.2	NUM
ejde-744	207	6	)	)	PUNCT
ejde-744	207	7	and	and	CCONJ
ejde-744	207	8	un	un	PROPN
ejde-744	207	9	x	x	VERB
ejde-744	207	10	is	be	AUX
ejde-744	207	11	a	a	DET
ejde-744	207	12	cauchy	cauchy	ADJ
ejde-744	207	13	sequence	sequence	NOUN
ejde-744	207	14	in	in	ADP
ejde-744	207	15	l2(0	l2(0	NOUN
ejde-744	207	16	,	,	PUNCT
ejde-744	207	17	1	1	NUM
ejde-744	207	18	)	)	PUNCT
ejde-744	207	19	.	.	PUNCT
ejde-744	208	1	therefore	therefore	ADV
ejde-744	208	2	,	,	PUNCT
ejde-744	208	3	we	we	PRON
ejde-744	208	4	extend	extend	VERB
ejde-744	208	5	(	(	PUNCT
ejde-744	208	6	4.2	4.2	NUM
ejde-744	208	7	)	)	PUNCT
ejde-744	208	8	to	to	ADP
ejde-744	208	9	the	the	DET
ejde-744	208	10	mild	mild	ADJ
ejde-744	208	11	solution	solution	NOUN
ejde-744	208	12	.	.	PUNCT
ejde-744	209	1	□	□	PUNCT
ejde-744	209	2	to	to	PART
ejde-744	209	3	facilitate	facilitate	VERB
ejde-744	209	4	the	the	DET
ejde-744	209	5	subsequent	subsequent	ADJ
ejde-744	209	6	proof	proof	NOUN
ejde-744	209	7	of	of	ADP
ejde-744	209	8	controllability	controllability	NOUN
ejde-744	209	9	results	result	NOUN
ejde-744	209	10	,	,	PUNCT
ejde-744	209	11	we	we	PRON
ejde-744	209	12	prove	prove	VERB
ejde-744	209	13	the	the	DET
ejde-744	209	14	following	follow	VERB
ejde-744	209	15	direct	direct	ADJ
ejde-744	209	16	inequality	inequality	NOUN
ejde-744	209	17	.	.	PUNCT
ejde-744	210	1	proposition	proposition	NOUN
ejde-744	210	2	4.3	4.3	NUM
ejde-744	210	3	.	.	PUNCT
ejde-744	211	1	under	under	ADP
ejde-744	211	2	assumption	assumption	NOUN
ejde-744	211	3	2.6	2.6	NUM
ejde-744	211	4	,	,	PUNCT
ejde-744	211	5	if	if	SCONJ
ejde-744	211	6	u	u	NOUN
ejde-744	211	7	is	be	AUX
ejde-744	211	8	a	a	DET
ejde-744	211	9	classical	classical	ADJ
ejde-744	211	10	solution	solution	NOUN
ejde-744	211	11	of	of	ADP
ejde-744	211	12	(	(	PUNCT
ejde-744	211	13	3.1	3.1	NUM
ejde-744	211	14	)	)	PUNCT
ejde-744	211	15	,	,	PUNCT
ejde-744	211	16	then	then	ADV
ejde-744	211	17	a(1	a(1	PROPN
ejde-744	211	18	)	)	PUNCT
ejde-744	211	19	∫	∫	PROPN
ejde-744	211	20	t	t	NOUN
ejde-744	211	21	0	0	NUM
ejde-744	211	22	u2	u2	PROPN
ejde-744	211	23	x(t	x(t	PROPN
ejde-744	211	24	,	,	PUNCT
ejde-744	211	25	1)dt	1)dt	NUM
ejde-744	212	1	=	=	SYM
ejde-744	212	2	∫	∫	PROPN
ejde-744	212	3	q	q	PROPN
ejde-744	213	1	(	(	PUNCT
ejde-744	213	2	u2	u2	PROPN
ejde-744	213	3	t	t	PROPN
ejde-744	213	4	+	+	CCONJ
ejde-744	213	5	(	(	PUNCT
ejde-744	213	6	a−	a−	PROPN
ejde-744	213	7	xa′)u2	xa′)u2	NUM
ejde-744	213	8	x	x	PUNCT
ejde-744	214	1	+	+	PUNCT
ejde-744	214	2	λ	λ	X
ejde-744	214	3	b−	b−	PROPN
ejde-744	214	4	xb′	xb′	PROPN
ejde-744	214	5	b2	b2	PROPN
ejde-744	214	6	)	)	PUNCT
ejde-744	214	7	dx	dx	PROPN
ejde-744	215	1	dt	dt	NOUN
ejde-744	216	1	+	+	CCONJ
ejde-744	216	2	2	2	NUM
ejde-744	216	3	[	[	X
ejde-744	216	4	∫	∫	PROPN
ejde-744	216	5	1	1	NUM
ejde-744	216	6	0	0	NUM
ejde-744	216	7	xuxutdx	xuxutdx	NOUN
ejde-744	216	8	]	]	X
ejde-744	216	9	t	t	NOUN
ejde-744	216	10	0	0	NUM
ejde-744	216	11	.	.	PUNCT
ejde-744	217	1	(	(	PUNCT
ejde-744	217	2	4.4	4.4	NUM
ejde-744	217	3	)	)	PUNCT
ejde-744	217	4	ejde-2025/04	ejde-2025/04	NOUN
ejde-744	217	5	exact	exact	ADJ
ejde-744	217	6	controllability	controllability	NOUN
ejde-744	217	7	9	9	NUM
ejde-744	217	8	as	as	ADP
ejde-744	217	9	a	a	DET
ejde-744	217	10	consequence	consequence	NOUN
ejde-744	217	11	,	,	PUNCT
ejde-744	217	12	if	if	SCONJ
ejde-744	217	13	u	u	NOUN
ejde-744	217	14	is	be	AUX
ejde-744	217	15	a	a	DET
ejde-744	217	16	mild	mild	ADJ
ejde-744	217	17	solution	solution	NOUN
ejde-744	217	18	,	,	PUNCT
ejde-744	217	19	then	then	ADV
ejde-744	217	20	ux	ux	PROPN
ejde-744	217	21	(	(	PUNCT
ejde-744	217	22	·	·	PUNCT
ejde-744	217	23	,	,	PUNCT
ejde-744	217	24	1	1	X
ejde-744	217	25	)	)	PUNCT
ejde-744	217	26	∈	∈	PROPN
ejde-744	217	27	l2(0	l2(0	NOUN
ejde-744	217	28	,	,	PUNCT
ejde-744	217	29	t	t	PROPN
ejde-744	217	30	)	)	PUNCT
ejde-744	217	31	for	for	ADP
ejde-744	217	32	every	every	DET
ejde-744	217	33	t	t	NOUN
ejde-744	217	34	>	>	X
ejde-744	217	35	0	0	PUNCT
ejde-744	217	36	and	and	CCONJ
ejde-744	217	37	a(1	a(1	ADJ
ejde-744	217	38	)	)	PUNCT
ejde-744	217	39	∫	∫	PROPN
ejde-744	218	1	t	t	NOUN
ejde-744	218	2	0	0	NUM
ejde-744	218	3	u2	u2	PROPN
ejde-744	218	4	x(t	x(t	PROPN
ejde-744	218	5	,	,	PUNCT
ejde-744	218	6	1)dt	1)dt	PROPN
ejde-744	218	7	≤	≤	NUM
ejde-744	218	8	4max	4max	NUM
ejde-744	218	9	{	{	PUNCT
ejde-744	218	10	1	1	NUM
ejde-744	218	11	a(1)cθ	a(1)cθ	NOUN
ejde-744	218	12	,	,	PUNCT
ejde-744	218	13	1}eu(0	1}eu(0	NUM
ejde-744	218	14	)	)	PUNCT
ejde-744	218	15	+	+	CCONJ
ejde-744	218	16	2	2	NUM
ejde-744	218	17	t	t	NOUN
ejde-744	218	18	cθ	cθ	ADP
ejde-744	218	19	(	(	PUNCT
ejde-744	218	20	1	1	NUM
ejde-744	218	21	+	+	NOUN
ejde-744	218	22	ka	ka	PROPN
ejde-744	218	23	+	+	X
ejde-744	218	24	ca	can	AUX
ejde-744	218	25	,	,	PUNCT
ejde-744	218	26	b|λ|(1	b|λ|(1	PROPN
ejde-744	218	27	+	+	ADJ
ejde-744	218	28	kb))eu(0	kb))eu(0	NOUN
ejde-744	218	29	)	)	PUNCT
ejde-744	218	30	.	.	PUNCT
ejde-744	219	1	(	(	PUNCT
ejde-744	219	2	4.5	4.5	X
ejde-744	219	3	)	)	PUNCT
ejde-744	219	4	proof	proof	NOUN
ejde-744	219	5	.	.	PUNCT
ejde-744	220	1	suppose	suppose	VERB
ejde-744	220	2	first	first	ADV
ejde-744	220	3	that	that	SCONJ
ejde-744	220	4	(	(	PUNCT
ejde-744	220	5	u0	u0	ADJ
ejde-744	220	6	,	,	PUNCT
ejde-744	220	7	u1	u1	NOUN
ejde-744	220	8	)	)	PUNCT
ejde-744	220	9	∈	∈	PROPN
ejde-744	220	10	h2	h2	PROPN
ejde-744	220	11	a	a	PRON
ejde-744	220	12	,	,	PUNCT
ejde-744	220	13	b(0	b(0	NOUN
ejde-744	220	14	,	,	PUNCT
ejde-744	220	15	1	1	NUM
ejde-744	220	16	)	)	PUNCT
ejde-744	220	17	×h1	×h1	PROPN
ejde-744	220	18	a	a	PRON
ejde-744	220	19	,	,	PUNCT
ejde-744	220	20	b(0	b(0	NOUN
ejde-744	220	21	,	,	PUNCT
ejde-744	220	22	1	1	NUM
ejde-744	220	23	)	)	PUNCT
ejde-744	220	24	,	,	PUNCT
ejde-744	220	25	so	so	SCONJ
ejde-744	220	26	that	that	SCONJ
ejde-744	220	27	u	u	NOUN
ejde-744	220	28	is	be	AUX
ejde-744	220	29	a	a	DET
ejde-744	220	30	classical	classical	ADJ
ejde-744	220	31	solution	solution	NOUN
ejde-744	220	32	of	of	ADP
ejde-744	220	33	(	(	PUNCT
ejde-744	220	34	3.1	3.1	NUM
ejde-744	220	35	)	)	PUNCT
ejde-744	220	36	.	.	PUNCT
ejde-744	221	1	then	then	ADV
ejde-744	221	2	,	,	PUNCT
ejde-744	221	3	multiplying	multiply	VERB
ejde-744	221	4	(	(	PUNCT
ejde-744	221	5	3.1	3.1	NUM
ejde-744	221	6	)	)	PUNCT
ejde-744	221	7	by	by	ADP
ejde-744	221	8	xux	xux	PROPN
ejde-744	221	9	and	and	CCONJ
ejde-744	221	10	integrating	integrate	VERB
ejde-744	221	11	over	over	ADP
ejde-744	221	12	q	q	NOUN
ejde-744	221	13	,	,	PUNCT
ejde-744	221	14	we	we	PRON
ejde-744	221	15	obtain	obtain	VERB
ejde-744	221	16	0	0	NUM
ejde-744	222	1	=	=	SYM
ejde-744	222	2	∫	∫	PROPN
ejde-744	222	3	q	q	X
ejde-744	222	4	xux	xux	X
ejde-744	222	5	(	(	PUNCT
ejde-744	222	6	utt	utt	PROPN
ejde-744	222	7	−	−	PROPN
ejde-744	222	8	(	(	PUNCT
ejde-744	222	9	aux)x	aux)x	PROPN
ejde-744	222	10	−	−	PROPN
ejde-744	222	11	λ	λ	X
ejde-744	222	12	b	b	PROPN
ejde-744	222	13	u	u	PROPN
ejde-744	222	14	)	)	PUNCT
ejde-744	222	15	dx	dx	PROPN
ejde-744	222	16	dt	dt	NOUN
ejde-744	223	1	=	=	PUNCT
ejde-744	224	1	[	[	X
ejde-744	224	2	∫	∫	PROPN
ejde-744	224	3	1	1	NUM
ejde-744	224	4	0	0	NUM
ejde-744	224	5	xuxutdx	xuxutdx	NOUN
ejde-744	224	6	]	]	X
ejde-744	224	7	t	t	NOUN
ejde-744	224	8	0	0	NUM
ejde-744	225	1	−	−	PROPN
ejde-744	226	1	∫	∫	PROPN
ejde-744	227	1	q	q	PROPN
ejde-744	228	1	xuxtut	xuxtut	PROPN
ejde-744	228	2	dx	dx	PROPN
ejde-744	229	1	dt	dt	X
ejde-744	230	1	−	−	PROPN
ejde-744	230	2	∫	∫	PROPN
ejde-744	230	3	q	q	PROPN
ejde-744	231	1	(	(	PUNCT
ejde-744	231	2	xa′u2	xa′u2	PROPN
ejde-744	231	3	x	x	PROPN
ejde-744	231	4	+	+	NUM
ejde-744	231	5	xauxuxx	xauxuxx	PROPN
ejde-744	231	6	+	+	X
ejde-744	231	7	λ	λ	PROPN
ejde-744	231	8	b	b	PROPN
ejde-744	231	9	xuux	xuux	PROPN
ejde-744	231	10	)	)	PUNCT
ejde-744	231	11	dx	dx	PROPN
ejde-744	232	1	dt	dt	NOUN
ejde-744	233	1	=	=	PUNCT
ejde-744	234	1	[	[	X
ejde-744	234	2	∫	∫	PROPN
ejde-744	234	3	1	1	NUM
ejde-744	234	4	0	0	NUM
ejde-744	234	5	xuxutdx	xuxutdx	NOUN
ejde-744	234	6	]	]	X
ejde-744	234	7	t	t	NOUN
ejde-744	234	8	0	0	NUM
ejde-744	235	1	−	−	PROPN
ejde-744	235	2	∫	∫	PROPN
ejde-744	235	3	q	q	PROPN
ejde-744	235	4	xa′u2	xa′u2	PROPN
ejde-744	235	5	x	x	SYM
ejde-744	235	6	dx	dx	PROPN
ejde-744	236	1	dt	dt	X
ejde-744	236	2	−	−	PROPN
ejde-744	237	1	∫	∫	PROPN
ejde-744	237	2	q	q	PROPN
ejde-744	238	1	(	(	PUNCT
ejde-744	238	2	x	x	X
ejde-744	238	3	(	(	PUNCT
ejde-744	238	4	u2	u2	PROPN
ejde-744	238	5	t	t	PROPN
ejde-744	238	6	2	2	NUM
ejde-744	238	7	)	)	PUNCT
ejde-744	238	8	x	x	PUNCT
ejde-744	239	1	+	+	NUM
ejde-744	239	2	xa	xa	PROPN
ejde-744	239	3	(	(	PUNCT
ejde-744	239	4	u2	u2	NOUN
ejde-744	239	5	x	x	NOUN
ejde-744	239	6	2	2	NUM
ejde-744	239	7	)	)	PUNCT
ejde-744	239	8	x	x	PUNCT
ejde-744	240	1	+	+	PUNCT
ejde-744	240	2	λ	λ	X
ejde-744	240	3	2	2	NUM
ejde-744	240	4	x(u2)x	x(u2)x	PROPN
ejde-744	240	5	b	b	PROPN
ejde-744	240	6	)	)	PUNCT
ejde-744	240	7	dx	dx	PROPN
ejde-744	241	1	dt	dt	PROPN
ejde-744	241	2	.	.	PUNCT
ejde-744	242	1	(	(	PUNCT
ejde-744	242	2	4.6	4.6	NUM
ejde-744	242	3	)	)	PUNCT
ejde-744	242	4	arguing	argue	VERB
ejde-744	242	5	as	as	ADP
ejde-744	242	6	in	in	ADP
ejde-744	242	7	the	the	DET
ejde-744	242	8	proof	proof	NOUN
ejde-744	242	9	of	of	ADP
ejde-744	242	10	alabau	alabau	NOUN
ejde-744	242	11	-	-	PUNCT
ejde-744	242	12	boussouira	boussouira	NOUN
ejde-744	242	13	et	et	PROPN
ejde-744	242	14	al	al	PROPN
ejde-744	242	15	.	.	PUNCT
ejde-744	243	1	[	[	X
ejde-744	243	2	1	1	NUM
ejde-744	243	3	,	,	PUNCT
ejde-744	243	4	lemma	lemma	PROPN
ejde-744	243	5	3.2	3.2	NUM
ejde-744	243	6	]	]	PUNCT
ejde-744	243	7	,	,	PUNCT
ejde-744	243	8	[	[	PUNCT
ejde-744	243	9	x	x	X
ejde-744	243	10	u2	u2	PROPN
ejde-744	243	11	t	t	PROPN
ejde-744	243	12	2	2	NUM
ejde-744	243	13	]	]	SYM
ejde-744	243	14	1	1	NUM
ejde-744	243	15	0	0	NUM
ejde-744	243	16	=	=	SYM
ejde-744	243	17	0	0	NUM
ejde-744	243	18	,	,	PUNCT
ejde-744	243	19	[	[	X
ejde-744	243	20	xau2	xau2	NOUN
ejde-744	243	21	x	x	X
ejde-744	243	22	]	]	X
ejde-744	243	23	1	1	NUM
ejde-744	243	24	0	0	NUM
ejde-744	243	25	=	=	NOUN
ejde-744	243	26	a(1)u2	a(1)u2	NOUN
ejde-744	243	27	x(t	x(t	PROPN
ejde-744	243	28	,	,	PUNCT
ejde-744	243	29	1	1	NUM
ejde-744	243	30	)	)	PUNCT
ejde-744	243	31	.	.	PUNCT
ejde-744	244	1	from	from	ADP
ejde-744	244	2	the	the	DET
ejde-744	244	3	boundary	boundary	ADJ
ejde-744	244	4	conditions	condition	NOUN
ejde-744	244	5	and	and	CCONJ
ejde-744	244	6	lemma	lemma	PROPN
ejde-744	244	7	2.8	2.8	NUM
ejde-744	244	8	,	,	PUNCT
ejde-744	244	9	we	we	PRON
ejde-744	244	10	have[xu2	have[xu2	VERB
ejde-744	245	1	b	b	X
ejde-744	245	2	]	]	SYM
ejde-744	245	3	1	1	NUM
ejde-744	245	4	0	0	NUM
ejde-744	245	5	=	=	SYM
ejde-744	245	6	0	0	NUM
ejde-744	245	7	.	.	PUNCT
ejde-744	246	1	hence	hence	ADV
ejde-744	246	2	,	,	PUNCT
ejde-744	246	3	∫	∫	PROPN
ejde-744	246	4	q	q	X
ejde-744	247	1	x	x	X
ejde-744	247	2	(	(	PUNCT
ejde-744	247	3	u2	u2	PROPN
ejde-744	247	4	t	t	PROPN
ejde-744	247	5	2	2	NUM
ejde-744	247	6	)	)	PUNCT
ejde-744	247	7	x	x	X
ejde-744	247	8	dx	dx	PROPN
ejde-744	247	9	dt	dt	NOUN
ejde-744	247	10	=	=	SYM
ejde-744	247	11	−1	−1	NOUN
ejde-744	247	12	2	2	NUM
ejde-744	247	13	∫	∫	PROPN
ejde-744	247	14	q	q	PROPN
ejde-744	247	15	u2	u2	PROPN
ejde-744	247	16	t	t	PROPN
ejde-744	247	17	dx	dx	PROPN
ejde-744	247	18	dt	dt	X
ejde-744	247	19	,	,	PUNCT
ejde-744	247	20	(	(	PUNCT
ejde-744	247	21	4.7)∫	4.7)∫	NUM
ejde-744	247	22	q	q	X
ejde-744	247	23	xa	xa	PROPN
ejde-744	247	24	(	(	PUNCT
ejde-744	247	25	u2	u2	NOUN
ejde-744	247	26	x	x	NOUN
ejde-744	247	27	2	2	NUM
ejde-744	247	28	)	)	PUNCT
ejde-744	247	29	x	x	SYM
ejde-744	247	30	dx	dx	PROPN
ejde-744	247	31	dt	dt	NOUN
ejde-744	247	32	=	=	SYM
ejde-744	247	33	−1	−1	NOUN
ejde-744	247	34	2	2	NUM
ejde-744	247	35	∫	∫	NOUN
ejde-744	247	36	q	q	PROPN
ejde-744	247	37	(	(	PUNCT
ejde-744	247	38	a+	a+	SYM
ejde-744	247	39	xa′)u2	xa′)u2	NUM
ejde-744	247	40	x	x	SYM
ejde-744	247	41	dx	dx	PROPN
ejde-744	247	42	dt+	dt+	NOUN
ejde-744	247	43	1	1	NUM
ejde-744	247	44	2	2	NUM
ejde-744	247	45	a(1	a(1	NUM
ejde-744	247	46	)	)	PUNCT
ejde-744	247	47	∫	∫	PROPN
ejde-744	247	48	t	t	NOUN
ejde-744	247	49	0	0	NUM
ejde-744	247	50	u2	u2	PROPN
ejde-744	247	51	x(t	x(t	PROPN
ejde-744	247	52	,	,	PUNCT
ejde-744	247	53	1)dt	1)dt	NUM
ejde-744	247	54	,	,	PUNCT
ejde-744	247	55	(	(	PUNCT
ejde-744	247	56	4.8	4.8	NUM
ejde-744	247	57	)	)	PUNCT
ejde-744	247	58	λ	λ	NOUN
ejde-744	247	59	2	2	NUM
ejde-744	247	60	∫	∫	NOUN
ejde-744	247	61	q	q	PROPN
ejde-744	247	62	x(u2)x	x(u2)x	PROPN
ejde-744	247	63	b	b	PROPN
ejde-744	247	64	dx	dx	PROPN
ejde-744	247	65	dt	dt	PROPN
ejde-744	248	1	=	=	PUNCT
ejde-744	248	2	−λ	−λ	PROPN
ejde-744	248	3	2	2	NUM
ejde-744	248	4	∫	∫	NOUN
ejde-744	248	5	q	q	PROPN
ejde-744	248	6	b−	b−	PROPN
ejde-744	248	7	xb′	xb′	PROPN
ejde-744	248	8	b2	b2	PROPN
ejde-744	248	9	u2	u2	PROPN
ejde-744	248	10	dx	dx	PROPN
ejde-744	248	11	dt	dt	X
ejde-744	248	12	.	.	PUNCT
ejde-744	249	1	(	(	PUNCT
ejde-744	249	2	4.9	4.9	NUM
ejde-744	249	3	)	)	PUNCT
ejde-744	249	4	then	then	ADV
ejde-744	249	5	(	(	PUNCT
ejde-744	249	6	4.4	4.4	NUM
ejde-744	249	7	)	)	PUNCT
ejde-744	249	8	follows	follow	VERB
ejde-744	249	9	by	by	ADP
ejde-744	249	10	inserting	insert	VERB
ejde-744	249	11	(	(	PUNCT
ejde-744	249	12	4.7)–(4.9	4.7)–(4.9	NUM
ejde-744	249	13	)	)	PUNCT
ejde-744	249	14	into	into	ADP
ejde-744	249	15	(	(	PUNCT
ejde-744	249	16	4.6	4.6	NUM
ejde-744	249	17	)	)	PUNCT
ejde-744	249	18	.	.	PUNCT
ejde-744	250	1	next	next	ADV
ejde-744	250	2	,	,	PUNCT
ejde-744	250	3	we	we	PRON
ejde-744	250	4	estimate	estimate	VERB
ejde-744	250	5	the	the	DET
ejde-744	250	6	term	term	NOUN
ejde-744	250	7	on	on	ADP
ejde-744	250	8	the	the	DET
ejde-744	250	9	right	right	ADJ
ejde-744	250	10	side	side	NOUN
ejde-744	250	11	of	of	ADP
ejde-744	250	12	equation	equation	NOUN
ejde-744	250	13	(	(	PUNCT
ejde-744	250	14	4.4	4.4	NUM
ejde-744	250	15	)	)	PUNCT
ejde-744	250	16	separately	separately	ADV
ejde-744	250	17	.	.	PUNCT
ejde-744	251	1	according	accord	VERB
ejde-744	251	2	the	the	DET
ejde-744	251	3	hölder	hölder	NOUN
ejde-744	251	4	inequality	inequality	NOUN
ejde-744	251	5	and	and	CCONJ
ejde-744	251	6	lemma	lemma	PROPN
ejde-744	251	7	2.5	2.5	NUM
ejde-744	251	8	,	,	PUNCT
ejde-744	251	9	we	we	PRON
ejde-744	251	10	have	have	VERB
ejde-744	251	11	2	2	NUM
ejde-744	251	12	∫	∫	NOUN
ejde-744	251	13	1	1	NUM
ejde-744	251	14	0	0	NUM
ejde-744	251	15	xuxutdx	xuxutdx	PROPN
ejde-744	251	16	≤	≤	NUM
ejde-744	251	17	∫	∫	PROPN
ejde-744	251	18	1	1	NUM
ejde-744	251	19	0	0	NUM
ejde-744	251	20	(	(	PUNCT
ejde-744	251	21	x2u2	x2u2	PUNCT
ejde-744	251	22	x	x	SYM
ejde-744	252	1	+	+	CCONJ
ejde-744	252	2	u2	u2	PROPN
ejde-744	252	3	t	t	PROPN
ejde-744	252	4	)	)	PUNCT
ejde-744	252	5	dx	dx	PROPN
ejde-744	252	6	≤	≤	NUM
ejde-744	252	7	1	1	NUM
ejde-744	253	1	a(1)cθ	a(1)cθ	NOUN
ejde-744	253	2	∫	∫	PROPN
ejde-744	253	3	1	1	NUM
ejde-744	253	4	0	0	NUM
ejde-744	254	1	(	(	PUNCT
ejde-744	254	2	au2	au2	NOUN
ejde-744	254	3	x	x	X
ejde-744	254	4	−	−	PUNCT
ejde-744	254	5	λ	λ	X
ejde-744	254	6	b	b	PROPN
ejde-744	254	7	u2	u2	PROPN
ejde-744	254	8	)	)	PUNCT
ejde-744	254	9	dx+	dx+	NOUN
ejde-744	254	10	∫	∫	PROPN
ejde-744	255	1	1	1	NUM
ejde-744	255	2	0	0	NUM
ejde-744	255	3	u2	u2	PROPN
ejde-744	255	4	t	t	PROPN
ejde-744	255	5	dx	dx	PROPN
ejde-744	255	6	≤	≤	PROPN
ejde-744	255	7	2max	2max	NUM
ejde-744	255	8	{	{	PUNCT
ejde-744	255	9	1	1	NUM
ejde-744	255	10	a(1)cθ	a(1)cθ	NOUN
ejde-744	255	11	,	,	PUNCT
ejde-744	255	12	1	1	X
ejde-744	255	13	}	}	PUNCT
ejde-744	255	14	eu(0	eu(0	PROPN
ejde-744	255	15	)	)	PUNCT
ejde-744	255	16	.	.	PUNCT
ejde-744	256	1	(	(	PUNCT
ejde-744	256	2	4.10	4.10	NUM
ejde-744	256	3	)	)	PUNCT
ejde-744	256	4	10	10	NUM
ejde-744	256	5	g.	g.	PROPN
ejde-744	256	6	zhang	zhang	PROPN
ejde-744	256	7	,	,	PUNCT
ejde-744	256	8	s.	s.	PROPN
ejde-744	256	9	chai	chai	PROPN
ejde-744	256	10	ejde-2025/04	ejde-2025/04	PROPN
ejde-744	256	11	moreover	moreover	ADV
ejde-744	256	12	,	,	PUNCT
ejde-744	256	13	using	use	VERB
ejde-744	256	14	the	the	DET
ejde-744	256	15	definition	definition	NOUN
ejde-744	256	16	of	of	ADP
ejde-744	256	17	kg	kg	NOUN
ejde-744	256	18	and	and	CCONJ
ejde-744	256	19	hardy	hardy	ADJ
ejde-744	256	20	’s	’s	NOUN
ejde-744	256	21	inequality	inequality	NOUN
ejde-744	256	22	(	(	PUNCT
ejde-744	256	23	2.3	2.3	NUM
ejde-744	256	24	)	)	PUNCT
ejde-744	256	25	,	,	PUNCT
ejde-744	256	26	we	we	PRON
ejde-744	256	27	have∫	have∫	VERB
ejde-744	256	28	1	1	NUM
ejde-744	256	29	0	0	NUM
ejde-744	256	30	(	(	PUNCT
ejde-744	256	31	a+	a+	PUNCT
ejde-744	256	32	xa′)u2	xa′)u2	NUM
ejde-744	256	33	xdx	xdx	PROPN
ejde-744	256	34	≤	≤	PROPN
ejde-744	256	35	(	(	PUNCT
ejde-744	256	36	1	1	NUM
ejde-744	256	37	+	+	NOUN
ejde-744	256	38	ka	ka	PROPN
ejde-744	256	39	)	)	PUNCT
ejde-744	256	40	∫	∫	PROPN
ejde-744	256	41	1	1	NUM
ejde-744	256	42	0	0	NUM
ejde-744	256	43	au2	au2	NOUN
ejde-744	256	44	x	x	X
ejde-744	256	45	dx	dx	PROPN
ejde-744	256	46	≤	≤	NUM
ejde-744	256	47	(	(	PUNCT
ejde-744	256	48	1	1	NUM
ejde-744	256	49	+	+	PROPN
ejde-744	256	50	ka	ka	NOUN
ejde-744	256	51	)	)	PUNCT
ejde-744	256	52	cθ	cθ	ADP
ejde-744	256	53	eu(0	eu(0	PROPN
ejde-744	256	54	)	)	PUNCT
ejde-744	256	55	,	,	PUNCT
ejde-744	256	56	(	(	PUNCT
ejde-744	256	57	4.11	4.11	NUM
ejde-744	256	58	)	)	PUNCT
ejde-744	256	59	and	and	CCONJ
ejde-744	256	60	λ	λ	X
ejde-744	256	61	∫	∫	PROPN
ejde-744	257	1	1	1	NUM
ejde-744	257	2	0	0	NUM
ejde-744	257	3	b−	b−	PROPN
ejde-744	257	4	xb′	xb′	PROPN
ejde-744	257	5	b2	b2	PROPN
ejde-744	257	6	u2	u2	PROPN
ejde-744	257	7	dx	dx	PROPN
ejde-744	257	8	≤	≤	PROPN
ejde-744	257	9	∫	∫	PROPN
ejde-744	258	1	1	1	NUM
ejde-744	258	2	0	0	NUM
ejde-744	258	3	λ	λ	X
ejde-744	258	4	b	b	PROPN
ejde-744	258	5	(	(	PUNCT
ejde-744	258	6	1−	1−	NUM
ejde-744	258	7	xb′	xb′	SYM
ejde-744	258	8	b	b	X
ejde-744	258	9	)	)	PUNCT
ejde-744	258	10	u2dx	u2dx	PROPN
ejde-744	258	11	≤	≤	NUM
ejde-744	258	12	∫	∫	PROPN
ejde-744	258	13	1	1	NUM
ejde-744	258	14	0	0	NUM
ejde-744	258	15	|λ|	|λ|	PROPN
ejde-744	258	16	b	b	PROPN
ejde-744	258	17	(	(	PUNCT
ejde-744	258	18	1	1	NUM
ejde-744	258	19	+	+	NOUN
ejde-744	258	20	kb)u	kb)u	PROPN
ejde-744	258	21	2dx	2dx	ADJ
ejde-744	258	22	≤	≤	PROPN
ejde-744	258	23	2ca	2ca	NOUN
ejde-744	258	24	,	,	PUNCT
ejde-744	258	25	b|λ|(1	b|λ|(1	PROPN
ejde-744	258	26	+	+	PROPN
ejde-744	258	27	kb	kb	PROPN
ejde-744	258	28	)	)	PUNCT
ejde-744	258	29	cθ	cθ	ADP
ejde-744	258	30	eu(0	eu(0	PROPN
ejde-744	258	31	)	)	PUNCT
ejde-744	258	32	.	.	PUNCT
ejde-744	259	1	(	(	PUNCT
ejde-744	259	2	4.12	4.12	NUM
ejde-744	259	3	)	)	PUNCT
ejde-744	259	4	hence	hence	ADV
ejde-744	259	5	,	,	PUNCT
ejde-744	259	6	by	by	ADP
ejde-744	259	7	(	(	PUNCT
ejde-744	259	8	4.4	4.4	NUM
ejde-744	259	9	)	)	PUNCT
ejde-744	259	10	and	and	CCONJ
ejde-744	259	11	the	the	DET
ejde-744	259	12	inequalities	inequality	NOUN
ejde-744	259	13	(	(	PUNCT
ejde-744	259	14	4.10)–(4.12	4.10)–(4.12	NOUN
ejde-744	259	15	)	)	PUNCT
ejde-744	259	16	,	,	PUNCT
ejde-744	259	17	we	we	PRON
ejde-744	259	18	obtain	obtain	VERB
ejde-744	259	19	(	(	PUNCT
ejde-744	259	20	4.5	4.5	NUM
ejde-744	259	21	)	)	PUNCT
ejde-744	259	22	.	.	PUNCT
ejde-744	260	1	as	as	ADP
ejde-744	260	2	before	before	ADV
ejde-744	260	3	,	,	PUNCT
ejde-744	260	4	to	to	PART
ejde-744	260	5	extend	extend	VERB
ejde-744	260	6	(	(	PUNCT
ejde-744	260	7	4.5	4.5	NUM
ejde-744	260	8	)	)	PUNCT
ejde-744	260	9	to	to	ADP
ejde-744	260	10	the	the	DET
ejde-744	260	11	mild	mild	ADJ
ejde-744	260	12	solution	solution	NOUN
ejde-744	260	13	associated	associate	VERB
ejde-744	260	14	with	with	ADP
ejde-744	260	15	the	the	DET
ejde-744	260	16	initial	initial	ADJ
ejde-744	260	17	data	datum	NOUN
ejde-744	260	18	(	(	PUNCT
ejde-744	260	19	u0	u0	ADJ
ejde-744	260	20	,	,	PUNCT
ejde-744	260	21	u1	u1	NOUN
ejde-744	260	22	)	)	PUNCT
ejde-744	260	23	∈	∈	PROPN
ejde-744	260	24	h1	h1	VERB
ejde-744	260	25	a	a	DET
ejde-744	260	26	,	,	PUNCT
ejde-744	260	27	b(0	b(0	NOUN
ejde-744	260	28	,	,	PUNCT
ejde-744	260	29	1)×l2(0	1)×l2(0	NUM
ejde-744	260	30	,	,	PUNCT
ejde-744	260	31	1	1	NUM
ejde-744	260	32	)	)	PUNCT
ejde-744	260	33	,	,	PUNCT
ejde-744	260	34	it	it	PRON
ejde-744	260	35	suffices	suffice	VERB
ejde-744	260	36	to	to	PART
ejde-744	260	37	approximate	approximate	VERB
ejde-744	260	38	such	such	ADJ
ejde-744	260	39	data	datum	NOUN
ejde-744	260	40	by	by	ADP
ejde-744	260	41	(	(	PUNCT
ejde-744	260	42	un	un	PROPN
ejde-744	260	43	0	0	PROPN
ejde-744	260	44	,	,	PUNCT
ejde-744	260	45	u	u	NOUN
ejde-744	260	46	n	n	ADV
ejde-744	260	47	1	1	NUM
ejde-744	260	48	)	)	PUNCT
ejde-744	260	49	∈	∈	PROPN
ejde-744	260	50	h2	h2	PROPN
ejde-744	260	51	a	a	PRON
ejde-744	260	52	,	,	PUNCT
ejde-744	260	53	b(0	b(0	NOUN
ejde-744	260	54	,	,	PUNCT
ejde-744	260	55	1)×	1)×	NUM
ejde-744	260	56	h1	h1	VERB
ejde-744	260	57	a	a	PRON
ejde-744	260	58	,	,	PUNCT
ejde-744	260	59	b(0	b(0	NOUN
ejde-744	260	60	,	,	PUNCT
ejde-744	260	61	1	1	NUM
ejde-744	260	62	)	)	PUNCT
ejde-744	260	63	,	,	PUNCT
ejde-744	260	64	and	and	CCONJ
ejde-744	260	65	thanks	thank	NOUN
ejde-744	260	66	to	to	ADP
ejde-744	260	67	(	(	PUNCT
ejde-744	260	68	4.5	4.5	NUM
ejde-744	260	69	)	)	PUNCT
ejde-744	260	70	,	,	PUNCT
ejde-744	260	71	we	we	PRON
ejde-744	260	72	can	can	AUX
ejde-744	260	73	show	show	VERB
ejde-744	260	74	that	that	SCONJ
ejde-744	260	75	the	the	DET
ejde-744	260	76	normal	normal	ADJ
ejde-744	260	77	derivatives	derivative	NOUN
ejde-744	260	78	of	of	ADP
ejde-744	260	79	the	the	DET
ejde-744	260	80	corresponding	corresponding	ADJ
ejde-744	260	81	classical	classical	ADJ
ejde-744	260	82	solutions	solution	NOUN
ejde-744	260	83	give	give	VERB
ejde-744	260	84	a	a	DET
ejde-744	260	85	cauchy	cauchy	ADJ
ejde-744	260	86	sequence	sequence	NOUN
ejde-744	260	87	in	in	ADP
ejde-744	260	88	l2(0	l2(0	NOUN
ejde-744	260	89	,	,	PUNCT
ejde-744	260	90	1	1	NUM
ejde-744	260	91	)	)	PUNCT
ejde-744	260	92	.	.	PUNCT
ejde-744	261	1	□	□	PUNCT
ejde-744	261	2	5	5	X
ejde-744	261	3	.	.	PUNCT
ejde-744	261	4	boundary	boundary	ADJ
ejde-744	261	5	observability	observability	NOUN
ejde-744	261	6	lemma	lemma	PROPN
ejde-744	261	7	5.1	5.1	NUM
ejde-744	261	8	.	.	PUNCT
ejde-744	262	1	under	under	ADP
ejde-744	262	2	assumption	assumption	NOUN
ejde-744	262	3	2.6	2.6	NUM
ejde-744	262	4	,	,	PUNCT
ejde-744	262	5	for	for	ADP
ejde-744	262	6	any	any	DET
ejde-744	262	7	mild	mild	ADJ
ejde-744	262	8	solution	solution	NOUN
ejde-744	262	9	u	u	NOUN
ejde-744	262	10	of	of	ADP
ejde-744	262	11	(	(	PUNCT
ejde-744	262	12	3.1	3.1	NUM
ejde-744	262	13	)	)	PUNCT
ejde-744	262	14	and	and	CCONJ
ejde-744	262	15	every	every	DET
ejde-744	262	16	t	t	NOUN
ejde-744	262	17	≥	≥	NOUN
ejde-744	262	18	0	0	NUM
ejde-744	262	19	,	,	PUNCT
ejde-744	262	20	we	we	PRON
ejde-744	262	21	have∫	have∫	VERB
ejde-744	262	22	q	q	NOUN
ejde-744	262	23	(	(	PUNCT
ejde-744	262	24	a(x)u2	a(x)u2	NOUN
ejde-744	262	25	x	x	PUNCT
ejde-744	262	26	−	−	PROPN
ejde-744	262	27	u2	u2	PROPN
ejde-744	262	28	t	t	PROPN
ejde-744	262	29	−	−	PROPN
ejde-744	263	1	λ	λ	NOUN
ejde-744	263	2	b(x	b(x	NOUN
ejde-744	263	3	)	)	PUNCT
ejde-744	263	4	u2	u2	NOUN
ejde-744	263	5	)	)	PUNCT
ejde-744	263	6	dx	dx	PROPN
ejde-744	263	7	dt+	dt+	NOUN
ejde-744	263	8	[	[	X
ejde-744	263	9	∫	∫	PROPN
ejde-744	263	10	1	1	NUM
ejde-744	263	11	0	0	NUM
ejde-744	263	12	uut	uut	PROPN
ejde-744	263	13	dx	dx	PROPN
ejde-744	264	1	]	]	X
ejde-744	264	2	t	t	X
ejde-744	264	3	0	0	NUM
ejde-744	265	1	=	=	SYM
ejde-744	265	2	0	0	PROPN
ejde-744	265	3	.	.	PUNCT
ejde-744	266	1	(	(	PUNCT
ejde-744	266	2	5.1	5.1	NUM
ejde-744	266	3	)	)	PUNCT
ejde-744	266	4	proof	proof	NOUN
ejde-744	266	5	.	.	PUNCT
ejde-744	267	1	as	as	ADP
ejde-744	267	2	before	before	ADV
ejde-744	267	3	,	,	PUNCT
ejde-744	267	4	suppose	suppose	VERB
ejde-744	267	5	that	that	SCONJ
ejde-744	267	6	u	u	PROPN
ejde-744	267	7	is	be	AUX
ejde-744	267	8	the	the	DET
ejde-744	267	9	classical	classical	ADJ
ejde-744	267	10	solution	solution	NOUN
ejde-744	267	11	of	of	ADP
ejde-744	267	12	(	(	PUNCT
ejde-744	267	13	3.1	3.1	NUM
ejde-744	267	14	)	)	PUNCT
ejde-744	267	15	.	.	PUNCT
ejde-744	268	1	multiplying	multiply	VERB
ejde-744	268	2	(	(	PUNCT
ejde-744	268	3	3.1	3.1	NUM
ejde-744	268	4	)	)	PUNCT
ejde-744	268	5	by	by	ADP
ejde-744	268	6	u	u	NOUN
ejde-744	268	7	and	and	CCONJ
ejde-744	268	8	integrating	integrate	VERB
ejde-744	268	9	over	over	ADP
ejde-744	268	10	the	the	DET
ejde-744	268	11	domain	domain	NOUN
ejde-744	268	12	q	q	NOUN
ejde-744	269	1	=	=	SYM
ejde-744	269	2	(	(	PUNCT
ejde-744	269	3	0	0	NUM
ejde-744	269	4	,	,	PUNCT
ejde-744	269	5	t	t	NOUN
ejde-744	269	6	)	)	PUNCT
ejde-744	269	7	×	×	NOUN
ejde-744	269	8	(	(	PUNCT
ejde-744	269	9	0	0	NUM
ejde-744	269	10	,	,	PUNCT
ejde-744	269	11	1	1	NUM
ejde-744	269	12	)	)	PUNCT
ejde-744	269	13	,	,	PUNCT
ejde-744	269	14	we	we	PRON
ejde-744	269	15	obtain	obtain	VERB
ejde-744	269	16	0	0	NUM
ejde-744	270	1	=	=	SYM
ejde-744	270	2	∫	∫	PROPN
ejde-744	270	3	1	1	NUM
ejde-744	270	4	0	0	NUM
ejde-744	270	5	u	u	NOUN
ejde-744	270	6	(	(	PUNCT
ejde-744	270	7	utt	utt	NOUN
ejde-744	270	8	−	−	PROPN
ejde-744	270	9	(	(	PUNCT
ejde-744	270	10	a(x)ux)x	a(x)ux)x	VERB
ejde-744	270	11	−	−	PROPN
ejde-744	270	12	λ	λ	NOUN
ejde-744	270	13	b(x	b(x	NOUN
ejde-744	270	14	)	)	PUNCT
ejde-744	270	15	u	u	NOUN
ejde-744	270	16	)	)	PUNCT
ejde-744	270	17	dx	dx	PROPN
ejde-744	270	18	=	=	SYM
ejde-744	270	19	∫	∫	PROPN
ejde-744	270	20	q	q	PROPN
ejde-744	270	21	(	(	PUNCT
ejde-744	270	22	a(x)u2	a(x)u2	NOUN
ejde-744	270	23	x	x	PUNCT
ejde-744	270	24	−	−	PROPN
ejde-744	270	25	u2	u2	PROPN
ejde-744	270	26	t	t	PROPN
ejde-744	270	27	−	−	PROPN
ejde-744	270	28	λ	λ	NOUN
ejde-744	270	29	b(x	b(x	NOUN
ejde-744	270	30	)	)	PUNCT
ejde-744	270	31	u2	u2	NOUN
ejde-744	270	32	)	)	PUNCT
ejde-744	270	33	dx	dx	PROPN
ejde-744	270	34	dt+	dt+	NOUN
ejde-744	270	35	[	[	X
ejde-744	270	36	∫	∫	PROPN
ejde-744	270	37	1	1	NUM
ejde-744	270	38	0	0	NUM
ejde-744	270	39	uut	uut	PROPN
ejde-744	270	40	dx	dx	PROPN
ejde-744	270	41	]	]	X
ejde-744	270	42	t	t	PROPN
ejde-744	270	43	0	0	NUM
ejde-744	271	1	−	−	NUM
ejde-744	271	2	∫	∫	PROPN
ejde-744	271	3	t	t	NOUN
ejde-744	271	4	0	0	NUM
ejde-744	272	1	[	[	X
ejde-744	272	2	a(x)uux	a(x)uux	ADJ
ejde-744	272	3	]	]	X
ejde-744	272	4	1	1	NUM
ejde-744	272	5	0	0	NUM
ejde-744	272	6	dt	dt	NOUN
ejde-744	272	7	.	.	PUNCT
ejde-744	273	1	(	(	PUNCT
ejde-744	273	2	5.2	5.2	NUM
ejde-744	273	3	)	)	PUNCT
ejde-744	273	4	thanks	thank	NOUN
ejde-744	273	5	to	to	ADP
ejde-744	273	6	the	the	DET
ejde-744	273	7	boundary	boundary	ADJ
ejde-744	273	8	conditions	condition	NOUN
ejde-744	273	9	and	and	CCONJ
ejde-744	273	10	alabau	alabau	NOUN
ejde-744	273	11	-	-	PUNCT
ejde-744	273	12	boussouira	boussouira	PROPN
ejde-744	273	13	et	et	PROPN
ejde-744	273	14	al	al	PROPN
ejde-744	273	15	.	.	PUNCT
ejde-744	274	1	[	[	X
ejde-744	274	2	12	12	NUM
ejde-744	274	3	,	,	PUNCT
ejde-744	274	4	proposition	proposition	NOUN
ejde-744	274	5	2.5	2.5	NUM
ejde-744	274	6	]	]	PUNCT
ejde-744	274	7	,	,	PUNCT
ejde-744	274	8	we	we	PRON
ejde-744	274	9	have	have	VERB
ejde-744	274	10	that	that	SCONJ
ejde-744	274	11	auux	auux	PROPN
ejde-744	274	12	also	also	ADV
ejde-744	274	13	vanishes	vanish	VERB
ejde-744	274	14	at	at	ADP
ejde-744	274	15	x	x	X
ejde-744	274	16	=	=	SYM
ejde-744	274	17	0	0	NUM
ejde-744	274	18	and	and	CCONJ
ejde-744	274	19	at	at	ADP
ejde-744	274	20	x	x	X
ejde-744	274	21	=	=	SYM
ejde-744	274	22	1	1	X
ejde-744	274	23	.	.	PUNCT
ejde-744	275	1	the	the	DET
ejde-744	275	2	conclusion	conclusion	NOUN
ejde-744	275	3	can	can	AUX
ejde-744	275	4	be	be	AUX
ejde-744	275	5	extended	extend	VERB
ejde-744	275	6	to	to	ADP
ejde-744	275	7	mild	mild	ADJ
ejde-744	275	8	solution	solution	NOUN
ejde-744	275	9	by	by	ADP
ejde-744	275	10	an	an	DET
ejde-744	275	11	approximation	approximation	NOUN
ejde-744	275	12	argument	argument	NOUN
ejde-744	275	13	.	.	PUNCT
ejde-744	276	1	□	□	PUNCT
ejde-744	276	2	theorem	theorem	VERB
ejde-744	276	3	5.2	5.2	NUM
ejde-744	276	4	.	.	PUNCT
ejde-744	277	1	under	under	ADP
ejde-744	277	2	assumption	assumption	NOUN
ejde-744	277	3	2.6	2.6	NUM
ejde-744	277	4	,	,	PUNCT
ejde-744	277	5	let	let	VERB
ejde-744	277	6	u	u	PRON
ejde-744	277	7	be	be	AUX
ejde-744	277	8	a	a	DET
ejde-744	277	9	mild	mild	ADJ
ejde-744	277	10	solution	solution	NOUN
ejde-744	277	11	of	of	ADP
ejde-744	277	12	(	(	PUNCT
ejde-744	277	13	3.1	3.1	NUM
ejde-744	277	14	)	)	PUNCT
ejde-744	277	15	.	.	PUNCT
ejde-744	278	1	then	then	ADV
ejde-744	278	2	,	,	PUNCT
ejde-744	278	3	for	for	ADP
ejde-744	278	4	every	every	DET
ejde-744	278	5	t	t	PROPN
ejde-744	278	6	≥	≥	NOUN
ejde-744	278	7	0	0	NUM
ejde-744	278	8	,	,	PUNCT
ejde-744	278	9	a(1	a(1	ADJ
ejde-744	278	10	)	)	PUNCT
ejde-744	278	11	∫	∫	PROPN
ejde-744	278	12	t	t	NOUN
ejde-744	278	13	0	0	NUM
ejde-744	278	14	u2	u2	PROPN
ejde-744	278	15	x(t	x(t	PROPN
ejde-744	278	16	,	,	PUNCT
ejde-744	278	17	1)dt	1)dt	PROPN
ejde-744	278	18	≥	≥	NOUN
ejde-744	278	19	−	−	PROPN
ejde-744	278	20	(	(	PUNCT
ejde-744	278	21	4max	4max	PROPN
ejde-744	278	22	{	{	PUNCT
ejde-744	278	23	1	1	NUM
ejde-744	278	24	a(1)θ	a(1)θ	INTJ
ejde-744	278	25	,	,	PUNCT
ejde-744	278	26	1}+	1}+	NUM
ejde-744	278	27	2ka	2ka	ADJ
ejde-744	278	28	1√	1√	PROPN
ejde-744	278	29	θa(1	θa(1	PROPN
ejde-744	278	30	)	)	PUNCT
ejde-744	278	31	)	)	PUNCT
ejde-744	278	32	eu(0	eu(0	PROPN
ejde-744	278	33	)	)	PUNCT
ejde-744	278	34	+	+	X
ejde-744	278	35	t{(2−ka	t{(2−ka	NUM
ejde-744	278	36	)	)	PUNCT
ejde-744	278	37	+	+	CCONJ
ejde-744	278	38	λca	λca	INTJ
ejde-744	278	39	,	,	PUNCT
ejde-744	278	40	bθ	bθ	PROPN
ejde-744	278	41	(	(	PUNCT
ejde-744	278	42	2−ka	2−ka	NUM
ejde-744	278	43	−kb)}eu(0	−kb)}eu(0	NOUN
ejde-744	278	44	)	)	PUNCT
ejde-744	278	45	,	,	PUNCT
ejde-744	278	46	(	(	PUNCT
ejde-744	278	47	5.3	5.3	NUM
ejde-744	278	48	)	)	PUNCT
ejde-744	278	49	for	for	ADP
ejde-744	278	50	λ	λ	PROPN
ejde-744	278	51	∈	∈	PROPN
ejde-744	278	52	(	(	PUNCT
ejde-744	278	53	0	0	NUM
ejde-744	278	54	,	,	PUNCT
ejde-744	278	55	1	1	NUM
ejde-744	278	56	ca	ca	NOUN
ejde-744	278	57	,	,	PUNCT
ejde-744	278	58	b	b	NOUN
ejde-744	278	59	)	)	PUNCT
ejde-744	278	60	,	,	PUNCT
ejde-744	278	61	and	and	CCONJ
ejde-744	278	62	a(1	a(1	ADJ
ejde-744	278	63	)	)	PUNCT
ejde-744	278	64	∫	∫	PROPN
ejde-744	278	65	t	t	NOUN
ejde-744	278	66	0	0	NUM
ejde-744	278	67	u2	u2	PROPN
ejde-744	278	68	x(t	x(t	PROPN
ejde-744	278	69	,	,	PUNCT
ejde-744	278	70	1)dt	1)dt	PROPN
ejde-744	278	71	≥	≥	NOUN
ejde-744	278	72	−	−	PROPN
ejde-744	278	73	(	(	PUNCT
ejde-744	278	74	4max	4max	PROPN
ejde-744	278	75	{	{	PUNCT
ejde-744	278	76	1	1	NUM
ejde-744	278	77	a(1	a(1	NUM
ejde-744	278	78	)	)	PUNCT
ejde-744	278	79	,	,	PUNCT
ejde-744	278	80	1}+	1}+	NUM
ejde-744	278	81	2ka	2ka	ADJ
ejde-744	278	82	1√	1√	ADJ
ejde-744	278	83	a(1	a(1	NOUN
ejde-744	278	84	)	)	PUNCT
ejde-744	278	85	)	)	PUNCT
ejde-744	278	86	eu(0	eu(0	PROPN
ejde-744	278	87	)	)	PUNCT
ejde-744	278	88	+	+	PUNCT
ejde-744	278	89	t{(2−ka)−	t{(2−ka)−	SYM
ejde-744	278	90	|λ|ca	|λ|ca	PROPN
ejde-744	278	91	,	,	PUNCT
ejde-744	278	92	b	b	PROPN
ejde-744	278	93	(	(	PUNCT
ejde-744	278	94	2−ka	2−ka	NUM
ejde-744	278	95	−kb)}eu(0	−kb)}eu(0	NOUN
ejde-744	278	96	)	)	PUNCT
ejde-744	278	97	,	,	PUNCT
ejde-744	278	98	(	(	PUNCT
ejde-744	278	99	5.4	5.4	NUM
ejde-744	278	100	)	)	PUNCT
ejde-744	278	101	for	for	ADP
ejde-744	278	102	λ	λ	PROPN
ejde-744	278	103	∈	∈	PROPN
ejde-744	278	104	(	(	PUNCT
ejde-744	278	105	−∞	−∞	NOUN
ejde-744	278	106	,	,	PUNCT
ejde-744	278	107	0	0	NUM
ejde-744	278	108	]	]	PUNCT
ejde-744	278	109	,	,	PUNCT
ejde-744	278	110	where	where	SCONJ
ejde-744	278	111	the	the	DET
ejde-744	278	112	constants	constant	NOUN
ejde-744	278	113	ca	can	AUX
ejde-744	278	114	,	,	PUNCT
ejde-744	278	115	b	b	PROPN
ejde-744	278	116	and	and	CCONJ
ejde-744	278	117	θ	θ	PROPN
ejde-744	278	118	are	be	AUX
ejde-744	278	119	given	give	VERB
ejde-744	278	120	in	in	ADP
ejde-744	278	121	(	(	PUNCT
ejde-744	278	122	2.3	2.3	NUM
ejde-744	278	123	)	)	PUNCT
ejde-744	278	124	and	and	CCONJ
ejde-744	278	125	(	(	PUNCT
ejde-744	278	126	2.6	2.6	NUM
ejde-744	278	127	)	)	PUNCT
ejde-744	278	128	,	,	PUNCT
ejde-744	278	129	respectively	respectively	ADV
ejde-744	278	130	.	.	PUNCT
ejde-744	279	1	ejde-2025/04	ejde-2025/04	X
ejde-744	279	2	exact	exact	ADJ
ejde-744	279	3	controllability	controllability	NOUN
ejde-744	279	4	11	11	NUM
ejde-744	279	5	proof	proof	NOUN
ejde-744	279	6	.	.	PUNCT
ejde-744	280	1	as	as	ADP
ejde-744	280	2	usual	usual	ADJ
ejde-744	280	3	,	,	PUNCT
ejde-744	280	4	let	let	VERB
ejde-744	280	5	us	we	PRON
ejde-744	280	6	suppose	suppose	VERB
ejde-744	280	7	that	that	SCONJ
ejde-744	280	8	u	u	PROPN
ejde-744	280	9	is	be	AUX
ejde-744	280	10	a	a	DET
ejde-744	280	11	classical	classical	ADJ
ejde-744	280	12	solution	solution	NOUN
ejde-744	280	13	of	of	ADP
ejde-744	280	14	(	(	PUNCT
ejde-744	280	15	3.1	3.1	NUM
ejde-744	280	16	)	)	PUNCT
ejde-744	280	17	.	.	PUNCT
ejde-744	281	1	multiplying	multiply	VERB
ejde-744	281	2	both	both	DET
ejde-744	281	3	sides	side	NOUN
ejde-744	281	4	of	of	ADP
ejde-744	281	5	equation	equation	NOUN
ejde-744	281	6	(	(	PUNCT
ejde-744	281	7	5.1	5.1	NUM
ejde-744	281	8	)	)	PUNCT
ejde-744	281	9	by	by	ADP
ejde-744	281	10	ka	ka	PROPN
ejde-744	281	11	2	2	NUM
ejde-744	281	12	and	and	CCONJ
ejde-744	281	13	summing	sum	VERB
ejde-744	281	14	the	the	DET
ejde-744	281	15	corresponding	correspond	VERB
ejde-744	281	16	ones	one	NOUN
ejde-744	281	17	to	to	ADP
ejde-744	281	18	both	both	DET
ejde-744	281	19	side	side	NOUN
ejde-744	281	20	of	of	ADP
ejde-744	281	21	equation	equation	NOUN
ejde-744	281	22	(	(	PUNCT
ejde-744	281	23	4.4	4.4	NUM
ejde-744	281	24	)	)	PUNCT
ejde-744	281	25	,	,	PUNCT
ejde-744	281	26	we	we	PRON
ejde-744	281	27	obtain	obtain	VERB
ejde-744	281	28	a(1	a(1	NOUN
ejde-744	281	29	)	)	PUNCT
ejde-744	282	1	∫	∫	PROPN
ejde-744	282	2	t	t	NOUN
ejde-744	282	3	0	0	NUM
ejde-744	283	1	u2	u2	PROPN
ejde-744	283	2	x(t	x(t	PROPN
ejde-744	283	3	,	,	PUNCT
ejde-744	283	4	1)dt	1)dt	NUM
ejde-744	283	5	=	=	SYM
ejde-744	283	6	2	2	NUM
ejde-744	284	1	[	[	X
ejde-744	284	2	∫	∫	PROPN
ejde-744	284	3	1	1	NUM
ejde-744	284	4	0	0	NUM
ejde-744	284	5	xuxutdx	xuxutdx	NOUN
ejde-744	284	6	]	]	X
ejde-744	284	7	t	t	NOUN
ejde-744	284	8	0	0	PUNCT
ejde-744	285	1	+	+	CCONJ
ejde-744	285	2	ka	ka	PROPN
ejde-744	285	3	2	2	NUM
ejde-744	285	4	[	[	X
ejde-744	285	5	∫	∫	PROPN
ejde-744	285	6	1	1	NUM
ejde-744	285	7	0	0	NUM
ejde-744	285	8	uut	uut	PROPN
ejde-744	285	9	dx	dx	PROPN
ejde-744	285	10	]	]	X
ejde-744	285	11	t	t	X
ejde-744	285	12	0	0	PUNCT
ejde-744	286	1	+	+	CCONJ
ejde-744	286	2	(	(	PUNCT
ejde-744	286	3	1−	1−	NUM
ejde-744	286	4	ka	ka	NOUN
ejde-744	286	5	2	2	NUM
ejde-744	286	6	)	)	PUNCT
ejde-744	286	7	∫	∫	PROPN
ejde-744	286	8	q	q	PROPN
ejde-744	286	9	u2	u2	PROPN
ejde-744	286	10	t	t	PROPN
ejde-744	286	11	dx	dx	PROPN
ejde-744	287	1	dt	dt	PROPN
ejde-744	288	1	+	+	CCONJ
ejde-744	288	2	∫	∫	PROPN
ejde-744	288	3	q	q	X
ejde-744	289	1	[	[	X
ejde-744	289	2	(	(	PUNCT
ejde-744	289	3	1	1	NUM
ejde-744	289	4	+	+	CCONJ
ejde-744	289	5	ka	ka	X
ejde-744	289	6	2	2	NUM
ejde-744	289	7	)	)	PUNCT
ejde-744	289	8	a−	a−	PROPN
ejde-744	289	9	xa′	xa′	PROPN
ejde-744	289	10	]	]	X
ejde-744	290	1	u2	u2	PROPN
ejde-744	290	2	x	x	SYM
ejde-744	290	3	dx	dx	PROPN
ejde-744	290	4	dt+	dt+	NOUN
ejde-744	290	5	∫	∫	PROPN
ejde-744	291	1	q	q	PROPN
ejde-744	291	2	λ	λ	X
ejde-744	291	3	b	b	PROPN
ejde-744	291	4	(	(	PUNCT
ejde-744	291	5	b−	b−	PROPN
ejde-744	291	6	xb′	xb′	PUNCT
ejde-744	291	7	b	b	PROPN
ejde-744	291	8	−	−	PROPN
ejde-744	291	9	ka	ka	PROPN
ejde-744	291	10	2	2	X
ejde-744	291	11	)	)	PUNCT
ejde-744	291	12	u2	u2	PROPN
ejde-744	291	13	dx	dx	PROPN
ejde-744	291	14	dt	dt	PROPN
ejde-744	292	1	=	=	PUNCT
ejde-744	292	2	∫	∫	PROPN
ejde-744	292	3	q	q	PROPN
ejde-744	293	1	(	(	PUNCT
ejde-744	293	2	1−	1−	NUM
ejde-744	293	3	ka	ka	NOUN
ejde-744	293	4	2	2	X
ejde-744	293	5	)	)	PUNCT
ejde-744	293	6	u2	u2	PROPN
ejde-744	293	7	t	t	PROPN
ejde-744	293	8	+	+	CCONJ
ejde-744	294	1	[	[	X
ejde-744	294	2	(	(	PUNCT
ejde-744	294	3	1	1	NUM
ejde-744	294	4	+	+	CCONJ
ejde-744	294	5	ka	ka	X
ejde-744	294	6	2	2	NUM
ejde-744	294	7	)	)	PUNCT
ejde-744	294	8	a−	a−	PROPN
ejde-744	294	9	xa′	xa′	PROPN
ejde-744	294	10	]	]	X
ejde-744	295	1	u2	u2	NOUN
ejde-744	295	2	x	x	X
ejde-744	295	3	+	+	CCONJ
ejde-744	295	4	(	(	PUNCT
ejde-744	295	5	1−	1−	NUM
ejde-744	295	6	ka	ka	NOUN
ejde-744	295	7	2	2	NUM
ejde-744	295	8	)	)	PUNCT
ejde-744	295	9	λ	λ	PROPN
ejde-744	295	10	b	b	PROPN
ejde-744	295	11	u2	u2	PROPN
ejde-744	295	12	dx	dx	PROPN
ejde-744	295	13	dt	dt	PROPN
ejde-744	296	1	+	+	CCONJ
ejde-744	296	2	2	2	NUM
ejde-744	296	3	[	[	X
ejde-744	296	4	∫	∫	PROPN
ejde-744	296	5	1	1	NUM
ejde-744	296	6	0	0	NUM
ejde-744	296	7	xuxutdx	xuxutdx	NOUN
ejde-744	296	8	]	]	X
ejde-744	297	1	t	t	NOUN
ejde-744	297	2	0	0	PUNCT
ejde-744	298	1	+	+	CCONJ
ejde-744	298	2	ka	ka	PROPN
ejde-744	298	3	2	2	NUM
ejde-744	298	4	[	[	X
ejde-744	298	5	∫	∫	PROPN
ejde-744	298	6	1	1	NUM
ejde-744	298	7	0	0	NUM
ejde-744	298	8	uut	uut	PROPN
ejde-744	298	9	dx	dx	PROPN
ejde-744	298	10	]	]	X
ejde-744	298	11	t	t	X
ejde-744	298	12	0	0	PUNCT
ejde-744	299	1	+	+	CCONJ
ejde-744	299	2	∫	∫	PROPN
ejde-744	299	3	q	q	X
ejde-744	299	4	λ	λ	X
ejde-744	299	5	b	b	PROPN
ejde-744	299	6	(	(	PUNCT
ejde-744	299	7	2−	2−	NUM
ejde-744	299	8	xb′	xb′	NOUN
ejde-744	300	1	b	b	NUM
ejde-744	300	2	−ka	−ka	NOUN
ejde-744	300	3	)	)	PUNCT
ejde-744	301	1	u2	u2	PROPN
ejde-744	301	2	dx	dx	PROPN
ejde-744	301	3	dt	dt	PROPN
ejde-744	301	4	.	.	PUNCT
ejde-744	302	1	(	(	PUNCT
ejde-744	302	2	5.5	5.5	NUM
ejde-744	302	3	)	)	PUNCT
ejde-744	302	4	using	use	VERB
ejde-744	302	5	remark	remark	NOUN
ejde-744	302	6	2.2	2.2	NUM
ejde-744	302	7	and	and	CCONJ
ejde-744	302	8	the	the	DET
ejde-744	302	9	inequality	inequality	NOUN
ejde-744	302	10	xa′	xa′	PROPN
ejde-744	303	1	≤	≤	PROPN
ejde-744	303	2	kaa	kaa	PROPN
ejde-744	303	3	,	,	PUNCT
ejde-744	303	4	we	we	PRON
ejde-744	303	5	have∫	have∫	VERB
ejde-744	303	6	q	q	X
ejde-744	304	1	(	(	PUNCT
ejde-744	304	2	1−	1−	NUM
ejde-744	304	3	ka	ka	NOUN
ejde-744	304	4	2	2	X
ejde-744	304	5	)	)	PUNCT
ejde-744	304	6	u2	u2	PROPN
ejde-744	304	7	t	t	PROPN
ejde-744	304	8	+	+	CCONJ
ejde-744	305	1	[	[	X
ejde-744	305	2	(	(	PUNCT
ejde-744	305	3	1	1	NUM
ejde-744	305	4	+	+	CCONJ
ejde-744	305	5	ka	ka	X
ejde-744	305	6	2	2	NUM
ejde-744	305	7	)	)	PUNCT
ejde-744	305	8	a−	a−	PROPN
ejde-744	305	9	xa′	xa′	PROPN
ejde-744	305	10	]	]	X
ejde-744	306	1	u2	u2	NOUN
ejde-744	306	2	x	x	X
ejde-744	306	3	+	+	CCONJ
ejde-744	306	4	(	(	PUNCT
ejde-744	306	5	1−	1−	NUM
ejde-744	306	6	ka	ka	NOUN
ejde-744	306	7	2	2	NUM
ejde-744	306	8	)	)	PUNCT
ejde-744	307	1	λ	λ	PROPN
ejde-744	307	2	b	b	PROPN
ejde-744	307	3	u2	u2	PROPN
ejde-744	307	4	dx	dx	PROPN
ejde-744	307	5	dt	dt	X
ejde-744	307	6	≥	≥	PROPN
ejde-744	307	7	(	(	PUNCT
ejde-744	307	8	1−	1−	NUM
ejde-744	307	9	ka	ka	NOUN
ejde-744	307	10	2	2	NUM
ejde-744	307	11	)	)	PUNCT
ejde-744	307	12	∫	∫	PROPN
ejde-744	307	13	q	q	PROPN
ejde-744	308	1	(	(	PUNCT
ejde-744	308	2	u2	u2	PROPN
ejde-744	308	3	t	t	PROPN
ejde-744	308	4	+	+	CCONJ
ejde-744	308	5	au2	au2	NOUN
ejde-744	308	6	x	x	X
ejde-744	308	7	−	−	PUNCT
ejde-744	308	8	λ	λ	X
ejde-744	308	9	b	b	PROPN
ejde-744	308	10	u2	u2	PROPN
ejde-744	308	11	)	)	PUNCT
ejde-744	308	12	dx	dx	PROPN
ejde-744	309	1	dt	dt	X
ejde-744	309	2	≥	≥	X
ejde-744	309	3	(	(	PUNCT
ejde-744	309	4	2−ka)teu(0	2−ka)teu(0	NOUN
ejde-744	309	5	)	)	PUNCT
ejde-744	309	6	.	.	PUNCT
ejde-744	310	1	(	(	PUNCT
ejde-744	310	2	5.6	5.6	NUM
ejde-744	310	3	)	)	PUNCT
ejde-744	310	4	by	by	ADP
ejde-744	310	5	(	(	PUNCT
ejde-744	310	6	4.10	4.10	NUM
ejde-744	310	7	)	)	PUNCT
ejde-744	310	8	,	,	PUNCT
ejde-744	310	9	we	we	PRON
ejde-744	310	10	obtain	obtain	VERB
ejde-744	310	11	2	2	NUM
ejde-744	311	1	[	[	X
ejde-744	311	2	∫	∫	PROPN
ejde-744	311	3	1	1	NUM
ejde-744	311	4	0	0	NUM
ejde-744	311	5	xuxutdx	xuxutdx	NOUN
ejde-744	311	6	]	]	X
ejde-744	311	7	t	t	NOUN
ejde-744	311	8	0	0	NUM
ejde-744	311	9	≤	≤	NUM
ejde-744	311	10	4max	4max	NUM
ejde-744	311	11	{	{	PUNCT
ejde-744	311	12	1	1	NUM
ejde-744	311	13	a(1)cθ	a(1)cθ	NOUN
ejde-744	311	14	,	,	PUNCT
ejde-744	311	15	1}eu(0	1}eu(0	NUM
ejde-744	311	16	)	)	PUNCT
ejde-744	311	17	.	.	PUNCT
ejde-744	312	1	(	(	PUNCT
ejde-744	312	2	5.7	5.7	NUM
ejde-744	312	3	)	)	PUNCT
ejde-744	312	4	furthermore	furthermore	ADV
ejde-744	312	5	,	,	PUNCT
ejde-744	312	6	applying	apply	VERB
ejde-744	312	7	the	the	DET
ejde-744	312	8	hardy	hardy	ADJ
ejde-744	312	9	inequality	inequality	NOUN
ejde-744	312	10	in	in	ADP
ejde-744	312	11	its	its	PRON
ejde-744	312	12	pure	pure	ADJ
ejde-744	312	13	degenerate	degenerate	ADJ
ejde-744	312	14	form	form	NOUN
ejde-744	312	15	(	(	PUNCT
ejde-744	312	16	see	see	VERB
ejde-744	312	17	[	[	X
ejde-744	312	18	1	1	NUM
ejde-744	312	19	]	]	NUM
ejde-744	312	20	)	)	PUNCT
ejde-744	312	21	,	,	PUNCT
ejde-744	312	22	we	we	PRON
ejde-744	312	23	obtain	obtain	VERB
ejde-744	312	24	∫	∫	PROPN
ejde-744	312	25	1	1	NUM
ejde-744	312	26	0	0	NUM
ejde-744	312	27	u2dx	u2dx	PUNCT
ejde-744	312	28	≤	≤	NUM
ejde-744	312	29	4	4	NUM
ejde-744	312	30	a(1	a(1	NUM
ejde-744	312	31	)	)	PUNCT
ejde-744	312	32	∫	∫	PROPN
ejde-744	312	33	1	1	NUM
ejde-744	312	34	0	0	NUM
ejde-744	312	35	au2	au2	PROPN
ejde-744	312	36	xdx	xdx	PROPN
ejde-744	312	37	,	,	PUNCT
ejde-744	312	38	∀u	∀u	NOUN
ejde-744	312	39	∈	∈	NOUN
ejde-744	312	40	c∞	c∞	PROPN
ejde-744	312	41	c	c	X
ejde-744	312	42	(	(	PUNCT
ejde-744	312	43	0	0	NUM
ejde-744	312	44	,	,	PUNCT
ejde-744	312	45	1	1	NUM
ejde-744	312	46	)	)	PUNCT
ejde-744	312	47	,	,	PUNCT
ejde-744	312	48	from	from	ADP
ejde-744	312	49	which	which	PRON
ejde-744	312	50	we	we	PRON
ejde-744	312	51	can	can	AUX
ejde-744	312	52	deduce	deduce	VERB
ejde-744	312	53	that∫	that∫	NOUN
ejde-744	312	54	1	1	NUM
ejde-744	312	55	0	0	NUM
ejde-744	312	56	uut	uut	PROPN
ejde-744	312	57	dx	dx	PROPN
ejde-744	312	58	≤	≤	NUM
ejde-744	312	59	1	1	NUM
ejde-744	312	60	2	2	NUM
ejde-744	312	61	∫	∫	NOUN
ejde-744	312	62	1	1	NUM
ejde-744	312	63	0	0	NUM
ejde-744	312	64	(	(	PUNCT
ejde-744	312	65	√cθa(1	√cθa(1	NOUN
ejde-744	312	66	)	)	PUNCT
ejde-744	312	67	2	2	NUM
ejde-744	312	68	u2	u2	NOUN
ejde-744	312	69	+	+	CCONJ
ejde-744	312	70	2√	2√	PROPN
ejde-744	312	71	cθa(1	cθa(1	NOUN
ejde-744	312	72	)	)	PUNCT
ejde-744	312	73	u2	u2	PROPN
ejde-744	312	74	t	t	PROPN
ejde-744	312	75	)	)	PUNCT
ejde-744	312	76	dx	dx	PROPN
ejde-744	313	1	≤	≤	NUM
ejde-744	313	2	1	1	NUM
ejde-744	313	3	2	2	NUM
ejde-744	313	4	∫	∫	NOUN
ejde-744	313	5	1	1	NUM
ejde-744	313	6	0	0	NUM
ejde-744	314	1	(	(	PUNCT
ejde-744	314	2	2	2	NUM
ejde-744	314	3	√	√	NUM
ejde-744	314	4	cθ	cθ	ADP
ejde-744	314	5	a(1	a(1	PROPN
ejde-744	314	6	)	)	PUNCT
ejde-744	314	7	au2	au2	NOUN
ejde-744	314	8	x	x	X
ejde-744	314	9	+	+	NUM
ejde-744	314	10	2√	2√	PROPN
ejde-744	314	11	cθa(1	cθa(1	NOUN
ejde-744	314	12	)	)	PUNCT
ejde-744	314	13	u2	u2	PROPN
ejde-744	314	14	t	t	PROPN
ejde-744	314	15	)	)	PUNCT
ejde-744	314	16	dx	dx	PROPN
ejde-744	314	17	≤	≤	NUM
ejde-744	314	18	1√	1√	PROPN
ejde-744	314	19	cθa(1	cθa(1	NOUN
ejde-744	314	20	)	)	PUNCT
ejde-744	314	21	∫	∫	PROPN
ejde-744	315	1	1	1	NUM
ejde-744	315	2	0	0	NUM
ejde-744	315	3	(	(	PUNCT
ejde-744	315	4	u2	u2	PROPN
ejde-744	315	5	t	t	PROPN
ejde-744	315	6	+	+	CCONJ
ejde-744	315	7	au2	au2	NOUN
ejde-744	315	8	x	x	X
ejde-744	315	9	−	−	PUNCT
ejde-744	315	10	λ	λ	X
ejde-744	315	11	b	b	PROPN
ejde-744	315	12	u2	u2	PROPN
ejde-744	315	13	)	)	PUNCT
ejde-744	315	14	dx	dx	PROPN
ejde-744	315	15	=	=	SYM
ejde-744	315	16	2√	2√	PROPN
ejde-744	315	17	cθa(1	cθa(1	NOUN
ejde-744	315	18	)	)	PUNCT
ejde-744	315	19	eu(0	eu(0	PROPN
ejde-744	315	20	)	)	PUNCT
ejde-744	315	21	.	.	PUNCT
ejde-744	316	1	(	(	PUNCT
ejde-744	316	2	5.8	5.8	NUM
ejde-744	316	3	)	)	PUNCT
ejde-744	316	4	finally	finally	ADV
ejde-744	316	5	,	,	PUNCT
ejde-744	316	6	if	if	SCONJ
ejde-744	316	7	λ	λ	X
ejde-744	316	8	∈	∈	PROPN
ejde-744	316	9	(	(	PUNCT
ejde-744	316	10	0	0	NUM
ejde-744	316	11	,	,	PUNCT
ejde-744	316	12	1	1	NUM
ejde-744	316	13	ca	ca	NOUN
ejde-744	316	14	,	,	PUNCT
ejde-744	316	15	b	b	PROPN
ejde-744	316	16	)	)	PUNCT
ejde-744	316	17	,	,	PUNCT
ejde-744	316	18	we	we	PRON
ejde-744	316	19	have∫	have∫	VERB
ejde-744	316	20	q	q	X
ejde-744	316	21	λ	λ	X
ejde-744	316	22	b	b	PROPN
ejde-744	316	23	(	(	PUNCT
ejde-744	316	24	2−	2−	NUM
ejde-744	316	25	xb′	xb′	NOUN
ejde-744	317	1	b	b	NUM
ejde-744	317	2	−ka	−ka	NOUN
ejde-744	317	3	)	)	PUNCT
ejde-744	318	1	u2	u2	PROPN
ejde-744	318	2	dx	dx	PROPN
ejde-744	318	3	dt	dt	PROPN
ejde-744	318	4	≥	≥	PROPN
ejde-744	318	5	∫	∫	PROPN
ejde-744	318	6	q	q	PROPN
ejde-744	318	7	λ	λ	X
ejde-744	318	8	b	b	PROPN
ejde-744	318	9	(	(	PUNCT
ejde-744	318	10	2−kb	2−kb	NUM
ejde-744	318	11	−ka)u	−ka)u	NOUN
ejde-744	318	12	2	2	NUM
ejde-744	318	13	dx	dx	PROPN
ejde-744	318	14	dt	dt	X
ejde-744	318	15	≥	≥	PROPN
ejde-744	318	16	λca	λca	INTJ
ejde-744	318	17	,	,	PUNCT
ejde-744	318	18	bcθ	bcθ	INTJ
ejde-744	318	19	(	(	PUNCT
ejde-744	318	20	2−ka	2−ka	NUM
ejde-744	318	21	−kb)teu(0	−kb)teu(0	NOUN
ejde-744	318	22	)	)	PUNCT
ejde-744	318	23	;	;	PUNCT
ejde-744	318	24	(	(	PUNCT
ejde-744	318	25	5.9	5.9	NUM
ejde-744	318	26	)	)	PUNCT
ejde-744	318	27	12	12	NUM
ejde-744	318	28	g.	g.	PROPN
ejde-744	318	29	zhang	zhang	PROPN
ejde-744	318	30	,	,	PUNCT
ejde-744	318	31	s.	s.	PROPN
ejde-744	318	32	chai	chai	PROPN
ejde-744	318	33	ejde-2025/04	ejde-2025/04	PROPN
ejde-744	318	34	if	if	SCONJ
ejde-744	318	35	λ	λ	PROPN
ejde-744	318	36	∈	∈	PROPN
ejde-744	318	37	(	(	PUNCT
ejde-744	318	38	−∞	−∞	NOUN
ejde-744	318	39	,	,	PUNCT
ejde-744	318	40	0	0	NUM
ejde-744	318	41	]	]	PUNCT
ejde-744	318	42	,	,	PUNCT
ejde-744	318	43	then∫	then∫	NOUN
ejde-744	318	44	q	q	PROPN
ejde-744	319	1	λ	λ	X
ejde-744	319	2	b	b	PROPN
ejde-744	319	3	(	(	PUNCT
ejde-744	319	4	2−	2−	NUM
ejde-744	319	5	xb′	xb′	NOUN
ejde-744	320	1	b	b	NUM
ejde-744	320	2	−ka	−ka	NOUN
ejde-744	320	3	)	)	PUNCT
ejde-744	320	4	u2	u2	PROPN
ejde-744	320	5	dx	dx	PROPN
ejde-744	320	6	dt	dt	PROPN
ejde-744	321	1	≤	≤	NUM
ejde-744	321	2	∫	∫	PROPN
ejde-744	321	3	q	q	PROPN
ejde-744	321	4	|λ|	|λ|	PROPN
ejde-744	321	5	b	b	PROPN
ejde-744	321	6	(	(	PUNCT
ejde-744	321	7	2−kb	2−kb	NUM
ejde-744	321	8	−ka)u	−ka)u	NOUN
ejde-744	321	9	2	2	NUM
ejde-744	321	10	dx	dx	NOUN
ejde-744	321	11	dt	dt	PROPN
ejde-744	321	12	≤	≤	PROPN
ejde-744	321	13	|λ|ca	|λ|ca	NOUN
ejde-744	321	14	,	,	PUNCT
ejde-744	321	15	bcθ	bcθ	INTJ
ejde-744	321	16	(	(	PUNCT
ejde-744	321	17	2−kb	2−kb	NUM
ejde-744	321	18	−ka)teu(0	−ka)teu(0	NUM
ejde-744	321	19	)	)	PUNCT
ejde-744	321	20	.	.	PUNCT
ejde-744	322	1	(	(	PUNCT
ejde-744	322	2	5.10	5.10	NUM
ejde-744	322	3	)	)	PUNCT
ejde-744	322	4	notice	notice	NOUN
ejde-744	322	5	that	that	SCONJ
ejde-744	322	6	cθ	cθ	ADP
ejde-744	322	7	=	=	SYM
ejde-744	322	8	θ	θ	PROPN
ejde-744	322	9	if	if	SCONJ
ejde-744	322	10	λ	λ	X
ejde-744	322	11	∈	∈	PROPN
ejde-744	322	12	(	(	PUNCT
ejde-744	322	13	0	0	NUM
ejde-744	322	14	,	,	PUNCT
ejde-744	322	15	1	1	NUM
ejde-744	322	16	ca	ca	NOUN
ejde-744	322	17	,	,	PUNCT
ejde-744	322	18	b	b	PROPN
ejde-744	322	19	)	)	PUNCT
ejde-744	322	20	;	;	PUNCT
ejde-744	322	21	cθ	cθ	ADP
ejde-744	322	22	=	=	SYM
ejde-744	322	23	1	1	NUM
ejde-744	322	24	if	if	SCONJ
ejde-744	322	25	λ	λ	X
ejde-744	322	26	∈	∈	PROPN
ejde-744	322	27	(	(	PUNCT
ejde-744	322	28	−∞	−∞	NOUN
ejde-744	322	29	,	,	PUNCT
ejde-744	322	30	0	0	NUM
ejde-744	322	31	]	]	PUNCT
ejde-744	322	32	.	.	PUNCT
ejde-744	323	1	therefore	therefore	ADV
ejde-744	323	2	,	,	PUNCT
ejde-744	323	3	(	(	PUNCT
ejde-744	323	4	5.3	5.3	NUM
ejde-744	323	5	)	)	PUNCT
ejde-744	323	6	and	and	CCONJ
ejde-744	323	7	(	(	PUNCT
ejde-744	323	8	5.4	5.4	NUM
ejde-744	323	9	)	)	PUNCT
ejde-744	323	10	by	by	ADP
ejde-744	323	11	substituting	substitute	VERB
ejde-744	323	12	(	(	PUNCT
ejde-744	323	13	5.6)-(5.10	5.6)-(5.10	NOUN
ejde-744	323	14	)	)	PUNCT
ejde-744	323	15	into	into	ADP
ejde-744	323	16	(	(	PUNCT
ejde-744	323	17	5.5	5.5	NUM
ejde-744	323	18	)	)	PUNCT
ejde-744	323	19	.	.	PUNCT
ejde-744	324	1	□	□	PUNCT
ejde-744	324	2	we	we	PRON
ejde-744	324	3	recall	recall	VERB
ejde-744	324	4	that	that	SCONJ
ejde-744	324	5	(	(	PUNCT
ejde-744	324	6	3.1	3.1	NUM
ejde-744	324	7	)	)	PUNCT
ejde-744	324	8	is	be	AUX
ejde-744	324	9	said	say	VERB
ejde-744	324	10	to	to	PART
ejde-744	324	11	be	be	AUX
ejde-744	324	12	observable	observable	ADJ
ejde-744	324	13	in	in	ADP
ejde-744	324	14	time	time	NOUN
ejde-744	324	15	t	t	PROPN
ejde-744	324	16	≥	≥	NOUN
ejde-744	324	17	0	0	NUM
ejde-744	324	18	via	via	ADP
ejde-744	324	19	the	the	DET
ejde-744	324	20	normal	normal	ADJ
ejde-744	324	21	derivative	derivative	NOUN
ejde-744	324	22	at	at	ADP
ejde-744	324	23	x	x	X
ejde-744	324	24	=	=	SYM
ejde-744	324	25	1	1	NUM
ejde-744	324	26	,	,	PUNCT
ejde-744	324	27	if	if	SCONJ
ejde-744	324	28	there	there	PRON
ejde-744	324	29	exists	exist	VERB
ejde-744	324	30	a	a	DET
ejde-744	324	31	constant	constant	ADJ
ejde-744	324	32	c	c	NOUN
ejde-744	324	33	>	>	X
ejde-744	324	34	0	0	NUM
ejde-744	324	35	such	such	ADJ
ejde-744	324	36	that	that	PRON
ejde-744	324	37	for	for	ADP
ejde-744	324	38	any	any	DET
ejde-744	324	39	(	(	PUNCT
ejde-744	324	40	u0	u0	ADJ
ejde-744	324	41	,	,	PUNCT
ejde-744	324	42	u1	u1	NOUN
ejde-744	324	43	)	)	PUNCT
ejde-744	324	44	∈	∈	PROPN
ejde-744	324	45	h1	h1	NOUN
ejde-744	324	46	a(0	a(0	PROPN
ejde-744	324	47	,	,	PUNCT
ejde-744	324	48	1)×	1)×	NUM
ejde-744	324	49	l2(0	l2(0	NOUN
ejde-744	324	50	,	,	PUNCT
ejde-744	324	51	1	1	NUM
ejde-744	324	52	)	)	PUNCT
ejde-744	324	53	,	,	PUNCT
ejde-744	324	54	the	the	DET
ejde-744	324	55	mild	mild	ADJ
ejde-744	324	56	solution	solution	NOUN
ejde-744	324	57	of	of	ADP
ejde-744	324	58	(	(	PUNCT
ejde-744	324	59	3.1	3.1	NUM
ejde-744	324	60	)	)	PUNCT
ejde-744	324	61	satisfies∫	satisfies∫	NOUN
ejde-744	324	62	t	t	NOUN
ejde-744	324	63	0	0	NUM
ejde-744	324	64	u2	u2	PROPN
ejde-744	324	65	x(t	x(t	PROPN
ejde-744	324	66	,	,	PUNCT
ejde-744	324	67	1	1	NUM
ejde-744	324	68	)	)	PUNCT
ejde-744	324	69	dt	dt	X
ejde-744	324	70	≥	≥	PROPN
ejde-744	324	71	ceu(0	ceu(0	PROPN
ejde-744	324	72	)	)	PUNCT
ejde-744	324	73	.	.	PUNCT
ejde-744	325	1	(	(	PUNCT
ejde-744	325	2	5.11	5.11	NUM
ejde-744	325	3	)	)	PUNCT
ejde-744	325	4	moreover	moreover	ADV
ejde-744	325	5	,	,	PUNCT
ejde-744	325	6	any	any	DET
ejde-744	325	7	constant	constant	ADJ
ejde-744	325	8	satisfying	satisfying	NOUN
ejde-744	325	9	(	(	PUNCT
ejde-744	325	10	5.11	5.11	NUM
ejde-744	325	11	)	)	PUNCT
ejde-744	325	12	is	be	AUX
ejde-744	325	13	called	call	VERB
ejde-744	325	14	an	an	DET
ejde-744	325	15	observability	observability	NOUN
ejde-744	325	16	constant	constant	ADJ
ejde-744	325	17	for	for	ADP
ejde-744	325	18	(	(	PUNCT
ejde-744	325	19	3.1	3.1	NUM
ejde-744	325	20	)	)	PUNCT
ejde-744	325	21	in	in	ADP
ejde-744	325	22	time	time	NOUN
ejde-744	325	23	t	t	PROPN
ejde-744	325	24	≥	≥	NOUN
ejde-744	325	25	0	0	NUM
ejde-744	325	26	.	.	PUNCT
ejde-744	326	1	the	the	DET
ejde-744	326	2	supremum	supremum	NOUN
ejde-744	326	3	of	of	ADP
ejde-744	326	4	all	all	DET
ejde-744	326	5	observability	observability	NOUN
ejde-744	326	6	constants	constant	NOUN
ejde-744	326	7	for	for	ADP
ejde-744	326	8	(	(	PUNCT
ejde-744	326	9	3.1	3.1	NUM
ejde-744	326	10	)	)	PUNCT
ejde-744	326	11	is	be	AUX
ejde-744	326	12	denoted	denote	VERB
ejde-744	326	13	by	by	ADP
ejde-744	326	14	ct	ct	PRON
ejde-744	326	15	,	,	PUNCT
ejde-744	326	16	namely	namely	ADV
ejde-744	326	17	,	,	PUNCT
ejde-744	326	18	ct	ct	PRON
ejde-744	326	19	:	:	PUNCT
ejde-744	326	20	=	=	NOUN
ejde-744	326	21	sup{c	sup{c	X
ejde-744	326	22	>	>	X
ejde-744	326	23	0	0	NUM
ejde-744	326	24	,	,	PUNCT
ejde-744	326	25	c	c	NOUN
ejde-744	326	26	satisfies	satisfie	NOUN
ejde-744	326	27	(	(	PUNCT
ejde-744	326	28	5.11	5.11	NUM
ejde-744	326	29	)	)	PUNCT
ejde-744	326	30	}	}	PUNCT
ejde-744	326	31	.	.	PUNCT
ejde-744	327	1	we	we	PRON
ejde-744	327	2	said	say	VERB
ejde-744	327	3	that	that	SCONJ
ejde-744	327	4	(	(	PUNCT
ejde-744	327	5	3.1	3.1	NUM
ejde-744	327	6	)	)	PUNCT
ejde-744	327	7	is	be	AUX
ejde-744	327	8	observable	observable	ADJ
ejde-744	327	9	if	if	SCONJ
ejde-744	327	10	ct	ct	PROPN
ejde-744	327	11	=	=	PUNCT
ejde-744	327	12	inf	inf	PROPN
ejde-744	327	13	(	(	PUNCT
ejde-744	327	14	u0,u1	u0,u1	PROPN
ejde-744	327	15	)	)	PUNCT
ejde-744	327	16	̸=(0,0	̸=(0,0	NOUN
ejde-744	327	17	)	)	PUNCT
ejde-744	327	18	∫	∫	PROPN
ejde-744	328	1	t	t	NOUN
ejde-744	328	2	0	0	NUM
ejde-744	328	3	u2	u2	PROPN
ejde-744	328	4	x(t	x(t	PROPN
ejde-744	328	5	,	,	PUNCT
ejde-744	328	6	1)dt	1)dt	PROPN
ejde-744	328	7	eu(0	eu(0	PROPN
ejde-744	328	8	)	)	PUNCT
ejde-744	328	9	>	>	X
ejde-744	328	10	0	0	X
ejde-744	328	11	.	.	PUNCT
ejde-744	328	12	(	(	PUNCT
ejde-744	328	13	5.12	5.12	NUM
ejde-744	328	14	)	)	PUNCT
ejde-744	328	15	from	from	ADP
ejde-744	328	16	the	the	DET
ejde-744	328	17	definition	definition	NOUN
ejde-744	328	18	of	of	ADP
ejde-744	328	19	observability	observability	NOUN
ejde-744	328	20	,	,	PUNCT
ejde-744	328	21	and	and	CCONJ
ejde-744	328	22	theorem	theorem	VERB
ejde-744	328	23	5.2	5.2	NUM
ejde-744	328	24	,	,	PUNCT
ejde-744	328	25	we	we	PRON
ejde-744	328	26	have	have	AUX
ejde-744	328	27	following	follow	VERB
ejde-744	328	28	corollary	corollary	ADJ
ejde-744	328	29	.	.	PUNCT
ejde-744	329	1	corollary	corollary	ADJ
ejde-744	329	2	5.3	5.3	NUM
ejde-744	329	3	.	.	PUNCT
ejde-744	330	1	under	under	ADP
ejde-744	330	2	assumption	assumption	NOUN
ejde-744	330	3	2.6	2.6	NUM
ejde-744	330	4	and	and	CCONJ
ejde-744	330	5	λ	λ	X
ejde-744	330	6	>	>	X
ejde-744	330	7	0	0	NUM
ejde-744	330	8	,	,	PUNCT
ejde-744	330	9	(	(	PUNCT
ejde-744	330	10	3.1	3.1	NUM
ejde-744	330	11	)	)	PUNCT
ejde-744	330	12	is	be	AUX
ejde-744	330	13	observable	observable	ADJ
ejde-744	330	14	in	in	ADP
ejde-744	330	15	time	time	NOUN
ejde-744	330	16	t	t	PROPN
ejde-744	330	17	,	,	PUNCT
ejde-744	330	18	provided	provide	VERB
ejde-744	330	19	that	that	SCONJ
ejde-744	330	20	t	t	PROPN
ejde-744	330	21	>	>	X
ejde-744	330	22	ta	ta	PROPN
ejde-744	330	23	,	,	PUNCT
ejde-744	330	24	b	b	NOUN
ejde-744	330	25	:	:	PUNCT
ejde-744	330	26	=	=	PROPN
ejde-744	330	27	c2	c2	PROPN
ejde-744	330	28	c1	c1	PROPN
ejde-744	330	29	.	.	PUNCT
ejde-744	331	1	in	in	ADP
ejde-744	331	2	this	this	DET
ejde-744	331	3	case	case	NOUN
ejde-744	331	4	,	,	PUNCT
ejde-744	331	5	ct	ct	NUM
ejde-744	331	6	≥	≥	NUM
ejde-744	331	7	1	1	NUM
ejde-744	331	8	a(1	a(1	NUM
ejde-744	331	9	)	)	PUNCT
ejde-744	331	10	(	(	PUNCT
ejde-744	331	11	c1	c1	PROPN
ejde-744	331	12	t	t	PROPN
ejde-744	331	13	−	−	PROPN
ejde-744	331	14	c2	c2	PROPN
ejde-744	331	15	)	)	PUNCT
ejde-744	331	16	,	,	PUNCT
ejde-744	331	17	where	where	SCONJ
ejde-744	331	18	c1	c1	PROPN
ejde-744	331	19	=	=	SYM
ejde-744	331	20	(	(	PUNCT
ejde-744	331	21	2−ka	2−ka	NUM
ejde-744	331	22	)	)	PUNCT
ejde-744	332	1	+	+	CCONJ
ejde-744	332	2	λca	λca	INTJ
ejde-744	332	3	,	,	PUNCT
ejde-744	332	4	bθ	bθ	PROPN
ejde-744	332	5	(	(	PUNCT
ejde-744	332	6	2−ka	2−ka	PROPN
ejde-744	332	7	−kb	−kb	PROPN
ejde-744	332	8	)	)	PUNCT
ejde-744	332	9	,	,	PUNCT
ejde-744	332	10	c2	c2	PROPN
ejde-744	332	11	=	=	SYM
ejde-744	332	12	(	(	PUNCT
ejde-744	332	13	4max	4max	PROPN
ejde-744	332	14	{	{	PUNCT
ejde-744	332	15	1	1	NUM
ejde-744	332	16	a(1)θ	a(1)θ	INTJ
ejde-744	332	17	,	,	PUNCT
ejde-744	332	18	1}+	1}+	NUM
ejde-744	332	19	2ka	2ka	ADJ
ejde-744	332	20	1√	1√	PROPN
ejde-744	332	21	θa(1	θa(1	NOUN
ejde-744	332	22	)	)	PUNCT
ejde-744	332	23	)	)	PUNCT
ejde-744	332	24	.	.	PUNCT
ejde-744	333	1	corollary	corollary	ADJ
ejde-744	333	2	5.4	5.4	NUM
ejde-744	333	3	.	.	PUNCT
ejde-744	334	1	under	under	ADP
ejde-744	334	2	assumption	assumption	NOUN
ejde-744	334	3	2.6	2.6	NUM
ejde-744	334	4	,	,	PUNCT
ejde-744	334	5	ka−2	ka−2	PROPN
ejde-744	334	6	ca	can	AUX
ejde-744	334	7	,	,	PUNCT
ejde-744	334	8	b(2−kb−ka	b(2−kb−ka	PROPN
ejde-744	334	9	)	)	PUNCT
ejde-744	335	1	<	<	X
ejde-744	335	2	λ	λ	X
ejde-744	335	3	≤	≤	NOUN
ejde-744	335	4	0	0	NUM
ejde-744	335	5	,	,	PUNCT
ejde-744	335	6	and	and	CCONJ
ejde-744	335	7	λ	λ	X
ejde-744	335	8	≤	≤	NOUN
ejde-744	335	9	0	0	PUNCT
ejde-744	336	1	if	if	SCONJ
ejde-744	336	2	ka	ka	PROPN
ejde-744	336	3	+	+	PROPN
ejde-744	336	4	kb	kb	PROPN
ejde-744	336	5	=	=	SYM
ejde-744	336	6	2	2	X
ejde-744	336	7	.	.	X
ejde-744	337	1	we	we	PRON
ejde-744	337	2	have	have	VERB
ejde-744	337	3	that	that	PRON
ejde-744	337	4	(	(	PUNCT
ejde-744	337	5	3.1	3.1	NUM
ejde-744	337	6	)	)	PUNCT
ejde-744	337	7	is	be	AUX
ejde-744	337	8	observable	observable	ADJ
ejde-744	337	9	in	in	ADP
ejde-744	337	10	time	time	NOUN
ejde-744	337	11	t	t	PROPN
ejde-744	337	12	,	,	PUNCT
ejde-744	337	13	provided	provide	VERB
ejde-744	337	14	that	that	SCONJ
ejde-744	337	15	t	t	PROPN
ejde-744	337	16	>	>	X
ejde-744	337	17	ta	ta	PROPN
ejde-744	337	18	,	,	PUNCT
ejde-744	337	19	b	b	NOUN
ejde-744	337	20	:	:	PUNCT
ejde-744	337	21	=	=	SYM
ejde-744	337	22	c4	c4	PROPN
ejde-744	337	23	c3	c3	PROPN
ejde-744	337	24	.	.	PUNCT
ejde-744	338	1	moreover	moreover	ADV
ejde-744	338	2	,	,	PUNCT
ejde-744	338	3	ct	ct	NUM
ejde-744	338	4	≥	≥	NOUN
ejde-744	338	5	1	1	NUM
ejde-744	338	6	a(1	a(1	NUM
ejde-744	338	7	)	)	PUNCT
ejde-744	338	8	(	(	PUNCT
ejde-744	338	9	c3	c3	PROPN
ejde-744	338	10	t	t	PROPN
ejde-744	338	11	−	−	PROPN
ejde-744	338	12	c4	c4	PROPN
ejde-744	338	13	)	)	PUNCT
ejde-744	338	14	,	,	PUNCT
ejde-744	338	15	where	where	SCONJ
ejde-744	338	16	c3	c3	PROPN
ejde-744	338	17	=	=	PRON
ejde-744	338	18	{	{	PUNCT
ejde-744	338	19	(	(	PUNCT
ejde-744	338	20	2−ka)−	2−ka)−	NOUN
ejde-744	338	21	|λ|ca	|λ|ca	PROPN
ejde-744	338	22	,	,	PUNCT
ejde-744	338	23	b	b	PROPN
ejde-744	338	24	(	(	PUNCT
ejde-744	338	25	2−ka	2−ka	PROPN
ejde-744	338	26	−kb	−kb	PROPN
ejde-744	338	27	)	)	PUNCT
ejde-744	338	28	}	}	PUNCT
ejde-744	338	29	,	,	PUNCT
ejde-744	338	30	c4	c4	NOUN
ejde-744	338	31	=	=	SYM
ejde-744	338	32	(	(	PUNCT
ejde-744	338	33	4max	4max	PROPN
ejde-744	338	34	{	{	PUNCT
ejde-744	338	35	1	1	NUM
ejde-744	338	36	a(1	a(1	NUM
ejde-744	338	37	)	)	PUNCT
ejde-744	338	38	,	,	PUNCT
ejde-744	338	39	1}+	1}+	NUM
ejde-744	338	40	2ka	2ka	ADJ
ejde-744	338	41	1√	1√	PROPN
ejde-744	338	42	θa(1	θa(1	NOUN
ejde-744	338	43	)	)	PUNCT
ejde-744	338	44	)	)	PUNCT
ejde-744	338	45	.	.	PUNCT
ejde-744	339	1	ejde-2025/04	ejde-2025/04	X
ejde-744	339	2	exact	exact	ADJ
ejde-744	339	3	controllability	controllability	NOUN
ejde-744	339	4	13	13	NUM
ejde-744	339	5	6	6	NUM
ejde-744	339	6	.	.	PUNCT
ejde-744	340	1	controllability	controllability	NOUN
ejde-744	340	2	in	in	ADP
ejde-744	340	3	this	this	DET
ejde-744	340	4	section	section	NOUN
ejde-744	340	5	,	,	PUNCT
ejde-744	340	6	we	we	PRON
ejde-744	340	7	study	study	VERB
ejde-744	340	8	the	the	DET
ejde-744	340	9	problem	problem	NOUN
ejde-744	340	10	of	of	ADP
ejde-744	340	11	exact	exact	ADJ
ejde-744	340	12	controllability	controllability	NOUN
ejde-744	340	13	for	for	ADP
ejde-744	340	14	(	(	PUNCT
ejde-744	340	15	1.7	1.7	NUM
ejde-744	340	16	)	)	PUNCT
ejde-744	340	17	.	.	PUNCT
ejde-744	341	1	by	by	ADP
ejde-744	341	2	its	its	PRON
ejde-744	341	3	linearity	linearity	NOUN
ejde-744	341	4	and	and	CCONJ
ejde-744	341	5	reversibility	reversibility	NOUN
ejde-744	341	6	,	,	PUNCT
ejde-744	341	7	it	it	PRON
ejde-744	341	8	is	be	AUX
ejde-744	341	9	straightforward	straightforward	ADJ
ejde-744	341	10	to	to	PART
ejde-744	341	11	verify	verify	VERB
ejde-744	341	12	that	that	DET
ejde-744	341	13	exact	exact	ADJ
ejde-744	341	14	controllability	controllability	NOUN
ejde-744	341	15	will	will	AUX
ejde-744	341	16	hold	hold	VERB
ejde-744	341	17	as	as	ADV
ejde-744	341	18	long	long	ADV
ejde-744	341	19	as	as	SCONJ
ejde-744	341	20	it	it	PRON
ejde-744	341	21	is	be	AUX
ejde-744	341	22	valid	valid	ADJ
ejde-744	341	23	for	for	ADP
ejde-744	341	24	any	any	DET
ejde-744	341	25	initial	initial	ADJ
ejde-744	341	26	data	datum	NOUN
ejde-744	341	27	(	(	PUNCT
ejde-744	341	28	y0	y0	NOUN
ejde-744	341	29	,	,	PUNCT
ejde-744	341	30	y1	y1	NOUN
ejde-744	341	31	)	)	PUNCT
ejde-744	341	32	and	and	CCONJ
ejde-744	341	33	a	a	DET
ejde-744	341	34	zero	zero	NUM
ejde-744	341	35	final	final	ADJ
ejde-744	341	36	state	state	NOUN
ejde-744	341	37	.	.	PUNCT
ejde-744	342	1	equivalently	equivalently	ADV
ejde-744	342	2	,	,	PUNCT
ejde-744	342	3	given	give	VERB
ejde-744	342	4	(	(	PUNCT
ejde-744	342	5	y0	y0	NOUN
ejde-744	342	6	,	,	PUNCT
ejde-744	342	7	y1	y1	ADJ
ejde-744	342	8	)	)	PUNCT
ejde-744	342	9	∈	∈	PROPN
ejde-744	342	10	h1	h1	VERB
ejde-744	342	11	a	a	PRON
ejde-744	342	12	,	,	PUNCT
ejde-744	342	13	b(0	b(0	NOUN
ejde-744	342	14	,	,	PUNCT
ejde-744	342	15	1	1	NUM
ejde-744	342	16	)	)	PUNCT
ejde-744	342	17	×	×	NOUN
ejde-744	342	18	l2(0	l2(0	NOUN
ejde-744	342	19	,	,	PUNCT
ejde-744	342	20	1	1	NUM
ejde-744	342	21	)	)	PUNCT
ejde-744	342	22	,	,	PUNCT
ejde-744	342	23	we	we	PRON
ejde-744	342	24	seek	seek	VERB
ejde-744	342	25	a	a	DET
ejde-744	342	26	control	control	NOUN
ejde-744	342	27	function	function	NOUN
ejde-744	342	28	f	f	PROPN
ejde-744	342	29	∈	∈	PROPN
ejde-744	342	30	l2(0	l2(0	PROPN
ejde-744	342	31	,	,	PUNCT
ejde-744	342	32	t	t	PROPN
ejde-744	342	33	)	)	PUNCT
ejde-744	342	34	such	such	ADJ
ejde-744	342	35	that	that	SCONJ
ejde-744	342	36	the	the	DET
ejde-744	342	37	solution	solution	NOUN
ejde-744	342	38	of	of	ADP
ejde-744	342	39	(	(	PUNCT
ejde-744	342	40	1.7	1.7	NUM
ejde-744	342	41	)	)	PUNCT
ejde-744	342	42	satisfies	satisfie	NOUN
ejde-744	342	43	(	(	PUNCT
ejde-744	342	44	y	y	NOUN
ejde-744	342	45	,	,	PUNCT
ejde-744	342	46	y′)(t	y′)(t	NOUN
ejde-744	342	47	,	,	PUNCT
ejde-744	342	48	·	·	PUNCT
ejde-744	342	49	)	)	PUNCT
ejde-744	342	50	≡	≡	PROPN
ejde-744	342	51	0	0	PUNCT
ejde-744	342	52	.	.	PUNCT
ejde-744	343	1	definition	definition	NOUN
ejde-744	343	2	6.1	6.1	NUM
ejde-744	343	3	.	.	PUNCT
ejde-744	344	1	let	let	VERB
ejde-744	344	2	f	f	PROPN
ejde-744	344	3	∈	∈	PROPN
ejde-744	344	4	l2	l2	NOUN
ejde-744	344	5	loc(0	loc(0	VERB
ejde-744	344	6	,	,	PUNCT
ejde-744	344	7	t	t	PROPN
ejde-744	344	8	)	)	PUNCT
ejde-744	344	9	and	and	CCONJ
ejde-744	344	10	(	(	PUNCT
ejde-744	344	11	y0	y0	NOUN
ejde-744	344	12	,	,	PUNCT
ejde-744	344	13	y1	y1	ADJ
ejde-744	344	14	)	)	PUNCT
ejde-744	344	15	∈	∈	PROPN
ejde-744	344	16	l2(0	l2(0	NOUN
ejde-744	344	17	,	,	PUNCT
ejde-744	344	18	1)×h−1	1)×h−1	X
ejde-744	344	19	a	a	PRON
ejde-744	344	20	,	,	PUNCT
ejde-744	344	21	b	b	PROPN
ejde-744	344	22	(	(	PUNCT
ejde-744	344	23	0	0	NUM
ejde-744	344	24	,	,	PUNCT
ejde-744	344	25	1	1	NUM
ejde-744	344	26	)	)	PUNCT
ejde-744	344	27	be	be	AUX
ejde-744	344	28	arbitrarily	arbitrarily	ADV
ejde-744	344	29	fixed	fix	VERB
ejde-744	344	30	.	.	PUNCT
ejde-744	345	1	we	we	PRON
ejde-744	345	2	say	say	VERB
ejde-744	345	3	that	that	SCONJ
ejde-744	345	4	y	y	PROPN
ejde-744	345	5	is	be	AUX
ejde-744	345	6	a	a	DET
ejde-744	345	7	solution	solution	NOUN
ejde-744	345	8	by	by	ADP
ejde-744	345	9	transposition	transposition	NOUN
ejde-744	345	10	of	of	ADP
ejde-744	345	11	(	(	PUNCT
ejde-744	345	12	1.7	1.7	NUM
ejde-744	345	13	)	)	PUNCT
ejde-744	345	14	if	if	SCONJ
ejde-744	345	15	y	y	PROPN
ejde-744	345	16	∈	∈	PROPN
ejde-744	345	17	c1([0	c1([0	PROPN
ejde-744	345	18	,	,	PUNCT
ejde-744	345	19	t	t	X
ejde-744	345	20	]	]	PUNCT
ejde-744	345	21	;	;	PUNCT
ejde-744	345	22	h−1	h−1	PROPN
ejde-744	345	23	a	a	PRON
ejde-744	345	24	,	,	PUNCT
ejde-744	345	25	b	b	PROPN
ejde-744	345	26	(	(	PUNCT
ejde-744	345	27	0	0	NUM
ejde-744	345	28	,	,	PUNCT
ejde-744	345	29	1	1	NUM
ejde-744	345	30	)	)	PUNCT
ejde-744	345	31	∩	∩	NOUN
ejde-744	345	32	c([0	c([0	NOUN
ejde-744	345	33	,	,	PUNCT
ejde-744	345	34	t	t	X
ejde-744	345	35	]	]	PUNCT
ejde-744	345	36	;	;	PUNCT
ejde-744	345	37	l2(0	l2(0	NOUN
ejde-744	345	38	,	,	PUNCT
ejde-744	345	39	1	1	NUM
ejde-744	345	40	)	)	PUNCT
ejde-744	345	41	)	)	PUNCT
ejde-744	345	42	and	and	CCONJ
ejde-744	345	43	for	for	ADP
ejde-744	345	44	any	any	DET
ejde-744	345	45	t	t	PROPN
ejde-744	345	46	>	>	X
ejde-744	345	47	0	0	NUM
ejde-744	345	48	,	,	PUNCT
ejde-744	345	49	⟨yt(t	⟨yt(t	NOUN
ejde-744	345	50	)	)	PUNCT
ejde-744	345	51	,	,	PUNCT
ejde-744	345	52	w0	w0	PROPN
ejde-744	345	53	t	t	PROPN
ejde-744	345	54	⟩h−1	⟩h−1	PROPN
ejde-744	345	55	a	a	DET
ejde-744	345	56	(	(	PUNCT
ejde-744	345	57	0,1)×h1	0,1)×h1	NUM
ejde-744	345	58	a(0,1	a(0,1	NOUN
ejde-744	345	59	)	)	PUNCT
ejde-744	345	60	−	−	NOUN
ejde-744	346	1	∫	∫	PROPN
ejde-744	346	2	1	1	NUM
ejde-744	346	3	0	0	NUM
ejde-744	347	1	y(t	y(t	NOUN
ejde-744	347	2	)	)	PUNCT
ejde-744	348	1	w1	w1	PROPN
ejde-744	348	2	t	t	PROPN
ejde-744	348	3	dx	dx	PROPN
ejde-744	348	4	=	=	SYM
ejde-744	348	5	⟨y1	⟨y1	PROPN
ejde-744	348	6	,	,	PUNCT
ejde-744	348	7	w(0)⟩h−1	w(0)⟩h−1	ADP
ejde-744	348	8	a	a	PRON
ejde-744	348	9	,	,	PUNCT
ejde-744	348	10	b(0,1)×h1	b(0,1)×h1	ADP
ejde-744	348	11	a	a	DET
ejde-744	348	12	,	,	PUNCT
ejde-744	348	13	b(0,1	b(0,1	NOUN
ejde-744	348	14	)	)	PUNCT
ejde-744	348	15	−	−	NOUN
ejde-744	349	1	∫	∫	PROPN
ejde-744	349	2	1	1	NUM
ejde-744	349	3	0	0	NUM
ejde-744	349	4	y(0)w′(0	y(0)w′(0	NOUN
ejde-744	349	5	)	)	PUNCT
ejde-744	349	6	dx+	dx+	NOUN
ejde-744	349	7	a(1	a(1	NOUN
ejde-744	349	8	)	)	PUNCT
ejde-744	350	1	∫	∫	PROPN
ejde-744	350	2	t	t	NOUN
ejde-744	350	3	0	0	NUM
ejde-744	350	4	f(t)wx(t	f(t)wx(t	PROPN
ejde-744	350	5	,	,	PUNCT
ejde-744	350	6	1	1	NUM
ejde-744	350	7	)	)	PUNCT
ejde-744	350	8	dt	dt	NOUN
ejde-744	350	9	(	(	PUNCT
ejde-744	350	10	6.1	6.1	NUM
ejde-744	350	11	)	)	PUNCT
ejde-744	350	12	for	for	ADP
ejde-744	350	13	all	all	PRON
ejde-744	350	14	(	(	PUNCT
ejde-744	350	15	w0	w0	PROPN
ejde-744	350	16	t	t	PROPN
ejde-744	350	17	,	,	PUNCT
ejde-744	350	18	w	w	PROPN
ejde-744	350	19	1	1	NUM
ejde-744	350	20	t	t	NOUN
ejde-744	350	21	)	)	PUNCT
ejde-744	350	22	∈	∈	PROPN
ejde-744	350	23	h1	h1	VERB
ejde-744	350	24	a	a	PRON
ejde-744	350	25	,	,	PUNCT
ejde-744	350	26	b(0	b(0	NOUN
ejde-744	350	27	,	,	PUNCT
ejde-744	350	28	1	1	NUM
ejde-744	350	29	)	)	PUNCT
ejde-744	350	30	×	×	NOUN
ejde-744	350	31	l2(0	l2(0	NOUN
ejde-744	350	32	,	,	PUNCT
ejde-744	350	33	1	1	NUM
ejde-744	350	34	)	)	PUNCT
ejde-744	350	35	,	,	PUNCT
ejde-744	350	36	where	where	SCONJ
ejde-744	350	37	w	w	NOUN
ejde-744	350	38	is	be	AUX
ejde-744	350	39	the	the	DET
ejde-744	350	40	solution	solution	NOUN
ejde-744	350	41	of	of	ADP
ejde-744	350	42	the	the	DET
ejde-744	350	43	backward	backward	ADJ
ejde-744	350	44	equation	equation	NOUN
ejde-744	350	45	wtt	wtt	PROPN
ejde-744	351	1	−	−	PROPN
ejde-744	351	2	(	(	PUNCT
ejde-744	351	3	a(x)wx)x	a(x)wx)x	PROPN
ejde-744	351	4	−	−	PROPN
ejde-744	351	5	λ	λ	NOUN
ejde-744	351	6	b(x	b(x	NOUN
ejde-744	351	7	)	)	PUNCT
ejde-744	351	8	w	w	NOUN
ejde-744	351	9	=	=	SYM
ejde-744	351	10	0	0	NUM
ejde-744	351	11	,	,	PUNCT
ejde-744	351	12	(	(	PUNCT
ejde-744	351	13	t	t	PROPN
ejde-744	351	14	,	,	PUNCT
ejde-744	351	15	x	x	NOUN
ejde-744	351	16	)	)	PUNCT
ejde-744	351	17	∈	∈	PROPN
ejde-744	351	18	(	(	PUNCT
ejde-744	351	19	0	0	NUM
ejde-744	351	20	,	,	PUNCT
ejde-744	351	21	t	t	NOUN
ejde-744	351	22	)	)	PUNCT
ejde-744	351	23	×	×	NOUN
ejde-744	351	24	(	(	PUNCT
ejde-744	351	25	0	0	NUM
ejde-744	351	26	,	,	PUNCT
ejde-744	351	27	1	1	NUM
ejde-744	351	28	)	)	PUNCT
ejde-744	351	29	,	,	PUNCT
ejde-744	351	30	w(t	w(t	PROPN
ejde-744	351	31	,	,	PUNCT
ejde-744	351	32	1	1	X
ejde-744	351	33	)	)	PUNCT
ejde-744	351	34	=	=	SYM
ejde-744	351	35	0	0	NUM
ejde-744	351	36	,	,	PUNCT
ejde-744	351	37	t	t	PROPN
ejde-744	351	38	∈	∈	PROPN
ejde-744	351	39	(	(	PUNCT
ejde-744	351	40	0	0	NUM
ejde-744	351	41	,	,	PUNCT
ejde-744	351	42	t	t	NOUN
ejde-744	351	43	)	)	PUNCT
ejde-744	351	44	,	,	PUNCT
ejde-744	351	45	w(t	w(t	PROPN
ejde-744	351	46	,	,	PUNCT
ejde-744	351	47	0	0	NUM
ejde-744	351	48	)	)	PUNCT
ejde-744	351	49	=	=	SYM
ejde-744	351	50	0	0	NUM
ejde-744	351	51	,	,	PUNCT
ejde-744	351	52	ka	ka	PROPN
ejde-744	351	53	∈	∈	PROPN
ejde-744	352	1	[	[	X
ejde-744	352	2	0	0	NUM
ejde-744	352	3	,	,	PUNCT
ejde-744	352	4	1	1	NUM
ejde-744	352	5	)	)	PUNCT
ejde-744	352	6	,	,	PUNCT
ejde-744	352	7	t	t	PROPN
ejde-744	352	8	∈	∈	PROPN
ejde-744	352	9	(	(	PUNCT
ejde-744	352	10	0	0	NUM
ejde-744	352	11	,	,	PUNCT
ejde-744	352	12	t	t	NOUN
ejde-744	352	13	)	)	PUNCT
ejde-744	352	14	,	,	PUNCT
ejde-744	352	15	a(x)wx(t	a(x)wx(t	PROPN
ejde-744	352	16	,	,	PUNCT
ejde-744	352	17	0	0	NUM
ejde-744	352	18	)	)	PUNCT
ejde-744	352	19	=	=	SYM
ejde-744	352	20	0	0	NUM
ejde-744	352	21	,	,	PUNCT
ejde-744	352	22	ka	ka	PROPN
ejde-744	352	23	∈	∈	PROPN
ejde-744	352	24	(	(	PUNCT
ejde-744	352	25	1	1	NUM
ejde-744	352	26	,	,	PUNCT
ejde-744	352	27	2	2	NUM
ejde-744	352	28	)	)	PUNCT
ejde-744	352	29	,	,	PUNCT
ejde-744	352	30	]	]	X
ejde-744	352	31	;	;	PUNCT
ejde-744	352	32	t	t	PROPN
ejde-744	352	33	∈	∈	PROPN
ejde-744	352	34	(	(	PUNCT
ejde-744	352	35	0	0	NUM
ejde-744	352	36	,	,	PUNCT
ejde-744	352	37	t	t	NOUN
ejde-744	352	38	)	)	PUNCT
ejde-744	352	39	,	,	PUNCT
ejde-744	352	40	w(t	w(t	PROPN
ejde-744	352	41	,	,	PUNCT
ejde-744	352	42	x	x	X
ejde-744	352	43	)	)	PUNCT
ejde-744	352	44	=	=	SYM
ejde-744	352	45	w0	w0	PROPN
ejde-744	352	46	t	t	PROPN
ejde-744	352	47	(	(	PUNCT
ejde-744	352	48	x	x	NOUN
ejde-744	352	49	)	)	PUNCT
ejde-744	352	50	,	,	PUNCT
ejde-744	352	51	wt(t	wt(t	NOUN
ejde-744	352	52	,	,	PUNCT
ejde-744	352	53	x	x	X
ejde-744	352	54	)	)	PUNCT
ejde-744	352	55	=	=	SYM
ejde-744	352	56	w1	w1	PROPN
ejde-744	352	57	t	t	PROPN
ejde-744	352	58	(	(	PUNCT
ejde-744	352	59	x	x	NOUN
ejde-744	352	60	)	)	PUNCT
ejde-744	352	61	,	,	PUNCT
ejde-744	352	62	x	x	PUNCT
ejde-744	352	63	∈	∈	PROPN
ejde-744	352	64	(	(	PUNCT
ejde-744	352	65	0	0	NUM
ejde-744	352	66	,	,	PUNCT
ejde-744	352	67	1	1	NUM
ejde-744	352	68	)	)	PUNCT
ejde-744	352	69	.	.	PUNCT
ejde-744	353	1	(	(	PUNCT
ejde-744	353	2	6.2	6.2	NUM
ejde-744	353	3	)	)	PUNCT
ejde-744	353	4	by	by	ADP
ejde-744	353	5	setting	set	VERB
ejde-744	353	6	y(t	y(t	PROPN
ejde-744	353	7	,	,	PUNCT
ejde-744	353	8	x	x	NOUN
ejde-744	353	9	)	)	PUNCT
ejde-744	353	10	=	=	SYM
ejde-744	353	11	w(t	w(t	PROPN
ejde-744	353	12	−	−	PROPN
ejde-744	353	13	t	t	PROPN
ejde-744	353	14	,	,	PUNCT
ejde-744	353	15	x	x	NOUN
ejde-744	353	16	)	)	PUNCT
ejde-744	353	17	,	,	PUNCT
ejde-744	353	18	we	we	PRON
ejde-744	353	19	leverage	leverage	VERB
ejde-744	353	20	the	the	DET
ejde-744	353	21	time	time	NOUN
ejde-744	353	22	reversibility	reversibility	NOUN
ejde-744	353	23	of	of	ADP
ejde-744	353	24	the	the	DET
ejde-744	353	25	wave	wave	NOUN
ejde-744	353	26	equation	equation	NOUN
ejde-744	353	27	to	to	PART
ejde-744	353	28	assert	assert	VERB
ejde-744	353	29	that	that	SCONJ
ejde-744	353	30	the	the	DET
ejde-744	353	31	solution	solution	NOUN
ejde-744	353	32	y	y	PROPN
ejde-744	353	33	maintains	maintain	VERB
ejde-744	353	34	the	the	DET
ejde-744	353	35	same	same	ADJ
ejde-744	353	36	regularity	regularity	NOUN
ejde-744	353	37	as	as	ADP
ejde-744	353	38	w	w	NOUN
ejde-744	353	39	for	for	ADP
ejde-744	353	40	t	t	NOUN
ejde-744	353	41	≤	≤	NUM
ejde-744	353	42	0	0	NUM
ejde-744	353	43	.	.	PUNCT
ejde-744	354	1	consequently	consequently	ADV
ejde-744	354	2	,	,	PUNCT
ejde-744	354	3	the	the	DET
ejde-744	354	4	backward	backward	ADJ
ejde-744	354	5	equation	equation	NOUN
ejde-744	354	6	(	(	PUNCT
ejde-744	354	7	6.2	6.2	NUM
ejde-744	354	8	)	)	PUNCT
ejde-744	354	9	admits	admit	VERB
ejde-744	354	10	a	a	DET
ejde-744	354	11	unique	unique	ADJ
ejde-744	354	12	solution	solution	NOUN
ejde-744	354	13	w	w	PROPN
ejde-744	354	14	∈	∈	PROPN
ejde-744	354	15	c1([0	c1([0	PROPN
ejde-744	354	16	,	,	PUNCT
ejde-744	354	17	t	t	X
ejde-744	354	18	]	]	PUNCT
ejde-744	354	19	;	;	PUNCT
ejde-744	354	20	l2(0	l2(0	NOUN
ejde-744	354	21	,	,	PUNCT
ejde-744	354	22	1	1	NUM
ejde-744	354	23	)	)	PUNCT
ejde-744	354	24	∩	∩	NOUN
ejde-744	354	25	c([0	c([0	NOUN
ejde-744	354	26	,	,	PUNCT
ejde-744	354	27	t	t	X
ejde-744	354	28	]	]	PUNCT
ejde-744	354	29	;	;	PUNCT
ejde-744	354	30	h1	h1	VERB
ejde-744	354	31	a	a	DET
ejde-744	354	32	,	,	PUNCT
ejde-744	354	33	b(0	b(0	NOUN
ejde-744	354	34	,	,	PUNCT
ejde-744	354	35	1	1	NUM
ejde-744	354	36	)	)	PUNCT
ejde-744	354	37	)	)	PUNCT
ejde-744	354	38	which	which	PRON
ejde-744	354	39	depends	depend	VERB
ejde-744	354	40	continuously	continuously	ADV
ejde-744	354	41	on	on	ADP
ejde-744	354	42	the	the	DET
ejde-744	354	43	initial	initial	ADJ
ejde-744	354	44	data	datum	NOUN
ejde-744	354	45	wt	wt	NOUN
ejde-744	354	46	=	=	SYM
ejde-744	354	47	(	(	PUNCT
ejde-744	354	48	w0	w0	PROPN
ejde-744	354	49	t	t	PROPN
ejde-744	354	50	,	,	PUNCT
ejde-744	355	1	w	w	PROPN
ejde-744	355	2	1	1	NUM
ejde-744	355	3	t	t	NOUN
ejde-744	355	4	)	)	PUNCT
ejde-744	355	5	∈	∈	PROPN
ejde-744	355	6	h.	h.	PROPN
ejde-744	355	7	by	by	ADP
ejde-744	355	8	proposition	proposition	NOUN
ejde-744	355	9	4.2	4.2	NUM
ejde-744	355	10	,	,	PUNCT
ejde-744	355	11	the	the	DET
ejde-744	355	12	energy	energy	NOUN
ejde-744	355	13	ew(t	ew(t	NOUN
ejde-744	355	14	)	)	PUNCT
ejde-744	355	15	of	of	ADP
ejde-744	355	16	w	w	PROPN
ejde-744	355	17	is	be	AUX
ejde-744	355	18	conserved	conserve	VERB
ejde-744	355	19	through	through	ADP
ejde-744	355	20	time	time	NOUN
ejde-744	355	21	,	,	PUNCT
ejde-744	355	22	which	which	PRON
ejde-744	355	23	implies	imply	VERB
ejde-744	355	24	that	that	SCONJ
ejde-744	355	25	the	the	DET
ejde-744	355	26	direct	direct	ADJ
ejde-744	355	27	inequality	inequality	NOUN
ejde-744	355	28	(	(	PUNCT
ejde-744	355	29	4.5	4.5	NUM
ejde-744	355	30	)	)	PUNCT
ejde-744	355	31	and	and	CCONJ
ejde-744	355	32	observabilities	observabilities	ADP
ejde-744	355	33	inequality	inequality	NOUN
ejde-744	355	34	(	(	PUNCT
ejde-744	355	35	5.3	5.3	NUM
ejde-744	355	36	)	)	PUNCT
ejde-744	355	37	and	and	CCONJ
ejde-744	355	38	(	(	PUNCT
ejde-744	355	39	5.4	5.4	NUM
ejde-744	355	40	)	)	PUNCT
ejde-744	355	41	remain	remain	VERB
ejde-744	355	42	valid	valid	ADJ
ejde-744	355	43	for	for	ADP
ejde-744	355	44	w.	w.	PROPN
ejde-744	355	45	therefore	therefore	ADV
ejde-744	355	46	,	,	PUNCT
ejde-744	355	47	there	there	PRON
ejde-744	355	48	is	be	VERB
ejde-744	355	49	a	a	DET
ejde-744	355	50	unique	unique	ADJ
ejde-744	355	51	solution	solution	NOUN
ejde-744	355	52	by	by	ADP
ejde-744	355	53	transposition	transposition	NOUN
ejde-744	355	54	w	w	PROPN
ejde-744	355	55	∈	∈	PROPN
ejde-744	355	56	c1([0	c1([0	PROPN
ejde-744	355	57	,	,	PUNCT
ejde-744	355	58	t	t	X
ejde-744	355	59	]	]	PUNCT
ejde-744	355	60	;	;	PUNCT
ejde-744	355	61	h−1	h−1	PROPN
ejde-744	355	62	a	a	PRON
ejde-744	355	63	,	,	PUNCT
ejde-744	355	64	b	b	PROPN
ejde-744	355	65	(	(	PUNCT
ejde-744	355	66	0	0	NUM
ejde-744	355	67	,	,	PUNCT
ejde-744	355	68	1	1	NUM
ejde-744	355	69	)	)	PUNCT
ejde-744	355	70	)	)	PUNCT
ejde-744	355	71	∩	∩	NOUN
ejde-744	355	72	c([0	c([0	NOUN
ejde-744	355	73	,	,	PUNCT
ejde-744	355	74	t	t	X
ejde-744	355	75	]	]	PUNCT
ejde-744	355	76	;	;	PUNCT
ejde-744	355	77	l2(0	l2(0	NOUN
ejde-744	355	78	,	,	PUNCT
ejde-744	355	79	1	1	NUM
ejde-744	355	80	)	)	PUNCT
ejde-744	355	81	)	)	PUNCT
ejde-744	355	82	.	.	PUNCT
ejde-744	356	1	now	now	ADV
ejde-744	356	2	,	,	PUNCT
ejde-744	356	3	consider	consider	VERB
ejde-744	356	4	the	the	DET
ejde-744	356	5	bilinear	bilinear	NOUN
ejde-744	356	6	form	form	NOUN
ejde-744	356	7	λ	λ	X
ejde-744	356	8	:	:	PUNCT
ejde-744	356	9	h×h	h×h	NOUN
ejde-744	356	10	→	→	SYM
ejde-744	356	11	r	r	NOUN
ejde-744	356	12	defined	define	VERB
ejde-744	356	13	as	as	ADP
ejde-744	356	14	λ(wt	λ(wt	PROPN
ejde-744	356	15	,	,	PUNCT
ejde-744	356	16	w̃t	w̃t	NOUN
ejde-744	356	17	)	)	PUNCT
ejde-744	356	18	=	=	PUNCT
ejde-744	356	19	a(1	a(1	PROPN
ejde-744	356	20	)	)	PUNCT
ejde-744	356	21	∫	∫	PROPN
ejde-744	356	22	t	t	PROPN
ejde-744	356	23	0	0	NUM
ejde-744	356	24	wx(t	wx(t	PROPN
ejde-744	356	25	,	,	PUNCT
ejde-744	356	26	1)w̃x(t	1)w̃x(t	NUM
ejde-744	356	27	,	,	PUNCT
ejde-744	356	28	1)dt	1)dt	NUM
ejde-744	356	29	,	,	PUNCT
ejde-744	356	30	(	(	PUNCT
ejde-744	356	31	6.3	6.3	NUM
ejde-744	356	32	)	)	PUNCT
ejde-744	356	33	where	where	SCONJ
ejde-744	356	34	wx	wx	PROPN
ejde-744	356	35	,	,	PUNCT
ejde-744	356	36	w̃x	w̃x	PROPN
ejde-744	356	37	are	be	AUX
ejde-744	356	38	the	the	DET
ejde-744	356	39	solution	solution	NOUN
ejde-744	356	40	of	of	ADP
ejde-744	356	41	(	(	PUNCT
ejde-744	356	42	1.7	1.7	NUM
ejde-744	356	43	)	)	PUNCT
ejde-744	356	44	associated	associate	VERB
ejde-744	356	45	with	with	ADP
ejde-744	356	46	the	the	DET
ejde-744	356	47	final	final	ADJ
ejde-744	356	48	datawt	datawt	NOUN
ejde-744	356	49	:	:	PUNCT
ejde-744	356	50	=	=	SYM
ejde-744	356	51	(	(	PUNCT
ejde-744	356	52	w0	w0	PROPN
ejde-744	356	53	t	t	PROPN
ejde-744	356	54	,	,	PUNCT
ejde-744	356	55	w	w	PROPN
ejde-744	356	56	1	1	NUM
ejde-744	356	57	t	t	NOUN
ejde-744	356	58	)	)	PUNCT
ejde-744	356	59	,	,	PUNCT
ejde-744	356	60	w̃t	w̃t	ADP
ejde-744	356	61	:	:	PUNCT
ejde-744	356	62	=	=	SYM
ejde-744	356	63	(	(	PUNCT
ejde-744	356	64	w̃0	w̃0	PROPN
ejde-744	356	65	t	t	PROPN
ejde-744	356	66	,	,	PUNCT
ejde-744	356	67	w̃	w̃	PROPN
ejde-744	356	68	1	1	NUM
ejde-744	356	69	t	t	NOUN
ejde-744	356	70	)	)	PUNCT
ejde-744	356	71	,	,	PUNCT
ejde-744	356	72	respectively	respectively	ADV
ejde-744	356	73	.	.	PUNCT
ejde-744	357	1	to	to	PART
ejde-744	357	2	prove	prove	VERB
ejde-744	357	3	that	that	SCONJ
ejde-744	357	4	(	(	PUNCT
ejde-744	357	5	1.7	1.7	NUM
ejde-744	357	6	)	)	PUNCT
ejde-744	357	7	is	be	AUX
ejde-744	357	8	exactly	exactly	ADV
ejde-744	357	9	controllable	controllable	ADJ
ejde-744	357	10	,	,	PUNCT
ejde-744	357	11	the	the	DET
ejde-744	357	12	following	follow	VERB
ejde-744	357	13	lemma	lemma	PROPN
ejde-744	357	14	is	be	AUX
ejde-744	357	15	key	key	ADJ
ejde-744	357	16	.	.	PUNCT
ejde-744	358	1	lemma	lemma	PROPN
ejde-744	358	2	6.2	6.2	NUM
ejde-744	358	3	.	.	PUNCT
ejde-744	359	1	under	under	ADP
ejde-744	359	2	assumption	assumption	NOUN
ejde-744	359	3	2.6	2.6	NUM
ejde-744	359	4	,	,	PUNCT
ejde-744	359	5	the	the	DET
ejde-744	359	6	bilinear	bilinear	NOUN
ejde-744	359	7	form	form	NOUN
ejde-744	359	8	λ	λ	PROPN
ejde-744	359	9	is	be	AUX
ejde-744	359	10	continuous	continuous	ADJ
ejde-744	359	11	and	and	CCONJ
ejde-744	359	12	coercive	coercive	ADJ
ejde-744	359	13	.	.	PUNCT
ejde-744	360	1	14	14	NUM
ejde-744	360	2	g.	g.	PROPN
ejde-744	360	3	zhang	zhang	PROPN
ejde-744	360	4	,	,	PUNCT
ejde-744	360	5	s.	s.	PROPN
ejde-744	360	6	chai	chai	NOUN
ejde-744	360	7	ejde-2025/04	ejde-2025/04	NOUN
ejde-744	360	8	proof	proof	NOUN
ejde-744	360	9	.	.	PUNCT
ejde-744	361	1	by	by	ADP
ejde-744	361	2	the	the	DET
ejde-744	361	3	direct	direct	ADJ
ejde-744	361	4	inequality	inequality	NOUN
ejde-744	361	5	and	and	CCONJ
ejde-744	361	6	the	the	DET
ejde-744	361	7	result	result	NOUN
ejde-744	361	8	of	of	ADP
ejde-744	361	9	energy	energy	NOUN
ejde-744	361	10	conservation	conservation	NOUN
ejde-744	361	11	,	,	PUNCT
ejde-744	361	12	|λ(wt	|λ(wt	PROPN
ejde-744	361	13	,	,	PUNCT
ejde-744	361	14	w̃t	w̃t	NOUN
ejde-744	361	15	)	)	PUNCT
ejde-744	362	1	|	|	ADV
ejde-744	362	2	≤	≤	NUM
ejde-744	362	3	a(1	a(1	ADJ
ejde-744	362	4	)	)	PUNCT
ejde-744	362	5	∫	∫	PROPN
ejde-744	362	6	t	t	NOUN
ejde-744	362	7	0	0	NUM
ejde-744	362	8	|wx(t	|wx(t	PROPN
ejde-744	362	9	,	,	PUNCT
ejde-744	362	10	1)w̃x(t	1)w̃x(t	NUM
ejde-744	362	11	,	,	PUNCT
ejde-744	362	12	1)|	1)|	NUM
ejde-744	362	13	dt	dt	NOUN
ejde-744	362	14	≤	≤	PROPN
ejde-744	362	15	(	(	PUNCT
ejde-744	362	16	a(1	a(1	PROPN
ejde-744	362	17	)	)	PUNCT
ejde-744	362	18	∫	∫	PROPN
ejde-744	362	19	t	t	PROPN
ejde-744	362	20	0	0	NUM
ejde-744	362	21	w2	w2	PROPN
ejde-744	362	22	x(t	x(t	PROPN
ejde-744	362	23	,	,	PUNCT
ejde-744	362	24	1	1	NUM
ejde-744	362	25	)	)	PUNCT
ejde-744	362	26	dt	dt	NOUN
ejde-744	362	27	)	)	PUNCT
ejde-744	362	28	1/2	1/2	NUM
ejde-744	362	29	(	(	PUNCT
ejde-744	362	30	a(1	a(1	PROPN
ejde-744	362	31	)	)	PUNCT
ejde-744	362	32	∫	∫	PROPN
ejde-744	362	33	t	t	PROPN
ejde-744	362	34	0	0	NUM
ejde-744	362	35	w̃2	w̃2	PROPN
ejde-744	362	36	x(t	x(t	PROPN
ejde-744	362	37	,	,	PUNCT
ejde-744	362	38	1	1	NUM
ejde-744	362	39	)	)	PUNCT
ejde-744	362	40	dt	dt	NOUN
ejde-744	362	41	)	)	PUNCT
ejde-744	362	42	1/2	1/2	NUM
ejde-744	362	43	≤	≤	NUM
ejde-744	363	1	ce1/2	ce1/2	PROPN
ejde-744	363	2	w	w	PROPN
ejde-744	363	3	(	(	PUNCT
ejde-744	363	4	t	t	NOUN
ejde-744	363	5	)	)	PUNCT
ejde-744	363	6	e	e	X
ejde-744	363	7	1/2	1/2	NUM
ejde-744	363	8	w̃	w̃	PROPN
ejde-744	363	9	(	(	PUNCT
ejde-744	363	10	t	t	PROPN
ejde-744	363	11	)	)	PUNCT
ejde-744	363	12	≤	≤	PUNCT
ejde-744	363	13	c∥wt	c∥wt	PROPN
ejde-744	363	14	∥h∥w̃t	∥h∥w̃t	PROPN
ejde-744	363	15	∥h	∥h	NOUN
ejde-744	363	16	.	.	PUNCT
ejde-744	364	1	(	(	PUNCT
ejde-744	364	2	6.4	6.4	NUM
ejde-744	364	3	)	)	PUNCT
ejde-744	364	4	by	by	ADP
ejde-744	364	5	the	the	DET
ejde-744	364	6	observability	observability	NOUN
ejde-744	364	7	inequality	inequality	NOUN
ejde-744	364	8	,	,	PUNCT
ejde-744	364	9	we	we	PRON
ejde-744	364	10	have	have	VERB
ejde-744	364	11	λ(wt	λ(wt	PROPN
ejde-744	364	12	,	,	PUNCT
ejde-744	364	13	wt	wt	PROPN
ejde-744	364	14	)	)	PUNCT
ejde-744	364	15	=	=	PUNCT
ejde-744	365	1	a(1	a(1	PROPN
ejde-744	365	2	)	)	PUNCT
ejde-744	365	3	∫	∫	PROPN
ejde-744	365	4	t	t	PROPN
ejde-744	365	5	0	0	NUM
ejde-744	365	6	w2	w2	PROPN
ejde-744	365	7	x(t	x(t	PROPN
ejde-744	365	8	,	,	PUNCT
ejde-744	365	9	1	1	NUM
ejde-744	365	10	)	)	PUNCT
ejde-744	365	11	dt	dt	X
ejde-744	365	12	≥	≥	PROPN
ejde-744	365	13	ctew(t	ctew(t	PROPN
ejde-744	365	14	)	)	PUNCT
ejde-744	365	15	=	=	PUNCT
ejde-744	366	1	c∥wt	c∥wt	VERB
ejde-744	366	2	∥2h	∥2h	NUM
ejde-744	366	3	.	.	PUNCT
ejde-744	367	1	(	(	PUNCT
ejde-744	367	2	6.5	6.5	NUM
ejde-744	367	3	)	)	PUNCT
ejde-744	367	4	□	□	PUNCT
ejde-744	367	5	theorem	theorem	VERB
ejde-744	367	6	6.3	6.3	NUM
ejde-744	367	7	.	.	PUNCT
ejde-744	368	1	under	under	ADP
ejde-744	368	2	assumption	assumption	NOUN
ejde-744	368	3	2.6	2.6	NUM
ejde-744	368	4	,	,	PUNCT
ejde-744	368	5	for	for	ADP
ejde-744	368	6	all	all	DET
ejde-744	368	7	t	t	NOUN
ejde-744	368	8	>	>	X
ejde-744	368	9	ta	ta	PROPN
ejde-744	368	10	,	,	PUNCT
ejde-744	368	11	b	b	PROPN
ejde-744	368	12	and	and	CCONJ
ejde-744	368	13	for	for	ADP
ejde-744	368	14	all	all	DET
ejde-744	368	15	(	(	PUNCT
ejde-744	368	16	y0	y0	NOUN
ejde-744	368	17	,	,	PUNCT
ejde-744	368	18	y1	y1	ADJ
ejde-744	368	19	)	)	PUNCT
ejde-744	368	20	∈	∈	PROPN
ejde-744	368	21	l2(0	l2(0	NOUN
ejde-744	368	22	,	,	PUNCT
ejde-744	368	23	1	1	NUM
ejde-744	368	24	)	)	PUNCT
ejde-744	368	25	×h−1	×h−1	PROPN
ejde-744	368	26	a	a	DET
ejde-744	368	27	,	,	PUNCT
ejde-744	368	28	b	b	PROPN
ejde-744	368	29	(	(	PUNCT
ejde-744	368	30	0	0	NUM
ejde-744	368	31	,	,	PUNCT
ejde-744	368	32	1	1	NUM
ejde-744	368	33	)	)	PUNCT
ejde-744	368	34	,	,	PUNCT
ejde-744	368	35	there	there	PRON
ejde-744	368	36	exists	exist	VERB
ejde-744	368	37	a	a	DET
ejde-744	368	38	control	control	NOUN
ejde-744	368	39	f	f	PROPN
ejde-744	368	40	∈	∈	PROPN
ejde-744	368	41	l2(0	l2(0	PROPN
ejde-744	368	42	,	,	PUNCT
ejde-744	368	43	t	t	PROPN
ejde-744	368	44	)	)	PUNCT
ejde-744	368	45	such	such	ADJ
ejde-744	368	46	that	that	SCONJ
ejde-744	368	47	the	the	DET
ejde-744	368	48	solution	solution	NOUN
ejde-744	368	49	(	(	PUNCT
ejde-744	368	50	in	in	ADP
ejde-744	368	51	the	the	DET
ejde-744	368	52	sense	sense	NOUN
ejde-744	368	53	of	of	ADP
ejde-744	368	54	transposition	transposition	NOUN
ejde-744	368	55	)	)	PUNCT
ejde-744	368	56	satisfies	satisfy	VERB
ejde-744	368	57	y(t	y(t	NUM
ejde-744	368	58	,	,	PUNCT
ejde-744	368	59	x	x	NOUN
ejde-744	368	60	)	)	PUNCT
ejde-744	368	61	=	=	SYM
ejde-744	368	62	yt(t	yt(t	NOUN
ejde-744	368	63	,	,	PUNCT
ejde-744	368	64	x	x	X
ejde-744	368	65	)	)	PUNCT
ejde-744	368	66	=	=	SYM
ejde-744	368	67	0	0	NUM
ejde-744	368	68	,	,	PUNCT
ejde-744	368	69	∀x	∀x	X
ejde-744	368	70	∈	∈	PROPN
ejde-744	368	71	(	(	PUNCT
ejde-744	368	72	0	0	NUM
ejde-744	368	73	,	,	PUNCT
ejde-744	368	74	1	1	NUM
ejde-744	368	75	)	)	PUNCT
ejde-744	368	76	.	.	PUNCT
ejde-744	369	1	proof	proof	NOUN
ejde-744	369	2	.	.	PUNCT
ejde-744	370	1	we	we	PRON
ejde-744	370	2	define	define	VERB
ejde-744	370	3	the	the	DET
ejde-744	370	4	continuous	continuous	ADJ
ejde-744	370	5	linear	linear	NOUN
ejde-744	370	6	map	map	NOUN
ejde-744	370	7	l(wt	l(wt	PRON
ejde-744	370	8	)	)	PUNCT
ejde-744	371	1	=	=	PUNCT
ejde-744	371	2	∫	∫	PROPN
ejde-744	371	3	1	1	NUM
ejde-744	371	4	0	0	NUM
ejde-744	371	5	y0wt(0	y0wt(0	PROPN
ejde-744	371	6	)	)	PUNCT
ejde-744	371	7	dx−⟨y1	dx−⟨y1	ADJ
ejde-744	371	8	,	,	PUNCT
ejde-744	371	9	w(0)⟩h−1	w(0)⟩h−1	ADP
ejde-744	371	10	a	a	PRON
ejde-744	371	11	,	,	PUNCT
ejde-744	371	12	b(0,1)×h1	b(0,1)×h1	ADP
ejde-744	371	13	a	a	DET
ejde-744	371	14	,	,	PUNCT
ejde-744	371	15	b(0,1	b(0,1	NOUN
ejde-744	371	16	)	)	PUNCT
ejde-744	371	17	,	,	PUNCT
ejde-744	371	18	∀wt	∀wt	NUM
ejde-744	371	19	∈	∈	PROPN
ejde-744	371	20	h1	h1	VERB
ejde-744	371	21	a	a	DET
ejde-744	371	22	,	,	PUNCT
ejde-744	371	23	b(0	b(0	NOUN
ejde-744	371	24	,	,	PUNCT
ejde-744	371	25	1)×l2(0	1)×l2(0	NUM
ejde-744	371	26	,	,	PUNCT
ejde-744	371	27	1	1	NUM
ejde-744	371	28	)	)	PUNCT
ejde-744	371	29	.	.	PUNCT
ejde-744	372	1	thanks	thank	NOUN
ejde-744	372	2	to	to	AUX
ejde-744	372	3	lemma	lemma	PROPN
ejde-744	372	4	6.2	6.2	NUM
ejde-744	372	5	and	and	CCONJ
ejde-744	372	6	the	the	DET
ejde-744	372	7	lax	lax	PROPN
ejde-744	372	8	-	-	PUNCT
ejde-744	372	9	milgram	milgram	NOUN
ejde-744	372	10	theorem	theorem	NOUN
ejde-744	372	11	,	,	PUNCT
ejde-744	372	12	there	there	PRON
ejde-744	372	13	exists	exist	VERB
ejde-744	372	14	a	a	DET
ejde-744	372	15	unique	unique	ADJ
ejde-744	372	16	wt	wt	NOUN
ejde-744	372	17	∈	∈	PROPN
ejde-744	372	18	h1	h1	PROPN
ejde-744	372	19	a	a	PRON
ejde-744	372	20	,	,	PUNCT
ejde-744	372	21	b(0	b(0	NOUN
ejde-744	372	22	,	,	PUNCT
ejde-744	372	23	1)×	1)×	NUM
ejde-744	372	24	l2(0	l2(0	NOUN
ejde-744	372	25	,	,	PUNCT
ejde-744	372	26	1	1	NUM
ejde-744	372	27	)	)	PUNCT
ejde-744	372	28	such	such	ADJ
ejde-744	372	29	that	that	DET
ejde-744	372	30	λ(wt	λ(wt	PROPN
ejde-744	372	31	,	,	PUNCT
ejde-744	372	32	w̃t	w̃t	NOUN
ejde-744	372	33	)	)	PUNCT
ejde-744	372	34	=	=	SYM
ejde-744	372	35	l(w̃t	l(w̃t	PROPN
ejde-744	372	36	)	)	PUNCT
ejde-744	372	37	,	,	PUNCT
ejde-744	372	38	∀w̃t	∀w̃t	PROPN
ejde-744	372	39	∈	∈	PROPN
ejde-744	372	40	h1	h1	PROPN
ejde-744	372	41	a	a	PRON
ejde-744	372	42	,	,	PUNCT
ejde-744	372	43	b(0	b(0	NOUN
ejde-744	372	44	,	,	PUNCT
ejde-744	372	45	1)×	1)×	NUM
ejde-744	372	46	l2(0	l2(0	NOUN
ejde-744	372	47	,	,	PUNCT
ejde-744	372	48	1	1	NUM
ejde-744	372	49	)	)	PUNCT
ejde-744	372	50	.	.	PUNCT
ejde-744	373	1	(	(	PUNCT
ejde-744	373	2	6.6	6.6	NUM
ejde-744	373	3	)	)	PUNCT
ejde-744	373	4	we	we	PRON
ejde-744	373	5	set	set	VERB
ejde-744	373	6	f	f	PROPN
ejde-744	373	7	=	=	SYM
ejde-744	373	8	wx(t	wx(t	PROPN
ejde-744	373	9	,	,	PUNCT
ejde-744	373	10	1	1	NUM
ejde-744	373	11	)	)	PUNCT
ejde-744	373	12	and	and	CCONJ
ejde-744	373	13	denote	denote	VERB
ejde-744	373	14	by	by	ADP
ejde-744	373	15	y	y	PROPN
ejde-744	373	16	the	the	DET
ejde-744	373	17	solution	solution	NOUN
ejde-744	373	18	by	by	ADP
ejde-744	373	19	transposition	transposition	NOUN
ejde-744	373	20	of	of	ADP
ejde-744	373	21	(	(	PUNCT
ejde-744	373	22	1.7	1.7	NUM
ejde-744	373	23	)	)	PUNCT
ejde-744	373	24	.	.	PUNCT
ejde-744	374	1	then	then	ADV
ejde-744	374	2	we	we	PRON
ejde-744	374	3	have	have	VERB
ejde-744	374	4	a(1	a(1	NOUN
ejde-744	374	5	)	)	PUNCT
ejde-744	374	6	∫	∫	PROPN
ejde-744	374	7	t	t	NOUN
ejde-744	374	8	0	0	NUM
ejde-744	374	9	f(t)w̃x(t	f(t)w̃x(t	X
ejde-744	374	10	,	,	PUNCT
ejde-744	374	11	1	1	NUM
ejde-744	374	12	)	)	PUNCT
ejde-744	374	13	dt	dt	NOUN
ejde-744	375	1	=	=	PUNCT
ejde-744	375	2	a(1	a(1	PROPN
ejde-744	375	3	)	)	PUNCT
ejde-744	375	4	∫	∫	PROPN
ejde-744	375	5	t	t	PROPN
ejde-744	375	6	0	0	NUM
ejde-744	375	7	wx(t	wx(t	PROPN
ejde-744	375	8	,	,	PUNCT
ejde-744	375	9	1)w̃x(t	1)w̃x(t	NUM
ejde-744	375	10	,	,	PUNCT
ejde-744	375	11	1)dt	1)dt	PROPN
ejde-744	375	12	=	=	SYM
ejde-744	375	13	λ(wt	λ(wt	PROPN
ejde-744	375	14	,	,	PUNCT
ejde-744	375	15	w̃t	w̃t	NOUN
ejde-744	375	16	)	)	PUNCT
ejde-744	376	1	=	=	SYM
ejde-744	376	2	l(w̃t	l(w̃t	PROPN
ejde-744	376	3	)	)	PUNCT
ejde-744	377	1	=	=	PUNCT
ejde-744	377	2	∫	∫	PROPN
ejde-744	377	3	1	1	NUM
ejde-744	377	4	0	0	NUM
ejde-744	377	5	u0w̃t(0	u0w̃t(0	NOUN
ejde-744	377	6	)	)	PUNCT
ejde-744	377	7	dx−	dx−	PROPN
ejde-744	378	1	⟨u1	⟨u1	PROPN
ejde-744	378	2	,	,	PUNCT
ejde-744	378	3	w̃(0)⟩h−1	w̃(0)⟩h−1	ADV
ejde-744	378	4	a	a	PRON
ejde-744	378	5	,	,	PUNCT
ejde-744	378	6	b(0,1)×h1	b(0,1)×h1	ADP
ejde-744	378	7	a	a	DET
ejde-744	378	8	,	,	PUNCT
ejde-744	378	9	b(0,1	b(0,1	NOUN
ejde-744	378	10	)	)	PUNCT
ejde-744	378	11	,	,	PUNCT
ejde-744	378	12	(	(	PUNCT
ejde-744	378	13	6.7	6.7	NUM
ejde-744	378	14	)	)	PUNCT
ejde-744	378	15	for	for	ADP
ejde-744	378	16	all	all	PRON
ejde-744	378	17	(	(	PUNCT
ejde-744	378	18	w̃0	w̃0	PROPN
ejde-744	378	19	t	t	PROPN
ejde-744	378	20	,	,	PUNCT
ejde-744	379	1	w̃	w̃	PROPN
ejde-744	379	2	1	1	NUM
ejde-744	379	3	t	t	NOUN
ejde-744	379	4	)	)	PUNCT
ejde-744	379	5	∈	∈	PROPN
ejde-744	379	6	h1	h1	VERB
ejde-744	379	7	a	a	DET
ejde-744	379	8	,	,	PUNCT
ejde-744	379	9	b(0	b(0	NOUN
ejde-744	379	10	,	,	PUNCT
ejde-744	379	11	1)×l2(0	1)×l2(0	NUM
ejde-744	379	12	,	,	PUNCT
ejde-744	379	13	1	1	NUM
ejde-744	379	14	)	)	PUNCT
ejde-744	379	15	.	.	PUNCT
ejde-744	380	1	on	on	ADP
ejde-744	380	2	the	the	DET
ejde-744	380	3	other	other	ADJ
ejde-744	380	4	hand	hand	NOUN
ejde-744	380	5	,	,	PUNCT
ejde-744	380	6	by	by	ADP
ejde-744	380	7	the	the	DET
ejde-744	380	8	definition	definition	NOUN
ejde-744	380	9	of	of	ADP
ejde-744	380	10	the	the	DET
ejde-744	380	11	solution	solution	NOUN
ejde-744	380	12	by	by	ADP
ejde-744	380	13	transposition	transposition	NOUN
ejde-744	380	14	,	,	PUNCT
ejde-744	380	15	for	for	ADP
ejde-744	380	16	all	all	PRON
ejde-744	380	17	(	(	PUNCT
ejde-744	380	18	w̃0	w̃0	PROPN
ejde-744	380	19	t	t	PROPN
ejde-744	380	20	,	,	PUNCT
ejde-744	380	21	w̃	w̃	PROPN
ejde-744	380	22	1	1	NUM
ejde-744	380	23	t	t	NOUN
ejde-744	380	24	)	)	PUNCT
ejde-744	380	25	∈	∈	PROPN
ejde-744	380	26	h1	h1	VERB
ejde-744	380	27	a	a	PRON
ejde-744	380	28	,	,	PUNCT
ejde-744	380	29	b(0	b(0	NOUN
ejde-744	380	30	,	,	PUNCT
ejde-744	380	31	1)×	1)×	NUM
ejde-744	380	32	l2(0	l2(0	NOUN
ejde-744	380	33	,	,	PUNCT
ejde-744	380	34	1	1	NUM
ejde-744	380	35	)	)	PUNCT
ejde-744	380	36	we	we	PRON
ejde-744	380	37	have	have	VERB
ejde-744	380	38	a(1	a(1	NOUN
ejde-744	380	39	)	)	PUNCT
ejde-744	381	1	∫	∫	PROPN
ejde-744	381	2	t	t	NOUN
ejde-744	381	3	0	0	NUM
ejde-744	381	4	f(t)w̃x(t	f(t)w̃x(t	X
ejde-744	381	5	,	,	PUNCT
ejde-744	381	6	1	1	NUM
ejde-744	381	7	)	)	PUNCT
ejde-744	381	8	dt	dt	NOUN
ejde-744	382	1	=	=	SYM
ejde-744	382	2	∫	∫	PROPN
ejde-744	382	3	t	t	NOUN
ejde-744	382	4	0	0	NUM
ejde-744	383	1	y(t	y(t	NOUN
ejde-744	383	2	)	)	PUNCT
ejde-744	384	1	w̃1	w̃1	PROPN
ejde-744	384	2	t	t	PROPN
ejde-744	384	3	dt−	dt−	PROPN
ejde-744	384	4	⟨yt(t	⟨yt(t	PROPN
ejde-744	384	5	)	)	PUNCT
ejde-744	384	6	,	,	PUNCT
ejde-744	384	7	w̃0	w̃0	PROPN
ejde-744	385	1	t	t	PROPN
ejde-744	385	2	⟩h−1	⟩h−1	PROPN
ejde-744	385	3	a	a	PRON
ejde-744	385	4	,	,	PUNCT
ejde-744	385	5	b(0,1)×h1	b(0,1)×h1	VERB
ejde-744	385	6	a	a	DET
ejde-744	385	7	,	,	PUNCT
ejde-744	385	8	b(0,1	b(0,1	NOUN
ejde-744	385	9	)	)	PUNCT
ejde-744	385	10	+	+	CCONJ
ejde-744	386	1	⟨y1	⟨y1	ADJ
ejde-744	386	2	,	,	PUNCT
ejde-744	386	3	w̃(0)⟩h−1	w̃(0)⟩h−1	DET
ejde-744	386	4	a	a	DET
ejde-744	386	5	(	(	PUNCT
ejde-744	386	6	0,1)×h1	0,1)×h1	NUM
ejde-744	386	7	a	a	DET
ejde-744	386	8	,	,	PUNCT
ejde-744	386	9	b(0,1	b(0,1	NOUN
ejde-744	386	10	)	)	PUNCT
ejde-744	386	11	−	−	NOUN
ejde-744	386	12	∫	∫	PROPN
ejde-744	386	13	1	1	NUM
ejde-744	386	14	0	0	NUM
ejde-744	386	15	y(0)wt(0	y(0)wt(0	NOUN
ejde-744	386	16	)	)	PUNCT
ejde-744	386	17	dx	dx	PROPN
ejde-744	386	18	.	.	PUNCT
ejde-744	387	1	(	(	PUNCT
ejde-744	387	2	6.8	6.8	NUM
ejde-744	387	3	)	)	PUNCT
ejde-744	387	4	by	by	ADP
ejde-744	387	5	equations	equation	NOUN
ejde-744	387	6	(	(	PUNCT
ejde-744	387	7	6.7	6.7	NUM
ejde-744	387	8	)	)	PUNCT
ejde-744	387	9	and	and	CCONJ
ejde-744	387	10	(	(	PUNCT
ejde-744	387	11	6.8	6.8	NUM
ejde-744	387	12	)	)	PUNCT
ejde-744	387	13	,	,	PUNCT
ejde-744	387	14	we	we	PRON
ejde-744	387	15	deduce	deduce	VERB
ejde-744	387	16	that	that	PRON
ejde-744	387	17	⟨yt(t	⟨yt(t	X
ejde-744	387	18	)	)	PUNCT
ejde-744	387	19	,	,	PUNCT
ejde-744	387	20	w̃0	w̃0	PROPN
ejde-744	388	1	t	t	PROPN
ejde-744	388	2	⟩h−1	⟩h−1	PROPN
ejde-744	388	3	a	a	PRON
ejde-744	388	4	,	,	PUNCT
ejde-744	388	5	b(0,1)×h1	b(0,1)×h1	VERB
ejde-744	388	6	a	a	DET
ejde-744	388	7	,	,	PUNCT
ejde-744	388	8	b(0,1	b(0,1	NOUN
ejde-744	388	9	)	)	PUNCT
ejde-744	388	10	=	=	SYM
ejde-744	389	1	∫	∫	PROPN
ejde-744	389	2	t	t	NOUN
ejde-744	389	3	0	0	NUM
ejde-744	390	1	y(t	y(t	NOUN
ejde-744	390	2	)	)	PUNCT
ejde-744	391	1	w̃1	w̃1	PROPN
ejde-744	391	2	t	t	PROPN
ejde-744	391	3	dt	dt	PROPN
ejde-744	391	4	,	,	PUNCT
ejde-744	391	5	∀(w̃0	∀(w̃0	PROPN
ejde-744	391	6	t	t	NOUN
ejde-744	391	7	,	,	PUNCT
ejde-744	391	8	w̃	w̃	PROPN
ejde-744	391	9	1	1	NUM
ejde-744	391	10	t	t	NOUN
ejde-744	391	11	)	)	PUNCT
ejde-744	391	12	∈	∈	PROPN
ejde-744	391	13	h1	h1	VERB
ejde-744	391	14	a	a	PRON
ejde-744	391	15	,	,	PUNCT
ejde-744	391	16	b(0	b(0	NOUN
ejde-744	391	17	,	,	PUNCT
ejde-744	391	18	1)×	1)×	NUM
ejde-744	391	19	l2(0	l2(0	NOUN
ejde-744	391	20	,	,	PUNCT
ejde-744	391	21	1	1	NUM
ejde-744	391	22	)	)	PUNCT
ejde-744	391	23	.	.	PUNCT
ejde-744	392	1	ejde-2025/04	ejde-2025/04	X
ejde-744	392	2	exact	exact	ADJ
ejde-744	392	3	controllability	controllability	NOUN
ejde-744	392	4	15	15	NUM
ejde-744	392	5	hence	hence	ADV
ejde-744	392	6	,	,	PUNCT
ejde-744	392	7	we	we	PRON
ejde-744	392	8	conclude	conclude	VERB
ejde-744	392	9	that	that	PRON
ejde-744	392	10	y(t	y(t	PROPN
ejde-744	392	11	,	,	PUNCT
ejde-744	392	12	x	x	NOUN
ejde-744	392	13	)	)	PUNCT
ejde-744	392	14	=	=	SYM
ejde-744	392	15	yt(t	yt(t	NOUN
ejde-744	392	16	,	,	PUNCT
ejde-744	392	17	x	x	X
ejde-744	392	18	)	)	PUNCT
ejde-744	392	19	=	=	SYM
ejde-744	392	20	0	0	NUM
ejde-744	392	21	,	,	PUNCT
ejde-744	392	22	∀x	∀x	X
ejde-744	392	23	∈	∈	PROPN
ejde-744	392	24	(	(	PUNCT
ejde-744	392	25	0	0	NUM
ejde-744	392	26	,	,	PUNCT
ejde-744	392	27	1	1	NUM
ejde-744	392	28	)	)	PUNCT
ejde-744	392	29	.	.	PUNCT
ejde-744	393	1	□	□	PUNCT
ejde-744	393	2	7	7	X
ejde-744	393	3	.	.	X
ejde-744	393	4	conclusion	conclusion	NOUN
ejde-744	393	5	in	in	ADP
ejde-744	393	6	this	this	DET
ejde-744	393	7	article	article	NOUN
ejde-744	393	8	,	,	PUNCT
ejde-744	393	9	we	we	PRON
ejde-744	393	10	have	have	AUX
ejde-744	393	11	considered	consider	VERB
ejde-744	393	12	the	the	DET
ejde-744	393	13	controllability	controllability	NOUN
ejde-744	393	14	of	of	ADP
ejde-744	393	15	degenerate	degenerate	ADJ
ejde-744	393	16	and	and	CCONJ
ejde-744	393	17	singular	singular	ADJ
ejde-744	393	18	wave	wave	NOUN
ejde-744	393	19	equations	equation	NOUN
ejde-744	393	20	,	,	PUNCT
ejde-744	393	21	and	and	CCONJ
ejde-744	393	22	obtained	obtain	VERB
ejde-744	393	23	the	the	DET
ejde-744	393	24	exact	exact	ADJ
ejde-744	393	25	controllability	controllability	NOUN
ejde-744	393	26	of	of	ADP
ejde-744	393	27	the	the	DET
ejde-744	393	28	system	system	NOUN
ejde-744	393	29	under	under	ADP
ejde-744	393	30	certain	certain	ADJ
ejde-744	393	31	assumptions	assumption	NOUN
ejde-744	393	32	.	.	PUNCT
ejde-744	394	1	we	we	PRON
ejde-744	394	2	have	have	AUX
ejde-744	394	3	adopted	adopt	VERB
ejde-744	394	4	the	the	DET
ejde-744	394	5	coefficient	coefficient	NOUN
ejde-744	394	6	settings	setting	NOUN
ejde-744	394	7	from	from	ADP
ejde-744	394	8	reference	reference	NOUN
ejde-744	394	9	[	[	X
ejde-744	394	10	1	1	NUM
ejde-744	394	11	]	]	PUNCT
ejde-744	394	12	,	,	PUNCT
ejde-744	394	13	enhanced	enhance	VERB
ejde-744	394	14	the	the	DET
ejde-744	394	15	equation	equation	NOUN
ejde-744	394	16	system	system	NOUN
ejde-744	394	17	from	from	ADP
ejde-744	394	18	reference	reference	NOUN
ejde-744	394	19	[	[	X
ejde-744	394	20	3	3	NUM
ejde-744	394	21	]	]	PUNCT
ejde-744	394	22	,	,	PUNCT
ejde-744	394	23	and	and	CCONJ
ejde-744	394	24	replaced	replace	VERB
ejde-744	394	25	the	the	DET
ejde-744	394	26	particular	particular	ADJ
ejde-744	394	27	exponential	exponential	ADJ
ejde-744	394	28	form	form	NOUN
ejde-744	394	29	of	of	ADP
ejde-744	394	30	the	the	DET
ejde-744	394	31	degenerate	degenerate	ADJ
ejde-744	394	32	and	and	CCONJ
ejde-744	394	33	singular	singular	ADJ
ejde-744	394	34	coefficients	coefficient	NOUN
ejde-744	394	35	presented	present	VERB
ejde-744	394	36	in	in	ADP
ejde-744	394	37	[	[	X
ejde-744	394	38	3	3	NUM
ejde-744	394	39	]	]	PUNCT
ejde-744	394	40	with	with	ADP
ejde-744	394	41	a	a	DET
ejde-744	394	42	more	more	ADV
ejde-744	394	43	generalized	generalized	ADJ
ejde-744	394	44	form	form	NOUN
ejde-744	394	45	.	.	PUNCT
ejde-744	395	1	furthermore	furthermore	ADV
ejde-744	395	2	,	,	PUNCT
ejde-744	395	3	we	we	PRON
ejde-744	395	4	do	do	AUX
ejde-744	395	5	not	not	PART
ejde-744	395	6	necessitate	necessitate	VERB
ejde-744	395	7	the	the	DET
ejde-744	395	8	relationship	relationship	NOUN
ejde-744	395	9	between	between	ADP
ejde-744	395	10	the	the	DET
ejde-744	395	11	generalized	generalized	ADJ
ejde-744	395	12	coefficients	coefficient	NOUN
ejde-744	395	13	to	to	PART
ejde-744	395	14	be	be	AUX
ejde-744	395	15	as	as	ADV
ejde-744	395	16	stringent	stringent	ADJ
ejde-744	395	17	as	as	ADP
ejde-744	395	18	in	in	ADP
ejde-744	395	19	[	[	X
ejde-744	395	20	3	3	NUM
ejde-744	395	21	]	]	PUNCT
ejde-744	395	22	;	;	PUNCT
ejde-744	395	23	it	it	PRON
ejde-744	395	24	is	be	AUX
ejde-744	395	25	only	only	ADV
ejde-744	395	26	necessary	necessary	ADJ
ejde-744	395	27	that	that	SCONJ
ejde-744	395	28	their	their	PRON
ejde-744	395	29	sum	sum	NOUN
ejde-744	395	30	falls	fall	VERB
ejde-744	395	31	within	within	ADP
ejde-744	395	32	a	a	DET
ejde-744	395	33	certain	certain	ADJ
ejde-744	395	34	range	range	NOUN
ejde-744	395	35	(	(	PUNCT
ejde-744	395	36	ka	ka	PROPN
ejde-744	396	1	+	+	PROPN
ejde-744	396	2	kb	kb	PROPN
ejde-744	396	3	≤	≤	ADJ
ejde-744	396	4	2	2	NUM
ejde-744	396	5	)	)	PUNCT
ejde-744	396	6	.	.	PUNCT
ejde-744	397	1	this	this	PRON
ejde-744	397	2	allows	allow	VERB
ejde-744	397	3	for	for	ADP
ejde-744	397	4	the	the	DET
ejde-744	397	5	application	application	NOUN
ejde-744	397	6	of	of	ADP
ejde-744	397	7	diverse	diverse	ADJ
ejde-744	397	8	technical	technical	ADJ
ejde-744	397	9	approaches	approach	NOUN
ejde-744	397	10	when	when	SCONJ
ejde-744	397	11	addressing	address	VERB
ejde-744	397	12	the	the	DET
ejde-744	397	13	singular	singular	ADJ
ejde-744	397	14	term	term	NOUN
ejde-744	397	15	,	,	PUNCT
ejde-744	397	16	leading	lead	VERB
ejde-744	397	17	to	to	ADP
ejde-744	397	18	various	various	ADJ
ejde-744	397	19	trade	trade	NOUN
ejde-744	397	20	-	-	PUNCT
ejde-744	397	21	offs	off	NOUN
ejde-744	397	22	in	in	ADP
ejde-744	397	23	parameter	parameter	NOUN
ejde-744	397	24	λ	λ	PROPN
ejde-744	397	25	selection	selection	NOUN
ejde-744	397	26	.	.	PUNCT
ejde-744	398	1	from	from	ADP
ejde-744	398	2	this	this	DET
ejde-744	398	3	perspective	perspective	NOUN
ejde-744	398	4	,	,	PUNCT
ejde-744	398	5	article	article	NOUN
ejde-744	398	6	[	[	X
ejde-744	398	7	3	3	X
ejde-744	398	8	]	]	PUNCT
ejde-744	398	9	can	can	AUX
ejde-744	398	10	be	be	AUX
ejde-744	398	11	viewed	view	VERB
ejde-744	398	12	as	as	ADP
ejde-744	398	13	a	a	DET
ejde-744	398	14	special	special	ADJ
ejde-744	398	15	case	case	NOUN
ejde-744	398	16	of	of	ADP
ejde-744	398	17	this	this	DET
ejde-744	398	18	paper	paper	NOUN
ejde-744	398	19	,	,	PUNCT
ejde-744	398	20	which	which	PRON
ejde-744	398	21	is	be	AUX
ejde-744	398	22	also	also	ADV
ejde-744	398	23	one	one	NUM
ejde-744	398	24	of	of	ADP
ejde-744	398	25	the	the	DET
ejde-744	398	26	novel	novel	ADJ
ejde-744	398	27	aspects	aspect	NOUN
ejde-744	398	28	of	of	ADP
ejde-744	398	29	this	this	DET
ejde-744	398	30	work	work	NOUN
ejde-744	398	31	.	.	PUNCT
ejde-744	399	1	to	to	PART
ejde-744	399	2	more	more	ADV
ejde-744	399	3	intuitively	intuitively	ADV
ejde-744	399	4	illustrate	illustrate	VERB
ejde-744	399	5	this	this	DET
ejde-744	399	6	point	point	NOUN
ejde-744	399	7	,	,	PUNCT
ejde-744	399	8	we	we	PRON
ejde-744	399	9	have	have	AUX
ejde-744	399	10	provided	provide	VERB
ejde-744	399	11	an	an	DET
ejde-744	399	12	example	example	NOUN
ejde-744	399	13	other	other	ADJ
ejde-744	399	14	than	than	ADP
ejde-744	399	15	xα	xα	INTJ
ejde-744	399	16	as	as	SCONJ
ejde-744	399	17	follows	follow	VERB
ejde-744	399	18	.	.	PUNCT
ejde-744	400	1	example	example	NOUN
ejde-744	400	2	.	.	PUNCT
ejde-744	401	1	let	let	VERB
ejde-744	401	2	θ	θ	PROPN
ejde-744	401	3	∈	∈	PROPN
ejde-744	401	4	(	(	PUNCT
ejde-744	401	5	0	0	NUM
ejde-744	401	6	,	,	PUNCT
ejde-744	401	7	2	2	NUM
ejde-744	401	8	)	)	PUNCT
ejde-744	401	9	be	be	AUX
ejde-744	401	10	given	give	VERB
ejde-744	401	11	,	,	PUNCT
ejde-744	401	12	we	we	PRON
ejde-744	401	13	construct	construct	VERB
ejde-744	401	14	the	the	DET
ejde-744	401	15	functions	function	NOUN
ejde-744	401	16	a(x	a(x	NOUN
ejde-744	401	17	)	)	PUNCT
ejde-744	401	18	,	,	PUNCT
ejde-744	401	19	b(x	b(x	NOUN
ejde-744	401	20	)	)	PUNCT
ejde-744	401	21	as	as	SCONJ
ejde-744	401	22	follows	follow	VERB
ejde-744	401	23	a(x	a(x	NOUN
ejde-744	401	24	)	)	PUNCT
ejde-744	401	25	=	=	PRON
ejde-744	401	26	{	{	PUNCT
ejde-744	401	27	xθ	xθ	PROPN
ejde-744	402	1	(	(	PUNCT
ejde-744	402	2	1	1	NUM
ejde-744	402	3	+	+	CCONJ
ejde-744	402	4	sin2(lnxα	sin2(lnxα	X
ejde-744	402	5	)	)	PUNCT
ejde-744	402	6	)	)	PUNCT
ejde-744	403	1	if	if	SCONJ
ejde-744	403	2	x	x	PUNCT
ejde-744	403	3	∈	∈	PROPN
ejde-744	403	4	(	(	PUNCT
ejde-744	403	5	0	0	NUM
ejde-744	403	6	,	,	PUNCT
ejde-744	403	7	1	1	NUM
ejde-744	403	8	]	]	PUNCT
ejde-744	403	9	0	0	PUNCT
ejde-744	404	1	if	if	SCONJ
ejde-744	404	2	x	x	X
ejde-744	404	3	=	=	SYM
ejde-744	404	4	0	0	NUM
ejde-744	404	5	,	,	PUNCT
ejde-744	404	6	where	where	SCONJ
ejde-744	404	7	α	α	PRON
ejde-744	404	8	∈	∈	PROPN
ejde-744	404	9	(	(	PUNCT
ejde-744	404	10	0	0	NUM
ejde-744	404	11	,	,	PUNCT
ejde-744	404	12	1−	1−	NUM
ejde-744	404	13	θ/2	θ/2	NOUN
ejde-744	404	14	)	)	PUNCT
ejde-744	404	15	;	;	PUNCT
ejde-744	404	16	b(x	b(x	X
ejde-744	404	17	)	)	PUNCT
ejde-744	404	18	=	=	PRON
ejde-744	404	19	{	{	PUNCT
ejde-744	404	20	x2−θ	x2−θ	PROPN
ejde-744	404	21	(	(	PUNCT
ejde-744	404	22	1	1	NUM
ejde-744	404	23	+	+	NUM
ejde-744	404	24	sin2(lnxβ	sin2(lnxβ	NOUN
ejde-744	404	25	)	)	PUNCT
ejde-744	404	26	)	)	PUNCT
ejde-744	404	27	if	if	SCONJ
ejde-744	404	28	x	x	PUNCT
ejde-744	404	29	∈	∈	PROPN
ejde-744	404	30	(	(	PUNCT
ejde-744	404	31	0	0	NUM
ejde-744	404	32	,	,	PUNCT
ejde-744	404	33	1	1	NUM
ejde-744	404	34	]	]	PUNCT
ejde-744	404	35	0	0	PUNCT
ejde-744	405	1	if	if	SCONJ
ejde-744	405	2	x	x	X
ejde-744	405	3	=	=	SYM
ejde-744	405	4	0	0	NUM
ejde-744	405	5	,	,	PUNCT
ejde-744	405	6	where	where	SCONJ
ejde-744	405	7	β	β	X
ejde-744	405	8	∈	∈	PROPN
ejde-744	406	1	[	[	X
ejde-744	406	2	θ/2	θ/2	X
ejde-744	406	3	−	−	PROPN
ejde-744	406	4	1,−α	1,−α	NOUN
ejde-744	406	5	]	]	X
ejde-744	406	6	.	.	PUNCT
ejde-744	407	1	then	then	ADV
ejde-744	407	2	the	the	DET
ejde-744	407	3	functions	function	NOUN
ejde-744	407	4	a(x	a(x	NOUN
ejde-744	407	5	)	)	PUNCT
ejde-744	407	6	,	,	PUNCT
ejde-744	407	7	b(x	b(x	NOUN
ejde-744	407	8	)	)	PUNCT
ejde-744	407	9	satisfy	satisfy	NOUN
ejde-744	407	10	assumption	assumption	NOUN
ejde-744	407	11	2.1	2.1	NUM
ejde-744	407	12	.	.	PUNCT
ejde-744	408	1	indeed	indeed	ADV
ejde-744	408	2	,	,	PUNCT
ejde-744	408	3	a′(x	a′(x	PROPN
ejde-744	408	4	)	)	PUNCT
ejde-744	408	5	=	=	SYM
ejde-744	409	1	θxθ−1	θxθ−1	PROPN
ejde-744	409	2	(	(	PUNCT
ejde-744	409	3	1	1	NUM
ejde-744	409	4	+	+	CCONJ
ejde-744	409	5	sin2(lnxα	sin2(lnxα	X
ejde-744	409	6	)	)	PUNCT
ejde-744	409	7	)	)	PUNCT
ejde-744	410	1	+	+	CCONJ
ejde-744	411	1	2αxθ−1	2αxθ−1	NUM
ejde-744	411	2	sin(lnxα	sin(lnxα	NOUN
ejde-744	411	3	)	)	PUNCT
ejde-744	411	4	cos(lnxα	cos(lnxα	PROPN
ejde-744	411	5	)	)	PUNCT
ejde-744	411	6	so	so	SCONJ
ejde-744	411	7	that	that	SCONJ
ejde-744	411	8	ka	ka	PROPN
ejde-744	411	9	≤	≤	PROPN
ejde-744	411	10	θ+	θ+	PUNCT
ejde-744	411	11	2α	2α	NOUN
ejde-744	411	12	<	<	X
ejde-744	411	13	2	2	X
ejde-744	411	14	.	.	PUNCT
ejde-744	412	1	it	it	PRON
ejde-744	412	2	is	be	AUX
ejde-744	412	3	also	also	ADV
ejde-744	412	4	easy	easy	ADJ
ejde-744	412	5	to	to	PART
ejde-744	412	6	discover	discover	VERB
ejde-744	412	7	that	that	SCONJ
ejde-744	412	8	ka	ka	PROPN
ejde-744	412	9	is	be	AUX
ejde-744	412	10	not	not	PART
ejde-744	412	11	always	always	ADV
ejde-744	412	12	equal	equal	ADJ
ejde-744	412	13	to	to	ADP
ejde-744	412	14	1	1	NUM
ejde-744	412	15	.	.	PUNCT
ejde-744	412	16	b′(x	b′(x	PRON
ejde-744	412	17	)	)	PUNCT
ejde-744	412	18	=	=	PUNCT
ejde-744	413	1	(	(	PUNCT
ejde-744	413	2	2−	2−	NUM
ejde-744	413	3	θ)x1−θ	θ)x1−θ	PROPN
ejde-744	413	4	(	(	PUNCT
ejde-744	413	5	1	1	NUM
ejde-744	413	6	+	+	NUM
ejde-744	413	7	sin2(lnxβ	sin2(lnxβ	NOUN
ejde-744	413	8	)	)	PUNCT
ejde-744	413	9	)	)	PUNCT
ejde-744	414	1	+	+	CCONJ
ejde-744	414	2	2βx1−θ	2βx1−θ	NUM
ejde-744	414	3	sin(lnxβ	sin(lnxβ	NOUN
ejde-744	414	4	)	)	PUNCT
ejde-744	414	5	cos(lnxβ	cos(lnxβ	ADJ
ejde-744	414	6	)	)	PUNCT
ejde-744	414	7	so	so	SCONJ
ejde-744	414	8	that	that	SCONJ
ejde-744	414	9	kb	kb	PROPN
ejde-744	414	10	≤	≤	NUM
ejde-744	414	11	2−	2−	NUM
ejde-744	414	12	θ	θ	NOUN
ejde-744	414	13	+	+	CCONJ
ejde-744	414	14	2β	2β	NOUN
ejde-744	414	15	≤	≤	NUM
ejde-744	414	16	2	2	NUM
ejde-744	414	17	,	,	PUNCT
ejde-744	414	18	and	and	CCONJ
ejde-744	414	19	ka	ka	PROPN
ejde-744	415	1	+	+	PROPN
ejde-744	415	2	kb	kb	PROPN
ejde-744	415	3	≤	≤	ADV
ejde-744	415	4	2	2	NUM
ejde-744	415	5	+	+	CCONJ
ejde-744	415	6	2α+	2α+	NUM
ejde-744	415	7	2β	2β	NOUN
ejde-744	415	8	≤	≤	NUM
ejde-744	415	9	2	2	NUM
ejde-744	415	10	.	.	PUNCT
ejde-744	416	1	we	we	PRON
ejde-744	416	2	summarize	summarize	VERB
ejde-744	416	3	the	the	DET
ejde-744	416	4	comparison	comparison	NOUN
ejde-744	416	5	between	between	ADP
ejde-744	416	6	this	this	DET
ejde-744	416	7	paper	paper	NOUN
ejde-744	416	8	and	and	CCONJ
ejde-744	416	9	existing	exist	VERB
ejde-744	416	10	literature	literature	NOUN
ejde-744	416	11	as	as	SCONJ
ejde-744	416	12	follows	follow	VERB
ejde-744	416	13	.	.	PUNCT
ejde-744	417	1	degenerate	degenerate	ADJ
ejde-744	417	2	term	term	NOUN
ejde-744	417	3	singular	singular	ADJ
ejde-744	417	4	term	term	NOUN
ejde-744	417	5	in	in	ADP
ejde-744	417	6	[	[	X
ejde-744	417	7	3	3	X
ejde-744	417	8	]	]	PUNCT
ejde-744	417	9	xα	xα	PROPN
ejde-744	417	10	,	,	PUNCT
ejde-744	417	11	α	α	PROPN
ejde-744	417	12	∈	∈	PROPN
ejde-744	418	1	[	[	X
ejde-744	418	2	0	0	NUM
ejde-744	418	3	,	,	PUNCT
ejde-744	418	4	2	2	NUM
ejde-744	418	5	)	)	PUNCT
ejde-744	418	6	\	\	NOUN
ejde-744	418	7	{	{	PUNCT
ejde-744	418	8	1	1	NUM
ejde-744	418	9	}	}	PUNCT
ejde-744	418	10	x2−α	x2−α	PROPN
ejde-744	418	11	µ	µ	ADJ
ejde-744	418	12	≤	≤	NOUN
ejde-744	418	13	(	(	PUNCT
ejde-744	418	14	1−α)2	1−α)2	NUM
ejde-744	418	15	4	4	NUM
ejde-744	418	16	here	here	ADV
ejde-744	418	17	ka	ka	PROPN
ejde-744	419	1	:	:	PUNCT
ejde-744	419	2	=	=	SYM
ejde-744	419	3	sup0	sup0	PROPN
ejde-744	419	4	<	<	X
ejde-744	419	5	x≤1	x≤1	PROPN
ejde-744	419	6	x|a′(x)|	x|a′(x)|	PROPN
ejde-744	419	7	a(x	a(x	PROPN
ejde-744	419	8	)	)	PUNCT
ejde-744	419	9	ka	ka	PROPN
ejde-744	419	10	∈	∈	PROPN
ejde-744	420	1	[	[	X
ejde-744	420	2	0	0	NUM
ejde-744	420	3	,	,	PUNCT
ejde-744	420	4	2	2	NUM
ejde-744	420	5	)	)	PUNCT
ejde-744	420	6	\	\	NOUN
ejde-744	420	7	{	{	PUNCT
ejde-744	420	8	1	1	NUM
ejde-744	420	9	}	}	PUNCT
ejde-744	420	10	kb	kb	NOUN
ejde-744	420	11	:	:	PUNCT
ejde-744	420	12	=	=	SYM
ejde-744	420	13	sup0	sup0	PROPN
ejde-744	420	14	<	<	X
ejde-744	420	15	x≤1	x≤1	PROPN
ejde-744	420	16	x|b′(x)|	x|b′(x)|	NOUN
ejde-744	420	17	b(x	b(x	NOUN
ejde-744	420	18	)	)	PUNCT
ejde-744	420	19	kb	kb	PROPN
ejde-744	420	20	∈	∈	PROPN
ejde-744	421	1	[	[	X
ejde-744	421	2	0	0	NUM
ejde-744	421	3	,	,	PUNCT
ejde-744	421	4	2	2	NUM
ejde-744	421	5	]	]	PUNCT
ejde-744	421	6	ka	ka	PROPN
ejde-744	422	1	+	+	PROPN
ejde-744	422	2	kb	kb	PROPN
ejde-744	422	3	≤	≤	ADJ
ejde-744	422	4	2	2	NUM
ejde-744	422	5	ka−2	ka−2	PROPN
ejde-744	422	6	ca	can	AUX
ejde-744	422	7	,	,	PUNCT
ejde-744	422	8	b(2−kb−ka	b(2−kb−ka	PROPN
ejde-744	422	9	)	)	PUNCT
ejde-744	422	10	<	<	X
ejde-744	423	1	λ	λ	X
ejde-744	423	2	<	<	X
ejde-744	423	3	1	1	NUM
ejde-744	423	4	ca	ca	NOUN
ejde-744	423	5	,	,	PUNCT
ejde-744	423	6	b	b	NOUN
ejde-744	423	7	or	or	CCONJ
ejde-744	423	8	λ	λ	X
ejde-744	423	9	<	<	X
ejde-744	423	10	1	1	NUM
ejde-744	423	11	ca	ca	NOUN
ejde-744	423	12	,	,	PUNCT
ejde-744	423	13	b	b	NOUN
ejde-744	423	14	acknowledgments	acknowledgment	NOUN
ejde-744	423	15	.	.	PUNCT
ejde-744	424	1	this	this	DET
ejde-744	424	2	work	work	NOUN
ejde-744	424	3	was	be	AUX
ejde-744	424	4	supported	support	VERB
ejde-744	424	5	by	by	ADP
ejde-744	424	6	the	the	DET
ejde-744	424	7	national	national	ADJ
ejde-744	424	8	natural	natural	PROPN
ejde-744	424	9	science	science	PROPN
ejde-744	424	10	foundation	foundation	PROPN
ejde-744	424	11	of	of	ADP
ejde-744	424	12	china	china	PROPN
ejde-744	424	13	(	(	PUNCT
ejde-744	424	14	grant	grant	PROPN
ejde-744	424	15	12271316	12271316	NUM
ejde-744	424	16	)	)	PUNCT
ejde-744	424	17	.	.	PUNCT
ejde-744	425	1	16	16	NUM
ejde-744	425	2	g.	g.	PROPN
ejde-744	425	3	zhang	zhang	PROPN
ejde-744	425	4	,	,	PUNCT
ejde-744	425	5	s.	s.	PROPN
ejde-744	425	6	chai	chai	PROPN
ejde-744	425	7	ejde-2025/04	ejde-2025/04	NOUN
ejde-744	425	8	references	reference	NOUN
ejde-744	425	9	[	[	X
ejde-744	425	10	1	1	NUM
ejde-744	425	11	]	]	PUNCT
ejde-744	425	12	f.	f.	PROPN
ejde-744	425	13	alabau	alabau	PROPN
ejde-744	425	14	-	-	PUNCT
ejde-744	425	15	boussouira	boussouira	PROPN
ejde-744	425	16	,	,	PUNCT
ejde-744	425	17	p.	p.	PROPN
ejde-744	425	18	cannarsa	cannarsa	PROPN
ejde-744	425	19	,	,	PUNCT
ejde-744	425	20	g.	g.	PROPN
ejde-744	425	21	leugering	leugering	PROPN
ejde-744	425	22	;	;	PUNCT
ejde-744	425	23	control	control	NOUN
ejde-744	425	24	and	and	CCONJ
ejde-744	425	25	stabilization	stabilization	NOUN
ejde-744	425	26	of	of	ADP
ejde-744	425	27	degenerate	degenerate	ADJ
ejde-744	425	28	wave	wave	NOUN
ejde-744	425	29	equations	equation	NOUN
ejde-744	425	30	,	,	PUNCT
ejde-744	425	31	siam	siam	ADJ
ejde-744	425	32	journal	journal	NOUN
ejde-744	425	33	on	on	ADP
ejde-744	425	34	control	control	NOUN
ejde-744	425	35	and	and	CCONJ
ejde-744	425	36	optimization	optimization	NOUN
ejde-744	425	37	,	,	PUNCT
ejde-744	425	38	2017	2017	NUM
ejde-744	425	39	,	,	PUNCT
ejde-744	425	40	55(3	55(3	NUM
ejde-744	425	41	):	):	PUNCT
ejde-744	425	42	2052	2052	NUM
ejde-744	425	43	-	-	SYM
ejde-744	425	44	2087	2087	NUM
ejde-744	425	45	.	.	PUNCT
ejde-744	426	1	[	[	X
ejde-744	426	2	2	2	NUM
ejde-744	426	3	]	]	PUNCT
ejde-744	426	4	f.	f.	PROPN
ejde-744	426	5	alabau	alabau	PROPN
ejde-744	426	6	-	-	PUNCT
ejde-744	426	7	boussouira	boussouira	PROPN
ejde-744	426	8	,	,	PUNCT
ejde-744	426	9	p.	p.	PROPN
ejde-744	426	10	cannarsa	cannarsa	PROPN
ejde-744	426	11	,	,	PUNCT
ejde-744	426	12	g.	g.	PROPN
ejde-744	426	13	fragnelli	fragnelli	PROPN
ejde-744	426	14	;	;	PUNCT
ejde-744	426	15	carleman	carleman	ADJ
ejde-744	426	16	estimates	estimate	NOUN
ejde-744	426	17	for	for	ADP
ejde-744	426	18	degenerate	degenerate	ADJ
ejde-744	426	19	parabolic	parabolic	NOUN
ejde-744	426	20	operators	operator	NOUN
ejde-744	426	21	with	with	ADP
ejde-744	426	22	applications	application	NOUN
ejde-744	426	23	to	to	ADP
ejde-744	426	24	null	null	ADJ
ejde-744	426	25	controllability	controllability	NOUN
ejde-744	426	26	,	,	PUNCT
ejde-744	426	27	journal	journal	NOUN
ejde-744	426	28	of	of	ADP
ejde-744	426	29	evolution	evolution	NOUN
ejde-744	426	30	equations	equation	NOUN
ejde-744	426	31	,	,	PUNCT
ejde-744	426	32	2006	2006	NUM
ejde-744	426	33	,	,	PUNCT
ejde-744	426	34	6	6	NUM
ejde-744	426	35	:	:	SYM
ejde-744	426	36	161	161	NUM
ejde-744	426	37	-	-	SYM
ejde-744	426	38	204	204	NUM
ejde-744	426	39	.	.	PUNCT
ejde-744	427	1	[	[	X
ejde-744	427	2	3	3	X
ejde-744	427	3	]	]	X
ejde-744	427	4	b.	b.	PROPN
ejde-744	427	5	allal	allal	PROPN
ejde-744	427	6	,	,	PUNCT
ejde-744	427	7	m.	m.	NOUN
ejde-744	427	8	alhabib	alhabib	PROPN
ejde-744	427	9	,	,	PUNCT
ejde-744	427	10	s.	s.	PROPN
ejde-744	427	11	jawad	jawad	PROPN
ejde-744	427	12	;	;	PUNCT
ejde-744	427	13	boundary	boundary	ADJ
ejde-744	427	14	controllability	controllability	NOUN
ejde-744	427	15	for	for	ADP
ejde-744	427	16	a	a	DET
ejde-744	427	17	degenerate	degenerate	ADJ
ejde-744	427	18	and	and	CCONJ
ejde-744	427	19	singular	singular	ADJ
ejde-744	427	20	wave	wave	NOUN
ejde-744	427	21	equation	equation	NOUN
ejde-744	427	22	,	,	PUNCT
ejde-744	427	23	mathematical	mathematical	ADJ
ejde-744	427	24	methods	method	NOUN
ejde-744	427	25	in	in	ADP
ejde-744	427	26	the	the	DET
ejde-744	427	27	applied	apply	VERB
ejde-744	427	28	sciences	science	NOUN
ejde-744	427	29	,	,	PUNCT
ejde-744	427	30	2022	2022	NUM
ejde-744	427	31	,	,	PUNCT
ejde-744	427	32	45(17	45(17	NUM
ejde-744	427	33	):	):	PUNCT
ejde-744	427	34	11526	11526	NUM
ejde-744	427	35	-	-	SYM
ejde-744	427	36	11544	11544	NUM
ejde-744	427	37	.	.	PUNCT
ejde-744	428	1	[	[	X
ejde-744	428	2	4	4	X
ejde-744	428	3	]	]	PUNCT
ejde-744	428	4	j.	j.	PROPN
ejde-744	428	5	bai	bai	PROPN
ejde-744	428	6	,	,	PUNCT
ejde-744	428	7	s.	s.	PROPN
ejde-744	428	8	chai	chai	PROPN
ejde-744	428	9	;	;	PUNCT
ejde-744	428	10	exact	exact	ADJ
ejde-744	428	11	controllability	controllability	NOUN
ejde-744	428	12	for	for	ADP
ejde-744	428	13	a	a	DET
ejde-744	428	14	one	one	NUM
ejde-744	428	15	-	-	PUNCT
ejde-744	428	16	dimensional	dimensional	ADJ
ejde-744	428	17	degenerate	degenerate	ADJ
ejde-744	428	18	wave	wave	NOUN
ejde-744	428	19	equation	equation	NOUN
ejde-744	428	20	in	in	ADP
ejde-744	428	21	domains	domain	NOUN
ejde-744	428	22	with	with	ADP
ejde-744	428	23	moving	move	VERB
ejde-744	428	24	boundary	boundary	ADJ
ejde-744	428	25	,	,	PUNCT
ejde-744	428	26	applied	apply	VERB
ejde-744	428	27	mathematics	mathematics	NOUN
ejde-744	428	28	letters	letter	NOUN
ejde-744	428	29	,	,	PUNCT
ejde-744	428	30	2021	2021	NUM
ejde-744	428	31	,	,	PUNCT
ejde-744	428	32	119	119	NUM
ejde-744	428	33	:	:	SYM
ejde-744	428	34	107235	107235	NUM
ejde-744	428	35	.	.	PUNCT
ejde-744	429	1	[	[	X
ejde-744	429	2	5	5	X
ejde-744	429	3	]	]	PUNCT
ejde-744	429	4	j.	j.	PROPN
ejde-744	429	5	bai	bai	PROPN
ejde-744	429	6	,	,	PUNCT
ejde-744	429	7	s.	s.	PROPN
ejde-744	429	8	chai	chai	PROPN
ejde-744	429	9	;	;	PUNCT
ejde-744	429	10	exact	exact	ADJ
ejde-744	429	11	controllability	controllability	NOUN
ejde-744	429	12	of	of	ADP
ejde-744	429	13	wave	wave	NOUN
ejde-744	429	14	equations	equation	NOUN
ejde-744	429	15	with	with	ADP
ejde-744	429	16	interior	interior	ADJ
ejde-744	429	17	degeneracy	degeneracy	NOUN
ejde-744	429	18	and	and	CCONJ
ejde-744	429	19	onesided	oneside	VERB
ejde-744	429	20	boundary	boundary	ADJ
ejde-744	429	21	control	control	NOUN
ejde-744	429	22	,	,	PUNCT
ejde-744	429	23	journal	journal	NOUN
ejde-744	429	24	of	of	ADP
ejde-744	429	25	systems	system	NOUN
ejde-744	429	26	science	science	NOUN
ejde-744	429	27	and	and	CCONJ
ejde-744	429	28	complexity	complexity	NOUN
ejde-744	429	29	,	,	PUNCT
ejde-744	429	30	2023	2023	NUM
ejde-744	429	31	,	,	PUNCT
ejde-744	429	32	36(2	36(2	NUM
ejde-744	429	33	):	):	PUNCT
ejde-744	429	34	656	656	NUM
ejde-744	429	35	-	-	SYM
ejde-744	429	36	671	671	NUM
ejde-744	429	37	.	.	PUNCT
ejde-744	430	1	[	[	X
ejde-744	430	2	6	6	NUM
ejde-744	430	3	]	]	PUNCT
ejde-744	430	4	i.	i.	NOUN
ejde-744	430	5	boutaayamou	boutaayamou	PROPN
ejde-744	430	6	,	,	PUNCT
ejde-744	430	7	g.	g.	PROPN
ejde-744	430	8	fragnelli	fragnelli	PROPN
ejde-744	430	9	,	,	PUNCT
ejde-744	430	10	l.	l.	PROPN
ejde-744	430	11	maniar	maniar	PROPN
ejde-744	430	12	;	;	PUNCT
ejde-744	430	13	carleman	carleman	ADJ
ejde-744	430	14	estimates	estimate	NOUN
ejde-744	430	15	for	for	ADP
ejde-744	430	16	parabolic	parabolic	ADJ
ejde-744	430	17	equations	equation	NOUN
ejde-744	430	18	with	with	ADP
ejde-744	430	19	interior	interior	ADJ
ejde-744	430	20	degeneracy	degeneracy	NOUN
ejde-744	430	21	and	and	CCONJ
ejde-744	430	22	neumann	neumann	PROPN
ejde-744	430	23	boundary	boundary	ADJ
ejde-744	430	24	conditions	condition	NOUN
ejde-744	430	25	.	.	PUNCT
ejde-744	431	1	journal	journal	PROPN
ejde-744	431	2	d’analyse	d’analyse	PROPN
ejde-744	431	3	mathématique	mathématique	PROPN
ejde-744	431	4	,	,	PUNCT
ejde-744	431	5	2018	2018	NUM
ejde-744	431	6	,	,	PUNCT
ejde-744	431	7	135	135	NUM
ejde-744	431	8	:	:	SYM
ejde-744	431	9	1	1	NUM
ejde-744	431	10	-	-	SYM
ejde-744	431	11	35	35	NUM
ejde-744	431	12	.	.	PUNCT
ejde-744	432	1	[	[	X
ejde-744	432	2	7	7	NUM
ejde-744	432	3	]	]	X
ejde-744	432	4	i.	i.	NOUN
ejde-744	432	5	boutaayamou	boutaayamou	PROPN
ejde-744	432	6	,	,	PUNCT
ejde-744	432	7	g.	g.	PROPN
ejde-744	432	8	fragnelli	fragnelli	PROPN
ejde-744	432	9	,	,	PUNCT
ejde-744	432	10	d.	d.	PROPN
ejde-744	432	11	mugnai	mugnai	PROPN
ejde-744	432	12	;	;	PUNCT
ejde-744	432	13	boundary	boundary	ADJ
ejde-744	432	14	controllability	controllability	NOUN
ejde-744	432	15	for	for	ADP
ejde-744	432	16	a	a	DET
ejde-744	432	17	degenerate	degenerate	ADJ
ejde-744	432	18	wave	wave	NOUN
ejde-744	432	19	equation	equation	NOUN
ejde-744	432	20	in	in	ADP
ejde-744	432	21	nondivergence	nondivergence	NOUN
ejde-744	432	22	form	form	NOUN
ejde-744	432	23	with	with	ADP
ejde-744	432	24	drift	drift	NOUN
ejde-744	432	25	,	,	PUNCT
ejde-744	432	26	siam	siam	ADJ
ejde-744	432	27	journal	journal	NOUN
ejde-744	432	28	on	on	ADP
ejde-744	432	29	control	control	NOUN
ejde-744	432	30	and	and	CCONJ
ejde-744	432	31	optimization	optimization	NOUN
ejde-744	432	32	,	,	PUNCT
ejde-744	432	33	2023	2023	NUM
ejde-744	432	34	,	,	PUNCT
ejde-744	432	35	61(4	61(4	NUM
ejde-744	432	36	):	):	PUNCT
ejde-744	432	37	1934	1934	NUM
ejde-744	432	38	-	-	SYM
ejde-744	432	39	1954	1954	NUM
ejde-744	432	40	.	.	PUNCT
ejde-744	433	1	[	[	X
ejde-744	433	2	8	8	NUM
ejde-744	433	3	]	]	PUNCT
ejde-744	433	4	p.	p.	NOUN
ejde-744	433	5	cannarsa	cannarsa	PROPN
ejde-744	433	6	,	,	PUNCT
ejde-744	433	7	p.	p.	PROPN
ejde-744	433	8	martinez	martinez	PROPN
ejde-744	433	9	,	,	PUNCT
ejde-744	433	10	j.	j.	PROPN
ejde-744	433	11	vancostenoble	vancostenoble	PROPN
ejde-744	433	12	;	;	PUNCT
ejde-744	433	13	null	null	ADJ
ejde-744	433	14	controllability	controllability	NOUN
ejde-744	433	15	of	of	ADP
ejde-744	433	16	degenerate	degenerate	ADJ
ejde-744	433	17	heat	heat	NOUN
ejde-744	433	18	equations	equation	NOUN
ejde-744	433	19	,	,	PUNCT
ejde-744	433	20	adv	adv	PROPN
ejde-744	433	21	.	.	PUNCT
ejde-744	433	22	differential	differential	PROPN
ejde-744	433	23	equations	equation	NOUN
ejde-744	433	24	,	,	PUNCT
ejde-744	433	25	2005	2005	NUM
ejde-744	433	26	,	,	PUNCT
ejde-744	433	27	10	10	NUM
ejde-744	433	28	:	:	SYM
ejde-744	433	29	153	153	NUM
ejde-744	433	30	-	-	SYM
ejde-744	433	31	190	190	NUM
ejde-744	433	32	.	.	PUNCT
ejde-744	434	1	[	[	X
ejde-744	434	2	9	9	NUM
ejde-744	434	3	]	]	PUNCT
ejde-744	434	4	p.	p.	NOUN
ejde-744	434	5	cannarsa	cannarsa	PROPN
ejde-744	434	6	,	,	PUNCT
ejde-744	434	7	p.	p.	PROPN
ejde-744	434	8	martinez	martinez	PROPN
ejde-744	434	9	,	,	PUNCT
ejde-744	434	10	j.	j.	PROPN
ejde-744	434	11	vancostenoble	vancostenoble	PROPN
ejde-744	434	12	;	;	PUNCT
ejde-744	434	13	carleman	carleman	ADJ
ejde-744	434	14	estimates	estimate	NOUN
ejde-744	434	15	for	for	ADP
ejde-744	434	16	a	a	DET
ejde-744	434	17	class	class	NOUN
ejde-744	434	18	of	of	ADP
ejde-744	434	19	degenerate	degenerate	ADJ
ejde-744	434	20	parabolic	parabolic	NOUN
ejde-744	434	21	operators	operator	NOUN
ejde-744	434	22	,	,	PUNCT
ejde-744	434	23	siam	siam	PROPN
ejde-744	434	24	journal	journal	NOUN
ejde-744	434	25	on	on	ADP
ejde-744	434	26	control	control	NOUN
ejde-744	434	27	and	and	CCONJ
ejde-744	434	28	optimization	optimization	NOUN
ejde-744	434	29	,	,	PUNCT
ejde-744	434	30	2008	2008	NUM
ejde-744	434	31	,	,	PUNCT
ejde-744	434	32	47(1	47(1	NUM
ejde-744	434	33	):	):	PUNCT
ejde-744	434	34	1	1	NUM
ejde-744	434	35	-	-	SYM
ejde-744	434	36	19	19	NUM
ejde-744	434	37	.	.	PUNCT
ejde-744	435	1	[	[	X
ejde-744	435	2	10	10	NUM
ejde-744	435	3	]	]	X
ejde-744	435	4	l.	l.	PROPN
ejde-744	435	5	cui	cui	PROPN
ejde-744	435	6	,	,	PUNCT
ejde-744	435	7	x.	x.	PROPN
ejde-744	435	8	liu	liu	PROPN
ejde-744	435	9	,	,	PUNCT
ejde-744	435	10	h.	h.	PROPN
ejde-744	435	11	gao	gao	PROPN
ejde-744	435	12	;	;	PUNCT
ejde-744	435	13	exact	exact	ADJ
ejde-744	435	14	controllability	controllability	NOUN
ejde-744	435	15	for	for	ADP
ejde-744	435	16	a	a	DET
ejde-744	435	17	one	one	NUM
ejde-744	435	18	-	-	PUNCT
ejde-744	435	19	dimensional	dimensional	ADJ
ejde-744	435	20	wave	wave	NOUN
ejde-744	435	21	equation	equation	NOUN
ejde-744	435	22	in	in	ADP
ejde-744	435	23	noncylindrical	noncylindrical	ADJ
ejde-744	435	24	domains	domain	NOUN
ejde-744	435	25	,	,	PUNCT
ejde-744	435	26	journal	journal	NOUN
ejde-744	435	27	of	of	ADP
ejde-744	435	28	mathematical	mathematical	ADJ
ejde-744	435	29	analysis	analysis	NOUN
ejde-744	435	30	and	and	CCONJ
ejde-744	435	31	applications	application	NOUN
ejde-744	435	32	,	,	PUNCT
ejde-744	435	33	2013	2013	NUM
ejde-744	435	34	,	,	PUNCT
ejde-744	435	35	402(2	402(2	NUM
ejde-744	435	36	):	):	PUNCT
ejde-744	435	37	612625	612625	NUM
ejde-744	435	38	.	.	PUNCT
ejde-744	436	1	[	[	X
ejde-744	436	2	11	11	NUM
ejde-744	436	3	]	]	X
ejde-744	436	4	e.	e.	PROPN
ejde-744	436	5	b.	b.	PROPN
ejde-744	436	6	davies	davies	PROPN
ejde-744	436	7	;	;	PUNCT
ejde-744	436	8	spectral	spectral	ADJ
ejde-744	436	9	theory	theory	NOUN
ejde-744	436	10	and	and	CCONJ
ejde-744	436	11	differential	differential	ADJ
ejde-744	436	12	operators	operator	NOUN
ejde-744	436	13	,	,	PUNCT
ejde-744	436	14	cambridge	cambridge	PROPN
ejde-744	436	15	university	university	PROPN
ejde-744	436	16	press	press	NOUN
ejde-744	436	17	,	,	PUNCT
ejde-744	436	18	1995	1995	NUM
ejde-744	436	19	.	.	PUNCT
ejde-744	437	1	[	[	X
ejde-744	437	2	12	12	NUM
ejde-744	437	3	]	]	PUNCT
ejde-744	437	4	s.	s.	PROPN
ejde-744	437	5	n.	n.	PROPN
ejde-744	437	6	ethier	ethier	PROPN
ejde-744	437	7	;	;	PUNCT
ejde-744	437	8	a	a	DET
ejde-744	437	9	class	class	NOUN
ejde-744	437	10	of	of	ADP
ejde-744	437	11	degenerate	degenerate	ADJ
ejde-744	437	12	diffusion	diffusion	NOUN
ejde-744	437	13	processes	process	NOUN
ejde-744	437	14	occurring	occur	VERB
ejde-744	437	15	in	in	ADP
ejde-744	437	16	population	population	NOUN
ejde-744	437	17	genetics	genetic	NOUN
ejde-744	437	18	,	,	PUNCT
ejde-744	437	19	journal	journal	NOUN
ejde-744	437	20	of	of	ADP
ejde-744	437	21	communications	communication	NOUN
ejde-744	437	22	on	on	ADP
ejde-744	437	23	pure	pure	ADJ
ejde-744	437	24	applied	applied	ADJ
ejde-744	437	25	mathematics	mathematic	NOUN
ejde-744	437	26	,	,	PUNCT
ejde-744	437	27	2010	2010	NUM
ejde-744	437	28	,	,	PUNCT
ejde-744	437	29	29(5	29(5	NUM
ejde-744	437	30	):	):	PUNCT
ejde-744	437	31	483	483	NUM
ejde-744	437	32	-	-	SYM
ejde-744	437	33	493	493	NUM
ejde-744	437	34	.	.	PUNCT
ejde-744	438	1	[	[	X
ejde-744	438	2	13	13	NUM
ejde-744	438	3	]	]	X
ejde-744	438	4	g.	g.	PROPN
ejde-744	438	5	fragnelli	fragnelli	PROPN
ejde-744	438	6	;	;	PUNCT
ejde-744	438	7	interior	interior	ADJ
ejde-744	438	8	degenerate	degenerate	ADJ
ejde-744	438	9	/	/	SYM
ejde-744	438	10	singular	singular	ADJ
ejde-744	438	11	parabolic	parabolic	NOUN
ejde-744	438	12	equations	equation	NOUN
ejde-744	438	13	in	in	ADP
ejde-744	438	14	nondivergence	nondivergence	NOUN
ejde-744	438	15	form	form	NOUN
ejde-744	438	16	:	:	PUNCT
ejde-744	438	17	wellposedness	wellposedness	NOUN
ejde-744	438	18	and	and	CCONJ
ejde-744	438	19	carleman	carleman	ADJ
ejde-744	438	20	estimates	estimate	NOUN
ejde-744	438	21	,	,	PUNCT
ejde-744	438	22	journal	journal	NOUN
ejde-744	438	23	of	of	ADP
ejde-744	438	24	differential	differential	ADJ
ejde-744	438	25	equations	equation	NOUN
ejde-744	438	26	,	,	PUNCT
ejde-744	438	27	2016	2016	NUM
ejde-744	438	28	,	,	PUNCT
ejde-744	438	29	260	260	NUM
ejde-744	438	30	:	:	SYM
ejde-744	438	31	1314	1314	NUM
ejde-744	438	32	-	-	SYM
ejde-744	438	33	1371	1371	NUM
ejde-744	438	34	.	.	PUNCT
ejde-744	439	1	[	[	X
ejde-744	439	2	14	14	NUM
ejde-744	439	3	]	]	X
ejde-744	439	4	g.	g.	PROPN
ejde-744	439	5	fragnelli	fragnelli	PROPN
ejde-744	439	6	,	,	PUNCT
ejde-744	439	7	d.	d.	PROPN
ejde-744	439	8	mugnai	mugnai	PROPN
ejde-744	439	9	;	;	PUNCT
ejde-744	439	10	carleman	carleman	ADJ
ejde-744	439	11	estimates	estimate	NOUN
ejde-744	439	12	for	for	ADP
ejde-744	439	13	singular	singular	ADJ
ejde-744	439	14	parabolic	parabolic	NOUN
ejde-744	439	15	equations	equation	NOUN
ejde-744	439	16	with	with	ADP
ejde-744	439	17	interior	interior	ADJ
ejde-744	439	18	degeneracy	degeneracy	NOUN
ejde-744	439	19	and	and	CCONJ
ejde-744	439	20	non	non	ADJ
ejde-744	439	21	smooth	smooth	ADJ
ejde-744	439	22	coefficients	coefficient	NOUN
ejde-744	439	23	.	.	PUNCT
ejde-744	440	1	advances	advance	NOUN
ejde-744	440	2	in	in	ADP
ejde-744	440	3	nonlinear	nonlinear	ADJ
ejde-744	440	4	analysis	analysis	NOUN
ejde-744	440	5	,	,	PUNCT
ejde-744	440	6	2017	2017	NUM
ejde-744	440	7	,	,	PUNCT
ejde-744	440	8	6	6	NUM
ejde-744	440	9	:	:	SYM
ejde-744	440	10	61	61	NUM
ejde-744	440	11	-	-	SYM
ejde-744	440	12	84	84	NUM
ejde-744	440	13	.	.	PUNCT
ejde-744	441	1	[	[	X
ejde-744	441	2	15	15	NUM
ejde-744	441	3	]	]	X
ejde-744	441	4	g.	g.	PROPN
ejde-744	441	5	fragnelli	fragnelli	PROPN
ejde-744	441	6	,	,	PUNCT
ejde-744	441	7	d.	d.	PROPN
ejde-744	441	8	mugnai	mugnai	PROPN
ejde-744	441	9	;	;	PUNCT
ejde-744	441	10	singular	singular	PROPN
ejde-744	441	11	parabolic	parabolic	NOUN
ejde-744	441	12	equations	equation	NOUN
ejde-744	441	13	with	with	ADP
ejde-744	441	14	interior	interior	ADJ
ejde-744	441	15	degeneracy	degeneracy	NOUN
ejde-744	441	16	and	and	CCONJ
ejde-744	441	17	non	non	ADJ
ejde-744	441	18	smooth	smooth	ADJ
ejde-744	441	19	coefficients	coefficient	NOUN
ejde-744	441	20	:	:	PUNCT
ejde-744	441	21	the	the	DET
ejde-744	441	22	neumann	neumann	PROPN
ejde-744	441	23	case	case	NOUN
ejde-744	441	24	,	,	PUNCT
ejde-744	441	25	discrete	discrete	VERB
ejde-744	441	26	continuous	continuous	ADJ
ejde-744	441	27	dynamical	dynamical	ADJ
ejde-744	441	28	systems	system	NOUN
ejde-744	441	29	,	,	PUNCT
ejde-744	441	30	series	series	NOUN
ejde-744	441	31	s	s	PROPN
ejde-744	441	32	,	,	PUNCT
ejde-744	441	33	2020	2020	NUM
ejde-744	441	34	,	,	PUNCT
ejde-744	441	35	13(4	13(4	NUM
ejde-744	441	36	):	):	PUNCT
ejde-744	441	37	1495	1495	NUM
ejde-744	441	38	-	-	SYM
ejde-744	441	39	1511	1511	NUM
ejde-744	441	40	.	.	PUNCT
ejde-744	442	1	[	[	X
ejde-744	442	2	16	16	NUM
ejde-744	442	3	]	]	PUNCT
ejde-744	442	4	m.	m.	NOUN
ejde-744	442	5	fotouhi	fotouhi	PROPN
ejde-744	442	6	,	,	PUNCT
ejde-744	442	7	l.	l.	PROPN
ejde-744	442	8	salimi	salimi	PROPN
ejde-744	442	9	;	;	PUNCT
ejde-744	442	10	null	null	ADJ
ejde-744	442	11	controllability	controllability	NOUN
ejde-744	442	12	of	of	ADP
ejde-744	442	13	degenerate	degenerate	ADJ
ejde-744	442	14	/	/	SYM
ejde-744	442	15	singular	singular	ADJ
ejde-744	442	16	parabolic	parabolic	NOUN
ejde-744	442	17	equations	equation	NOUN
ejde-744	442	18	.	.	PUNCT
ejde-744	443	1	journal	journal	NOUN
ejde-744	443	2	of	of	ADP
ejde-744	443	3	dynamical	dynamical	ADJ
ejde-744	443	4	and	and	CCONJ
ejde-744	443	5	control	control	NOUN
ejde-744	443	6	systems	system	NOUN
ejde-744	443	7	,	,	PUNCT
ejde-744	443	8	2012	2012	NUM
ejde-744	443	9	,	,	PUNCT
ejde-744	443	10	18	18	NUM
ejde-744	443	11	:	:	SYM
ejde-744	443	12	573	573	NUM
ejde-744	443	13	-	-	SYM
ejde-744	443	14	602	602	NUM
ejde-744	443	15	.	.	PUNCT
ejde-744	444	1	[	[	X
ejde-744	444	2	17	17	NUM
ejde-744	444	3	]	]	PUNCT
ejde-744	444	4	m.	m.	NOUN
ejde-744	444	5	ghil	ghil	NOUN
ejde-744	444	6	;	;	PUNCT
ejde-744	444	7	climate	climate	NOUN
ejde-744	444	8	stability	stability	NOUN
ejde-744	444	9	for	for	ADP
ejde-744	444	10	a	a	DET
ejde-744	444	11	sellers	seller	NOUN
ejde-744	444	12	type	type	NOUN
ejde-744	444	13	model	model	NOUN
ejde-744	444	14	.	.	PUNCT
ejde-744	445	1	journal	journal	NOUN
ejde-744	445	2	of	of	ADP
ejde-744	445	3	the	the	DET
ejde-744	445	4	atmospheric	atmospheric	ADJ
ejde-744	445	5	science	science	NOUN
ejde-744	445	6	,	,	PUNCT
ejde-744	445	7	2015	2015	NUM
ejde-744	445	8	,	,	PUNCT
ejde-744	445	9	16(1	16(1	NUM
ejde-744	445	10	):	):	PUNCT
ejde-744	445	11	3	3	NUM
ejde-744	445	12	-	-	SYM
ejde-744	445	13	20	20	NUM
ejde-744	445	14	.	.	PUNCT
ejde-744	446	1	[	[	X
ejde-744	446	2	18	18	NUM
ejde-744	446	3	]	]	PUNCT
ejde-744	446	4	a.	a.	NOUN
ejde-744	446	5	greenleaf	greenleaf	PROPN
ejde-744	446	6	,	,	PUNCT
ejde-744	446	7	y.	y.	PROPN
ejde-744	446	8	kurylev	kurylev	PROPN
ejde-744	446	9	,	,	PUNCT
ejde-744	446	10	m.	m.	NOUN
ejde-744	446	11	lassas	lassas	PROPN
ejde-744	446	12	,	,	PUNCT
ejde-744	446	13	et	et	PROPN
ejde-744	446	14	al	al	PROPN
ejde-744	446	15	.	.	PROPN
ejde-744	446	16	;	;	PUNCT
ejde-744	446	17	cloaking	cloaking	NOUN
ejde-744	446	18	devices	device	NOUN
ejde-744	446	19	,	,	PUNCT
ejde-744	446	20	electromagnetic	electromagnetic	ADJ
ejde-744	446	21	wormholes	wormhole	NOUN
ejde-744	446	22	,	,	PUNCT
ejde-744	446	23	and	and	CCONJ
ejde-744	446	24	transformation	transformation	NOUN
ejde-744	446	25	optics	optic	NOUN
ejde-744	446	26	,	,	PUNCT
ejde-744	446	27	siam	siam	PROPN
ejde-744	446	28	review	review	NOUN
ejde-744	446	29	,	,	PUNCT
ejde-744	446	30	2009	2009	NUM
ejde-744	446	31	,	,	PUNCT
ejde-744	446	32	51(1	51(1	NUM
ejde-744	446	33	):	):	PUNCT
ejde-744	446	34	3	3	NUM
ejde-744	446	35	-	-	SYM
ejde-744	446	36	33	33	NUM
ejde-744	446	37	.	.	PUNCT
ejde-744	447	1	[	[	X
ejde-744	447	2	19	19	NUM
ejde-744	447	3	]	]	PUNCT
ejde-744	447	4	m.	m.	NOUN
ejde-744	447	5	gueye	gueye	NOUN
ejde-744	447	6	;	;	PUNCT
ejde-744	447	7	exact	exact	ADJ
ejde-744	447	8	boundary	boundary	ADJ
ejde-744	447	9	controllability	controllability	NOUN
ejde-744	447	10	of	of	ADP
ejde-744	447	11	1	1	NUM
ejde-744	447	12	-	-	PUNCT
ejde-744	447	13	d	d	NOUN
ejde-744	447	14	parabolic	parabolic	ADJ
ejde-744	447	15	and	and	CCONJ
ejde-744	447	16	hyperbolic	hyperbolic	ADJ
ejde-744	447	17	degenerate	degenerate	ADJ
ejde-744	447	18	equations	equation	NOUN
ejde-744	447	19	,	,	PUNCT
ejde-744	447	20	siam	siam	ADJ
ejde-744	447	21	journal	journal	NOUN
ejde-744	447	22	on	on	ADP
ejde-744	447	23	control	control	NOUN
ejde-744	447	24	and	and	CCONJ
ejde-744	447	25	optimization	optimization	NOUN
ejde-744	447	26	,	,	PUNCT
ejde-744	447	27	2014	2014	NUM
ejde-744	447	28	,	,	PUNCT
ejde-744	447	29	52(4	52(4	NUM
ejde-744	447	30	):	):	PUNCT
ejde-744	447	31	2037	2037	NUM
ejde-744	447	32	-	-	SYM
ejde-744	447	33	2054	2054	NUM
ejde-744	447	34	.	.	PUNCT
ejde-744	448	1	[	[	X
ejde-744	448	2	20	20	NUM
ejde-744	448	3	]	]	PUNCT
ejde-744	448	4	v.	v.	X
ejde-744	449	1	komornik	komornik	ADJ
ejde-744	449	2	;	;	PUNCT
ejde-744	449	3	exact	exact	ADJ
ejde-744	449	4	controllability	controllability	NOUN
ejde-744	449	5	and	and	CCONJ
ejde-744	449	6	stabilization	stabilization	NOUN
ejde-744	449	7	:	:	PUNCT
ejde-744	449	8	the	the	DET
ejde-744	449	9	multiplier	multipli	ADJ
ejde-744	449	10	method	method	NOUN
ejde-744	449	11	.	.	PUNCT
ejde-744	450	1	elsevier	elsevier	PROPN
ejde-744	450	2	masson	masson	PROPN
ejde-744	450	3	,	,	PUNCT
ejde-744	450	4	1994	1994	NUM
ejde-744	450	5	.	.	PUNCT
ejde-744	451	1	[	[	X
ejde-744	451	2	21	21	NUM
ejde-744	451	3	]	]	PUNCT
ejde-744	451	4	k.	k.	PROPN
ejde-744	451	5	h.	h.	PROPN
ejde-744	451	6	karlsen	karlsen	PROPN
ejde-744	451	7	,	,	PUNCT
ejde-744	451	8	n.	n.	PROPN
ejde-744	451	9	h.	h.	PROPN
ejde-744	451	10	risebro	risebro	PROPN
ejde-744	451	11	;	;	PUNCT
ejde-744	451	12	on	on	ADP
ejde-744	451	13	the	the	DET
ejde-744	451	14	uniqueness	uniqueness	NOUN
ejde-744	451	15	and	and	CCONJ
ejde-744	451	16	stability	stability	NOUN
ejde-744	451	17	of	of	ADP
ejde-744	451	18	entropy	entropy	PROPN
ejde-744	451	19	solutions	solution	NOUN
ejde-744	451	20	of	of	ADP
ejde-744	451	21	nonlinear	nonlinear	ADJ
ejde-744	451	22	degenerate	degenerate	ADJ
ejde-744	451	23	parabolic	parabolic	ADJ
ejde-744	451	24	equations	equation	NOUN
ejde-744	451	25	with	with	ADP
ejde-744	451	26	rough	rough	ADJ
ejde-744	451	27	coefficients	coefficient	NOUN
ejde-744	451	28	.	.	PUNCT
ejde-744	452	1	discrete	discrete	ADJ
ejde-744	452	2	continuous	continuous	ADJ
ejde-744	452	3	dynamical	dynamical	ADJ
ejde-744	452	4	systems	system	NOUN
ejde-744	452	5	,	,	PUNCT
ejde-744	452	6	series	series	NOUN
ejde-744	452	7	s	s	PROPN
ejde-744	452	8	,	,	PUNCT
ejde-744	452	9	2003	2003	NUM
ejde-744	452	10	,	,	PUNCT
ejde-744	452	11	9(1	9(1	NUM
ejde-744	452	12	):	):	PUNCT
ejde-744	452	13	1081	1081	NUM
ejde-744	452	14	-	-	SYM
ejde-744	452	15	1104	1104	NUM
ejde-744	452	16	.	.	PUNCT
ejde-744	453	1	[	[	X
ejde-744	453	2	22	22	NUM
ejde-744	453	3	]	]	PUNCT
ejde-744	453	4	j.	j.	PROPN
ejde-744	453	5	l.	l.	PROPN
ejde-744	453	6	lions	lions	PROPN
ejde-744	453	7	;	;	PUNCT
ejde-744	453	8	exact	exact	ADJ
ejde-744	453	9	controllability	controllability	NOUN
ejde-744	453	10	,	,	PUNCT
ejde-744	453	11	stabilization	stabilization	NOUN
ejde-744	453	12	and	and	CCONJ
ejde-744	453	13	perturbations	perturbation	NOUN
ejde-744	453	14	for	for	ADP
ejde-744	453	15	distributed	distributed	ADJ
ejde-744	453	16	systems	system	NOUN
ejde-744	453	17	.	.	PUNCT
ejde-744	454	1	siam	siam	PROPN
ejde-744	454	2	review	review	PROPN
ejde-744	454	3	,	,	PUNCT
ejde-744	454	4	1988	1988	NUM
ejde-744	454	5	,	,	PUNCT
ejde-744	454	6	30(1	30(1	NUM
ejde-744	454	7	):	):	PUNCT
ejde-744	454	8	1	1	NUM
ejde-744	454	9	-	-	SYM
ejde-744	454	10	68	68	NUM
ejde-744	454	11	.	.	PUNCT
ejde-744	455	1	[	[	X
ejde-744	455	2	23	23	NUM
ejde-744	455	3	]	]	PUNCT
ejde-744	455	4	x.	x.	PROPN
ejde-744	455	5	liu	liu	PROPN
ejde-744	455	6	,	,	PUNCT
ejde-744	455	7	x.	x.	PROPN
ejde-744	455	8	zhang	zhang	PROPN
ejde-744	455	9	;	;	PUNCT
ejde-744	455	10	local	local	ADJ
ejde-744	455	11	controllability	controllability	NOUN
ejde-744	455	12	of	of	ADP
ejde-744	455	13	multidimensional	multidimensional	ADJ
ejde-744	455	14	quasi	quasi	ADJ
ejde-744	455	15	-	-	ADJ
ejde-744	455	16	linear	linear	ADJ
ejde-744	455	17	parabolic	parabolic	NOUN
ejde-744	455	18	equations	equation	NOUN
ejde-744	455	19	,	,	PUNCT
ejde-744	455	20	siam	siam	ADJ
ejde-744	455	21	journal	journal	NOUN
ejde-744	455	22	on	on	ADP
ejde-744	455	23	control	control	NOUN
ejde-744	455	24	and	and	CCONJ
ejde-744	455	25	optimization	optimization	NOUN
ejde-744	455	26	,	,	PUNCT
ejde-744	455	27	2012	2012	NUM
ejde-744	455	28	,	,	PUNCT
ejde-744	455	29	50(4	50(4	NUM
ejde-744	455	30	):	):	PUNCT
ejde-744	455	31	2046	2046	NUM
ejde-744	455	32	-	-	SYM
ejde-744	455	33	2064	2064	NUM
ejde-744	455	34	.	.	PUNCT
ejde-744	456	1	[	[	X
ejde-744	456	2	24	24	NUM
ejde-744	456	3	]	]	PUNCT
ejde-744	456	4	p.	p.	PROPN
ejde-744	456	5	martinez	martinez	PROPN
ejde-744	456	6	,	,	PUNCT
ejde-744	456	7	j.	j.	PROPN
ejde-744	456	8	vancostenoble	vancostenoble	PROPN
ejde-744	456	9	;	;	PUNCT
ejde-744	456	10	carleman	carleman	ADJ
ejde-744	456	11	estimates	estimate	NOUN
ejde-744	456	12	for	for	ADP
ejde-744	456	13	one	one	NUM
ejde-744	456	14	-	-	PUNCT
ejde-744	456	15	dimensional	dimensional	ADJ
ejde-744	456	16	degenerate	degenerate	ADJ
ejde-744	456	17	heat	heat	NOUN
ejde-744	456	18	equations	equation	NOUN
ejde-744	456	19	,	,	PUNCT
ejde-744	456	20	journal	journal	NOUN
ejde-744	456	21	of	of	ADP
ejde-744	456	22	evolution	evolution	NOUN
ejde-744	456	23	equations	equation	NOUN
ejde-744	456	24	,	,	PUNCT
ejde-744	456	25	2006	2006	NUM
ejde-744	456	26	,	,	PUNCT
ejde-744	456	27	6	6	NUM
ejde-744	456	28	:	:	SYM
ejde-744	456	29	325	325	NUM
ejde-744	456	30	-	-	SYM
ejde-744	456	31	362	362	NUM
ejde-744	456	32	.	.	PUNCT
ejde-744	457	1	[	[	X
ejde-744	457	2	25	25	NUM
ejde-744	457	3	]	]	X
ejde-744	457	4	l.	l.	PROPN
ejde-744	457	5	rosier	rosier	PROPN
ejde-744	457	6	;	;	PUNCT
ejde-744	457	7	a	a	DET
ejde-744	457	8	survey	survey	NOUN
ejde-744	457	9	of	of	ADP
ejde-744	457	10	controllability	controllability	NOUN
ejde-744	457	11	and	and	CCONJ
ejde-744	457	12	stabilization	stabilization	NOUN
ejde-744	457	13	results	result	NOUN
ejde-744	457	14	for	for	ADP
ejde-744	457	15	partial	partial	ADJ
ejde-744	457	16	differential	differential	ADJ
ejde-744	457	17	equations	equation	NOUN
ejde-744	457	18	,	,	PUNCT
ejde-744	457	19	journal	journal	NOUN
ejde-744	457	20	européen	européen	PROPN
ejde-744	457	21	des	des	PROPN
ejde-744	457	22	systèmes	systèmes	PROPN
ejde-744	457	23	automatisés	automatisés	PROPN
ejde-744	457	24	,	,	PUNCT
ejde-744	457	25	2007	2007	NUM
ejde-744	457	26	,	,	PUNCT
ejde-744	457	27	41(3/4	41(3/4	NUM
ejde-744	457	28	):	):	PUNCT
ejde-744	457	29	365	365	NUM
ejde-744	457	30	.	.	PUNCT
ejde-744	458	1	ejde-2025/04	ejde-2025/04	X
ejde-744	458	2	exact	exact	ADJ
ejde-744	458	3	controllability	controllability	NOUN
ejde-744	458	4	17	17	NUM
ejde-744	459	1	[	[	SYM
ejde-744	459	2	26	26	NUM
ejde-744	459	3	]	]	X
ejde-744	459	4	l.	l.	PROPN
ejde-744	459	5	russel	russel	PROPN
ejde-744	459	6	;	;	PUNCT
ejde-744	459	7	controllability	controllability	NOUN
ejde-744	459	8	and	and	CCONJ
ejde-744	459	9	stabilization	stabilization	NOUN
ejde-744	459	10	theory	theory	NOUN
ejde-744	459	11	for	for	ADP
ejde-744	459	12	linear	linear	ADJ
ejde-744	459	13	partial	partial	ADJ
ejde-744	459	14	differential	differential	NOUN
ejde-744	459	15	equations	equation	NOUN
ejde-744	459	16	:	:	PUNCT
ejde-744	459	17	recent	recent	ADJ
ejde-744	459	18	progress	progress	NOUN
ejde-744	459	19	and	and	CCONJ
ejde-744	459	20	open	open	ADJ
ejde-744	459	21	questions	question	NOUN
ejde-744	459	22	,	,	PUNCT
ejde-744	459	23	siam	siam	PROPN
ejde-744	459	24	review	review	NOUN
ejde-744	459	25	,	,	PUNCT
ejde-744	459	26	1978	1978	NUM
ejde-744	459	27	,	,	PUNCT
ejde-744	459	28	20	20	NUM
ejde-744	459	29	:	:	PUNCT
ejde-744	459	30	639	639	NUM
ejde-744	459	31	-	-	SYM
ejde-744	459	32	739	739	NUM
ejde-744	459	33	.	.	PUNCT
ejde-744	460	1	[	[	X
ejde-744	460	2	27	27	NUM
ejde-744	460	3	]	]	X
ejde-744	460	4	j.	j.	PROPN
ejde-744	460	5	vancostenoble	vancostenoble	PROPN
ejde-744	460	6	;	;	PUNCT
ejde-744	460	7	improved	improve	VERB
ejde-744	460	8	hardy	hardy	ADJ
ejde-744	460	9	-	-	PUNCT
ejde-744	460	10	poincare	poincare	NOUN
ejde-744	460	11	inequalities	inequality	NOUN
ejde-744	460	12	and	and	CCONJ
ejde-744	460	13	sharp	sharp	ADJ
ejde-744	460	14	carleman	carleman	ADJ
ejde-744	460	15	estimates	estimate	NOUN
ejde-744	460	16	for	for	ADP
ejde-744	460	17	degenerate	degenerate	ADJ
ejde-744	460	18	/	/	SYM
ejde-744	460	19	singular	singular	ADJ
ejde-744	460	20	parabolic	parabolic	NOUN
ejde-744	460	21	problems	problem	NOUN
ejde-744	460	22	,	,	PUNCT
ejde-744	460	23	discrete	discrete	VERB
ejde-744	460	24	continuous	continuous	ADJ
ejde-744	460	25	dynamical	dynamical	ADJ
ejde-744	460	26	systems	system	NOUN
ejde-744	460	27	,	,	PUNCT
ejde-744	460	28	series	series	NOUN
ejde-744	460	29	s	s	PROPN
ejde-744	460	30	,	,	PUNCT
ejde-744	460	31	2011	2011	NUM
ejde-744	460	32	,	,	PUNCT
ejde-744	460	33	4(1	4(1	NOUN
ejde-744	460	34	):	):	PUNCT
ejde-744	460	35	761	761	NUM
ejde-744	460	36	-	-	SYM
ejde-744	460	37	790	790	NUM
ejde-744	460	38	.	.	PUNCT
ejde-744	461	1	[	[	X
ejde-744	461	2	28	28	NUM
ejde-744	461	3	]	]	X
ejde-744	461	4	p.	p.	NOUN
ejde-744	461	5	yao	yao	PROPN
ejde-744	461	6	;	;	PUNCT
ejde-744	461	7	on	on	ADP
ejde-744	461	8	the	the	DET
ejde-744	461	9	observability	observability	NOUN
ejde-744	461	10	inequalities	inequality	NOUN
ejde-744	461	11	for	for	ADP
ejde-744	461	12	exact	exact	ADJ
ejde-744	461	13	controllability	controllability	NOUN
ejde-744	461	14	of	of	ADP
ejde-744	461	15	wave	wave	NOUN
ejde-744	461	16	equations	equation	NOUN
ejde-744	461	17	with	with	ADP
ejde-744	461	18	variable	variable	ADJ
ejde-744	461	19	coefficients	coefficient	NOUN
ejde-744	461	20	.	.	PUNCT
ejde-744	462	1	siam	siam	PROPN
ejde-744	462	2	journal	journal	PROPN
ejde-744	462	3	on	on	ADP
ejde-744	462	4	control	control	NOUN
ejde-744	462	5	and	and	CCONJ
ejde-744	462	6	optimization	optimization	NOUN
ejde-744	462	7	,	,	PUNCT
ejde-744	462	8	1999	1999	NUM
ejde-744	462	9	,	,	PUNCT
ejde-744	462	10	37(5	37(5	NUM
ejde-744	462	11	):	):	PUNCT
ejde-744	462	12	1568	1568	NUM
ejde-744	462	13	-	-	SYM
ejde-744	462	14	1599	1599	NUM
ejde-744	462	15	.	.	PUNCT
ejde-744	463	1	[	[	X
ejde-744	463	2	29	29	NUM
ejde-744	463	3	]	]	X
ejde-744	463	4	g.	g.	PROPN
ejde-744	463	5	zhang	zhang	PROPN
ejde-744	463	6	,	,	PUNCT
ejde-744	463	7	s.	s.	PROPN
ejde-744	463	8	chai	chai	PROPN
ejde-744	463	9	;	;	PUNCT
ejde-744	463	10	stabilization	stabilization	NOUN
ejde-744	463	11	for	for	ADP
ejde-744	463	12	some	some	DET
ejde-744	463	13	degenerate	degenerate	ADJ
ejde-744	463	14	wave	wave	NOUN
ejde-744	463	15	equations	equation	NOUN
ejde-744	463	16	,	,	PUNCT
ejde-744	463	17	evolution	evolution	NOUN
ejde-744	463	18	equations	equation	NOUN
ejde-744	463	19	and	and	CCONJ
ejde-744	463	20	control	control	PROPN
ejde-744	463	21	theory	theory	NOUN
ejde-744	463	22	,	,	PUNCT
ejde-744	463	23	2024	2024	NUM
ejde-744	463	24	,	,	PUNCT
ejde-744	463	25	13(2	13(2	NUM
ejde-744	463	26	):	):	PUNCT
ejde-744	463	27	316	316	NUM
ejde-744	463	28	-	-	SYM
ejde-744	463	29	328	328	NUM
ejde-744	463	30	.	.	PUNCT
ejde-744	464	1	[	[	X
ejde-744	464	2	30	30	NUM
ejde-744	464	3	]	]	PUNCT
ejde-744	464	4	m.	m.	PROPN
ejde-744	464	5	zhang	zhang	PROPN
ejde-744	464	6	,	,	PUNCT
ejde-744	464	7	h.	h.	PROPN
ejde-744	464	8	gao	gao	PROPN
ejde-744	464	9	;	;	PUNCT
ejde-744	464	10	null	null	ADJ
ejde-744	464	11	controllability	controllability	NOUN
ejde-744	464	12	of	of	ADP
ejde-744	464	13	some	some	DET
ejde-744	464	14	degenerate	degenerate	ADJ
ejde-744	464	15	wave	wave	NOUN
ejde-744	464	16	equations	equation	NOUN
ejde-744	464	17	,	,	PUNCT
ejde-744	464	18	j.	j.	PROPN
ejde-744	464	19	syst	syst	PROPN
ejde-744	464	20	.	.	PUNCT
ejde-744	465	1	sci	sci	PROPN
ejde-744	465	2	.	.	PUNCT
ejde-744	465	3	complex	complex	PROPN
ejde-744	465	4	.	.	PUNCT
ejde-744	465	5	,	,	PUNCT
ejde-744	465	6	2017	2017	NUM
ejde-744	465	7	,	,	PUNCT
ejde-744	465	8	29	29	NUM
ejde-744	465	9	:	:	SYM
ejde-744	465	10	1	1	NUM
ejde-744	465	11	-	-	SYM
ejde-744	465	12	15	15	NUM
ejde-744	465	13	.	.	PUNCT
ejde-744	466	1	[	[	X
ejde-744	466	2	31	31	NUM
ejde-744	466	3	]	]	PUNCT
ejde-744	466	4	m.	m.	PROPN
ejde-744	466	5	zhang	zhang	PROPN
ejde-744	466	6	,	,	PUNCT
ejde-744	466	7	h.	h.	PROPN
ejde-744	466	8	gao	gao	PROPN
ejde-744	466	9	;	;	PUNCT
ejde-744	466	10	persistent	persistent	ADJ
ejde-744	466	11	regional	regional	ADJ
ejde-744	466	12	null	null	ADJ
ejde-744	466	13	controllability	controllability	NOUN
ejde-744	466	14	of	of	ADP
ejde-744	466	15	some	some	DET
ejde-744	466	16	degenerate	degenerate	ADJ
ejde-744	466	17	wave	wave	NOUN
ejde-744	466	18	equations	equation	NOUN
ejde-744	466	19	.	.	PUNCT
ejde-744	467	1	math	math	NOUN
ejde-744	467	2	.	.	PUNCT
ejde-744	468	1	methods	method	NOUN
ejde-744	468	2	in	in	ADP
ejde-744	468	3	the	the	DET
ejde-744	468	4	app	app	PROPN
ejde-744	468	5	.	.	PUNCT
ejde-744	469	1	sci	sci	PROPN
ejde-744	469	2	.	.	PROPN
ejde-744	469	3	,	,	PUNCT
ejde-744	469	4	2017	2017	NUM
ejde-744	469	5	,	,	PUNCT
ejde-744	469	6	40	40	NUM
ejde-744	469	7	:	:	SYM
ejde-744	469	8	5821	5821	NUM
ejde-744	469	9	-	-	SYM
ejde-744	469	10	5830	5830	NUM
ejde-744	469	11	.	.	PUNCT
ejde-744	470	1	[	[	X
ejde-744	470	2	32	32	NUM
ejde-744	470	3	]	]	X
ejde-744	470	4	e.	e.	PROPN
ejde-744	470	5	zuazua	zuazua	PROPN
ejde-744	470	6	;	;	PUNCT
ejde-744	470	7	controllability	controllability	NOUN
ejde-744	470	8	of	of	ADP
ejde-744	470	9	partial	partial	ADJ
ejde-744	470	10	differential	differential	NOUN
ejde-744	470	11	equations	equation	NOUN
ejde-744	470	12	,	,	PUNCT
ejde-744	470	13	optimization	optimization	NOUN
ejde-744	470	14	and	and	CCONJ
ejde-744	470	15	control	control	NOUN
ejde-744	470	16	,	,	PUNCT
ejde-744	470	17	2006	2006	NUM
ejde-744	470	18	.	.	PUNCT
ejde-744	471	1	guang	guang	PROPN
ejde-744	471	2	zhang	zhang	PROPN
ejde-744	471	3	school	school	PROPN
ejde-744	471	4	of	of	ADP
ejde-744	471	5	mathematics	mathematic	NOUN
ejde-744	471	6	and	and	CCONJ
ejde-744	471	7	statistics	statistic	NOUN
ejde-744	471	8	,	,	PUNCT
ejde-744	471	9	shanxi	shanxi	PROPN
ejde-744	471	10	university	university	PROPN
ejde-744	471	11	,	,	PUNCT
ejde-744	471	12	taiyuan	taiyuan	PROPN
ejde-744	471	13	030006	030006	NUM
ejde-744	471	14	,	,	PUNCT
ejde-744	471	15	china	china	PROPN
ejde-744	471	16	email	email	NOUN
ejde-744	471	17	address	address	NOUN
ejde-744	471	18	:	:	PUNCT
ejde-744	471	19	gzhang1231@163.com	gzhang1231@163.com	X
ejde-744	471	20	shugen	shugen	NOUN
ejde-744	471	21	chai	chai	NOUN
ejde-744	471	22	(	(	PUNCT
ejde-744	471	23	corresponding	correspond	VERB
ejde-744	471	24	author	author	NOUN
ejde-744	471	25	)	)	PUNCT
ejde-744	471	26	school	school	NOUN
ejde-744	471	27	of	of	ADP
ejde-744	471	28	mathematics	mathematic	NOUN
ejde-744	471	29	and	and	CCONJ
ejde-744	471	30	statistics	statistic	NOUN
ejde-744	471	31	,	,	PUNCT
ejde-744	471	32	shanxi	shanxi	PROPN
ejde-744	471	33	university	university	PROPN
ejde-744	471	34	,	,	PUNCT
ejde-744	471	35	taiyuan	taiyuan	PROPN
ejde-744	471	36	030006	030006	NUM
ejde-744	471	37	,	,	PUNCT
ejde-744	471	38	china	china	PROPN
ejde-744	471	39	email	email	NOUN
ejde-744	471	40	address	address	NOUN
ejde-744	471	41	:	:	PUNCT
ejde-744	471	42	sgchai@sxu.edu.cn	sgchai@sxu.edu.cn	NOUN
ejde-744	471	43	1	1	NUM
ejde-744	471	44	.	.	PUNCT
ejde-744	472	1	introduction	introduction	NOUN
ejde-744	472	2	2	2	NUM
ejde-744	472	3	.	.	PUNCT
ejde-744	472	4	preliminaries	preliminary	NOUN
ejde-744	472	5	3	3	NUM
ejde-744	472	6	.	.	PUNCT
ejde-744	473	1	well	well	ADJ
ejde-744	473	2	-	-	PUNCT
ejde-744	473	3	posedness	posedness	NOUN
ejde-744	473	4	4	4	NUM
ejde-744	473	5	.	.	PUNCT
ejde-744	473	6	energy	energy	NOUN
ejde-744	473	7	estimate	estimate	NOUN
ejde-744	473	8	5	5	NUM
ejde-744	473	9	.	.	PUNCT
ejde-744	473	10	boundary	boundary	ADJ
ejde-744	473	11	observability	observability	NOUN
ejde-744	473	12	6	6	NUM
ejde-744	473	13	.	.	PUNCT
ejde-744	474	1	controllability	controllability	PROPN
ejde-744	474	2	7	7	NUM
ejde-744	474	3	.	.	PUNCT
ejde-744	474	4	conclusion	conclusion	NOUN
ejde-744	474	5	example	example	NOUN
ejde-744	474	6	acknowledgments	acknowledgment	NOUN
ejde-744	474	7	references	reference	NOUN