id	sid	tid	token	lemma	pos
cana-3960	1	1	communications	communication	NOUN
cana-3960	1	2	on	on	ADP
cana-3960	1	3	applied	apply	VERB
cana-3960	1	4	nonlinear	nonlinear	ADJ
cana-3960	1	5	analysis	analysis	NOUN
cana-3960	1	6	issn	issn	NOUN
cana-3960	1	7	:	:	PUNCT
cana-3960	1	8	1074	1074	NUM
cana-3960	1	9	-	-	PUNCT
cana-3960	1	10	133x	133x	NUM
cana-3960	1	11	vol	vol	NOUN
cana-3960	1	12	32	32	NUM
cana-3960	1	13	no	no	NOUN
cana-3960	1	14	.	.	PUNCT
cana-3960	2	1	9s	9s	NUM
cana-3960	2	2	(	(	PUNCT
cana-3960	2	3	2025	2025	NUM
cana-3960	2	4	)	)	PUNCT
cana-3960	3	1	493	493	NUM
cana-3960	3	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-3960	3	3	controllability	controllability	NOUN
cana-3960	3	4	and	and	CCONJ
cana-3960	3	5	observability	observability	NOUN
cana-3960	3	6	of	of	ADP
cana-3960	3	7	matrix	matrix	NOUN
cana-3960	3	8	sylvester	sylvest	ADJ
cana-3960	3	9	adjoint	adjoint	PROPN
cana-3960	3	10	dynamic	dynamic	ADJ
cana-3960	3	11	impulsive	impulsive	ADJ
cana-3960	3	12	systems	system	NOUN
cana-3960	3	13	on	on	ADP
cana-3960	3	14	time	time	NOUN
cana-3960	3	15	scales	scale	VERB
cana-3960	3	16	a.	a.	NOUN
cana-3960	3	17	sreenivasulu1	sreenivasulu1	PROPN
cana-3960	3	18	*	*	PROPN
cana-3960	3	19	,	,	PUNCT
cana-3960	3	20	b.	b.	PROPN
cana-3960	4	1	v.	v.	PROPN
cana-3960	4	2	appa	appa	PROPN
cana-3960	4	3	rao1	rao1	PROPN
cana-3960	4	4	,	,	PUNCT
cana-3960	4	5	p.	p.	NOUN
cana-3960	4	6	lakshmi	lakshmi	PROPN
cana-3960	4	7	pallavi2	pallavi2	PROPN
cana-3960	4	8	,	,	PUNCT
cana-3960	4	9	gudala	gudala	PROPN
cana-3960	4	10	balaji	balaji	PROPN
cana-3960	4	11	prakash3	prakash3	PROPN
cana-3960	4	12	,	,	PUNCT
cana-3960	4	13	m.	m.	NOUN
cana-3960	4	14	srinivasa	srinivasa	PROPN
cana-3960	4	15	reddy4	reddy4	PROPN
cana-3960	4	16	,	,	PUNCT
cana-3960	4	17	j	j	PROPN
cana-3960	4	18	peter	peter	PROPN
cana-3960	4	19	praveen5	praveen5	PROPN
cana-3960	4	20	and	and	CCONJ
cana-3960	4	21	d.	d.	PROPN
cana-3960	4	22	ramesh1	ramesh1	PROPN
cana-3960	4	23	1department	1department	NUM
cana-3960	4	24	of	of	ADP
cana-3960	4	25	engineering	engineering	NOUN
cana-3960	4	26	mathematics	mathematic	NOUN
cana-3960	4	27	,	,	PUNCT
cana-3960	4	28	koneru	koneru	PROPN
cana-3960	4	29	lakshmaiah	lakshmaiah	PROPN
cana-3960	4	30	education	education	PROPN
cana-3960	4	31	foundation	foundation	PROPN
cana-3960	4	32	,	,	PUNCT
cana-3960	4	33	green	green	ADJ
cana-3960	4	34	fields	field	NOUN
cana-3960	4	35	,	,	PUNCT
cana-3960	4	36	vaddeswaram	vaddeswaram	NOUN
cana-3960	4	37	,	,	PUNCT
cana-3960	4	38	guntur-522302	guntur-522302	NOUN
cana-3960	4	39	,	,	PUNCT
cana-3960	4	40	andhra	andhra	PROPN
cana-3960	4	41	pradesh	pradesh	PROPN
cana-3960	4	42	,	,	PUNCT
cana-3960	4	43	india	india	PROPN
cana-3960	4	44	.	.	PUNCT
cana-3960	5	1	2department	2department	NUM
cana-3960	5	2	of	of	ADP
cana-3960	5	3	mathematics	mathematic	NOUN
cana-3960	5	4	,	,	PUNCT
cana-3960	5	5	b	b	PROPN
cana-3960	5	6	v	v	ADP
cana-3960	5	7	raju	raju	PROPN
cana-3960	5	8	institute	institute	PROPN
cana-3960	5	9	of	of	ADP
cana-3960	5	10	technology	technology	PROPN
cana-3960	5	11	,	,	PUNCT
cana-3960	5	12	narsapur	narsapur	NOUN
cana-3960	5	13	,	,	PUNCT
cana-3960	5	14	502313	502313	NUM
cana-3960	5	15	telangana	telangana	PROPN
cana-3960	5	16	,	,	PUNCT
cana-3960	5	17	india	india	PROPN
cana-3960	5	18	.	.	PUNCT
cana-3960	6	1	3department	3department	NUM
cana-3960	6	2	of	of	ADP
cana-3960	6	3	mathematics	mathematic	NOUN
cana-3960	6	4	,	,	PUNCT
cana-3960	6	5	aditya	aditya	PROPN
cana-3960	6	6	university	university	PROPN
cana-3960	6	7	,	,	PUNCT
cana-3960	6	8	surampalem	surampalem	NOUN
cana-3960	6	9	,	,	PUNCT
cana-3960	6	10	533437	533437	NUM
cana-3960	6	11	andhra	andhra	PROPN
cana-3960	6	12	pradesh	pradesh	PROPN
cana-3960	6	13	,	,	PUNCT
cana-3960	6	14	india	india	PROPN
cana-3960	6	15	.	.	PUNCT
cana-3960	7	1	4freshman	4freshman	NUM
cana-3960	7	2	engineering	engineering	NOUN
cana-3960	7	3	department	department	NOUN
cana-3960	7	4	,	,	PUNCT
cana-3960	7	5	lakireddy	lakireddy	PROPN
cana-3960	7	6	bali	bali	PROPN
cana-3960	7	7	reddy	reddy	PROPN
cana-3960	7	8	college	college	PROPN
cana-3960	7	9	of	of	ADP
cana-3960	7	10	engineering	engineering	PROPN
cana-3960	7	11	,	,	PUNCT
cana-3960	7	12	mylavaram-521230	mylavaram-521230	INTJ
cana-3960	7	13	,	,	PUNCT
cana-3960	7	14	ntr	ntr	NOUN
cana-3960	7	15	-	-	PUNCT
cana-3960	7	16	district	district	PROPN
cana-3960	7	17	,	,	PUNCT
cana-3960	7	18	india	india	PROPN
cana-3960	7	19	.	.	PUNCT
cana-3960	8	1	5vignan	5vignan	NUM
cana-3960	8	2	institute	institute	NOUN
cana-3960	8	3	of	of	ADP
cana-3960	8	4	information	information	NOUN
cana-3960	8	5	technology	technology	PROPN
cana-3960	8	6	,	,	PUNCT
cana-3960	8	7	duvvada	duvvada	PROPN
cana-3960	8	8	,	,	PUNCT
cana-3960	8	9	visakhapatnam,530049	visakhapatnam,530049	PROPN
cana-3960	8	10	,	,	PUNCT
cana-3960	8	11	inida	inida	PROPN
cana-3960	8	12	.	.	PUNCT
cana-3960	9	1	corresponding	correspond	VERB
cana-3960	9	2	author	author	NOUN
cana-3960	9	3	email	email	NOUN
cana-3960	9	4	:	:	PUNCT
cana-3960	9	5	asreenivasulu@kluniversity.in	asreenivasulu@kluniversity.in	ADV
cana-3960	9	6	;	;	PUNCT
cana-3960	9	7	author	author	NOUN
cana-3960	9	8	:	:	PUNCT
cana-3960	9	9	bvardr2010@kluniversity.in	bvardr2010@kluniversity.in	X
cana-3960	9	10	;	;	PUNCT
cana-3960	9	11	lakshmipallavi.p@bvrit.ac.in	lakshmipallavi.p@bvrit.ac.in	NUM
cana-3960	9	12	;	;	PUNCT
cana-3960	9	13	balajiprakashgudala@gmail.com	balajiprakashgudala@gmail.com	PROPN
cana-3960	9	14	;	;	PUNCT
cana-3960	9	15	maths4444@gmail.com	maths4444@gmail.com	X
cana-3960	9	16	;	;	PUNCT
cana-3960	10	1	jppraveen17@gmail.com	jppraveen17@gmail.com	PROPN
cana-3960	10	2	and	and	CCONJ
cana-3960	10	3	ram.fuzzy@gmail.com	ram.fuzzy@gmail.com	PROPN
cana-3960	10	4	article	article	NOUN
cana-3960	10	5	history	history	NOUN
cana-3960	10	6	:	:	PUNCT
cana-3960	10	7	received	receive	VERB
cana-3960	10	8	:	:	PUNCT
cana-3960	10	9	12	12	NUM
cana-3960	10	10	-	-	SYM
cana-3960	10	11	11	11	NUM
cana-3960	10	12	-	-	PUNCT
cana-3960	10	13	2024	2024	NUM
cana-3960	10	14	revised	revise	VERB
cana-3960	10	15	:	:	PUNCT
cana-3960	10	16	17	17	NUM
cana-3960	10	17	-	-	SYM
cana-3960	10	18	12	12	NUM
cana-3960	10	19	-	-	PUNCT
cana-3960	10	20	2024	2024	NUM
cana-3960	10	21	accepted	accept	VERB
cana-3960	10	22	:	:	PUNCT
cana-3960	10	23	06	06	NUM
cana-3960	10	24	-	-	SYM
cana-3960	10	25	01	01	NUM
cana-3960	10	26	-	-	PUNCT
cana-3960	10	27	2025	2025	NUM
cana-3960	10	28	abstract	abstract	NOUN
cana-3960	10	29	:	:	PUNCT
cana-3960	10	30	this	this	DET
cana-3960	10	31	paper	paper	NOUN
cana-3960	10	32	investigates	investigate	VERB
cana-3960	10	33	the	the	DET
cana-3960	10	34	controllability	controllability	NOUN
cana-3960	10	35	and	and	CCONJ
cana-3960	10	36	observability	observability	NOUN
cana-3960	10	37	of	of	ADP
cana-3960	10	38	matrix	matrix	NOUN
cana-3960	10	39	sylvester	sylvest	ADJ
cana-3960	10	40	adjoint	adjoint	PROPN
cana-3960	10	41	dynamic	dynamic	ADJ
cana-3960	10	42	impulsive	impulsive	ADJ
cana-3960	10	43	systems	system	NOUN
cana-3960	10	44	within	within	ADP
cana-3960	10	45	the	the	DET
cana-3960	10	46	framework	framework	NOUN
cana-3960	10	47	of	of	ADP
cana-3960	10	48	time	time	NOUN
cana-3960	10	49	scales	scale	NOUN
cana-3960	10	50	.	.	PUNCT
cana-3960	11	1	by	by	ADP
cana-3960	11	2	applying	apply	VERB
cana-3960	11	3	the	the	DET
cana-3960	11	4	vectorization	vectorization	NOUN
cana-3960	11	5	operator	operator	NOUN
cana-3960	11	6	,	,	PUNCT
cana-3960	11	7	the	the	DET
cana-3960	11	8	system	system	NOUN
cana-3960	11	9	is	be	AUX
cana-3960	11	10	reformulated	reformulate	VERB
cana-3960	11	11	into	into	ADP
cana-3960	11	12	an	an	DET
cana-3960	11	13	equivalent	equivalent	ADJ
cana-3960	11	14	kronecker	kronecker	NOUN
cana-3960	11	15	productbased	productbase	VERB
cana-3960	11	16	dynamic	dynamic	ADJ
cana-3960	11	17	impulsive	impulsive	ADJ
cana-3960	11	18	system	system	NOUN
cana-3960	11	19	,	,	PUNCT
cana-3960	11	20	enabling	enable	VERB
cana-3960	11	21	more	more	ADV
cana-3960	11	22	efficient	efficient	ADJ
cana-3960	11	23	analysis	analysis	NOUN
cana-3960	11	24	.	.	PUNCT
cana-3960	12	1	the	the	DET
cana-3960	12	2	study	study	NOUN
cana-3960	12	3	derives	derive	VERB
cana-3960	12	4	necessary	necessary	ADJ
cana-3960	12	5	and	and	CCONJ
cana-3960	12	6	sufficient	sufficient	ADJ
cana-3960	12	7	conditions	condition	NOUN
cana-3960	12	8	for	for	ADP
cana-3960	12	9	controllability	controllability	NOUN
cana-3960	12	10	and	and	CCONJ
cana-3960	12	11	observability	observability	NOUN
cana-3960	12	12	through	through	ADP
cana-3960	12	13	the	the	DET
cana-3960	12	14	adjoint	adjoint	NOUN
cana-3960	12	15	matrix	matrix	NOUN
cana-3960	12	16	approach	approach	NOUN
cana-3960	12	17	,	,	PUNCT
cana-3960	12	18	highlighting	highlight	VERB
cana-3960	12	19	its	its	PRON
cana-3960	12	20	analytical	analytical	ADJ
cana-3960	12	21	significance	significance	NOUN
cana-3960	12	22	.	.	PUNCT
cana-3960	13	1	additionally	additionally	ADV
cana-3960	13	2	,	,	PUNCT
cana-3960	13	3	the	the	DET
cana-3960	13	4	research	research	NOUN
cana-3960	13	5	incorporates	incorporate	VERB
cana-3960	13	6	the	the	DET
cana-3960	13	7	gramian	gramian	ADJ
cana-3960	13	8	matrix	matrix	NOUN
cana-3960	13	9	to	to	PART
cana-3960	13	10	establish	establish	VERB
cana-3960	13	11	results	result	NOUN
cana-3960	13	12	for	for	ADP
cana-3960	13	13	impulsive	impulsive	ADJ
cana-3960	13	14	dynamic	dynamic	ADJ
cana-3960	13	15	systems	system	NOUN
cana-3960	13	16	on	on	ADP
cana-3960	13	17	time	time	NOUN
cana-3960	13	18	scales	scale	NOUN
cana-3960	13	19	,	,	PUNCT
cana-3960	13	20	providing	provide	VERB
cana-3960	13	21	a	a	DET
cana-3960	13	22	comprehensive	comprehensive	ADJ
cana-3960	13	23	criterion	criterion	NOUN
cana-3960	13	24	for	for	ADP
cana-3960	13	25	evaluating	evaluate	VERB
cana-3960	13	26	system	system	NOUN
cana-3960	13	27	properties	property	NOUN
cana-3960	13	28	.	.	PUNCT
cana-3960	14	1	this	this	DET
cana-3960	14	2	unified	unify	VERB
cana-3960	14	3	framework	framework	NOUN
cana-3960	14	4	bridges	bridge	NOUN
cana-3960	14	5	discrete	discrete	ADJ
cana-3960	14	6	and	and	CCONJ
cana-3960	14	7	continuous	continuous	ADJ
cana-3960	14	8	-	-	PUNCT
cana-3960	14	9	time	time	NOUN
cana-3960	14	10	dynamics	dynamic	NOUN
cana-3960	14	11	,	,	PUNCT
cana-3960	14	12	enabling	enable	VERB
cana-3960	14	13	the	the	DET
cana-3960	14	14	modeling	modeling	NOUN
cana-3960	14	15	and	and	CCONJ
cana-3960	14	16	analysis	analysis	NOUN
cana-3960	14	17	of	of	ADP
cana-3960	14	18	hybrid	hybrid	ADJ
cana-3960	14	19	systems	system	NOUN
cana-3960	14	20	with	with	ADP
cana-3960	14	21	abrupt	abrupt	ADJ
cana-3960	14	22	state	state	NOUN
cana-3960	14	23	transitions	transition	NOUN
cana-3960	14	24	.	.	PUNCT
cana-3960	15	1	the	the	DET
cana-3960	15	2	findings	finding	NOUN
cana-3960	15	3	advance	advance	VERB
cana-3960	15	4	the	the	DET
cana-3960	15	5	mathematical	mathematical	ADJ
cana-3960	15	6	understanding	understanding	NOUN
cana-3960	15	7	of	of	ADP
cana-3960	15	8	sylvester	sylvester	ADJ
cana-3960	15	9	matrix	matrix	NOUN
cana-3960	15	10	systems	system	NOUN
cana-3960	15	11	and	and	CCONJ
cana-3960	15	12	offer	offer	VERB
cana-3960	15	13	practical	practical	ADJ
cana-3960	15	14	insights	insight	NOUN
cana-3960	15	15	into	into	ADP
cana-3960	15	16	the	the	DET
cana-3960	15	17	design	design	NOUN
cana-3960	15	18	and	and	CCONJ
cana-3960	15	19	control	control	NOUN
cana-3960	15	20	of	of	ADP
cana-3960	15	21	complex	complex	ADJ
cana-3960	15	22	impulsive	impulsive	ADJ
cana-3960	15	23	systems	system	NOUN
cana-3960	15	24	across	across	ADP
cana-3960	15	25	varied	varied	ADJ
cana-3960	15	26	applications	application	NOUN
cana-3960	15	27	.	.	PUNCT
cana-3960	16	1	this	this	DET
cana-3960	16	2	work	work	NOUN
cana-3960	16	3	contributes	contribute	VERB
cana-3960	16	4	to	to	ADP
cana-3960	16	5	the	the	DET
cana-3960	16	6	growing	grow	VERB
cana-3960	16	7	field	field	NOUN
cana-3960	16	8	of	of	ADP
cana-3960	16	9	time	time	NOUN
cana-3960	16	10	-	-	PUNCT
cana-3960	16	11	scale	scale	NOUN
cana-3960	16	12	calculus	calculus	NOUN
cana-3960	16	13	and	and	CCONJ
cana-3960	16	14	its	its	PRON
cana-3960	16	15	applications	application	NOUN
cana-3960	16	16	in	in	ADP
cana-3960	16	17	dynamic	dynamic	ADJ
cana-3960	16	18	system	system	NOUN
cana-3960	16	19	analysis	analysis	NOUN
cana-3960	16	20	.	.	PUNCT
cana-3960	17	1	keywords	keyword	NOUN
cana-3960	17	2	:	:	PUNCT
cana-3960	17	3	controllability	controllability	NOUN
cana-3960	17	4	,	,	PUNCT
cana-3960	17	5	observability	observability	NOUN
cana-3960	17	6	,	,	PUNCT
cana-3960	17	7	kronecker	kronecker	NOUN
cana-3960	17	8	product	product	NOUN
cana-3960	17	9	,	,	PUNCT
cana-3960	17	10	time	time	NOUN
cana-3960	17	11	scales	scale	NOUN
cana-3960	17	12	.	.	PUNCT
cana-3960	18	1	mathematics	mathematic	NOUN
cana-3960	18	2	subject	subject	ADJ
cana-3960	18	3	classification	classification	NOUN
cana-3960	18	4	:	:	PUNCT
cana-3960	18	5	93b05	93b05	NUM
cana-3960	18	6	,	,	PUNCT
cana-3960	18	7	93b07,39a12	93b07,39a12	NUM
cana-3960	18	8	,	,	PUNCT
cana-3960	18	9	18a40	18a40	NUM
cana-3960	18	10	,	,	PUNCT
cana-3960	18	11	34n05	34n05	NUM
cana-3960	18	12	.	.	PUNCT
cana-3960	19	1	1	1	X
cana-3960	19	2	.	.	X
cana-3960	19	3	introduction	introduction	NOUN
cana-3960	19	4	the	the	DET
cana-3960	19	5	study	study	NOUN
cana-3960	19	6	of	of	ADP
cana-3960	19	7	adjoint	adjoint	PROPN
cana-3960	19	8	matrix	matrix	NOUN
cana-3960	19	9	sylvester	sylvest	ADJ
cana-3960	19	10	dynamic	dynamic	ADJ
cana-3960	19	11	impulsive	impulsive	ADJ
cana-3960	19	12	systems	system	NOUN
cana-3960	19	13	on	on	ADP
cana-3960	19	14	time	time	NOUN
cana-3960	19	15	scales	scale	NOUN
cana-3960	19	16	combines	combine	VERB
cana-3960	19	17	the	the	DET
cana-3960	19	18	strengths	strength	NOUN
cana-3960	19	19	of	of	ADP
cana-3960	19	20	continuous	continuous	ADJ
cana-3960	19	21	and	and	CCONJ
cana-3960	19	22	discrete	discrete	ADJ
cana-3960	19	23	systems	system	NOUN
cana-3960	19	24	,	,	PUNCT
cana-3960	19	25	offering	offer	VERB
cana-3960	19	26	a	a	DET
cana-3960	19	27	unified	unified	ADJ
cana-3960	19	28	framework	framework	NOUN
cana-3960	19	29	to	to	PART
cana-3960	19	30	analyze	analyze	VERB
cana-3960	19	31	complex	complex	ADJ
cana-3960	19	32	realworld	realworld	PROPN
cana-3960	19	33	phenomena	phenomena	PROPN
cana-3960	19	34	.	.	PUNCT
cana-3960	20	1	this	this	DET
cana-3960	20	2	approach	approach	NOUN
cana-3960	20	3	effectively	effectively	ADV
cana-3960	20	4	models	model	VERB
cana-3960	20	5	systems	system	NOUN
cana-3960	20	6	that	that	PRON
cana-3960	20	7	exhibit	exhibit	VERB
cana-3960	20	8	both	both	CCONJ
cana-3960	20	9	abrupt	abrupt	ADJ
cana-3960	20	10	changes	change	NOUN
cana-3960	20	11	and	and	CCONJ
cana-3960	20	12	continuous	continuous	ADJ
cana-3960	20	13	evolution	evolution	NOUN
cana-3960	20	14	,	,	PUNCT
cana-3960	20	15	making	make	VERB
cana-3960	20	16	it	it	PRON
cana-3960	20	17	particularly	particularly	ADV
cana-3960	20	18	relevant	relevant	ADJ
cana-3960	20	19	in	in	ADP
cana-3960	20	20	fields	field	NOUN
cana-3960	20	21	such	such	ADJ
cana-3960	20	22	as	as	ADP
cana-3960	20	23	engineering	engineering	NOUN
cana-3960	20	24	,	,	PUNCT
cana-3960	20	25	biology	biology	NOUN
cana-3960	20	26	,	,	PUNCT
cana-3960	20	27	and	and	CCONJ
cana-3960	20	28	economics	economic	NOUN
cana-3960	20	29	.	.	PUNCT
cana-3960	21	1	the	the	DET
cana-3960	21	2	use	use	NOUN
cana-3960	21	3	of	of	ADP
cana-3960	21	4	adjoint	adjoint	NOUN
cana-3960	21	5	matrix	matrix	NOUN
cana-3960	21	6	techniques	technique	NOUN
cana-3960	21	7	brings	bring	VERB
cana-3960	21	8	significant	significant	ADJ
cana-3960	21	9	advantages	advantage	NOUN
cana-3960	21	10	,	,	PUNCT
cana-3960	21	11	such	such	ADJ
cana-3960	21	12	as	as	ADP
cana-3960	21	13	simplifying	simplify	VERB
cana-3960	21	14	the	the	DET
cana-3960	21	15	process	process	NOUN
cana-3960	21	16	of	of	ADP
cana-3960	21	17	solving	solve	VERB
cana-3960	21	18	linear	linear	ADJ
cana-3960	21	19	dynamic	dynamic	ADJ
cana-3960	21	20	equations	equation	NOUN
cana-3960	21	21	,	,	PUNCT
cana-3960	21	22	reducing	reduce	VERB
cana-3960	21	23	computational	computational	ADJ
cana-3960	21	24	complexity	complexity	NOUN
cana-3960	21	25	,	,	PUNCT
cana-3960	21	26	and	and	CCONJ
cana-3960	21	27	enhancing	enhance	VERB
cana-3960	21	28	analytical	analytical	ADJ
cana-3960	21	29	clarity	clarity	NOUN
cana-3960	21	30	.	.	PUNCT
cana-3960	22	1	matrices	matrix	NOUN
cana-3960	22	2	also	also	ADV
cana-3960	22	3	provide	provide	VERB
cana-3960	22	4	a	a	DET
cana-3960	22	5	compact	compact	ADJ
cana-3960	22	6	representation	representation	NOUN
cana-3960	22	7	of	of	ADP
cana-3960	22	8	system	system	NOUN
cana-3960	22	9	dynamics	dynamic	NOUN
cana-3960	22	10	,	,	PUNCT
cana-3960	22	11	facilitating	facilitate	VERB
cana-3960	22	12	efficient	efficient	ADJ
cana-3960	22	13	manipulation	manipulation	NOUN
cana-3960	22	14	,	,	PUNCT
cana-3960	22	15	scalability	scalability	NOUN
cana-3960	22	16	,	,	PUNCT
cana-3960	22	17	and	and	CCONJ
cana-3960	22	18	the	the	DET
cana-3960	22	19	application	application	NOUN
cana-3960	22	20	of	of	ADP
cana-3960	22	21	powerful	powerful	ADJ
cana-3960	22	22	algebraic	algebraic	ADJ
cana-3960	22	23	methods	method	NOUN
cana-3960	22	24	.	.	PUNCT
cana-3960	23	1	by	by	ADP
cana-3960	23	2	incorporating	incorporate	VERB
cana-3960	23	3	the	the	DET
cana-3960	23	4	time	time	NOUN
cana-3960	23	5	scales	scale	NOUN
cana-3960	23	6	framework	framework	NOUN
cana-3960	23	7	,	,	PUNCT
cana-3960	23	8	this	this	DET
cana-3960	23	9	method	method	NOUN
cana-3960	23	10	seamlessly	seamlessly	ADV
cana-3960	23	11	integrates	integrate	VERB
cana-3960	23	12	hybrid	hybrid	ADJ
cana-3960	23	13	systems	system	NOUN
cana-3960	23	14	,	,	PUNCT
cana-3960	23	15	enabling	enable	VERB
cana-3960	23	16	comprehensive	comprehensive	ADJ
cana-3960	23	17	analyses	analysis	NOUN
cana-3960	23	18	of	of	ADP
cana-3960	23	19	controllability	controllability	NOUN
cana-3960	23	20	,	,	PUNCT
cana-3960	23	21	stability	stability	NOUN
cana-3960	23	22	,	,	PUNCT
cana-3960	23	23	and	and	CCONJ
cana-3960	23	24	optimization	optimization	NOUN
cana-3960	23	25	under	under	ADP
cana-3960	23	26	impulsive	impulsive	ADJ
cana-3960	23	27	effects	effect	NOUN
cana-3960	23	28	.	.	PUNCT
cana-3960	24	1	consequently	consequently	ADV
cana-3960	24	2	,	,	PUNCT
cana-3960	24	3	this	this	DET
cana-3960	24	4	framework	framework	NOUN
cana-3960	24	5	bridges	bridge	VERB
cana-3960	24	6	the	the	DET
cana-3960	24	7	gap	gap	NOUN
cana-3960	24	8	between	between	ADP
cana-3960	24	9	discrete	discrete	ADJ
cana-3960	24	10	and	and	CCONJ
cana-3960	24	11	continuous	continuous	ADJ
cana-3960	24	12	dynamics	dynamic	NOUN
cana-3960	24	13	,	,	PUNCT
cana-3960	24	14	advancing	advance	VERB
cana-3960	24	15	both	both	CCONJ
cana-3960	24	16	theoretical	theoretical	ADJ
cana-3960	24	17	mathematics	mathematic	NOUN
cana-3960	24	18	and	and	CCONJ
cana-3960	24	19	practical	practical	ADJ
cana-3960	24	20	applications	application	NOUN
cana-3960	24	21	.	.	PUNCT
cana-3960	25	1	mailto:asreenivasulu@kluniversity.in	mailto:asreenivasulu@kluniversity.in	PROPN
cana-3960	25	2	;	;	PUNCT
cana-3960	25	3	mailto:bvardr2010@kluniversity.in	mailto:bvardr2010@kluniversity.in	PROPN
cana-3960	25	4	mailto:lakshmipallavi.p@bvrit.ac.in	mailto:lakshmipallavi.p@bvrit.ac.in	AUX
cana-3960	25	5	mailto:balajiprakashgudala@gmail.com	mailto:balajiprakashgudala@gmail.com	PROPN
cana-3960	25	6	mailto:maths4444@gmail.com	mailto:maths4444@gmail.com	NUM
cana-3960	25	7	mailto:jppraveen17@gmail.com	mailto:jppraveen17@gmail.com	NOUN
cana-3960	25	8	communications	communication	NOUN
cana-3960	25	9	on	on	ADP
cana-3960	25	10	applied	apply	VERB
cana-3960	25	11	nonlinear	nonlinear	ADJ
cana-3960	25	12	analysis	analysis	NOUN
cana-3960	25	13	issn	issn	NOUN
cana-3960	25	14	:	:	PUNCT
cana-3960	25	15	1074	1074	NUM
cana-3960	25	16	-	-	PUNCT
cana-3960	25	17	133x	133x	NUM
cana-3960	25	18	vol	vol	NOUN
cana-3960	25	19	32	32	NUM
cana-3960	26	1	no	no	NOUN
cana-3960	26	2	.	.	PUNCT
cana-3960	27	1	9s	9s	NUM
cana-3960	27	2	(	(	PUNCT
cana-3960	27	3	2025	2025	NUM
cana-3960	27	4	)	)	PUNCT
cana-3960	27	5	494	494	NUM
cana-3960	28	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3960	28	2	impulsive	impulsive	ADJ
cana-3960	28	3	differential	differential	ADJ
cana-3960	28	4	equations	equation	NOUN
cana-3960	28	5	play	play	VERB
cana-3960	28	6	a	a	DET
cana-3960	28	7	crucial	crucial	ADJ
cana-3960	28	8	role	role	NOUN
cana-3960	28	9	in	in	ADP
cana-3960	28	10	describing	describe	VERB
cana-3960	28	11	systems	system	NOUN
cana-3960	28	12	with	with	ADP
cana-3960	28	13	sudden	sudden	ADJ
cana-3960	28	14	changes	change	NOUN
cana-3960	28	15	in	in	ADP
cana-3960	28	16	state	state	NOUN
cana-3960	28	17	,	,	PUNCT
cana-3960	28	18	governed	govern	VERB
cana-3960	28	19	by	by	ADP
cana-3960	28	20	continuous	continuous	ADJ
cana-3960	28	21	dynamics	dynamic	NOUN
cana-3960	28	22	interspersed	intersperse	VERB
cana-3960	28	23	with	with	ADP
cana-3960	28	24	jump	jump	NOUN
cana-3960	28	25	criteria	criterion	NOUN
cana-3960	28	26	.	.	PUNCT
cana-3960	29	1	these	these	DET
cana-3960	29	2	equations	equation	NOUN
cana-3960	29	3	provide	provide	VERB
cana-3960	29	4	a	a	DET
cana-3960	29	5	logical	logical	ADJ
cana-3960	29	6	and	and	CCONJ
cana-3960	29	7	robust	robust	ADJ
cana-3960	29	8	framework	framework	NOUN
cana-3960	29	9	for	for	ADP
cana-3960	29	10	modeling	model	VERB
cana-3960	29	11	abrupt	abrupt	ADJ
cana-3960	29	12	transitions	transition	NOUN
cana-3960	29	13	commonly	commonly	ADV
cana-3960	29	14	observed	observe	VERB
cana-3960	29	15	in	in	ADP
cana-3960	29	16	real	real	ADJ
cana-3960	29	17	-	-	PUNCT
cana-3960	29	18	world	world	NOUN
cana-3960	29	19	processes	process	NOUN
cana-3960	29	20	.	.	PUNCT
cana-3960	30	1	their	their	PRON
cana-3960	30	2	versatility	versatility	NOUN
cana-3960	30	3	has	have	AUX
cana-3960	30	4	led	lead	VERB
cana-3960	30	5	to	to	ADP
cana-3960	30	6	extensive	extensive	ADJ
cana-3960	30	7	research	research	NOUN
cana-3960	30	8	and	and	CCONJ
cana-3960	30	9	development	development	NOUN
cana-3960	30	10	in	in	ADP
cana-3960	30	11	this	this	DET
cana-3960	30	12	area	area	NOUN
cana-3960	30	13	.	.	PUNCT
cana-3960	31	1	for	for	ADP
cana-3960	31	2	instance	instance	NOUN
cana-3960	31	3	,	,	PUNCT
cana-3960	31	4	in	in	ADP
cana-3960	31	5	[	[	PUNCT
cana-3960	31	6	4	4	NUM
cana-3960	31	7	]	]	PUNCT
cana-3960	31	8	,	,	PUNCT
cana-3960	31	9	solutions	solution	NOUN
cana-3960	31	10	to	to	ADP
cana-3960	31	11	fractional	fractional	ADJ
cana-3960	31	12	equations	equation	NOUN
cana-3960	31	13	such	such	ADJ
cana-3960	31	14	as	as	ADP
cana-3960	31	15	the	the	DET
cana-3960	31	16	fitzhugh	fitzhugh	PROPN
cana-3960	31	17	-	-	PUNCT
cana-3960	31	18	nagumo	nagumo	ADJ
cana-3960	31	19	equation	equation	NOUN
cana-3960	31	20	,	,	PUNCT
cana-3960	31	21	the	the	DET
cana-3960	31	22	newell	newell	PROPN
cana-3960	31	23	-	-	PUNCT
cana-3960	31	24	whitehead	whitehead	PROPN
cana-3960	31	25	-	-	PUNCT
cana-3960	31	26	segel	segel	NOUN
cana-3960	31	27	equation	equation	NOUN
cana-3960	31	28	,	,	PUNCT
cana-3960	31	29	and	and	CCONJ
cana-3960	31	30	the	the	DET
cana-3960	31	31	zeldovich	zeldovich	ADJ
cana-3960	31	32	equation	equation	NOUN
cana-3960	31	33	were	be	AUX
cana-3960	31	34	explored	explore	VERB
cana-3960	31	35	,	,	PUNCT
cana-3960	31	36	demonstrating	demonstrate	VERB
cana-3960	31	37	their	their	PRON
cana-3960	31	38	practical	practical	ADJ
cana-3960	31	39	significance	significance	NOUN
cana-3960	31	40	.	.	PUNCT
cana-3960	32	1	similarly	similarly	ADV
cana-3960	32	2	,	,	PUNCT
cana-3960	32	3	[	[	X
cana-3960	32	4	5	5	NUM
cana-3960	32	5	]	]	PUNCT
cana-3960	32	6	addressed	address	VERB
cana-3960	32	7	an	an	DET
cana-3960	32	8	inverse	inverse	ADJ
cana-3960	32	9	coefficient	coefficient	NOUN
cana-3960	32	10	problem	problem	NOUN
cana-3960	32	11	for	for	ADP
cana-3960	32	12	the	the	DET
cana-3960	32	13	conformable	conformable	ADJ
cana-3960	32	14	time	time	NOUN
cana-3960	32	15	-	-	PUNCT
cana-3960	32	16	diffusion	diffusion	NOUN
cana-3960	32	17	equation	equation	NOUN
cana-3960	32	18	,	,	PUNCT
cana-3960	32	19	retrieving	retrieve	VERB
cana-3960	32	20	time	time	NOUN
cana-3960	32	21	-	-	PUNCT
cana-3960	32	22	dependent	dependent	ADJ
cana-3960	32	23	diffusion	diffusion	NOUN
cana-3960	32	24	coefficients	coefficient	NOUN
cana-3960	32	25	with	with	ADP
cana-3960	32	26	precision	precision	NOUN
cana-3960	32	27	.	.	PUNCT
cana-3960	33	1	furthermore	furthermore	ADV
cana-3960	33	2	,	,	PUNCT
cana-3960	33	3	[	[	X
cana-3960	33	4	6	6	NUM
cana-3960	33	5	]	]	PUNCT
cana-3960	33	6	proposed	propose	VERB
cana-3960	33	7	a	a	DET
cana-3960	33	8	novel	novel	ADJ
cana-3960	33	9	mathematical	mathematical	ADJ
cana-3960	33	10	formation	formation	NOUN
cana-3960	33	11	model	model	NOUN
cana-3960	33	12	using	use	VERB
cana-3960	33	13	the	the	DET
cana-3960	33	14	fractional	fractional	ADJ
cana-3960	33	15	atangana	atangana	PROPN
cana-3960	33	16	-	-	PUNCT
cana-3960	33	17	baleanu	baleanu	PROPN
cana-3960	33	18	-	-	PUNCT
cana-3960	33	19	caputo	caputo	PROPN
cana-3960	33	20	derivative	derivative	NOUN
cana-3960	33	21	,	,	PUNCT
cana-3960	33	22	showcasing	showcase	VERB
cana-3960	33	23	the	the	DET
cana-3960	33	24	reliability	reliability	NOUN
cana-3960	33	25	and	and	CCONJ
cana-3960	33	26	computational	computational	ADJ
cana-3960	33	27	efficiency	efficiency	NOUN
cana-3960	33	28	of	of	ADP
cana-3960	33	29	this	this	DET
cana-3960	33	30	method	method	NOUN
cana-3960	33	31	.	.	PUNCT
cana-3960	34	1	applications	application	NOUN
cana-3960	34	2	of	of	ADP
cana-3960	34	3	impulsive	impulsive	ADJ
cana-3960	34	4	systems	system	NOUN
cana-3960	34	5	extend	extend	VERB
cana-3960	34	6	to	to	ADP
cana-3960	34	7	biological	biological	ADJ
cana-3960	34	8	models	model	NOUN
cana-3960	34	9	,	,	PUNCT
cana-3960	34	10	as	as	SCONJ
cana-3960	34	11	demonstrated	demonstrate	VERB
cana-3960	34	12	in	in	ADP
cana-3960	34	13	[	[	X
cana-3960	34	14	8	8	NUM
cana-3960	34	15	]	]	PUNCT
cana-3960	34	16	,	,	PUNCT
cana-3960	34	17	where	where	SCONJ
cana-3960	34	18	insect	insect	NOUN
cana-3960	34	19	population	population	NOUN
cana-3960	34	20	dynamics	dynamic	NOUN
cana-3960	34	21	were	be	AUX
cana-3960	34	22	analyzed	analyze	VERB
cana-3960	34	23	using	use	VERB
cana-3960	34	24	exponential	exponential	NOUN
cana-3960	34	25	,	,	PUNCT
cana-3960	34	26	hyperbolic	hyperbolic	ADJ
cana-3960	34	27	,	,	PUNCT
cana-3960	34	28	and	and	CCONJ
cana-3960	34	29	trigonometric	trigonometric	ADJ
cana-3960	34	30	functions	function	NOUN
cana-3960	34	31	to	to	PART
cana-3960	34	32	solve	solve	VERB
cana-3960	34	33	second	second	ADJ
cana-3960	34	34	-	-	PUNCT
cana-3960	34	35	order	order	NOUN
cana-3960	34	36	linear	linear	ADJ
cana-3960	34	37	dynamic	dynamic	ADJ
cana-3960	34	38	equations	equation	NOUN
cana-3960	34	39	.	.	PUNCT
cana-3960	35	1	additionally	additionally	ADV
cana-3960	35	2	,	,	PUNCT
cana-3960	35	3	[	[	X
cana-3960	35	4	9	9	NUM
cana-3960	35	5	]	]	PUNCT
cana-3960	35	6	discussed	discuss	VERB
cana-3960	35	7	stability	stability	NOUN
cana-3960	35	8	conditions	condition	NOUN
cana-3960	35	9	that	that	PRON
cana-3960	35	10	ensure	ensure	VERB
cana-3960	35	11	input	input	NOUN
cana-3960	35	12	-	-	PUNCT
cana-3960	35	13	to	to	ADP
cana-3960	35	14	-	-	PUNCT
cana-3960	35	15	state	state	NOUN
cana-3960	35	16	stability	stability	NOUN
cana-3960	35	17	for	for	ADP
cana-3960	35	18	hybrid	hybrid	ADJ
cana-3960	35	19	systems	system	NOUN
cana-3960	35	20	,	,	PUNCT
cana-3960	35	21	even	even	ADV
cana-3960	35	22	under	under	ADP
cana-3960	35	23	instability	instability	NOUN
cana-3960	35	24	.	.	PUNCT
cana-3960	36	1	the	the	DET
cana-3960	36	2	concept	concept	NOUN
cana-3960	36	3	of	of	ADP
cana-3960	36	4	controllability	controllability	NOUN
cana-3960	36	5	further	far	ADV
cana-3960	36	6	enhances	enhance	VERB
cana-3960	36	7	system	system	NOUN
cana-3960	36	8	stabilization	stabilization	NOUN
cana-3960	36	9	by	by	ADP
cana-3960	36	10	constraining	constrain	VERB
cana-3960	36	11	behavior	behavior	NOUN
cana-3960	36	12	through	through	ADP
cana-3960	36	13	the	the	DET
cana-3960	36	14	analysis	analysis	NOUN
cana-3960	36	15	of	of	ADP
cana-3960	36	16	linear	linear	ADJ
cana-3960	36	17	and	and	CCONJ
cana-3960	36	18	nonlinear	nonlinear	ADJ
cana-3960	36	19	operators	operator	NOUN
cana-3960	36	20	[	[	X
cana-3960	36	21	10	10	NUM
cana-3960	36	22	]	]	PUNCT
cana-3960	36	23	.	.	PUNCT
cana-3960	37	1	matrix	matrix	NOUN
cana-3960	37	2	-	-	PUNCT
cana-3960	37	3	based	base	VERB
cana-3960	37	4	approaches	approach	NOUN
cana-3960	37	5	further	far	ADV
cana-3960	37	6	bolster	bolster	VERB
cana-3960	37	7	the	the	DET
cana-3960	37	8	study	study	NOUN
cana-3960	37	9	of	of	ADP
cana-3960	37	10	impulsive	impulsive	ADJ
cana-3960	37	11	systems	system	NOUN
cana-3960	37	12	.	.	PUNCT
cana-3960	38	1	they	they	PRON
cana-3960	38	2	allow	allow	VERB
cana-3960	38	3	for	for	ADP
cana-3960	38	4	efficient	efficient	ADJ
cana-3960	38	5	representation	representation	NOUN
cana-3960	38	6	of	of	ADP
cana-3960	38	7	multi	multi	ADJ
cana-3960	38	8	-	-	ADJ
cana-3960	38	9	dimensional	dimensional	ADJ
cana-3960	38	10	systems	system	NOUN
cana-3960	38	11	and	and	CCONJ
cana-3960	38	12	enable	enable	VERB
cana-3960	38	13	the	the	DET
cana-3960	38	14	use	use	NOUN
cana-3960	38	15	of	of	ADP
cana-3960	38	16	spectral	spectral	ADJ
cana-3960	38	17	analysis	analysis	NOUN
cana-3960	38	18	,	,	PUNCT
cana-3960	38	19	eigenvalue	eigenvalue	NOUN
cana-3960	38	20	computation	computation	NOUN
cana-3960	38	21	,	,	PUNCT
cana-3960	38	22	and	and	CCONJ
cana-3960	38	23	matrix	matrix	NOUN
cana-3960	38	24	decompositions	decomposition	NOUN
cana-3960	38	25	to	to	PART
cana-3960	38	26	study	study	VERB
cana-3960	38	27	system	system	NOUN
cana-3960	38	28	properties	property	NOUN
cana-3960	38	29	like	like	ADP
cana-3960	38	30	stability	stability	NOUN
cana-3960	38	31	and	and	CCONJ
cana-3960	38	32	controllability	controllability	NOUN
cana-3960	38	33	.	.	PUNCT
cana-3960	39	1	the	the	DET
cana-3960	39	2	study	study	NOUN
cana-3960	39	3	of	of	ADP
cana-3960	39	4	impulsive	impulsive	ADJ
cana-3960	39	5	systems	system	NOUN
cana-3960	39	6	on	on	ADP
cana-3960	39	7	time	time	NOUN
cana-3960	39	8	scales	scale	NOUN
cana-3960	39	9	has	have	AUX
cana-3960	39	10	also	also	ADV
cana-3960	39	11	proven	prove	VERB
cana-3960	39	12	the	the	DET
cana-3960	39	13	existence	existence	NOUN
cana-3960	39	14	and	and	CCONJ
cana-3960	39	15	uniqueness	uniqueness	NOUN
cana-3960	39	16	of	of	ADP
cana-3960	39	17	solutions	solution	NOUN
cana-3960	39	18	for	for	ADP
cana-3960	39	19	nonlinear	nonlinear	ADJ
cana-3960	39	20	impulsive	impulsive	ADJ
cana-3960	39	21	dynamic	dynamic	ADJ
cana-3960	39	22	equations	equation	NOUN
cana-3960	39	23	[	[	X
cana-3960	39	24	11	11	NUM
cana-3960	39	25	]	]	PUNCT
cana-3960	39	26	.	.	PUNCT
cana-3960	40	1	in	in	ADP
cana-3960	40	2	[	[	X
cana-3960	40	3	13	13	NUM
cana-3960	40	4	]	]	PUNCT
cana-3960	40	5	,	,	PUNCT
cana-3960	40	6	the	the	DET
cana-3960	40	7	properties	property	NOUN
cana-3960	40	8	of	of	ADP
cana-3960	40	9	impulsive	impulsive	ADJ
cana-3960	40	10	dirac	dirac	NOUN
cana-3960	40	11	systems	system	NOUN
cana-3960	40	12	on	on	ADP
cana-3960	40	13	sturmian	sturmian	NOUN
cana-3960	40	14	time	time	NOUN
cana-3960	40	15	scales	scale	NOUN
cana-3960	40	16	were	be	AUX
cana-3960	40	17	examined	examine	VERB
cana-3960	40	18	,	,	PUNCT
cana-3960	40	19	including	include	VERB
cana-3960	40	20	the	the	DET
cana-3960	40	21	construction	construction	NOUN
cana-3960	40	22	of	of	ADP
cana-3960	40	23	self	self	NOUN
cana-3960	40	24	-	-	PUNCT
cana-3960	40	25	adjoint	adjoint	NOUN
cana-3960	40	26	operators	operator	NOUN
cana-3960	40	27	.	.	PUNCT
cana-3960	41	1	additionally	additionally	ADV
cana-3960	41	2	,	,	PUNCT
cana-3960	41	3	[	[	X
cana-3960	41	4	14	14	NUM
cana-3960	41	5	]	]	PUNCT
cana-3960	41	6	introduced	introduce	VERB
cana-3960	41	7	a	a	DET
cana-3960	41	8	new	new	ADJ
cana-3960	41	9	transition	transition	NOUN
cana-3960	41	10	matrix	matrix	NOUN
cana-3960	41	11	to	to	PART
cana-3960	41	12	analyze	analyze	VERB
cana-3960	41	13	the	the	DET
cana-3960	41	14	controllability	controllability	NOUN
cana-3960	41	15	and	and	CCONJ
cana-3960	41	16	observability	observability	NOUN
cana-3960	41	17	of	of	ADP
cana-3960	41	18	impulsive	impulsive	ADJ
cana-3960	41	19	systems	system	NOUN
cana-3960	41	20	on	on	ADP
cana-3960	41	21	time	time	NOUN
cana-3960	41	22	scales	scale	NOUN
cana-3960	41	23	.	.	PUNCT
cana-3960	42	1	this	this	DET
cana-3960	42	2	research	research	NOUN
cana-3960	42	3	highlights	highlight	VERB
cana-3960	42	4	the	the	DET
cana-3960	42	5	flexibility	flexibility	NOUN
cana-3960	42	6	and	and	CCONJ
cana-3960	42	7	adaptability	adaptability	NOUN
cana-3960	42	8	of	of	ADP
cana-3960	42	9	the	the	DET
cana-3960	42	10	impulsive	impulsive	ADJ
cana-3960	42	11	framework	framework	NOUN
cana-3960	42	12	,	,	PUNCT
cana-3960	42	13	enabling	enable	VERB
cana-3960	42	14	its	its	PRON
cana-3960	42	15	application	application	NOUN
cana-3960	42	16	to	to	ADP
cana-3960	42	17	nonuniform	nonuniform	ADJ
cana-3960	42	18	time	time	NOUN
cana-3960	42	19	domains	domain	NOUN
cana-3960	42	20	[	[	X
cana-3960	42	21	15	15	NUM
cana-3960	42	22	]	]	PUNCT
cana-3960	42	23	.	.	PUNCT
cana-3960	43	1	in	in	ADP
cana-3960	43	2	this	this	DET
cana-3960	43	3	paper	paper	NOUN
cana-3960	43	4	,	,	PUNCT
cana-3960	43	5	we	we	PRON
cana-3960	43	6	address	address	VERB
cana-3960	43	7	the	the	DET
cana-3960	43	8	sufficient	sufficient	ADJ
cana-3960	43	9	and	and	CCONJ
cana-3960	43	10	necessary	necessary	ADJ
cana-3960	43	11	controllability	controllability	NOUN
cana-3960	43	12	and	and	CCONJ
cana-3960	43	13	observability	observability	NOUN
cana-3960	43	14	conditions	condition	NOUN
cana-3960	43	15	for	for	ADP
cana-3960	43	16	matrix	matrix	NOUN
cana-3960	43	17	sylvester	sylvest	ADJ
cana-3960	43	18	adjoint	adjoint	PROPN
cana-3960	43	19	dynamic	dynamic	ADJ
cana-3960	43	20	impulsive	impulsive	ADJ
cana-3960	43	21	systems	system	NOUN
cana-3960	43	22	over	over	ADP
cana-3960	43	23	various	various	ADJ
cana-3960	43	24	time	time	NOUN
cana-3960	43	25	scales	scale	NOUN
cana-3960	43	26	.	.	PUNCT
cana-3960	44	1	{	{	PUNCT
cana-3960	44	2	𝑋𝛥(𝑡	𝑋𝛥(𝑡	NOUN
cana-3960	44	3	)	)	PUNCT
cana-3960	44	4	=	=	SYM
cana-3960	44	5	𝑃(𝑡)𝑋(𝑡	𝑃(𝑡)𝑋(𝑡	NOUN
cana-3960	44	6	)	)	PUNCT
cana-3960	44	7	+	+	CCONJ
cana-3960	44	8	𝑋(𝑡)𝑄(𝑡	𝑋(𝑡)𝑄(𝑡	X
cana-3960	44	9	)	)	PUNCT
cana-3960	44	10	+	+	CCONJ
cana-3960	44	11	𝜇(𝑡)𝑃(𝑡)𝑋(𝑡)𝑄(𝑡	𝜇(𝑡)𝑃(𝑡)𝑋(𝑡)𝑄(𝑡	NOUN
cana-3960	44	12	)	)	PUNCT
cana-3960	44	13	+	+	CCONJ
cana-3960	44	14	𝑇1(𝑡)𝑈(𝑡)𝑇2	𝑇1(𝑡)𝑈(𝑡)𝑇2	ADJ
cana-3960	44	15	∗(𝑡	∗(𝑡	NOUN
cana-3960	44	16	)	)	PUNCT
cana-3960	44	17	𝑋(𝑡𝑘	𝑋(𝑡𝑘	ADP
cana-3960	44	18	+	+	PROPN
cana-3960	44	19	)	)	PUNCT
cana-3960	44	20	=	=	SYM
cana-3960	44	21	(	(	PUNCT
cana-3960	44	22	𝐼	𝐼	PROPN
cana-3960	44	23	+	+	CCONJ
cana-3960	44	24	𝐿𝑘)𝑋(𝑡𝑘	𝐿𝑘)𝑋(𝑡𝑘	NOUN
cana-3960	44	25	)	)	PUNCT
cana-3960	44	26	,	,	PUNCT
cana-3960	44	27	𝑡	𝑡	PROPN
cana-3960	45	1	=	=	VERB
cana-3960	45	2	𝑡𝑘	𝑡𝑘	ADV
cana-3960	45	3	k	k	NOUN
cana-3960	45	4	=	=	PUNCT
cana-3960	45	5	1,2,3	1,2,3	NUM
cana-3960	45	6	....	....	PUNCT
cana-3960	45	7	𝑌(𝑡	𝑌(𝑡	X
cana-3960	45	8	)	)	PUNCT
cana-3960	45	9	=	=	SYM
cana-3960	45	10	𝐶(𝑡)𝑋(𝑡	𝐶(𝑡)𝑋(𝑡	NUM
cana-3960	45	11	)	)	PUNCT
cana-3960	45	12	+	+	NUM
cana-3960	45	13	𝐷(𝑡)𝑈(𝑡	𝐷(𝑡)𝑈(𝑡	X
cana-3960	45	14	)	)	PUNCT
cana-3960	45	15	𝑋(𝑡0	𝑋(𝑡0	NOUN
cana-3960	45	16	)	)	PUNCT
cana-3960	45	17	=	=	NOUN
cana-3960	45	18	𝑋0	𝑋0	NOUN
cana-3960	45	19	.	.	PUNCT
cana-3960	46	1	(	(	PUNCT
cana-3960	46	2	1.1	1.1	NUM
cana-3960	46	3	)	)	PUNCT
cana-3960	46	4	where	where	SCONJ
cana-3960	46	5	𝑋(𝑡	𝑋(𝑡	NOUN
cana-3960	46	6	)	)	PUNCT
cana-3960	46	7	is	be	AUX
cana-3960	46	8	an	an	DET
cana-3960	46	9	𝑛	𝑛	PRON
cana-3960	46	10	×	×	NOUN
cana-3960	46	11	𝑛	𝑛	DET
cana-3960	46	12	matrix	matrix	NOUN
cana-3960	46	13	,	,	PUNCT
cana-3960	46	14	𝑈(𝑡	𝑈(𝑡	NUM
cana-3960	46	15	)	)	PUNCT
cana-3960	46	16	is	be	AUX
cana-3960	46	17	m×	m×	PROPN
cana-3960	46	18	𝑛	𝑛	ADP
cana-3960	46	19	input	input	NOUN
cana-3960	46	20	pricewise	pricewise	PROPN
cana-3960	46	21	rd	rd	PROPN
cana-3960	46	22	-	-	ADJ
cana-3960	46	23	continuous	continuous	ADJ
cana-3960	46	24	matrix	matrix	NOUN
cana-3960	46	25	called	call	VERB
cana-3960	46	26	control	control	NOUN
cana-3960	46	27	input	input	NOUN
cana-3960	46	28	and	and	CCONJ
cana-3960	46	29	𝑌(𝑡	𝑌(𝑡	X
cana-3960	46	30	)	)	PUNCT
cana-3960	46	31	is	be	AUX
cana-3960	46	32	𝑝	𝑝	PROPN
cana-3960	46	33	×	×	NOUN
cana-3960	46	34	𝑛	𝑛	DET
cana-3960	46	35	output	output	NOUN
cana-3960	46	36	rd	rd	NOUN
cana-3960	46	37	-	-	NOUN
cana-3960	46	38	continuous	continuous	ADJ
cana-3960	46	39	.	.	PUNCT
cana-3960	47	1	here	here	ADV
cana-3960	47	2	𝑃	𝑃	PROPN
cana-3960	47	3	(	(	PUNCT
cana-3960	47	4	𝑡	𝑡	NOUN
cana-3960	47	5	)	)	PUNCT
cana-3960	47	6	,	,	PUNCT
cana-3960	47	7	𝑄(𝑡	𝑄(𝑡	PRON
cana-3960	47	8	)	)	PUNCT
cana-3960	47	9	,	,	PUNCT
cana-3960	47	10	𝑇1(𝑡	𝑇1(𝑡	NUM
cana-3960	47	11	)	)	PUNCT
cana-3960	47	12	,	,	PUNCT
cana-3960	47	13	𝑇2(𝑡	𝑇2(𝑡	PROPN
cana-3960	47	14	)	)	PUNCT
cana-3960	47	15	and	and	CCONJ
cana-3960	47	16	𝐿𝑘	𝐿𝑘	PROPN
cana-3960	47	17	are	be	AUX
cana-3960	47	18	𝑛	𝑛	DET
cana-3960	47	19	×	×	PROPN
cana-3960	47	20	𝑛	𝑛	PROPN
cana-3960	47	21	,	,	PUNCT
cana-3960	47	22	𝑛	𝑛	DET
cana-3960	47	23	×	×	PROPN
cana-3960	47	24	𝑛	𝑛	PROPN
cana-3960	47	25	,	,	PUNCT
cana-3960	47	26	𝑛	𝑛	DET
cana-3960	47	27	×	×	PROPN
cana-3960	47	28	𝑛	𝑛	PROPN
cana-3960	47	29	,	,	PUNCT
cana-3960	47	30	𝑛	𝑛	DET
cana-3960	47	31	×	×	NOUN
cana-3960	47	32	𝑚	𝑚	NOUN
cana-3960	47	33	and	and	CCONJ
cana-3960	47	34	𝑛	𝑛	DET
cana-3960	47	35	×	×	NOUN
cana-3960	47	36	𝑛	𝑛	PROPN
cana-3960	47	37	rd	rd	NOUN
cana-3960	47	38	-	-	ADJ
cana-3960	47	39	continuous	continuous	ADJ
cana-3960	47	40	matrices	matrix	NOUN
cana-3960	47	41	respectively	respectively	ADV
cana-3960	47	42	.	.	PUNCT
cana-3960	48	1	c(t	c(t	PROPN
cana-3960	48	2	)	)	PUNCT
cana-3960	48	3	,	,	PUNCT
cana-3960	48	4	d(t	d(t	PROPN
cana-3960	48	5	)	)	PUNCT
cana-3960	48	6	are	be	AUX
cana-3960	48	7	rdcontinuous	rdcontinuous	ADJ
cana-3960	48	8	matrices	matrix	NOUN
cana-3960	48	9	of	of	ADP
cana-3960	48	10	order	order	NOUN
cana-3960	48	11	𝑝	𝑝	ADP
cana-3960	48	12	×	×	NOUN
cana-3960	48	13	𝑛	𝑛	PROPN
cana-3960	48	14	and	and	CCONJ
cana-3960	48	15	𝑝	𝑝	NOUN
cana-3960	48	16	×	×	NOUN
cana-3960	48	17	𝑚	𝑚	ADP
cana-3960	48	18	respectively	respectively	ADV
cana-3960	48	19	.	.	PUNCT
cana-3960	49	1	𝑋∆(𝑡	𝑋∆(𝑡	VERB
cana-3960	49	2	)	)	PUNCT
cana-3960	49	3	is	be	AUX
cana-3960	49	4	the	the	DET
cana-3960	49	5	generalized	generalized	ADJ
cana-3960	49	6	delta	delta	NOUN
cana-3960	49	7	derivative	derivative	NOUN
cana-3960	49	8	of	of	ADP
cana-3960	49	9	x	x	PRON
cana-3960	49	10	,	,	PUNCT
cana-3960	49	11	and	and	CCONJ
cana-3960	49	12	t	t	PROPN
cana-3960	49	13	is	be	AUX
cana-3960	49	14	from	from	ADP
cana-3960	49	15	a	a	DET
cana-3960	49	16	time	time	NOUN
cana-3960	49	17	scales	scale	NOUN
cana-3960	49	18	𝕋	𝕋	NOUN
cana-3960	49	19	,	,	PUNCT
cana-3960	49	20	which	which	PRON
cana-3960	49	21	is	be	AUX
cana-3960	49	22	a	a	DET
cana-3960	49	23	non	non	ADJ
cana-3960	49	24	-	-	ADJ
cana-3960	49	25	empty	empty	ADJ
cana-3960	49	26	closed	closed	ADJ
cana-3960	49	27	subset	subset	NOUN
cana-3960	49	28	of	of	ADP
cana-3960	49	29	ℝ	ℝ	PROPN
cana-3960	49	30	and	and	CCONJ
cana-3960	49	31	𝜇	𝜇	X
cana-3960	49	32	is	be	AUX
cana-3960	49	33	a	a	DET
cana-3960	49	34	graininess	graininess	NOUN
cana-3960	49	35	function	function	NOUN
cana-3960	49	36	.	.	PUNCT
cana-3960	50	1	when	when	SCONJ
cana-3960	50	2	𝑄	𝑄	PROPN
cana-3960	50	3	=	=	SYM
cana-3960	50	4	𝑃∗	𝑃∗	PROPN
cana-3960	50	5	(	(	PUNCT
cana-3960	50	6	*	*	PUNCT
cana-3960	50	7	denotes	denote	VERB
cana-3960	50	8	the	the	DET
cana-3960	50	9	transpose	transpose	NOUN
cana-3960	50	10	of	of	ADP
cana-3960	50	11	matrix	matrix	NOUN
cana-3960	50	12	)	)	PUNCT
cana-3960	50	13	equation	equation	NOUN
cana-3960	50	14	(	(	PUNCT
cana-3960	50	15	1.1	1.1	NUM
cana-3960	50	16	)	)	PUNCT
cana-3960	50	17	is	be	AUX
cana-3960	50	18	called	call	VERB
cana-3960	50	19	matrix	matrix	NOUN
cana-3960	50	20	lyapunov	lyapunov	NOUN
cana-3960	50	21	dynamical	dynamical	ADJ
cana-3960	50	22	system	system	NOUN
cana-3960	50	23	on	on	ADP
cana-3960	50	24	time	time	NOUN
cana-3960	50	25	scale	scale	NOUN
cana-3960	50	26	.	.	PUNCT
cana-3960	51	1	2	2	X
cana-3960	51	2	.	.	X
cana-3960	51	3	preliminaries	preliminary	NOUN
cana-3960	51	4	we	we	PRON
cana-3960	51	5	provide	provide	VERB
cana-3960	51	6	some	some	DET
cana-3960	51	7	preliminary	preliminary	ADJ
cana-3960	51	8	information	information	NOUN
cana-3960	51	9	to	to	PART
cana-3960	51	10	help	help	VERB
cana-3960	51	11	you	you	PRON
cana-3960	51	12	understand	understand	VERB
cana-3960	51	13	the	the	DET
cana-3960	51	14	notation	notation	NOUN
cana-3960	51	15	used	use	VERB
cana-3960	51	16	in	in	ADP
cana-3960	51	17	this	this	DET
cana-3960	51	18	paper	paper	NOUN
cana-3960	51	19	.	.	PUNCT
cana-3960	52	1	a	a	DET
cana-3960	52	2	summary	summary	NOUN
cana-3960	52	3	of	of	ADP
cana-3960	52	4	the	the	DET
cana-3960	52	5	time	time	NOUN
cana-3960	52	6	scales	scale	NOUN
cana-3960	52	7	can	can	AUX
cana-3960	52	8	be	be	AUX
cana-3960	52	9	found	find	VERB
cana-3960	52	10	in	in	ADP
cana-3960	52	11	[	[	X
cana-3960	52	12	6	6	NUM
cana-3960	52	13	,	,	PUNCT
cana-3960	52	14	7	7	NUM
cana-3960	52	15	]	]	PUNCT
cana-3960	52	16	.	.	PUNCT
cana-3960	53	1	a	a	DET
cana-3960	53	2	nonempty	nonempty	ADV
cana-3960	53	3	closed	close	VERB
cana-3960	53	4	subset	subset	NOUN
cana-3960	53	5	of	of	ADP
cana-3960	53	6	the	the	DET
cana-3960	53	7	r	r	NOUN
cana-3960	53	8	real	real	ADJ
cana-3960	53	9	line	line	NOUN
cana-3960	53	10	is	be	AUX
cana-3960	53	11	called	call	VERB
cana-3960	53	12	a	a	DET
cana-3960	53	13	time	time	NOUN
cana-3960	53	14	scale	scale	NOUN
cana-3960	53	15	𝕋.	𝕋.	NOUN
cana-3960	53	16	we	we	PRON
cana-3960	53	17	usually	usually	ADV
cana-3960	53	18	write	write	VERB
cana-3960	53	19	𝕋𝑘	𝕋𝑘	PROPN
cana-3960	53	20	=	=	SYM
cana-3960	53	21	𝕋{𝑚𝑎𝑥𝕋	𝕋{𝑚𝑎𝑥𝕋	PROPN
cana-3960	53	22	}	}	PUNCT
cana-3960	53	23	if	if	SCONJ
cana-3960	53	24	𝑚𝑎𝑥𝕋	𝑚𝑎𝑥𝕋	X
cana-3960	53	25	<	<	X
cana-3960	53	26	∞	∞	PROPN
cana-3960	53	27	,	,	PUNCT
cana-3960	53	28	otherwise	otherwise	ADV
cana-3960	53	29	𝕋𝑘	𝕋𝑘	PROPN
cana-3960	53	30	=	=	SYM
cana-3960	53	31	𝕋.	𝕋.	NOUN
cana-3960	53	32	definition	definition	NOUN
cana-3960	53	33	2.1[6	2.1[6	NOUN
cana-3960	53	34	]	]	PUNCT
cana-3960	53	35	let	let	VERB
cana-3960	53	36	𝑓	𝑓	PRON
cana-3960	53	37	:	:	PUNCT
cana-3960	53	38	𝕋	𝕋	PROPN
cana-3960	53	39	→	→	SYM
cana-3960	53	40	ℝ	ℝ	PROPN
cana-3960	53	41	and	and	CCONJ
cana-3960	53	42	𝑡	𝑡	NOUN
cana-3960	53	43	∈	∈	NOUN
cana-3960	54	1	𝕋𝑘	𝕋𝑘	PROPN
cana-3960	54	2	the	the	DET
cana-3960	54	3	delta	delta	NOUN
cana-3960	54	4	derivative	derivative	NOUN
cana-3960	54	5	of	of	ADP
cana-3960	54	6	𝑓𝛥(𝑡	𝑓𝛥(𝑡	PROPN
cana-3960	54	7	)	)	PUNCT
cana-3960	54	8	is	be	AUX
cana-3960	54	9	the	the	DET
cana-3960	54	10	number	number	NOUN
cana-3960	54	11	(	(	PUNCT
cana-3960	54	12	when	when	SCONJ
cana-3960	54	13	it	it	PRON
cana-3960	54	14	exists	exist	VERB
cana-3960	54	15	)	)	PUNCT
cana-3960	54	16	,	,	PUNCT
cana-3960	54	17	with	with	ADP
cana-3960	54	18	the	the	DET
cana-3960	54	19	property	property	NOUN
cana-3960	54	20	that	that	PRON
cana-3960	54	21	,	,	PUNCT
cana-3960	54	22	for	for	ADP
cana-3960	54	23	any	any	DET
cana-3960	54	24	𝜀	𝜀	NOUN
cana-3960	54	25	>	>	X
cana-3960	54	26	0	0	NUM
cana-3960	54	27	,	,	PUNCT
cana-3960	54	28	there	there	PRON
cana-3960	54	29	is	be	VERB
cana-3960	54	30	a	a	DET
cana-3960	54	31	neighbourhood	neighbourhood	NOUN
cana-3960	54	32	𝑈	𝑈	NOUN
cana-3960	54	33	of	of	ADP
cana-3960	54	34	𝜏	𝜏	PRON
cana-3960	54	35	such	such	ADJ
cana-3960	54	36	that	that	SCONJ
cana-3960	54	37	communications	communication	NOUN
cana-3960	54	38	on	on	ADP
cana-3960	54	39	applied	apply	VERB
cana-3960	54	40	nonlinear	nonlinear	ADJ
cana-3960	54	41	analysis	analysis	NOUN
cana-3960	54	42	issn	issn	NOUN
cana-3960	54	43	:	:	PUNCT
cana-3960	54	44	1074	1074	NUM
cana-3960	54	45	-	-	PUNCT
cana-3960	54	46	133x	133x	NUM
cana-3960	54	47	vol	vol	NOUN
cana-3960	54	48	32	32	NUM
cana-3960	54	49	no	no	NOUN
cana-3960	54	50	.	.	PUNCT
cana-3960	55	1	9s	9s	NUM
cana-3960	55	2	(	(	PUNCT
cana-3960	55	3	2025	2025	NUM
cana-3960	55	4	)	)	PUNCT
cana-3960	55	5	495	495	NUM
cana-3960	55	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3960	55	7	|[𝑓(𝜎(𝜏	|[𝑓(𝜎(𝜏	NOUN
cana-3960	55	8	)	)	PUNCT
cana-3960	55	9	)	)	PUNCT
cana-3960	56	1	−	−	PROPN
cana-3960	56	2	𝑓(𝑠	𝑓(𝑠	NOUN
cana-3960	56	3	)	)	PUNCT
cana-3960	56	4	]	]	PUNCT
cana-3960	57	1	−	−	PROPN
cana-3960	57	2	𝑓𝛥(𝜏)[𝜎(𝜏	𝑓𝛥(𝜏)[𝜎(𝜏	NOUN
cana-3960	57	3	)	)	PUNCT
cana-3960	58	1	−	−	PROPN
cana-3960	58	2	𝑠]|	𝑠]|	PROPN
cana-3960	58	3	≤	≤	ADV
cana-3960	58	4	𝜀|𝜎(𝜏	𝜀|𝜎(𝜏	PROPN
cana-3960	58	5	)	)	PUNCT
cana-3960	58	6	−	−	PROPN
cana-3960	59	1	𝑠|	𝑠|	PROPN
cana-3960	59	2	,	,	PUNCT
cana-3960	59	3	for	for	ADP
cana-3960	59	4	all	all	PRON
cana-3960	59	5	𝑠	𝑠	PROPN
cana-3960	59	6	∈	∈	PROPN
cana-3960	59	7	𝑈	𝑈	PROPN
cana-3960	59	8	definition	definition	NOUN
cana-3960	59	9	2.2.[7	2.2.[7	NUM
cana-3960	59	10	]	]	X
cana-3960	59	11	:	:	PUNCT
cana-3960	59	12	the	the	DET
cana-3960	59	13	regressive	regressive	ADJ
cana-3960	59	14	function	function	NOUN
cana-3960	59	15	y(t	y(t	NUM
cana-3960	59	16	)	)	PUNCT
cana-3960	59	17	mapping	mapping	NOUN
cana-3960	59	18	from	from	ADP
cana-3960	59	19	𝕋	𝕋	PRON
cana-3960	59	20	to	to	ADP
cana-3960	59	21	ℝ	ℝ	PROPN
cana-3960	59	22	is	be	AUX
cana-3960	59	23	defined	define	VERB
cana-3960	59	24	as1	as1	NOUN
cana-3960	59	25	+	+	NOUN
cana-3960	59	26	𝜇(𝑡)𝑦(𝑡	𝜇(𝑡)𝑦(𝑡	ADJ
cana-3960	59	27	)	)	PUNCT
cana-3960	59	28	≠	≠	PROPN
cana-3960	59	29	0	0	NUM
cana-3960	59	30	∀	∀	NOUN
cana-3960	59	31	𝑡	𝑡	NOUN
cana-3960	59	32	∈	∈	NOUN
cana-3960	59	33	𝕋.	𝕋.	NOUN
cana-3960	59	34	the	the	DET
cana-3960	59	35	combination	combination	NOUN
cana-3960	59	36	of	of	ADP
cana-3960	59	37	all	all	DET
cana-3960	59	38	regressive	regressive	ADJ
cana-3960	59	39	and	and	CCONJ
cana-3960	59	40	right	right	ADV
cana-3960	59	41	dense	dense	ADJ
cana-3960	59	42	continuous	continuous	ADJ
cana-3960	59	43	function	function	NOUN
cana-3960	59	44	is	be	AUX
cana-3960	59	45	represented	represent	VERB
cana-3960	59	46	as	as	ADP
cana-3960	59	47	ℛ	ℛ	NOUN
cana-3960	59	48	=	=	SYM
cana-3960	59	49	ℛ(𝑡	ℛ(𝑡	NUM
cana-3960	59	50	)	)	PUNCT
cana-3960	59	51	=	=	SYM
cana-3960	59	52	ℛ(𝕋,ℝ	ℛ(𝕋,ℝ	PROPN
cana-3960	59	53	)	)	PUNCT
cana-3960	59	54	.	.	PUNCT
cana-3960	60	1	similarly	similarly	ADV
cana-3960	60	2	all	all	DET
cana-3960	60	3	positively	positively	ADV
cana-3960	60	4	regressive	regressive	ADJ
cana-3960	60	5	function	function	NOUN
cana-3960	60	6	is	be	AUX
cana-3960	60	7	denoted	denote	VERB
cana-3960	60	8	by	by	ADP
cana-3960	60	9	ℛ+	ℛ+	NOUN
cana-3960	60	10	=	=	SYM
cana-3960	60	11	ℛ+(𝕋,ℝ	ℛ+(𝕋,ℝ	ADJ
cana-3960	60	12	)	)	PUNCT
cana-3960	60	13	=	=	SYM
cana-3960	60	14	{	{	PUNCT
cana-3960	60	15	𝑦	𝑦	NOUN
cana-3960	60	16	∈	∈	PROPN
cana-3960	60	17	ℛ	ℛ	NOUN
cana-3960	60	18	:	:	PUNCT
cana-3960	60	19	1	1	NUM
cana-3960	60	20	+	+	CCONJ
cana-3960	60	21	𝜇(𝑡)𝑦(𝑡	𝜇(𝑡)𝑦(𝑡	NUM
cana-3960	60	22	)	)	PUNCT
cana-3960	60	23	>	>	X
cana-3960	61	1	0	0	NUM
cana-3960	61	2	,	,	PUNCT
cana-3960	61	3	∀	∀	NUM
cana-3960	61	4	𝑡	𝑡	NOUN
cana-3960	61	5	∈	∈	PROPN
cana-3960	61	6	𝕋	𝕋	PROPN
cana-3960	61	7	}	}	PUNCT
cana-3960	61	8	definition	definition	NOUN
cana-3960	61	9	2.3[6	2.3[6	NUM
cana-3960	61	10	]	]	PUNCT
cana-3960	61	11	if	if	SCONJ
cana-3960	61	12	𝐹	𝐹	PROPN
cana-3960	61	13	:	:	PUNCT
cana-3960	61	14	𝕋𝑘	𝕋𝑘	PROPN
cana-3960	61	15	→	→	PUNCT
cana-3960	61	16	ℝ	ℝ	PROPN
cana-3960	61	17	is	be	AUX
cana-3960	61	18	said	say	VERB
cana-3960	61	19	to	to	PART
cana-3960	61	20	be	be	AUX
cana-3960	61	21	anti	anti	ADJ
cana-3960	61	22	-	-	ADJ
cana-3960	61	23	derivative	derivative	ADJ
cana-3960	61	24	of	of	ADP
cana-3960	61	25	𝑓	𝑓	PRON
cana-3960	61	26	:	:	PUNCT
cana-3960	61	27	𝕋𝑘	𝕋𝑘	PROPN
cana-3960	61	28	→	→	PUNCT
cana-3960	61	29	ℝ	ℝ	PROPN
cana-3960	61	30	provided	provide	VERB
cana-3960	61	31	𝐹𝛥(𝑡	𝐹𝛥(𝑡	NUM
cana-3960	61	32	)	)	PUNCT
cana-3960	61	33	=	=	SYM
cana-3960	62	1	𝑓(𝑡	𝑓(𝑡	VERB
cana-3960	62	2	)	)	PUNCT
cana-3960	63	1	fulfilled	fulfil	VERB
cana-3960	63	2	,	,	PUNCT
cana-3960	63	3	for	for	ADP
cana-3960	63	4	all	all	DET
cana-3960	63	5	𝑡	𝑡	ADP
cana-3960	63	6	∈	∈	PROPN
cana-3960	63	7	𝕋𝑘	𝕋𝑘	PROPN
cana-3960	63	8	,	,	PUNCT
cana-3960	63	9	then	then	ADV
cana-3960	63	10	∫	∫	PROPN
cana-3960	63	11	𝑓(𝑠)𝛥𝑠	𝑓(𝑠)𝛥𝑠	PROPN
cana-3960	63	12	=	=	SYM
cana-3960	64	1	𝐹(𝑡	𝐹(𝑡	NUM
cana-3960	64	2	)	)	PUNCT
cana-3960	64	3	−	−	PROPN
cana-3960	64	4	𝐹(𝑎	𝐹(𝑎	NOUN
cana-3960	64	5	)	)	PUNCT
cana-3960	64	6	𝑡	𝑡	NOUN
cana-3960	64	7	𝑎	𝑎	PRON
cana-3960	64	8	definition	definition	NOUN
cana-3960	64	9	2.4	2.4	NUM
cana-3960	64	10	let	let	VERB
cana-3960	64	11	the	the	DET
cana-3960	64	12	matrices	matrix	NOUN
cana-3960	64	13	are	be	AUX
cana-3960	64	14	𝐴	𝐴	PROPN
cana-3960	64	15	∈	∈	PROPN
cana-3960	64	16	𝐶𝑚×𝑛(ℝ𝑚×𝑛	𝐶𝑚×𝑛(ℝ𝑚×𝑛	NOUN
cana-3960	64	17	)	)	PUNCT
cana-3960	64	18	and	and	CCONJ
cana-3960	64	19	𝐵	𝐵	PROPN
cana-3960	64	20	∈	∈	PROPN
cana-3960	64	21	𝐶𝑝×𝑞(ℝ𝑝×𝑞	𝐶𝑝×𝑞(ℝ𝑝×𝑞	NOUN
cana-3960	64	22	)	)	PUNCT
cana-3960	64	23	the	the	DET
cana-3960	64	24	the	the	DET
cana-3960	64	25	kronecker	kronecker	NOUN
cana-3960	64	26	product	product	NOUN
cana-3960	64	27	of	of	ADP
cana-3960	64	28	a	a	PRON
cana-3960	64	29	and	and	CCONJ
cana-3960	64	30	b.	b.	NOUN
cana-3960	64	31	we	we	PRON
cana-3960	64	32	have	have	AUX
cana-3960	64	33	defined	define	VERB
cana-3960	64	34	to	to	PART
cana-3960	64	35	be	be	AUX
cana-3960	64	36	the	the	DET
cana-3960	64	37	partitioned	partition	VERB
cana-3960	64	38	matrix	matrix	NOUN
cana-3960	64	39	written	write	VERB
cana-3960	64	40	(	(	PUNCT
cana-3960	64	41	𝐴	𝐴	PROPN
cana-3960	64	42	⊗	⊗	PROPN
cana-3960	64	43	𝐵	𝐵	PROPN
cana-3960	64	44	)	)	PUNCT
cana-3960	64	45	is	be	AUX
cana-3960	64	46	𝐴⊗	𝐴⊗	PROPN
cana-3960	64	47	𝐵	𝐵	NOUN
cana-3960	64	48	=	=	PUNCT
cana-3960	64	49	[	[	PUNCT
cana-3960	64	50	𝑎11𝐵	𝑎11𝐵	X
cana-3960	64	51	𝑎12𝐵	𝑎12𝐵	PROPN
cana-3960	64	52	⋯	⋯	PROPN
cana-3960	64	53	𝑎1𝑛𝐵	𝑎1𝑛𝐵	PROPN
cana-3960	64	54	𝑎21𝐵	𝑎21𝐵	PROPN
cana-3960	65	1	𝑎22𝐵	𝑎22𝐵	PROPN
cana-3960	65	2	⋯	⋯	PROPN
cana-3960	65	3	𝑎2𝑛2𝐵	𝑎2𝑛2𝐵	PROPN
cana-3960	65	4	⋯	⋯	PROPN
cana-3960	65	5	⋯	⋯	PROPN
cana-3960	65	6	⋯	⋯	PROPN
cana-3960	65	7	⋯	⋯	PROPN
cana-3960	65	8	𝑎11𝐵	𝑎11𝐵	PROPN
cana-3960	65	9	𝑎12𝐵	𝑎12𝐵	PROPN
cana-3960	65	10	⋯	⋯	X
cana-3960	65	11	𝑎1𝑛𝐵	𝑎1𝑛𝐵	PROPN
cana-3960	65	12	]	]	PUNCT
cana-3960	65	13	is	be	AUX
cana-3960	65	14	an	an	DET
cana-3960	65	15	𝑚𝑝	𝑚𝑝	NOUN
cana-3960	65	16	×	×	NOUN
cana-3960	65	17	𝑛𝑞	𝑛𝑞	SCONJ
cana-3960	65	18	matrix	matrix	NOUN
cana-3960	65	19	is	be	AUX
cana-3960	65	20	in	in	ADP
cana-3960	65	21	𝐶𝑚×𝑛(ℝ𝑚×𝑛	𝐶𝑚×𝑛(ℝ𝑚×𝑛	NOUN
cana-3960	65	22	)	)	PUNCT
cana-3960	65	23	.	.	PUNCT
cana-3960	66	1	definition	definition	NOUN
cana-3960	66	2	2.5	2.5	NUM
cana-3960	66	3	.	.	PUNCT
cana-3960	67	1	let	let	VERB
cana-3960	67	2	𝐴	𝐴	PROPN
cana-3960	67	3	=	=	PUNCT
cana-3960	68	1	[	[	X
cana-3960	68	2	𝑎𝑖𝑗	𝑎𝑖𝑗	X
cana-3960	68	3	]	]	X
cana-3960	68	4	∈	∈	PROPN
cana-3960	68	5	ℝ𝑚×𝑛	ℝ𝑚×𝑛	PROPN
cana-3960	68	6	,	,	PUNCT
cana-3960	68	7	we	we	PRON
cana-3960	68	8	denote	denote	VERB
cana-3960	68	9	�	�	PROPN
cana-3960	68	10	̂	̂	VERB
cana-3960	68	11	�	�	NOUN
cana-3960	68	12	=	=	SYM
cana-3960	68	13	𝑉𝑒𝑐𝐴	𝑉𝑒𝑐𝐴	NOUN
cana-3960	68	14	=	=	PUNCT
cana-3960	69	1	[	[	PUNCT
cana-3960	69	2	𝐴1	𝐴1	PROPN
cana-3960	69	3	𝐴2	𝐴2	PROPN
cana-3960	69	4	⋮	⋮	NOUN
cana-3960	69	5	𝐴𝑛	𝐴𝑛	PROPN
cana-3960	69	6	]	]	PUNCT
cana-3960	69	7	,	,	PUNCT
cana-3960	69	8	where	where	SCONJ
cana-3960	69	9	𝐴.𝑗	𝐴.𝑗	NOUN
cana-3960	69	10	=	=	PUNCT
cana-3960	69	11	[	[	PUNCT
cana-3960	69	12	𝑎1𝑗	𝑎1𝑗	X
cana-3960	69	13	𝑎2𝑗	𝑎2𝑗	ADP
cana-3960	69	14	⋮	⋮	NOUN
cana-3960	69	15	𝑎𝑚𝑗	𝑎𝑚𝑗	VERB
cana-3960	69	16	]	]	PUNCT
cana-3960	69	17	(	(	PUNCT
cana-3960	69	18	1	1	NUM
cana-3960	69	19	≤	≤	NUM
cana-3960	69	20	𝑗	𝑗	PRON
cana-3960	69	21	≤	≤	NUM
cana-3960	69	22	𝑛	𝑛	NOUN
cana-3960	69	23	)	)	PUNCT
cana-3960	69	24	here	here	ADV
cana-3960	69	25	we	we	PRON
cana-3960	69	26	converted	convert	VERB
cana-3960	69	27	the	the	DET
cana-3960	69	28	linear	linear	ADJ
cana-3960	69	29	matrix	matrix	NOUN
cana-3960	69	30	sylvester	sylvest	ADJ
cana-3960	69	31	dynamic	dynamic	ADJ
cana-3960	69	32	impulsive	impulsive	ADJ
cana-3960	69	33	system	system	NOUN
cana-3960	69	34	on	on	ADP
cana-3960	69	35	time	time	NOUN
cana-3960	69	36	scales	scale	NOUN
cana-3960	69	37	to	to	ADP
cana-3960	69	38	an	an	DET
cana-3960	69	39	equivalent	equivalent	ADJ
cana-3960	69	40	kp	kp	PROPN
cana-3960	69	41	dynamic	dynamic	ADJ
cana-3960	69	42	impulsive	impulsive	ADJ
cana-3960	69	43	system	system	NOUN
cana-3960	69	44	on	on	ADP
cana-3960	69	45	time	time	NOUN
cana-3960	69	46	scales	scale	NOUN
cana-3960	69	47	using	use	VERB
cana-3960	69	48	vectorization	vectorization	NOUN
cana-3960	69	49	operator	operator	NOUN
cana-3960	69	50	.	.	PUNCT
cana-3960	70	1	the	the	DET
cana-3960	70	2	dynamical	dynamical	ADJ
cana-3960	70	3	system	system	NOUN
cana-3960	70	4	is	be	AUX
cana-3960	70	5	{	{	PUNCT
cana-3960	70	6	𝑧	𝑧	PRON
cana-3960	70	7	𝛥(𝑡	𝛥(𝑡	NOUN
cana-3960	70	8	)	)	PUNCT
cana-3960	70	9	=	=	SYM
cana-3960	70	10	𝐺(𝑡)𝑧(𝑡	𝐺(𝑡)𝑧(𝑡	NOUN
cana-3960	70	11	)	)	PUNCT
cana-3960	71	1	+	+	PUNCT
cana-3960	71	2	𝐴(𝑡)𝑈(𝑡	𝐴(𝑡)𝑈(𝑡	NOUN
cana-3960	71	3	)	)	PUNCT
cana-3960	71	4	,	,	PUNCT
cana-3960	71	5	t	t	PROPN
cana-3960	71	6	∈	∈	PROPN
cana-3960	72	1	[	[	X
cana-3960	72	2	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	72	3	,	,	PUNCT
cana-3960	72	4	𝑡𝑘)𝕋	𝑡𝑘)𝕋	VERB
cana-3960	72	5	𝑧(𝑡𝑘	𝑧(𝑡𝑘	NUM
cana-3960	72	6	+	+	NOUN
cana-3960	72	7	)	)	PUNCT
cana-3960	72	8	=	=	PUNCT
cana-3960	73	1	[	[	X
cana-3960	73	2	𝐼𝑛⊗𝑅𝑘]𝑧(𝑡𝑘	𝐼𝑛⊗𝑅𝑘]𝑧(𝑡𝑘	NOUN
cana-3960	73	3	)	)	PUNCT
cana-3960	73	4	,	,	PUNCT
cana-3960	73	5	𝑡	𝑡	X
cana-3960	73	6	=	=	VERB
cana-3960	73	7	𝑡𝑘	𝑡𝑘	ADV
cana-3960	73	8	,	,	PUNCT
cana-3960	73	9	k	k	PROPN
cana-3960	73	10	=	=	SYM
cana-3960	73	11	1,2,3	1,2,3	NUM
cana-3960	73	12	…	…	SYM
cana-3960	73	13	�	�	NOUN
cana-3960	73	14	̂	̂	NOUN
cana-3960	73	15	�	�	NOUN
cana-3960	73	16	(𝑡	(𝑡	NOUN
cana-3960	73	17	)	)	PUNCT
cana-3960	73	18	=	=	SYM
cana-3960	73	19	(	(	PUNCT
cana-3960	73	20	i⊗𝐶)(𝑡)𝑧(𝑡	i⊗𝐶)(𝑡)𝑧(𝑡	NOUN
cana-3960	73	21	)	)	PUNCT
cana-3960	73	22	+	+	CCONJ
cana-3960	73	23	(	(	PUNCT
cana-3960	73	24	𝐼	𝐼	PROPN
cana-3960	73	25	⊗	⊗	PROPN
cana-3960	73	26	𝐷)	𝐷)	PROPN
cana-3960	73	27	�	�	PROPN
cana-3960	73	28	̂	̂	VERB
cana-3960	73	29	�	�	NOUN
cana-3960	73	30	(𝑡	(𝑡	SYM
cana-3960	73	31	)	)	PUNCT
cana-3960	73	32	𝑧(𝑡0	𝑧(𝑡0	NUM
cana-3960	73	33	)	)	PUNCT
cana-3960	73	34	=	=	SYM
cana-3960	73	35	𝑧0	𝑧0	PROPN
cana-3960	73	36	.	.	PUNCT
cana-3960	74	1	(	(	PUNCT
cana-3960	74	2	2.1	2.1	NUM
cana-3960	74	3	)	)	PUNCT
cana-3960	74	4	where	where	SCONJ
cana-3960	74	5	z(t	z(t	NOUN
cana-3960	74	6	)	)	PUNCT
cana-3960	74	7	=	=	SYM
cana-3960	74	8	vec	vec	PROPN
cana-3960	74	9	x(t	x(t	PROPN
cana-3960	74	10	)	)	PUNCT
cana-3960	74	11	,	,	PUNCT
cana-3960	74	12	�	�	PROPN
cana-3960	74	13	̂	̂	SYM
cana-3960	74	14	�	�	NOUN
cana-3960	74	15	(𝑡	(𝑡	NOUN
cana-3960	74	16	)	)	PUNCT
cana-3960	74	17	=	=	PROPN
cana-3960	74	18	vec	vec	NOUN
cana-3960	74	19	u(t	u(t	PROPN
cana-3960	74	20	)	)	PUNCT
cana-3960	74	21	,	,	PUNCT
cana-3960	74	22	�	�	PROPN
cana-3960	74	23	̂	̂	SYM
cana-3960	74	24	�	�	NOUN
cana-3960	74	25	(𝑡	(𝑡	NOUN
cana-3960	74	26	)	)	PUNCT
cana-3960	74	27	=	=	PROPN
cana-3960	75	1	vec	vec	PROPN
cana-3960	75	2	y(t	y(t	PROPN
cana-3960	75	3	)	)	PUNCT
cana-3960	75	4	,	,	PUNCT
cana-3960	75	5	𝑅𝑘	𝑅𝑘	PROPN
cana-3960	75	6	=	=	SYM
cana-3960	75	7	(	(	PUNCT
cana-3960	75	8	𝐼𝑛	𝐼𝑛	PROPN
cana-3960	75	9	+	+	CCONJ
cana-3960	75	10	𝐿𝑘	𝐿𝑘	PROPN
cana-3960	75	11	)	)	PUNCT
cana-3960	75	12	and	and	CCONJ
cana-3960	75	13	g(t)=	g(t)=	X
cana-3960	76	1	[	[	X
cana-3960	76	2	𝑄∗⊗	𝑄∗⊗	NOUN
cana-3960	76	3	𝐼	𝐼	NOUN
cana-3960	76	4	+	+	CCONJ
cana-3960	76	5	𝐼	𝐼	PROPN
cana-3960	76	6	⊗	⊗	ADJ
cana-3960	76	7	𝑃	𝑃	NOUN
cana-3960	76	8	+	+	NOUN
cana-3960	76	9	𝜇(𝑡)(𝑄∗⊗𝑃	𝜇(𝑡)(𝑄∗⊗𝑃	NOUN
cana-3960	76	10	)	)	PUNCT
cana-3960	76	11	]	]	PUNCT
cana-3960	77	1	a(t)=	a(t)=	PROPN
cana-3960	78	1	[	[	X
cana-3960	78	2	𝑇2	𝑇2	NOUN
cana-3960	78	3	∗⊗𝑇1	∗⊗𝑇1	NOUN
cana-3960	78	4	]	]	PUNCT
cana-3960	78	5	.	.	PUNCT
cana-3960	79	1	now	now	ADV
cana-3960	79	2	rearranged	rearrange	VERB
cana-3960	79	3	the	the	DET
cana-3960	79	4	linear	linear	ADJ
cana-3960	79	5	adjoint	adjoint	PROPN
cana-3960	79	6	dynamic	dynamic	ADJ
cana-3960	79	7	system	system	NOUN
cana-3960	79	8	(	(	PUNCT
cana-3960	79	9	2.1	2.1	NUM
cana-3960	79	10	)	)	PUNCT
cana-3960	79	11	as	as	SCONJ
cana-3960	79	12	follows	follow	VERB
cana-3960	79	13	{	{	PUNCT
cana-3960	79	14	𝑧	𝑧	PRON
cana-3960	79	15	𝛥(𝑡	𝛥(𝑡	NOUN
cana-3960	79	16	)	)	PUNCT
cana-3960	79	17	=	=	SYM
cana-3960	79	18	−𝐺𝑘	−𝐺𝑘	NOUN
cana-3960	79	19	𝑇(𝑡)𝑧𝜎(𝑡	𝑇(𝑡)𝑧𝜎(𝑡	NOUN
cana-3960	79	20	)	)	PUNCT
cana-3960	80	1	+	+	CCONJ
cana-3960	80	2	𝐴𝑘(𝑡)	𝐴𝑘(𝑡)	PROPN
cana-3960	80	3	�	�	PROPN
cana-3960	80	4	̂	̂	VERB
cana-3960	80	5	�	�	NOUN
cana-3960	80	6	(𝑡	(𝑡	NOUN
cana-3960	80	7	)	)	PUNCT
cana-3960	80	8	,	,	PUNCT
cana-3960	80	9	t	t	PROPN
cana-3960	80	10	∈	∈	PROPN
cana-3960	81	1	[	[	X
cana-3960	81	2	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	81	3	,	,	PUNCT
cana-3960	81	4	𝑡𝑘)𝕋	𝑡𝑘)𝕋	VERB
cana-3960	81	5	𝑧(𝑡𝑘	𝑧(𝑡𝑘	NUM
cana-3960	81	6	+	+	NOUN
cana-3960	81	7	)	)	PUNCT
cana-3960	81	8	=	=	PUNCT
cana-3960	82	1	[	[	X
cana-3960	82	2	𝐼𝑛⊗𝑅𝑘]𝑧(𝑡𝑘	𝐼𝑛⊗𝑅𝑘]𝑧(𝑡𝑘	NOUN
cana-3960	82	3	)	)	PUNCT
cana-3960	82	4	,	,	PUNCT
cana-3960	82	5	𝑡	𝑡	X
cana-3960	82	6	=	=	VERB
cana-3960	82	7	𝑡𝑘	𝑡𝑘	ADV
cana-3960	82	8	,	,	PUNCT
cana-3960	82	9	k	k	PROPN
cana-3960	82	10	=	=	SYM
cana-3960	82	11	1,2,3	1,2,3	NUM
cana-3960	82	12	…	…	SYM
cana-3960	82	13	�	�	NOUN
cana-3960	82	14	̂	̂	NOUN
cana-3960	82	15	�	�	NOUN
cana-3960	82	16	(𝑡	(𝑡	NOUN
cana-3960	82	17	)	)	PUNCT
cana-3960	82	18	=	=	PUNCT
cana-3960	82	19	(	(	PUNCT
cana-3960	82	20	i⊗𝐶𝑘)(𝑡)𝑧	i⊗𝐶𝑘)(𝑡)𝑧	PROPN
cana-3960	82	21	𝜎(𝑡	𝜎(𝑡	PROPN
cana-3960	82	22	)	)	PUNCT
cana-3960	82	23	+	+	CCONJ
cana-3960	82	24	(	(	PUNCT
cana-3960	82	25	𝐼	𝐼	PROPN
cana-3960	82	26	⊗	⊗	PROPN
cana-3960	82	27	𝐷𝑘)	𝐷𝑘)	PROPN
cana-3960	82	28	�	�	PROPN
cana-3960	82	29	̂	̂	NOUN
cana-3960	82	30	�	�	NOUN
cana-3960	82	31	(𝑡	(𝑡	SYM
cana-3960	82	32	)	)	PUNCT
cana-3960	82	33	𝑧(𝑡0	𝑧(𝑡0	NUM
cana-3960	82	34	)	)	PUNCT
cana-3960	82	35	=	=	SYM
cana-3960	82	36	𝑧0	𝑧0	PROPN
cana-3960	82	37	.	.	PUNCT
cana-3960	83	1	(	(	PUNCT
cana-3960	83	2	2.2	2.2	NUM
cana-3960	83	3	)	)	PUNCT
cana-3960	83	4	remark	remark	NOUN
cana-3960	83	5	2.1.[3	2.1.[3	NUM
cana-3960	83	6	]	]	PUNCT
cana-3960	83	7	.	.	PUNCT
cana-3960	84	1	clearly	clearly	ADV
cana-3960	84	2	observe	observe	VERB
cana-3960	84	3	that	that	SCONJ
cana-3960	84	4	,	,	PUNCT
cana-3960	84	5	the	the	DET
cana-3960	84	6	matrix	matrix	NOUN
cana-3960	84	7	valued	value	VERB
cana-3960	84	8	function	function	NOUN
cana-3960	84	9	x(t	x(t	PROPN
cana-3960	84	10	)	)	PUNCT
cana-3960	84	11	is	be	AUX
cana-3960	84	12	a	a	DET
cana-3960	84	13	solution	solution	NOUN
cana-3960	84	14	(	(	PUNCT
cana-3960	84	15	2.1	2.1	NUM
cana-3960	84	16	)	)	PUNCT
cana-3960	84	17	on	on	ADP
cana-3960	84	18	𝕋	𝕋	PROPN
cana-3960	84	19	if	if	SCONJ
cana-3960	84	20	and	and	CCONJ
cana-3960	84	21	only	only	ADV
cana-3960	84	22	if	if	SCONJ
cana-3960	84	23	the	the	DET
cana-3960	84	24	vector	vector	NOUN
cana-3960	84	25	valued	value	VERB
cana-3960	84	26	function	function	NOUN
cana-3960	84	27	𝑧(𝑡	𝑧(𝑡	PROPN
cana-3960	84	28	)	)	PUNCT
cana-3960	84	29	=	=	SYM
cana-3960	84	30	𝑉𝑒𝑐𝑋(𝑡	𝑉𝑒𝑐𝑋(𝑡	PROPN
cana-3960	84	31	)	)	PUNCT
cana-3960	84	32	is	be	AUX
cana-3960	84	33	a	a	DET
cana-3960	84	34	solution	solution	NOUN
cana-3960	84	35	of	of	ADP
cana-3960	84	36	the	the	DET
cana-3960	84	37	system	system	NOUN
cana-3960	84	38	(	(	PUNCT
cana-3960	84	39	2.1	2.1	NUM
cana-3960	84	40	)	)	PUNCT
cana-3960	84	41	on	on	ADP
cana-3960	84	42	𝕋.	𝕋.	NOUN
cana-3960	84	43	theorem	theorem	VERB
cana-3960	84	44	2.1.[7	2.1.[7	NUM
cana-3960	84	45	]	]	PUNCT
cana-3960	84	46	.	.	PUNCT
cana-3960	85	1	if	if	SCONJ
cana-3960	85	2	𝐺	𝐺	PROPN
cana-3960	85	3	∈	∈	PROPN
cana-3960	85	4	𝐶𝑟𝑑ℛ	𝐶𝑟𝑑ℛ	ADJ
cana-3960	85	5	(	(	PUNCT
cana-3960	85	6	𝕋+	𝕋+	ADJ
cana-3960	85	7	,	,	PUNCT
cana-3960	85	8	𝑀𝑛2×𝑛2(ℝ	𝑀𝑛2×𝑛2(ℝ	NOUN
cana-3960	85	9	)	)	PUNCT
cana-3960	85	10	)	)	PUNCT
cana-3960	86	1	and	and	CCONJ
cana-3960	86	2	𝑙	𝑙	DET
cana-3960	86	3	∈	∈	PROPN
cana-3960	86	4	𝐶𝑟𝑑	𝐶𝑟𝑑	PROPN
cana-3960	86	5	(	(	PUNCT
cana-3960	86	6	𝕋+	𝕋+	ADJ
cana-3960	86	7	,	,	PUNCT
cana-3960	86	8	𝑀𝑛2×1(ℝ	𝑀𝑛2×1(ℝ	ADJ
cana-3960	86	9	)	)	PUNCT
cana-3960	86	10	)	)	PUNCT
cana-3960	86	11	,	,	PUNCT
cana-3960	86	12	then	then	ADV
cana-3960	86	13	for	for	ADP
cana-3960	86	14	each	each	DET
cana-3960	86	15	(	(	PUNCT
cana-3960	86	16	𝜏	𝜏	PROPN
cana-3960	86	17	,	,	PUNCT
cana-3960	86	18	𝜂	𝜂	NOUN
cana-3960	86	19	)	)	PUNCT
cana-3960	86	20	∈	∈	PROPN
cana-3960	86	21	𝕋+	𝕋+	ADP
cana-3960	86	22	×	×	NOUN
cana-3960	86	23	ℝ𝑛	ℝ𝑛	ADP
cana-3960	86	24	2	2	NUM
cana-3960	86	25	the	the	DET
cana-3960	86	26	initial	initial	ADJ
cana-3960	86	27	value	value	NOUN
cana-3960	86	28	problem	problem	NOUN
cana-3960	86	29	communications	communication	NOUN
cana-3960	86	30	on	on	ADP
cana-3960	86	31	applied	apply	VERB
cana-3960	86	32	nonlinear	nonlinear	ADJ
cana-3960	86	33	analysis	analysis	NOUN
cana-3960	86	34	issn	issn	NOUN
cana-3960	86	35	:	:	PUNCT
cana-3960	86	36	1074	1074	NUM
cana-3960	86	37	-	-	PUNCT
cana-3960	86	38	133x	133x	NUM
cana-3960	86	39	vol	vol	NOUN
cana-3960	86	40	32	32	NUM
cana-3960	86	41	no	no	NOUN
cana-3960	86	42	.	.	PUNCT
cana-3960	87	1	9s	9s	NUM
cana-3960	87	2	(	(	PUNCT
cana-3960	87	3	2025	2025	NUM
cana-3960	87	4	)	)	PUNCT
cana-3960	87	5	496	496	NUM
cana-3960	87	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3960	87	7	𝑧𝛥(𝑡	𝑧𝛥(𝑡	NUM
cana-3960	87	8	)	)	PUNCT
cana-3960	87	9	=	=	SYM
cana-3960	87	10	𝐺(𝑡)𝑧(𝑡	𝐺(𝑡)𝑧(𝑡	NOUN
cana-3960	87	11	)	)	PUNCT
cana-3960	88	1	+	+	CCONJ
cana-3960	88	2	𝑙(𝑡	𝑙(𝑡	NOUN
cana-3960	88	3	)	)	PUNCT
cana-3960	88	4	,	,	PUNCT
cana-3960	88	5	z(τ	z(τ	NOUN
cana-3960	88	6	)	)	PUNCT
cana-3960	88	7	=	=	SYM
cana-3960	88	8	𝜂	𝜂	NOUN
cana-3960	88	9	,	,	PUNCT
cana-3960	88	10	has	have	VERB
cana-3960	88	11	a	a	DET
cana-3960	88	12	unique	unique	ADJ
cana-3960	88	13	solution	solution	NOUN
cana-3960	88	14	𝑧	𝑧	NOUN
cana-3960	88	15	:	:	PUNCT
cana-3960	88	16	𝕋(𝜏	𝕋(𝜏	X
cana-3960	88	17	)	)	PUNCT
cana-3960	88	18	→	→	PUNCT
cana-3960	89	1	ℝ𝑛	ℝ𝑛	ADP
cana-3960	89	2	2	2	NUM
cana-3960	89	3	.	.	PUNCT
cana-3960	90	1	lemma	lemma	PROPN
cana-3960	90	2	2.1.[7	2.1.[7	NUM
cana-3960	90	3	]	]	PUNCT
cana-3960	90	4	.	.	PUNCT
cana-3960	91	1	if	if	SCONJ
cana-3960	91	2	𝐺	𝐺	PROPN
cana-3960	91	3	∈	∈	PROPN
cana-3960	91	4	𝐶𝑟𝑑ℛ	𝐶𝑟𝑑ℛ	ADJ
cana-3960	91	5	(	(	PUNCT
cana-3960	91	6	𝕋+	𝕋+	ADJ
cana-3960	91	7	,	,	PUNCT
cana-3960	91	8	𝑀𝑛2(ℝ	𝑀𝑛2(ℝ	NOUN
cana-3960	91	9	)	)	PUNCT
cana-3960	91	10	)	)	PUNCT
cana-3960	91	11	and	and	CCONJ
cana-3960	91	12	𝑙	𝑙	DET
cana-3960	91	13	∈	∈	PROPN
cana-3960	91	14	𝐶𝑟𝑑	𝐶𝑟𝑑	PROPN
cana-3960	91	15	(	(	PUNCT
cana-3960	91	16	𝕋+	𝕋+	ADJ
cana-3960	91	17	,	,	PUNCT
cana-3960	91	18	𝑀𝑛2×1(ℝ	𝑀𝑛2×1(ℝ	ADJ
cana-3960	91	19	)	)	PUNCT
cana-3960	91	20	)	)	PUNCT
cana-3960	91	21	,	,	PUNCT
cana-3960	91	22	then	then	ADV
cana-3960	91	23	for	for	ADP
cana-3960	91	24	each	each	DET
cana-3960	91	25	(	(	PUNCT
cana-3960	91	26	𝜏	𝜏	PROPN
cana-3960	91	27	,	,	PUNCT
cana-3960	91	28	𝜂	𝜂	NOUN
cana-3960	91	29	)	)	PUNCT
cana-3960	91	30	∈	∈	PROPN
cana-3960	91	31	𝕋+	𝕋+	ADP
cana-3960	91	32	×	×	NOUN
cana-3960	91	33	ℝ𝑛	ℝ𝑛	ADP
cana-3960	91	34	2	2	NUM
cana-3960	91	35	the	the	DET
cana-3960	91	36	initial	initial	ADJ
cana-3960	91	37	value	value	NOUN
cana-3960	91	38	problem	problem	NOUN
cana-3960	91	39	𝑧𝛥(𝑡	𝑧𝛥(𝑡	NUM
cana-3960	91	40	)	)	PUNCT
cana-3960	91	41	=	=	SYM
cana-3960	91	42	𝐺(𝑡)𝑧(𝑡	𝐺(𝑡)𝑧(𝑡	NOUN
cana-3960	91	43	)	)	PUNCT
cana-3960	91	44	+	+	CCONJ
cana-3960	91	45	𝑙(𝑡	𝑙(𝑡	NOUN
cana-3960	91	46	)	)	PUNCT
cana-3960	91	47	,	,	PUNCT
cana-3960	91	48	z(τ	z(τ	NOUN
cana-3960	91	49	)	)	PUNCT
cana-3960	91	50	=	=	SYM
cana-3960	91	51	𝜂	𝜂	NOUN
cana-3960	91	52	,	,	PUNCT
cana-3960	91	53	has	have	VERB
cana-3960	91	54	one	one	NUM
cana-3960	91	55	and	and	CCONJ
cana-3960	91	56	only	only	ADV
cana-3960	91	57	one	one	NUM
cana-3960	91	58	solution	solution	NOUN
cana-3960	91	59	𝑧	𝑧	NOUN
cana-3960	91	60	:	:	PUNCT
cana-3960	91	61	𝕋(𝜏	𝕋(𝜏	X
cana-3960	91	62	)	)	PUNCT
cana-3960	91	63	→	→	PUNCT
cana-3960	91	64	ℝ𝑛	ℝ𝑛	ADP
cana-3960	91	65	2	2	NUM
cana-3960	91	66	is	be	AUX
cana-3960	91	67	given	give	VERB
cana-3960	91	68	by	by	ADP
cana-3960	91	69	𝑧(𝑡	𝑧(𝑡	NOUN
cana-3960	91	70	)	)	PUNCT
cana-3960	92	1	=	=	SYM
cana-3960	92	2	𝜓𝐺(𝑡	𝜓𝐺(𝑡	VERB
cana-3960	92	3	,	,	PUNCT
cana-3960	92	4	𝜏)𝜂	𝜏)𝜂	X
cana-3960	92	5	+	+	CCONJ
cana-3960	92	6	∫	∫	PROPN
cana-3960	92	7	𝜓𝐺(𝑡	𝜓𝐺(𝑡	PROPN
cana-3960	92	8	,	,	PUNCT
cana-3960	92	9	𝜎(𝑠))𝑙(𝑠)𝛥𝑠	𝜎(𝑠))𝑙(𝑠)𝛥𝑠	NUM
cana-3960	92	10	𝑡	𝑡	X
cana-3960	92	11	𝜏	𝜏	PROPN
cana-3960	92	12	,	,	PUNCT
cana-3960	92	13	𝑡	𝑡	PROPN
cana-3960	92	14	≥	≥	PROPN
cana-3960	92	15	𝜏.	𝜏.	NOUN
cana-3960	92	16	lemma	lemma	PROPN
cana-3960	92	17	2.2.[7	2.2.[7	NUM
cana-3960	92	18	]	]	PUNCT
cana-3960	92	19	.	.	PUNCT
cana-3960	93	1	if	if	SCONJ
cana-3960	93	2	𝐺	𝐺	PROPN
cana-3960	93	3	∈	∈	PROPN
cana-3960	93	4	𝐶𝑟𝑑ℛ	𝐶𝑟𝑑ℛ	ADJ
cana-3960	93	5	(	(	PUNCT
cana-3960	93	6	𝕋+	𝕋+	ADJ
cana-3960	93	7	,	,	PUNCT
cana-3960	93	8	𝑀𝑛2(ℝ	𝑀𝑛2(ℝ	NOUN
cana-3960	93	9	)	)	PUNCT
cana-3960	93	10	)	)	PUNCT
cana-3960	93	11	and	and	CCONJ
cana-3960	93	12	𝑙	𝑙	DET
cana-3960	93	13	∈	∈	PROPN
cana-3960	93	14	𝐶𝑟𝑑	𝐶𝑟𝑑	PROPN
cana-3960	93	15	(	(	PUNCT
cana-3960	93	16	𝕋+	𝕋+	ADJ
cana-3960	93	17	,	,	PUNCT
cana-3960	93	18	𝑀𝑛2×1(ℝ	𝑀𝑛2×1(ℝ	ADJ
cana-3960	93	19	)	)	PUNCT
cana-3960	93	20	)	)	PUNCT
cana-3960	93	21	,	,	PUNCT
cana-3960	93	22	then	then	ADV
cana-3960	93	23	for	for	ADP
cana-3960	93	24	each	each	DET
cana-3960	93	25	(	(	PUNCT
cana-3960	93	26	𝜏	𝜏	PROPN
cana-3960	93	27	,	,	PUNCT
cana-3960	93	28	𝜂	𝜂	NOUN
cana-3960	93	29	)	)	PUNCT
cana-3960	93	30	∈	∈	PROPN
cana-3960	93	31	𝕋+	𝕋+	ADP
cana-3960	93	32	×	×	NOUN
cana-3960	93	33	ℝ𝑛	ℝ𝑛	ADP
cana-3960	93	34	2	2	NUM
cana-3960	93	35	the	the	DET
cana-3960	93	36	initial	initial	ADJ
cana-3960	93	37	value	value	NOUN
cana-3960	93	38	problem	problem	NOUN
cana-3960	93	39	𝑧𝛥(𝑡	𝑧𝛥(𝑡	NUM
cana-3960	93	40	)	)	PUNCT
cana-3960	93	41	=	=	SYM
cana-3960	93	42	−𝐺𝑇(𝑡)𝑧𝜎(𝑡	−𝐺𝑇(𝑡)𝑧𝜎(𝑡	NOUN
cana-3960	93	43	)	)	PUNCT
cana-3960	93	44	+	+	CCONJ
cana-3960	93	45	𝑙(𝑡	𝑙(𝑡	NOUN
cana-3960	93	46	)	)	PUNCT
cana-3960	93	47	,	,	PUNCT
cana-3960	93	48	z(τ	z(τ	NOUN
cana-3960	93	49	)	)	PUNCT
cana-3960	93	50	=	=	SYM
cana-3960	93	51	𝜂	𝜂	NOUN
cana-3960	93	52	,	,	PUNCT
cana-3960	93	53	has	have	VERB
cana-3960	93	54	one	one	NUM
cana-3960	93	55	and	and	CCONJ
cana-3960	93	56	only	only	ADV
cana-3960	93	57	one	one	NUM
cana-3960	93	58	solution	solution	NOUN
cana-3960	93	59	𝑧	𝑧	NOUN
cana-3960	93	60	:	:	PUNCT
cana-3960	93	61	𝕋(𝜏	𝕋(𝜏	X
cana-3960	93	62	)	)	PUNCT
cana-3960	93	63	→	→	PUNCT
cana-3960	93	64	ℝ𝑛	ℝ𝑛	ADP
cana-3960	93	65	2	2	NUM
cana-3960	93	66	is	be	AUX
cana-3960	93	67	given	give	VERB
cana-3960	93	68	by	by	ADP
cana-3960	93	69	𝑧(𝑡	𝑧(𝑡	NOUN
cana-3960	93	70	)	)	PUNCT
cana-3960	93	71	=	=	SYM
cana-3960	93	72	𝜓⊝𝐺𝑇(𝑡	𝜓⊝𝐺𝑇(𝑡	NOUN
cana-3960	93	73	,	,	PUNCT
cana-3960	93	74	𝜏)𝜂	𝜏)𝜂	X
cana-3960	93	75	+	+	CCONJ
cana-3960	93	76	∫	∫	PROPN
cana-3960	93	77	𝜓⊝𝐺𝑇(𝑡	𝜓⊝𝐺𝑇(𝑡	NOUN
cana-3960	93	78	,	,	PUNCT
cana-3960	93	79	𝜎(𝑠))𝑙(𝑠)𝛥𝑠	𝜎(𝑠))𝑙(𝑠)𝛥𝑠	PROPN
cana-3960	93	80	𝑡	𝑡	PROPN
cana-3960	93	81	𝑡0	𝑡0	PROPN
cana-3960	93	82	,	,	PUNCT
cana-3960	93	83	𝑡	𝑡	PROPN
cana-3960	93	84	∈	∈	PROPN
cana-3960	93	85	𝕋(𝜏	𝕋(𝜏	X
cana-3960	93	86	)	)	PUNCT
cana-3960	93	87	.	.	PUNCT
cana-3960	94	1	proposition	proposition	NOUN
cana-3960	94	2	2.1	2.1	NUM
cana-3960	94	3	.	.	PUNCT
cana-3960	95	1	[	[	X
cana-3960	95	2	8	8	X
cana-3960	95	3	]	]	PUNCT
cana-3960	95	4	the	the	DET
cana-3960	95	5	system	system	NOUN
cana-3960	95	6	(	(	PUNCT
cana-3960	95	7	2.2	2.2	NUM
cana-3960	95	8	)	)	PUNCT
cana-3960	95	9	with	with	ADP
cana-3960	95	10	𝐺𝑘	𝐺𝑘	PROPN
cana-3960	95	11	∈	∈	PROPN
cana-3960	95	12	𝑀𝑛2(𝑅	𝑀𝑛2(𝑅	NOUN
cana-3960	95	13	)	)	PUNCT
cana-3960	95	14	constant	constant	ADJ
cana-3960	95	15	,	,	PUNCT
cana-3960	95	16	there	there	PRON
cana-3960	95	17	exist	exist	VERB
cana-3960	95	18	scalar	scalar	ADJ
cana-3960	95	19	functions	function	NOUN
cana-3960	95	20	𝜒0(𝑡	𝜒0(𝑡	PROPN
cana-3960	95	21	,	,	PUNCT
cana-3960	95	22	𝜏	𝜏	NOUN
cana-3960	95	23	)	)	PUNCT
cana-3960	95	24	,	,	PUNCT
cana-3960	95	25	𝜒1(𝑡	𝜒1(𝑡	PRON
cana-3960	95	26	,	,	PUNCT
cana-3960	95	27	𝜏	𝜏	NOUN
cana-3960	95	28	)	)	PUNCT
cana-3960	95	29	,	,	PUNCT
cana-3960	95	30	.	.	PUNCT
cana-3960	95	31	.	.	PUNCT
cana-3960	95	32	.	.	PUNCT
cana-3960	95	33	.	.	PUNCT
cana-3960	95	34	.	.	PUNCT
cana-3960	96	1	𝜒𝑛2−1(𝑡	𝜒𝑛2−1(𝑡	VERB
cana-3960	96	2	,	,	PUNCT
cana-3960	96	3	𝜏	𝜏	NOUN
cana-3960	96	4	)	)	PUNCT
cana-3960	96	5	∈	∈	PROPN
cana-3960	96	6	𝐶𝑟𝑑	𝐶𝑟𝑑	PROPN
cana-3960	96	7	∞	∞	PROPN
cana-3960	96	8	(	(	PUNCT
cana-3960	96	9	𝕋+,ℝ	𝕋+,ℝ	PROPN
cana-3960	96	10	)	)	PUNCT
cana-3960	96	11	such	such	ADJ
cana-3960	96	12	that	that	SCONJ
cana-3960	96	13	the	the	DET
cana-3960	96	14	one	one	NOUN
cana-3960	96	15	and	and	CCONJ
cana-3960	96	16	only	only	ADV
cana-3960	96	17	one	one	NUM
cana-3960	96	18	solution	solution	NOUN
cana-3960	96	19	is	be	AUX
cana-3960	96	20	given	give	VERB
cana-3960	96	21	by	by	ADP
cana-3960	96	22	𝑒𝐺𝑘	𝑒𝐺𝑘	NOUN
cana-3960	96	23	𝑇(𝑡	𝑇(𝑡	NUM
cana-3960	96	24	,	,	PUNCT
cana-3960	96	25	𝜏	𝜏	NOUN
cana-3960	96	26	)	)	PUNCT
cana-3960	96	27	=	=	PUNCT
cana-3960	96	28	∑	∑	PUNCT
cana-3960	96	29	𝜒𝑖	𝜒𝑖	PROPN
cana-3960	96	30	𝑛2−1	𝑛2−1	NOUN
cana-3960	96	31	𝑖=0	𝑖=0	PROPN
cana-3960	96	32	(	(	PUNCT
cana-3960	96	33	𝑡	𝑡	PROPN
cana-3960	96	34	,	,	PUNCT
cana-3960	96	35	𝜏)𝐺𝑖.	𝜏)𝐺𝑖.	PROPN
cana-3960	96	36	3	3	NUM
cana-3960	96	37	.	.	PUNCT
cana-3960	96	38	complete	complete	ADJ
cana-3960	96	39	controllability	controllability	NOUN
cana-3960	96	40	in	in	ADP
cana-3960	96	41	this	this	DET
cana-3960	96	42	section	section	NOUN
cana-3960	96	43	,	,	PUNCT
cana-3960	96	44	we	we	PRON
cana-3960	96	45	present	present	VERB
cana-3960	96	46	the	the	DET
cana-3960	96	47	controllability	controllability	NOUN
cana-3960	96	48	in	in	ADP
cana-3960	96	49	time	time	NOUN
cana-3960	96	50	variant	variant	NOUN
cana-3960	96	51	and	and	CCONJ
cana-3960	96	52	time	time	NOUN
cana-3960	96	53	invariant	invariant	PROPN
cana-3960	96	54	adjoint	adjoint	PROPN
cana-3960	96	55	dynamic	dynamic	ADJ
cana-3960	96	56	system	system	NOUN
cana-3960	96	57	(	(	PUNCT
cana-3960	96	58	3	3	NUM
cana-3960	96	59	)	)	PUNCT
cana-3960	96	60	on	on	ADP
cana-3960	96	61	time	time	NOUN
cana-3960	96	62	scales	scale	NOUN
cana-3960	96	63	.	.	PUNCT
cana-3960	97	1	lemma	lemma	PROPN
cana-3960	97	2	3.1	3.1	NUM
cana-3960	97	3	.	.	PUNCT
cana-3960	98	1	for	for	ADP
cana-3960	98	2	any	any	DET
cana-3960	98	3	∈	∈	PROPN
cana-3960	98	4	[	[	X
cana-3960	98	5	𝑡𝑙−1	𝑡𝑙−1	X
cana-3960	98	6	,	,	PUNCT
cana-3960	98	7	𝑡𝑙)𝕋	𝑡𝑙)𝕋	ADP
cana-3960	98	8	,	,	PUNCT
cana-3960	98	9	𝑙	𝑙	NOUN
cana-3960	98	10	=	=	SYM
cana-3960	98	11	1,2	1,2	NUM
cana-3960	98	12	,	,	PUNCT
cana-3960	98	13	.	.	PUNCT
cana-3960	98	14	.	.	PUNCT
cana-3960	98	15	.	.	PUNCT
cana-3960	99	1	,	,	PUNCT
cana-3960	99	2	𝑘	𝑘	X
cana-3960	99	3	the	the	DET
cana-3960	99	4	solution	solution	NOUN
cana-3960	99	5	of	of	ADP
cana-3960	99	6	initial	initial	ADJ
cana-3960	99	7	value	value	NOUN
cana-3960	99	8	problem	problem	NOUN
cana-3960	99	9	(	(	PUNCT
cana-3960	99	10	2.2	2.2	NUM
cana-3960	99	11	)	)	PUNCT
cana-3960	99	12	is	be	AUX
cana-3960	99	13	given	give	VERB
cana-3960	99	14	by	by	ADP
cana-3960	99	15	𝑧(𝑡	𝑧(𝑡	NOUN
cana-3960	99	16	)	)	PUNCT
cana-3960	100	1	=	=	PRON
cana-3960	100	2	{	{	PUNCT
cana-3960	100	3	𝜓𝐺1	𝜓𝐺1	NOUN
cana-3960	100	4	𝑇	𝑇	PROPN
cana-3960	100	5	(	(	PUNCT
cana-3960	100	6	𝑡0	𝑡0	NOUN
cana-3960	100	7	,	,	PUNCT
cana-3960	101	1	𝑡)𝑧0	𝑡)𝑧0	PROPN
cana-3960	101	2	+	+	NUM
cana-3960	101	3	∫	∫	PROPN
cana-3960	101	4	𝜓𝐺1	𝜓𝐺1	NOUN
cana-3960	101	5	𝑇	𝑇	PROPN
cana-3960	101	6	(	(	PUNCT
cana-3960	101	7	𝜏	𝜏	NOUN
cana-3960	101	8	,	,	PUNCT
cana-3960	101	9	𝑡)𝐴1(𝜏)	𝑡)𝐴1(𝜏)	PROPN
cana-3960	101	10	�	�	NOUN
cana-3960	101	11	̂	̂	NOUN
cana-3960	101	12	�	�	NOUN
cana-3960	101	13	(𝜏)∆𝜏	(𝜏)∆𝜏	PROPN
cana-3960	101	14	,	,	PUNCT
cana-3960	101	15	𝑙	𝑙	X
cana-3960	101	16	=	=	SYM
cana-3960	101	17	1	1	NUM
cana-3960	101	18	𝑡	𝑡	NOUN
cana-3960	101	19	𝑡0	𝑡0	NOUN
cana-3960	101	20	𝜓𝐺𝑙	𝜓𝐺𝑙	PROPN
cana-3960	101	21	𝑇	𝑇	PROPN
cana-3960	101	22	(	(	PUNCT
cana-3960	101	23	𝑡𝑙−1	𝑡𝑙−1	ADJ
cana-3960	101	24	,	,	PUNCT
cana-3960	101	25	𝑡){∏	𝑡){∏	PUNCT
cana-3960	102	1	[	[	X
cana-3960	102	2	𝐼𝑛⊗𝑅𝑗	𝐼𝑛⊗𝑅𝑗	X
cana-3960	102	3	]	]	X
cana-3960	102	4	∏	∏	PROPN
cana-3960	102	5	𝜓𝐺𝑗	𝜓𝐺𝑗	PROPN
cana-3960	102	6	𝑇	𝑇	PROPN
cana-3960	102	7	(	(	PUNCT
cana-3960	102	8	𝑡𝑗−1	𝑡𝑗−1	PROPN
cana-3960	102	9	,	,	PUNCT
cana-3960	102	10	𝑡𝑗)𝑧0	𝑡𝑗)𝑧0	NOUN
cana-3960	102	11	+	+	CCONJ
cana-3960	102	12	1	1	NUM
cana-3960	102	13	𝑗=𝑙−1	𝑗=𝑙−1	SYM
cana-3960	102	14	1	1	NUM
cana-3960	102	15	𝑗=𝑙−1	𝑗=𝑙−1	NUM
cana-3960	102	16	∑(∏[𝐼𝑛⊗𝑅𝑗	∑(∏[𝐼𝑛⊗𝑅𝑗	NOUN
cana-3960	102	17	]	]	PUNCT
cana-3960	102	18	∏	∏	PROPN
cana-3960	102	19	𝜓𝐺𝑖	𝜓𝐺𝑖	ADJ
cana-3960	102	20	𝑇	𝑇	PROPN
cana-3960	102	21	(	(	PUNCT
cana-3960	102	22	𝑡𝑖−1	𝑡𝑖−1	PROPN
cana-3960	102	23	,	,	PUNCT
cana-3960	102	24	𝑡𝑖)∫	𝑡𝑖)∫	NOUN
cana-3960	102	25	𝜓𝐺𝑗	𝜓𝐺𝑗	NOUN
cana-3960	102	26	𝑇	𝑇	PROPN
cana-3960	102	27	(	(	PUNCT
cana-3960	102	28	𝜏	𝜏	NOUN
cana-3960	102	29	,	,	PUNCT
cana-3960	102	30	𝑡𝑗)𝐴𝑗(𝜏)	𝑡𝑗)𝐴𝑗(𝜏)	NOUN
cana-3960	102	31	�	�	PROPN
cana-3960	102	32	̂	̂	NOUN
cana-3960	102	33	�	�	NOUN
cana-3960	102	34	(𝜏)∆𝜏	(𝜏)∆𝜏	SYM
cana-3960	102	35	𝑡𝑗	𝑡𝑗	PROPN
cana-3960	102	36	𝑡𝑗−1	𝑡𝑗−1	PROPN
cana-3960	102	37	𝑗+1	𝑗+1	X
cana-3960	102	38	𝑖=𝑙−1	𝑖=𝑙−1	PUNCT
cana-3960	102	39	𝑗	𝑗	X
cana-3960	102	40	𝑖=𝑙−1	𝑖=𝑙−1	PUNCT
cana-3960	102	41	)	)	PUNCT
cana-3960	102	42	𝑙−2	𝑙−2	PUNCT
cana-3960	103	1	𝑗=1	𝑗=1	PUNCT
cana-3960	103	2	+	+	PROPN
cana-3960	103	3	[	[	X
cana-3960	103	4	𝐼𝑛⊗𝑅𝑙−1]∫	𝐼𝑛⊗𝑅𝑙−1]∫	NOUN
cana-3960	103	5	𝜓𝐺𝑙−1	𝜓𝐺𝑙−1	X
cana-3960	103	6	𝑇	𝑇	PROPN
cana-3960	103	7	(	(	PUNCT
cana-3960	103	8	𝜏	𝜏	NOUN
cana-3960	103	9	,	,	PUNCT
cana-3960	103	10	𝑡𝑙−1)𝐴𝑙−1(𝜏)	𝑡𝑙−1)𝐴𝑙−1(𝜏)	NOUN
cana-3960	103	11	�	�	NOUN
cana-3960	103	12	̂	̂	SYM
cana-3960	103	13	�	�	NOUN
cana-3960	103	14	(𝜏)∆𝜏	(𝜏)∆𝜏	SYM
cana-3960	103	15	𝑡𝑙−1	𝑡𝑙−1	ADV
cana-3960	103	16	𝑡𝑙−2	𝑡𝑙−2	PROPN
cana-3960	103	17	}	}	PUNCT
cana-3960	103	18	+	+	NOUN
cana-3960	103	19	∫	∫	PROPN
cana-3960	103	20	𝜓𝐺𝑙	𝜓𝐺𝑙	PROPN
cana-3960	103	21	𝑇	𝑇	PROPN
cana-3960	103	22	(	(	PUNCT
cana-3960	103	23	𝜏	𝜏	NOUN
cana-3960	103	24	,	,	PUNCT
cana-3960	103	25	𝑡)𝐴𝑙(𝜏)	𝑡)𝐴𝑙(𝜏)	PROPN
cana-3960	103	26	�	�	PROPN
cana-3960	103	27	̂	̂	NOUN
cana-3960	103	28	�	�	NOUN
cana-3960	103	29	(𝜏)∆𝜏	(𝜏)∆𝜏	SYM
cana-3960	103	30	,	,	PUNCT
cana-3960	103	31	𝑙	𝑙	X
cana-3960	103	32	=	=	SYM
cana-3960	103	33	2,3	2,3	NUM
cana-3960	103	34	,	,	PUNCT
cana-3960	103	35	⋯	⋯	PROPN
cana-3960	103	36	,	,	PUNCT
cana-3960	103	37	𝑘.	𝑘.	NOUN
cana-3960	103	38	𝑡	𝑡	X
cana-3960	103	39	𝑡𝑙−1	𝑡𝑙−1	PROPN
cana-3960	103	40	(	(	PUNCT
cana-3960	103	41	3.1	3.1	NUM
cana-3960	103	42	)	)	PUNCT
cana-3960	103	43	proof	proof	NOUN
cana-3960	103	44	:	:	PUNCT
cana-3960	103	45	form	form	NOUN
cana-3960	103	46	lemma	lemma	PROPN
cana-3960	103	47	2.2	2.2	NUM
cana-3960	103	48	for	for	ADP
cana-3960	103	49	𝑡	𝑡	PROPN
cana-3960	103	50	∈	∈	PROPN
cana-3960	103	51	[	[	X
cana-3960	103	52	𝑡0	𝑡0	NOUN
cana-3960	103	53	,	,	PUNCT
cana-3960	103	54	𝑡1]𝕋	𝑡1]𝕋	PROPN
cana-3960	103	55	,	,	PUNCT
cana-3960	103	56	we	we	PRON
cana-3960	103	57	have	have	VERB
cana-3960	103	58	𝑧(𝑡	𝑧(𝑡	NOUN
cana-3960	103	59	)	)	PUNCT
cana-3960	104	1	=	=	SYM
cana-3960	104	2	𝜓𝐺1	𝜓𝐺1	PROPN
cana-3960	104	3	𝑇	𝑇	PROPN
cana-3960	104	4	(	(	PUNCT
cana-3960	104	5	𝑡0	𝑡0	NOUN
cana-3960	104	6	,	,	PUNCT
cana-3960	104	7	𝑡)𝑧0	𝑡)𝑧0	PROPN
cana-3960	104	8	+	+	NUM
cana-3960	104	9	∫	∫	PROPN
cana-3960	104	10	𝜓𝐺1	𝜓𝐺1	NOUN
cana-3960	104	11	𝑇	𝑇	PROPN
cana-3960	104	12	(	(	PUNCT
cana-3960	104	13	𝜏	𝜏	NOUN
cana-3960	104	14	,	,	PUNCT
cana-3960	104	15	𝑡)𝐴1(𝜏)	𝑡)𝐴1(𝜏)	PROPN
cana-3960	104	16	�	�	NOUN
cana-3960	104	17	̂	̂	VERB
cana-3960	104	18	�	�	NOUN
cana-3960	104	19	(𝜏)∆𝜏	(𝜏)∆𝜏	SYM
cana-3960	104	20	𝑡	𝑡	PROPN
cana-3960	104	21	𝑡0	𝑡0	PROPN
cana-3960	104	22	,	,	PUNCT
cana-3960	104	23	communications	communication	NOUN
cana-3960	104	24	on	on	ADP
cana-3960	104	25	applied	apply	VERB
cana-3960	104	26	nonlinear	nonlinear	ADJ
cana-3960	104	27	analysis	analysis	NOUN
cana-3960	104	28	issn	issn	NOUN
cana-3960	104	29	:	:	PUNCT
cana-3960	104	30	1074	1074	NUM
cana-3960	104	31	-	-	PUNCT
cana-3960	104	32	133x	133x	NUM
cana-3960	104	33	vol	vol	NOUN
cana-3960	104	34	32	32	NUM
cana-3960	104	35	no	no	NOUN
cana-3960	104	36	.	.	PUNCT
cana-3960	105	1	9s	9s	NUM
cana-3960	105	2	(	(	PUNCT
cana-3960	105	3	2025	2025	NUM
cana-3960	105	4	)	)	PUNCT
cana-3960	105	5	497	497	NUM
cana-3960	105	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3960	105	7	similarly	similarly	ADV
cana-3960	105	8	,	,	PUNCT
cana-3960	105	9	for	for	ADP
cana-3960	105	10	𝑡	𝑡	PROPN
cana-3960	105	11	∈	∈	PROPN
cana-3960	106	1	[	[	X
cana-3960	106	2	𝑡𝑙−1	𝑡𝑙−1	X
cana-3960	106	3	,	,	PUNCT
cana-3960	106	4	𝑡𝑙)𝕋	𝑡𝑙)𝕋	PROPN
cana-3960	106	5	,	,	PUNCT
cana-3960	106	6	we	we	PRON
cana-3960	106	7	have	have	VERB
cana-3960	106	8	𝑧(𝑡	𝑧(𝑡	NOUN
cana-3960	106	9	)	)	PUNCT
cana-3960	107	1	=	=	SYM
cana-3960	107	2	𝜓𝐺𝑙	𝜓𝐺𝑙	PROPN
cana-3960	107	3	𝑇	𝑇	PROPN
cana-3960	107	4	(	(	PUNCT
cana-3960	107	5	𝑡𝑙−1	𝑡𝑙−1	ADV
cana-3960	107	6	,	,	PUNCT
cana-3960	107	7	𝑡){𝑧(𝑡𝑙	𝑡){𝑧(𝑡𝑙	NOUN
cana-3960	107	8	)	)	PUNCT
cana-3960	107	9	}	}	PUNCT
cana-3960	108	1	+	+	CCONJ
cana-3960	108	2	∫	∫	PROPN
cana-3960	108	3	𝜓𝐺𝑙	𝜓𝐺𝑙	PROPN
cana-3960	108	4	𝑇	𝑇	PROPN
cana-3960	108	5	(	(	PUNCT
cana-3960	108	6	𝜏	𝜏	NOUN
cana-3960	108	7	,	,	PUNCT
cana-3960	108	8	𝑡)𝐴𝑙(𝜏)	𝑡)𝐴𝑙(𝜏)	PROPN
cana-3960	108	9	�	�	PROPN
cana-3960	108	10	̂	̂	NOUN
cana-3960	108	11	�	�	NOUN
cana-3960	108	12	(𝜏)∆𝜏	(𝜏)∆𝜏	SYM
cana-3960	108	13	,	,	PUNCT
cana-3960	108	14	𝑡	𝑡	X
cana-3960	108	15	𝑡𝑙−1	𝑡𝑙−1	ADV
cana-3960	108	16	also	also	ADV
cana-3960	108	17	,	,	PUNCT
cana-3960	108	18	form	form	VERB
cana-3960	108	19	the	the	DET
cana-3960	108	20	system	system	NOUN
cana-3960	108	21	(	(	PUNCT
cana-3960	108	22	2.2	2.2	NUM
cana-3960	108	23	)	)	PUNCT
cana-3960	108	24	,	,	PUNCT
cana-3960	108	25	we	we	PRON
cana-3960	108	26	have	have	VERB
cana-3960	108	27	𝑧(𝑡𝑙	𝑧(𝑡𝑙	NOUN
cana-3960	108	28	)	)	PUNCT
cana-3960	108	29	=	=	SYM
cana-3960	109	1	∏	∏	PROPN
cana-3960	110	1	[	[	X
cana-3960	110	2	𝐼𝑛⊗𝑅𝑗	𝐼𝑛⊗𝑅𝑗	X
cana-3960	110	3	]	]	X
cana-3960	110	4	∏	∏	PROPN
cana-3960	110	5	𝜓𝐺𝑗	𝜓𝐺𝑗	PROPN
cana-3960	110	6	𝑇	𝑇	PROPN
cana-3960	110	7	(	(	PUNCT
cana-3960	110	8	𝑡𝑗−1	𝑡𝑗−1	PROPN
cana-3960	110	9	,	,	PUNCT
cana-3960	110	10	𝑡𝑗)𝑧0	𝑡𝑗)𝑧0	NOUN
cana-3960	110	11	1	1	NUM
cana-3960	110	12	𝑗=𝑙−1	𝑗=𝑙−1	SYM
cana-3960	110	13	1	1	NUM
cana-3960	110	14	𝑗=𝑙−1	𝑗=𝑙−1	PUNCT
cana-3960	111	1	+	+	NOUN
cana-3960	111	2	∑	∑	PROPN
cana-3960	111	3	(	(	PUNCT
cana-3960	111	4	∏	∏	PROPN
cana-3960	111	5	[	[	X
cana-3960	111	6	𝐼𝐼⊗𝐼𝐼	𝐼𝐼⊗𝐼𝐼	PROPN
cana-3960	111	7	]	]	PUNCT
cana-3960	111	8	∏	∏	PROPN
cana-3960	111	9	𝐼𝐼𝐼	𝐼𝐼𝐼	NOUN
cana-3960	111	10	𝐼	𝐼	PROPN
cana-3960	111	11	(	(	PUNCT
cana-3960	111	12	𝐼𝐼−1,𝐼𝐼)∫	𝐼𝐼−1,𝐼𝐼)∫	NOUN
cana-3960	111	13	𝐼𝐼𝐼	𝐼𝐼𝐼	NOUN
cana-3960	111	14	𝐼	𝐼	PROPN
cana-3960	111	15	(	(	PUNCT
cana-3960	111	16	𝐼,𝐼𝐼)𝐼𝐼(𝐼)	𝐼,𝐼𝐼)𝐼𝐼(𝐼)	X
cana-3960	111	17	�	�	NOUN
cana-3960	111	18	̂	̂	SYM
cana-3960	111	19	�	�	NOUN
cana-3960	111	20	(𝐼)∆𝐼	(𝐼)∆𝐼	VERB
cana-3960	111	21	𝐼𝐼	𝐼𝐼	PROPN
cana-3960	111	22	𝐼𝐼−1	𝐼𝐼−1	NOUN
cana-3960	111	23	𝐼+1	𝐼+1	PUNCT
cana-3960	111	24	𝐼=𝐼−1	𝐼=𝐼−1	PROPN
cana-3960	111	25	𝐼	𝐼	PROPN
cana-3960	111	26	𝐼=𝐼−1	𝐼=𝐼−1	PROPN
cana-3960	111	27	)	)	PUNCT
cana-3960	111	28	𝐼−2	𝐼−2	PROPN
cana-3960	111	29	𝐼=1	𝐼=1	PROPN
cana-3960	112	1	+	+	PUNCT
cana-3960	113	1	[	[	X
cana-3960	113	2	𝐼𝑛⊗𝑅𝑙−1]∫	𝐼𝑛⊗𝑅𝑙−1]∫	NOUN
cana-3960	113	3	𝜓𝐺𝑙−1	𝜓𝐺𝑙−1	X
cana-3960	113	4	𝑇	𝑇	PROPN
cana-3960	113	5	(	(	PUNCT
cana-3960	113	6	𝜏	𝜏	NOUN
cana-3960	113	7	,	,	PUNCT
cana-3960	113	8	𝑡𝑙−1)𝐴𝑙−1(𝜏)	𝑡𝑙−1)𝐴𝑙−1(𝜏)	NOUN
cana-3960	113	9	�	�	NOUN
cana-3960	113	10	̂	̂	SYM
cana-3960	113	11	�	�	NOUN
cana-3960	113	12	(𝜏)∆𝜏	(𝜏)∆𝜏	SYM
cana-3960	113	13	𝑡𝑙−1	𝑡𝑙−1	ADV
cana-3960	113	14	𝑡𝑙−2	𝑡𝑙−2	PROPN
cana-3960	113	15	,	,	PUNCT
cana-3960	113	16	𝑙	𝑙	X
cana-3960	113	17	=	=	SYM
cana-3960	113	18	2,3	2,3	NUM
cana-3960	113	19	,	,	PUNCT
cana-3960	113	20	⋯	⋯	PROPN
cana-3960	113	21	,	,	PUNCT
cana-3960	113	22	𝑘	𝑘	PROPN
cana-3960	113	23	,	,	PUNCT
cana-3960	113	24	therefore	therefore	ADV
cana-3960	113	25	for	for	ADP
cana-3960	113	26	𝑡	𝑡	PROPN
cana-3960	113	27	∈	∈	PROPN
cana-3960	113	28	(	(	PUNCT
cana-3960	113	29	𝑡𝑙−1	𝑡𝑙−1	X
cana-3960	113	30	,	,	PUNCT
cana-3960	113	31	𝑡𝑙)𝕋	𝑡𝑙)𝕋	PROPN
cana-3960	113	32	,	,	PUNCT
cana-3960	113	33	we	we	PRON
cana-3960	113	34	have	have	VERB
cana-3960	113	35	𝑧(𝑡	𝑧(𝑡	NOUN
cana-3960	113	36	)	)	PUNCT
cana-3960	114	1	=	=	SYM
cana-3960	114	2	𝜓𝐺𝑙	𝜓𝐺𝑙	PROPN
cana-3960	114	3	𝑇	𝑇	PROPN
cana-3960	114	4	(	(	PUNCT
cana-3960	114	5	𝑡𝑙−1	𝑡𝑙−1	X
cana-3960	114	6	,	,	PUNCT
cana-3960	114	7	𝑡	𝑡	NOUN
cana-3960	114	8	)	)	PUNCT
cana-3960	114	9	{	{	PUNCT
cana-3960	114	10	∏	∏	PROPN
cana-3960	114	11	[	[	X
cana-3960	114	12	𝐼𝑛⊗𝑅𝑗	𝐼𝑛⊗𝑅𝑗	X
cana-3960	114	13	]	]	X
cana-3960	114	14	∏	∏	PROPN
cana-3960	114	15	𝜓𝐺𝑗	𝜓𝐺𝑗	PROPN
cana-3960	114	16	𝑇	𝑇	PROPN
cana-3960	114	17	(	(	PUNCT
cana-3960	114	18	𝑡𝑗−1	𝑡𝑗−1	PROPN
cana-3960	114	19	,	,	PUNCT
cana-3960	114	20	𝑡𝑗)𝑧0	𝑡𝑗)𝑧0	NOUN
cana-3960	114	21	1	1	NUM
cana-3960	114	22	𝑗=𝑙−1	𝑗=𝑙−1	SYM
cana-3960	114	23	1	1	NUM
cana-3960	114	24	𝑗=𝑙−1	𝑗=𝑙−1	PUNCT
cana-3960	115	1	+	+	NOUN
cana-3960	115	2	∑	∑	PROPN
cana-3960	115	3	(	(	PUNCT
cana-3960	115	4	∏	∏	PROPN
cana-3960	115	5	[	[	X
cana-3960	115	6	𝐼𝐼⊗𝐼𝐼	𝐼𝐼⊗𝐼𝐼	PROPN
cana-3960	115	7	]	]	PUNCT
cana-3960	115	8	∏	∏	PROPN
cana-3960	115	9	𝐼𝐼𝐼	𝐼𝐼𝐼	NOUN
cana-3960	115	10	𝐼	𝐼	PROPN
cana-3960	115	11	(	(	PUNCT
cana-3960	115	12	𝐼𝐼−1,𝐼𝐼)∫	𝐼𝐼−1,𝐼𝐼)∫	NOUN
cana-3960	115	13	𝐼𝐼𝐼	𝐼𝐼𝐼	NOUN
cana-3960	115	14	𝐼	𝐼	PROPN
cana-3960	115	15	(	(	PUNCT
cana-3960	115	16	𝐼,𝐼𝐼)𝐼𝐼(𝐼)	𝐼,𝐼𝐼)𝐼𝐼(𝐼)	X
cana-3960	115	17	�	�	NOUN
cana-3960	115	18	̂	̂	SYM
cana-3960	115	19	�	�	NOUN
cana-3960	115	20	(𝐼)∆𝐼	(𝐼)∆𝐼	VERB
cana-3960	115	21	𝐼𝐼	𝐼𝐼	PROPN
cana-3960	115	22	𝐼𝐼−1	𝐼𝐼−1	NOUN
cana-3960	115	23	𝐼+1	𝐼+1	PUNCT
cana-3960	115	24	𝐼=𝐼−1	𝐼=𝐼−1	PROPN
cana-3960	115	25	𝐼	𝐼	PROPN
cana-3960	115	26	𝐼=𝐼−1	𝐼=𝐼−1	PROPN
cana-3960	115	27	)	)	PUNCT
cana-3960	115	28	𝐼−2	𝐼−2	PROPN
cana-3960	115	29	𝐼=1	𝐼=1	PROPN
cana-3960	116	1	+	+	PUNCT
cana-3960	117	1	[	[	X
cana-3960	117	2	𝐼𝑛⊗𝑅𝑙−1]∫	𝐼𝑛⊗𝑅𝑙−1]∫	NOUN
cana-3960	117	3	𝜓𝐺𝑙−1	𝜓𝐺𝑙−1	X
cana-3960	117	4	𝑇	𝑇	PROPN
cana-3960	117	5	(	(	PUNCT
cana-3960	117	6	𝜏	𝜏	NOUN
cana-3960	117	7	,	,	PUNCT
cana-3960	117	8	𝑡𝑙−1)𝐴𝑙−1(𝜏)	𝑡𝑙−1)𝐴𝑙−1(𝜏)	NOUN
cana-3960	117	9	�	�	NOUN
cana-3960	117	10	̂	̂	SYM
cana-3960	117	11	�	�	NOUN
cana-3960	117	12	(𝜏)∆𝜏	(𝜏)∆𝜏	SYM
cana-3960	117	13	𝑡𝑙−1	𝑡𝑙−1	ADV
cana-3960	117	14	𝑡𝑙−2	𝑡𝑙−2	PROPN
cana-3960	117	15	}	}	PUNCT
cana-3960	117	16	+	+	NOUN
cana-3960	117	17	∫	∫	PROPN
cana-3960	117	18	𝜓𝐺𝑙	𝜓𝐺𝑙	PROPN
cana-3960	117	19	𝑇	𝑇	PROPN
cana-3960	117	20	(	(	PUNCT
cana-3960	117	21	𝜏	𝜏	NOUN
cana-3960	117	22	,	,	PUNCT
cana-3960	117	23	𝑡)𝐴𝑙(𝜏)	𝑡)𝐴𝑙(𝜏)	PROPN
cana-3960	117	24	�	�	PROPN
cana-3960	117	25	̂	̂	NOUN
cana-3960	117	26	�	�	NOUN
cana-3960	117	27	(𝜏)∆𝜏.	(𝜏)∆𝜏.	NOUN
cana-3960	117	28	𝑡	𝑡	NOUN
cana-3960	117	29	𝑡𝑙−1	𝑡𝑙−1	ADV
cana-3960	117	30	after	after	ADP
cana-3960	117	31	repeating	repeat	VERB
cana-3960	117	32	the	the	DET
cana-3960	117	33	above	above	ADJ
cana-3960	117	34	same	same	ADJ
cana-3960	117	35	process	process	NOUN
cana-3960	117	36	,	,	PUNCT
cana-3960	117	37	we	we	PRON
cana-3960	117	38	get	get	VERB
cana-3960	117	39	the	the	DET
cana-3960	117	40	desire	desire	NOUN
cana-3960	117	41	results	result	NOUN
cana-3960	117	42	.	.	PUNCT
cana-3960	118	1	theorem	theorem	VERB
cana-3960	118	2	3.1	3.1	NUM
cana-3960	118	3	.	.	PUNCT
cana-3960	118	4	i.	i.	PROPN
cana-3960	118	5	if	if	SCONJ
cana-3960	118	6	there	there	PRON
cana-3960	118	7	exist	exist	VERB
cana-3960	118	8	at	at	ADP
cana-3960	118	9	least	least	ADJ
cana-3960	118	10	k∈	k∈	PRON
cana-3960	118	11	{	{	PUNCT
cana-3960	118	12	1,2	1,2	NUM
cana-3960	118	13	,	,	PUNCT
cana-3960	118	14	…	…	PUNCT
cana-3960	118	15	,	,	PUNCT
cana-3960	118	16	𝑙	𝑙	X
cana-3960	118	17	}	}	PUNCT
cana-3960	118	18	such	such	ADJ
cana-3960	118	19	that	that	DET
cana-3960	118	20	𝑟𝑎𝑛𝑘	𝑟𝑎𝑛𝑘	PROPN
cana-3960	118	21	{	{	PUNCT
cana-3960	118	22	𝐻𝑘(𝑡𝑘−1	𝐻𝑘(𝑡𝑘−1	NOUN
cana-3960	118	23	,	,	PUNCT
cana-3960	118	24	𝑡𝑘	𝑡𝑘	ADV
cana-3960	118	25	,	,	PUNCT
cana-3960	118	26	𝑡𝑓	𝑡𝑓	NOUN
cana-3960	118	27	)	)	PUNCT
cana-3960	118	28	}	}	PUNCT
cana-3960	119	1	=	=	SYM
cana-3960	119	2	𝑛	𝑛	DET
cana-3960	119	3	2	2	NUM
cana-3960	119	4	then	then	ADV
cana-3960	119	5	the	the	DET
cana-3960	119	6	impulsive	impulsive	ADJ
cana-3960	119	7	system	system	NOUN
cana-3960	119	8	(	(	PUNCT
cana-3960	119	9	2.2	2.2	NUM
cana-3960	119	10	)	)	PUNCT
cana-3960	119	11	is	be	AUX
cana-3960	119	12	controllable	controllable	ADJ
cana-3960	119	13	on	on	ADP
cana-3960	119	14	[	[	X
cana-3960	119	15	𝑡0	𝑡0	NOUN
cana-3960	119	16	,	,	PUNCT
cana-3960	119	17	𝑡1]𝕋(𝑡𝑓	𝑡1]𝕋(𝑡𝑓	PROPN
cana-3960	119	18	∈	∈	PROPN
cana-3960	120	1	[	[	X
cana-3960	120	2	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	120	3	,	,	PUNCT
cana-3960	120	4	𝑡𝑘)𝕋	𝑡𝑘)𝕋	NOUN
cana-3960	120	5	)	)	PUNCT
cana-3960	120	6	.	.	PUNCT
cana-3960	121	1	ii	ii	PROPN
cana-3960	121	2	.	.	PROPN
cana-3960	121	3	suppose	suppose	VERB
cana-3960	121	4	that	that	SCONJ
cana-3960	121	5	(	(	PUNCT
cana-3960	121	6	𝐼𝑛⊗𝑅𝑗	𝐼𝑛⊗𝑅𝑗	NOUN
cana-3960	121	7	)	)	PUNCT
cana-3960	121	8	≠	≠	PROPN
cana-3960	121	9	−1	−1	NOUN
cana-3960	121	10	,	,	PUNCT
cana-3960	121	11	𝑗	𝑗	NOUN
cana-3960	121	12	=	=	SYM
cana-3960	121	13	1,2	1,2	NUM
cana-3960	121	14	,	,	PUNCT
cana-3960	121	15	…	…	PUNCT
cana-3960	121	16	,	,	PUNCT
cana-3960	121	17	𝑘.	𝑘.	VERB
cana-3960	121	18	if	if	SCONJ
cana-3960	121	19	impulsive	impulsive	ADJ
cana-3960	121	20	system	system	NOUN
cana-3960	121	21	(	(	PUNCT
cana-3960	121	22	2.2	2.2	NUM
cana-3960	121	23	)	)	PUNCT
cana-3960	121	24	is	be	AUX
cana-3960	121	25	controllable	controllable	ADJ
cana-3960	121	26	on	on	ADP
cana-3960	121	27	[	[	X
cana-3960	121	28	𝑡0	𝑡0	NOUN
cana-3960	121	29	,	,	PUNCT
cana-3960	121	30	𝑡1]𝕋(𝑡𝑓	𝑡1]𝕋(𝑡𝑓	PROPN
cana-3960	121	31	∈	∈	PROPN
cana-3960	121	32	[	[	X
cana-3960	121	33	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	121	34	,	,	PUNCT
cana-3960	121	35	𝑡𝑘)𝕋	𝑡𝑘)𝕋	NOUN
cana-3960	121	36	)	)	PUNCT
cana-3960	121	37	,	,	PUNCT
cana-3960	121	38	then	then	ADV
cana-3960	121	39	𝑟𝑎𝑛𝑘	𝑟𝑎𝑛𝑘	PROPN
cana-3960	121	40	{	{	PUNCT
cana-3960	121	41	𝐻1	𝐻1	PROPN
cana-3960	121	42	,	,	PUNCT
cana-3960	121	43	…	…	PUNCT
cana-3960	121	44	,	,	PUNCT
cana-3960	121	45	𝐻𝑙	𝐻𝑙	ADJ
cana-3960	121	46	}	}	PUNCT
cana-3960	121	47	=	=	NOUN
cana-3960	121	48	𝑛2	𝑛2	NOUN
cana-3960	121	49	.	.	PUNCT
cana-3960	122	1	proof	proof	NOUN
cana-3960	122	2	:	:	PUNCT
cana-3960	122	3	(	(	PUNCT
cana-3960	122	4	i	i	NOUN
cana-3960	122	5	)	)	PUNCT
cana-3960	122	6	.	.	PUNCT
cana-3960	123	1	let	let	VERB
cana-3960	123	2	k∈	k∈	X
cana-3960	123	3	{	{	PUNCT
cana-3960	123	4	1,2	1,2	NUM
cana-3960	123	5	,	,	PUNCT
cana-3960	123	6	…	…	PUNCT
cana-3960	123	7	,	,	PUNCT
cana-3960	123	8	𝑙	𝑙	X
cana-3960	123	9	}	}	PUNCT
cana-3960	123	10	such	such	ADJ
cana-3960	123	11	that	that	SCONJ
cana-3960	123	12	the	the	DET
cana-3960	123	13	𝑟𝑎𝑛𝑘	𝑟𝑎𝑛𝑘	PROPN
cana-3960	123	14	{	{	PUNCT
cana-3960	123	15	𝐻𝑘(𝑡𝑘−1	𝐻𝑘(𝑡𝑘−1	NOUN
cana-3960	123	16	,	,	PUNCT
cana-3960	123	17	𝑡𝑘	𝑡𝑘	ADV
cana-3960	123	18	,	,	PUNCT
cana-3960	123	19	𝑡𝑓	𝑡𝑓	VERB
cana-3960	123	20	)	)	PUNCT
cana-3960	123	21	}	}	PUNCT
cana-3960	124	1	=	=	SYM
cana-3960	124	2	𝑛2	𝑛2	NOUN
cana-3960	124	3	i.e.	i.e.	ADV
cana-3960	124	4	,	,	PUNCT
cana-3960	124	5	the	the	DET
cana-3960	124	6	matrix	matrix	NOUN
cana-3960	124	7	𝐻𝑘(𝑡𝑘−1	𝐻𝑘(𝑡𝑘−1	NOUN
cana-3960	124	8	,	,	PUNCT
cana-3960	124	9	𝑡𝑘	𝑡𝑘	ADV
cana-3960	124	10	,	,	PUNCT
cana-3960	124	11	𝑡𝑓	𝑡𝑓	X
cana-3960	124	12	)	)	PUNCT
cana-3960	124	13	is	be	AUX
cana-3960	124	14	invertible	invertible	ADJ
cana-3960	124	15	then	then	ADV
cana-3960	124	16	for	for	ADP
cana-3960	124	17	a	a	DET
cana-3960	124	18	given	give	VERB
cana-3960	124	19	𝑧0	𝑧0	PROPN
cana-3960	124	20	∈	∈	PROPN
cana-3960	125	1	ℝ𝑛	ℝ𝑛	PROPN
cana-3960	125	2	2	2	NUM
cana-3960	125	3	,	,	PUNCT
cana-3960	125	4	we	we	PRON
cana-3960	125	5	choose	choose	VERB
cana-3960	125	6	a	a	DET
cana-3960	125	7	control	control	NOUN
cana-3960	125	8	function	function	NOUN
cana-3960	125	9	given	give	VERB
cana-3960	125	10	as	as	ADP
cana-3960	125	11	communications	communication	NOUN
cana-3960	125	12	on	on	ADP
cana-3960	125	13	applied	apply	VERB
cana-3960	125	14	nonlinear	nonlinear	ADJ
cana-3960	125	15	analysis	analysis	NOUN
cana-3960	125	16	issn	issn	NOUN
cana-3960	125	17	:	:	PUNCT
cana-3960	125	18	1074	1074	NUM
cana-3960	125	19	-	-	PUNCT
cana-3960	125	20	133x	133x	NUM
cana-3960	125	21	vol	vol	NOUN
cana-3960	125	22	32	32	NUM
cana-3960	125	23	no	no	NOUN
cana-3960	125	24	.	.	PUNCT
cana-3960	126	1	9s	9s	NUM
cana-3960	126	2	(	(	PUNCT
cana-3960	126	3	2025	2025	NUM
cana-3960	126	4	)	)	PUNCT
cana-3960	126	5	498	498	NUM
cana-3960	126	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3960	126	7	�	�	PROPN
cana-3960	126	8	̂	̂	PROPN
cana-3960	126	9	�	�	NOUN
cana-3960	126	10	(𝑡	(𝑡	NOUN
cana-3960	126	11	)	)	PUNCT
cana-3960	126	12	=	=	SYM
cana-3960	126	13	{	{	PUNCT
cana-3960	126	14	−𝐴1	−𝐴1	PROPN
cana-3960	126	15	𝑇(𝑡)𝜓𝐺1(𝑡	𝑇(𝑡)𝜓𝐺1(𝑡	PROPN
cana-3960	126	16	,	,	PUNCT
cana-3960	126	17	𝑡𝑓)𝐻1	𝑡𝑓)𝐻1	PROPN
cana-3960	126	18	−1𝜓𝐺1	−1𝜓𝐺1	VERB
cana-3960	126	19	𝑇	𝑇	PROPN
cana-3960	126	20	(	(	PUNCT
cana-3960	126	21	𝑡0	𝑡0	PROPN
cana-3960	126	22	,	,	PUNCT
cana-3960	126	23	𝑡𝑓)𝑧0	𝑡𝑓)𝑧0	NOUN
cana-3960	126	24	,	,	PUNCT
cana-3960	126	25	𝑓𝑜𝑟	𝑓𝑜𝑟	X
cana-3960	126	26	𝑡	𝑡	PROPN
cana-3960	126	27	∈	∈	PROPN
cana-3960	127	1	[	[	X
cana-3960	127	2	𝑡0	𝑡0	NOUN
cana-3960	127	3	,	,	PUNCT
cana-3960	127	4	𝑡1]𝕋	𝑡1]𝕋	PROPN
cana-3960	127	5	,	,	PUNCT
cana-3960	127	6	2	2	NUM
cana-3960	127	7	≤	≤	NOUN
cana-3960	127	8	𝑘	𝑘	DET
cana-3960	127	9	≤	≤	NOUN
cana-3960	127	10	𝑙	𝑙	PRON
cana-3960	127	11	−	−	PROPN
cana-3960	127	12	1	1	NUM
cana-3960	127	13	,	,	PUNCT
cana-3960	127	14	−𝐴𝑘	−𝐴𝑘	ADP
cana-3960	127	15	𝑇(𝑡)𝜓𝐺𝑘(𝑡	𝑇(𝑡)𝜓𝐺𝑘(𝑡	NOUN
cana-3960	127	16	,	,	PUNCT
cana-3960	127	17	𝑡𝑓)𝐻𝑘	𝑡𝑓)𝐻𝑘	NOUN
cana-3960	127	18	−1𝜓𝐺𝑘	−1𝜓𝐺𝑘	PROPN
cana-3960	127	19	𝑇	𝑇	PROPN
cana-3960	127	20	(	(	PUNCT
cana-3960	127	21	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	127	22	,	,	PUNCT
cana-3960	127	23	𝑡𝑓	𝑡𝑓	NOUN
cana-3960	127	24	)	)	PUNCT
cana-3960	127	25	∏	∏	PROPN
cana-3960	128	1	[	[	X
cana-3960	128	2	𝐼𝑛⊗𝑅𝑗	𝐼𝑛⊗𝑅𝑗	X
cana-3960	128	3	]	]	X
cana-3960	128	4	1	1	NUM
cana-3960	128	5	𝑗=𝑘−1	𝑗=𝑘−1	PROPN
cana-3960	128	6	∏	∏	PROPN
cana-3960	128	7	𝜓𝐺𝑘	𝜓𝐺𝑘	PROPN
cana-3960	128	8	𝑇	𝑇	PROPN
cana-3960	128	9	(	(	PUNCT
cana-3960	128	10	𝑡𝑗−1	𝑡𝑗−1	PROPN
cana-3960	128	11	,	,	PUNCT
cana-3960	128	12	𝑡𝑗	𝑡𝑗	PROPN
cana-3960	128	13	)	)	PUNCT
cana-3960	128	14	1	1	NUM
cana-3960	128	15	𝑗=𝑘−1	𝑗=𝑘−1	PROPN
cana-3960	128	16	𝑧0	𝑧0	PROPN
cana-3960	128	17	,	,	PUNCT
cana-3960	128	18	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-3960	128	19	𝑡	𝑡	PROPN
cana-3960	128	20	∈	∈	PROPN
cana-3960	129	1	[	[	X
cana-3960	129	2	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	129	3	,	,	PUNCT
cana-3960	129	4	𝑡𝑘)𝕋	𝑡𝑘)𝕋	NOUN
cana-3960	129	5	0	0	NUM
cana-3960	129	6	𝑖𝑓	𝑖𝑓	NUM
cana-3960	129	7	𝑡	𝑡	PROPN
cana-3960	129	8	∈	∈	PROPN
cana-3960	130	1	[	[	X
cana-3960	130	2	𝑡0	𝑡0	NOUN
cana-3960	130	3	,	,	PUNCT
cana-3960	130	4	𝑡𝑓]𝕋\	𝑡𝑓]𝕋\	PUNCT
cana-3960	131	1	[	[	X
cana-3960	131	2	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	131	3	,	,	PUNCT
cana-3960	131	4	𝑡𝑘)𝕋	𝑡𝑘)𝕋	NOUN
cana-3960	131	5	,	,	PUNCT
cana-3960	131	6	−𝐴𝑙	−𝐴𝑙	PROPN
cana-3960	131	7	𝑇(𝑡)𝜓𝐺𝑙(𝑡	𝑇(𝑡)𝜓𝐺𝑙(𝑡	PROPN
cana-3960	131	8	,	,	PUNCT
cana-3960	131	9	𝑡𝑓)𝐻𝑙	𝑡𝑓)𝐻𝑙	PROPN
cana-3960	131	10	−1𝜓𝐺𝑙	−1𝜓𝐺𝑙	PROPN
cana-3960	131	11	𝑇	𝑇	PROPN
cana-3960	131	12	(	(	PUNCT
cana-3960	131	13	𝑡𝑙−1	𝑡𝑙−1	X
cana-3960	131	14	,	,	PUNCT
cana-3960	131	15	𝑡𝑓	𝑡𝑓	NOUN
cana-3960	131	16	)	)	PUNCT
cana-3960	131	17	∏	∏	PROPN
cana-3960	132	1	[	[	X
cana-3960	132	2	𝐼𝑛⊗𝑅𝑗	𝐼𝑛⊗𝑅𝑗	X
cana-3960	132	3	]	]	X
cana-3960	132	4	1	1	NUM
cana-3960	132	5	𝑗=𝑙−1	𝑗=𝑙−1	PROPN
cana-3960	132	6	∏	∏	PROPN
cana-3960	133	1	𝜓𝐺𝑗	𝜓𝐺𝑗	NOUN
cana-3960	133	2	𝑇	𝑇	PROPN
cana-3960	133	3	(	(	PUNCT
cana-3960	133	4	𝑡𝑗−1	𝑡𝑗−1	PROPN
cana-3960	133	5	,	,	PUNCT
cana-3960	133	6	𝑡𝑗	𝑡𝑗	PROPN
cana-3960	133	7	)	)	PUNCT
cana-3960	133	8	1	1	NUM
cana-3960	133	9	𝑗=𝑙−1	𝑗=𝑙−1	PROPN
cana-3960	133	10	𝑧0	𝑧0	NOUN
cana-3960	133	11	,	,	PUNCT
cana-3960	133	12	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-3960	133	13	𝑡	𝑡	PROPN
cana-3960	133	14	∈	∈	PROPN
cana-3960	134	1	[	[	X
cana-3960	134	2	𝑡𝑙−1	𝑡𝑙−1	X
cana-3960	134	3	,	,	PUNCT
cana-3960	134	4	𝑡𝑙)𝕋	𝑡𝑙)𝕋	PRON
cana-3960	134	5	0	0	NUM
cana-3960	134	6	𝑖𝑓	𝑖𝑓	NUM
cana-3960	134	7	𝑡	𝑡	PROPN
cana-3960	134	8	∈	∈	PROPN
cana-3960	135	1	[	[	X
cana-3960	135	2	𝑡0	𝑡0	NOUN
cana-3960	135	3	,	,	PUNCT
cana-3960	135	4	𝑡𝑓]𝕋\	𝑡𝑓]𝕋\	PUNCT
cana-3960	136	1	[	[	X
cana-3960	136	2	𝑡𝑙−1	𝑡𝑙−1	X
cana-3960	136	3	,	,	PUNCT
cana-3960	136	4	𝑡𝑙)𝕋	𝑡𝑙)𝕋	ADP
cana-3960	136	5	,	,	PUNCT
cana-3960	136	6	(	(	PUNCT
cana-3960	136	7	3.5	3.5	NUM
cana-3960	136	8	)	)	PUNCT
cana-3960	136	9	obviously	obviously	ADV
cana-3960	136	10	,	,	PUNCT
cana-3960	136	11	the	the	DET
cana-3960	136	12	control	control	NOUN
cana-3960	136	13	function	function	PROPN
cana-3960	136	14	�	�	PROPN
cana-3960	136	15	̂	̂	NOUN
cana-3960	136	16	�	�	NOUN
cana-3960	136	17	(𝑡	(𝑡	NOUN
cana-3960	136	18	)	)	PUNCT
cana-3960	136	19	is	be	AUX
cana-3960	136	20	a	a	DET
cana-3960	136	21	piecewise	piecewise	NOUN
cana-3960	136	22	rd	rd	NOUN
cana-3960	136	23	-	-	NOUN
cana-3960	136	24	continuous	continuous	ADJ
cana-3960	136	25	on	on	ADP
cana-3960	136	26	[	[	X
cana-3960	136	27	𝑡0	𝑡0	NOUN
cana-3960	136	28	,	,	PUNCT
cana-3960	136	29	𝑡1]𝕋.	𝑡1]𝕋.	PROPN
cana-3960	136	30	by	by	ADP
cana-3960	136	31	lemma	lemma	PROPN
cana-3960	136	32	3.1	3.1	NUM
cana-3960	136	33	.	.	PUNCT
cana-3960	137	1	we	we	PRON
cana-3960	137	2	obtain	obtain	VERB
cana-3960	137	3	𝑧(𝑡𝑓	𝑧(𝑡𝑓	NOUN
cana-3960	137	4	)	)	PUNCT
cana-3960	138	1	=	=	SYM
cana-3960	138	2	𝜓𝐺1	𝜓𝐺1	NOUN
cana-3960	138	3	𝑇	𝑇	PROPN
cana-3960	138	4	(	(	PUNCT
cana-3960	138	5	𝑡0	𝑡0	PROPN
cana-3960	138	6	,	,	PUNCT
cana-3960	138	7	𝑡𝑓)𝑧0	𝑡𝑓)𝑧0	NOUN
cana-3960	138	8	−	−	PROPN
cana-3960	138	9	∫	∫	PROPN
cana-3960	138	10	𝜓𝐺1	𝜓𝐺1	PROPN
cana-3960	138	11	𝑇	𝑇	PROPN
cana-3960	138	12	(	(	PUNCT
cana-3960	138	13	𝜏	𝜏	NOUN
cana-3960	138	14	,	,	PUNCT
cana-3960	138	15	𝑡𝑓	𝑡𝑓	X
cana-3960	138	16	)	)	PUNCT
cana-3960	138	17	𝑡𝑓	𝑡𝑓	VERB
cana-3960	138	18	𝑡0	𝑡0	PROPN
cana-3960	138	19	𝐴1(𝜏)𝐴1	𝐴1(𝜏)𝐴1	PROPN
cana-3960	138	20	𝑇(𝜏)𝜓𝐺1(𝜏	𝑇(𝜏)𝜓𝐺1(𝜏	PROPN
cana-3960	138	21	,	,	PUNCT
cana-3960	138	22	𝑡𝑓)𝐻1	𝑡𝑓)𝐻1	PROPN
cana-3960	138	23	−1𝜓𝐺1	−1𝜓𝐺1	VERB
cana-3960	138	24	𝑇	𝑇	PROPN
cana-3960	138	25	(	(	PUNCT
cana-3960	138	26	𝑡0	𝑡0	PROPN
cana-3960	138	27	,	,	PUNCT
cana-3960	138	28	𝑡𝑓)𝑧0∆𝜏	𝑡𝑓)𝑧0∆𝜏	ADJ
cana-3960	138	29	,	,	PUNCT
cana-3960	138	30	by	by	ADP
cana-3960	138	31	using	use	VERB
cana-3960	138	32	equation	equation	NOUN
cana-3960	138	33	(	(	PUNCT
cana-3960	138	34	3.2	3.2	NUM
cana-3960	138	35	)	)	PUNCT
cana-3960	138	36	,	,	PUNCT
cana-3960	138	37	we	we	PRON
cana-3960	138	38	have	have	VERB
cana-3960	138	39	𝑧(𝑡𝑓	𝑧(𝑡𝑓	NUM
cana-3960	138	40	)	)	PUNCT
cana-3960	139	1	=	=	SYM
cana-3960	139	2	𝜓𝐺1	𝜓𝐺1	NOUN
cana-3960	139	3	𝑇	𝑇	PROPN
cana-3960	139	4	(	(	PUNCT
cana-3960	139	5	𝑡0	𝑡0	PROPN
cana-3960	139	6	,	,	PUNCT
cana-3960	139	7	𝑡𝑓)𝑧0	𝑡𝑓)𝑧0	NOUN
cana-3960	139	8	−𝐻1𝐻1	−𝐻1𝐻1	NUM
cana-3960	139	9	−1𝜓𝐺1	−1𝜓𝐺1	VERB
cana-3960	139	10	𝑇	𝑇	PROPN
cana-3960	139	11	(	(	PUNCT
cana-3960	139	12	𝑡0	𝑡0	PROPN
cana-3960	139	13	,	,	PUNCT
cana-3960	139	14	𝑡𝑓)𝑧0	𝑡𝑓)𝑧0	NOUN
cana-3960	139	15	=	=	SYM
cana-3960	139	16	0	0	NUM
cana-3960	139	17	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
cana-3960	139	18	𝑡	𝑡	PROPN
cana-3960	139	19	∈	∈	PROPN
cana-3960	140	1	[	[	X
cana-3960	140	2	𝑡0	𝑡0	NOUN
cana-3960	140	3	,	,	PUNCT
cana-3960	140	4	𝑡1]𝕋	𝑡1]𝕋	PROPN
cana-3960	140	5	,	,	PUNCT
cana-3960	140	6	then	then	ADV
cana-3960	140	7	the	the	DET
cana-3960	140	8	system	system	NOUN
cana-3960	140	9	(	(	PUNCT
cana-3960	140	10	2.2	2.2	NUM
cana-3960	140	11	)	)	PUNCT
cana-3960	140	12	is	be	AUX
cana-3960	140	13	a	a	DET
cana-3960	140	14	controllable	controllable	ADJ
cana-3960	140	15	on	on	ADP
cana-3960	140	16	[	[	X
cana-3960	140	17	𝑡0	𝑡0	NOUN
cana-3960	140	18	,	,	PUNCT
cana-3960	140	19	𝑡1]𝕋.	𝑡1]𝕋.	PROPN
cana-3960	140	20	next	next	ADJ
cana-3960	140	21	,	,	PUNCT
cana-3960	140	22	for	for	ADP
cana-3960	140	23	2	2	NUM
cana-3960	140	24	≤	≤	NOUN
cana-3960	140	25	𝑘	𝑘	DET
cana-3960	140	26	≤	≤	NOUN
cana-3960	140	27	𝑙	𝑙	PRON
cana-3960	140	28	−	−	PROPN
cana-3960	140	29	1	1	NUM
cana-3960	140	30	,	,	PUNCT
cana-3960	140	31	and	and	CCONJ
cana-3960	140	32	𝑡	𝑡	PROPN
cana-3960	140	33	∈	∈	PROPN
cana-3960	140	34	[	[	X
cana-3960	140	35	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	140	36	,	,	PUNCT
cana-3960	140	37	𝑡𝑘)𝕋	𝑡𝑘)𝕋	NOUN
cana-3960	140	38	,	,	PUNCT
cana-3960	140	39	𝑧(𝑡𝑓	𝑧(𝑡𝑓	NUM
cana-3960	140	40	)	)	PUNCT
cana-3960	140	41	=	=	SYM
cana-3960	140	42	𝜓𝐺𝑘	𝜓𝐺𝑘	PROPN
cana-3960	140	43	𝑇	𝑇	PROPN
cana-3960	140	44	(	(	PUNCT
cana-3960	140	45	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	140	46	,	,	PUNCT
cana-3960	140	47	𝑡𝑓	𝑡𝑓	NOUN
cana-3960	140	48	)	)	PUNCT
cana-3960	140	49	∏	∏	PROPN
cana-3960	141	1	[	[	X
cana-3960	141	2	𝐼𝑛⊗𝑅𝑗	𝐼𝑛⊗𝑅𝑗	X
cana-3960	141	3	]	]	X
cana-3960	141	4	1	1	NUM
cana-3960	141	5	𝑗=𝑘−1	𝑗=𝑘−1	PROPN
cana-3960	141	6	∏	∏	PROPN
cana-3960	141	7	𝜓𝐺𝑗	𝜓𝐺𝑗	NOUN
cana-3960	141	8	𝑇	𝑇	PROPN
cana-3960	141	9	(	(	PUNCT
cana-3960	141	10	𝑡𝑗−1	𝑡𝑗−1	PROPN
cana-3960	141	11	,	,	PUNCT
cana-3960	141	12	𝑡𝑗	𝑡𝑗	PROPN
cana-3960	141	13	)	)	PUNCT
cana-3960	141	14	1	1	NUM
cana-3960	141	15	𝑗=𝑘−1	𝑗=𝑘−1	PROPN
cana-3960	141	16	𝑧0	𝑧0	PROPN
cana-3960	141	17	−	−	PROPN
cana-3960	141	18	∫	∫	PROPN
cana-3960	141	19	𝜓𝐺𝑘	𝜓𝐺𝑘	PROPN
cana-3960	141	20	𝑇	𝑇	PROPN
cana-3960	141	21	(	(	PUNCT
cana-3960	141	22	𝜏	𝜏	NOUN
cana-3960	141	23	,	,	PUNCT
cana-3960	141	24	𝑡𝑓	𝑡𝑓	NOUN
cana-3960	141	25	)	)	PUNCT
cana-3960	141	26	𝑡	𝑡	PROPN
cana-3960	141	27	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	141	28	𝐴𝑘(𝜏)𝐴𝑘	𝐴𝑘(𝜏)𝐴𝑘	ADJ
cana-3960	141	29	𝑇(𝜏)𝜓𝐺𝑘(𝜏	𝑇(𝜏)𝜓𝐺𝑘(𝜏	NOUN
cana-3960	141	30	,	,	PUNCT
cana-3960	141	31	𝑡𝑓)𝐻𝑘	𝑡𝑓)𝐻𝑘	NOUN
cana-3960	141	32	−1𝜓𝐺𝑘	−1𝜓𝐺𝑘	PROPN
cana-3960	141	33	𝑇	𝑇	PROPN
cana-3960	141	34	(	(	PUNCT
cana-3960	141	35	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	141	36	,	,	PUNCT
cana-3960	141	37	𝑡𝑓	𝑡𝑓	NOUN
cana-3960	141	38	)	)	PUNCT
cana-3960	141	39	∏	∏	PROPN
cana-3960	142	1	[	[	X
cana-3960	142	2	𝐼𝑛	𝐼𝑛	PROPN
cana-3960	142	3	1	1	NUM
cana-3960	142	4	𝑗=𝑘−1	𝑗=𝑘−1	PROPN
cana-3960	142	5	⊗𝑅𝑗	⊗𝑅𝑗	PROPN
cana-3960	142	6	]	]	X
cana-3960	142	7	∏	∏	PROPN
cana-3960	142	8	𝜓𝐺𝑗	𝜓𝐺𝑗	PROPN
cana-3960	142	9	𝑇	𝑇	PROPN
cana-3960	142	10	(	(	PUNCT
cana-3960	142	11	𝑡𝑗−1	𝑡𝑗−1	PROPN
cana-3960	142	12	,	,	PUNCT
cana-3960	142	13	𝑡𝑗	𝑡𝑗	PROPN
cana-3960	142	14	)	)	PUNCT
cana-3960	142	15	1	1	NUM
cana-3960	142	16	𝑗=𝑘−1	𝑗=𝑘−1	PROPN
cana-3960	142	17	𝑧0∆𝜏	𝑧0∆𝜏	NOUN
cana-3960	142	18	,	,	PUNCT
cana-3960	142	19	it	it	PRON
cana-3960	142	20	follows	follow	VERB
cana-3960	142	21	that	that	SCONJ
cana-3960	142	22	from	from	ADP
cana-3960	142	23	equation	equation	NOUN
cana-3960	142	24	(	(	PUNCT
cana-3960	142	25	3.3	3.3	NUM
cana-3960	142	26	)	)	PUNCT
cana-3960	142	27	𝑧(𝑡𝑓	𝑧(𝑡𝑓	NUM
cana-3960	142	28	)	)	PUNCT
cana-3960	142	29	=	=	PUNCT
cana-3960	142	30	0	0	NUM
cana-3960	143	1	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
cana-3960	143	2	𝑡	𝑡	PROPN
cana-3960	143	3	∈	∈	PROPN
cana-3960	143	4	[	[	X
cana-3960	143	5	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	143	6	,	,	PUNCT
cana-3960	143	7	𝑡𝑘)𝕋	𝑡𝑘)𝕋	NOUN
cana-3960	143	8	,	,	PUNCT
cana-3960	143	9	and	and	CCONJ
cana-3960	143	10	similarly	similarly	ADV
cana-3960	143	11	,	,	PUNCT
cana-3960	143	12	we	we	PRON
cana-3960	143	13	have	have	VERB
cana-3960	143	14	𝑧(𝑡𝑓	𝑧(𝑡𝑓	NUM
cana-3960	143	15	)	)	PUNCT
cana-3960	143	16	=	=	PUNCT
cana-3960	143	17	0	0	NUM
cana-3960	144	1	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
cana-3960	144	2	𝑡	𝑡	PROPN
cana-3960	144	3	∈	∈	PROPN
cana-3960	144	4	[	[	X
cana-3960	144	5	𝑡𝑙−1	𝑡𝑙−1	X
cana-3960	144	6	,	,	PUNCT
cana-3960	144	7	𝑡𝑙)𝕋	𝑡𝑙)𝕋	ADP
cana-3960	144	8	,	,	PUNCT
cana-3960	144	9	thus	thus	ADV
cana-3960	144	10	,	,	PUNCT
cana-3960	144	11	the	the	DET
cana-3960	144	12	system	system	NOUN
cana-3960	144	13	(	(	PUNCT
cana-3960	144	14	2.2	2.2	NUM
cana-3960	144	15	)	)	PUNCT
cana-3960	144	16	is	be	AUX
cana-3960	144	17	a	a	DET
cana-3960	144	18	controllable	controllable	ADJ
cana-3960	144	19	on	on	ADP
cana-3960	144	20	[	[	X
cana-3960	144	21	𝑡0	𝑡0	NOUN
cana-3960	144	22	,	,	PUNCT
cana-3960	144	23	𝑡𝑓]𝕋.	𝑡𝑓]𝕋.	PROPN
cana-3960	144	24	so	so	ADV
cana-3960	144	25	(	(	PUNCT
cana-3960	144	26	i	i	NOUN
cana-3960	144	27	)	)	PUNCT
cana-3960	144	28	holds	hold	VERB
cana-3960	144	29	.	.	PUNCT
cana-3960	145	1	(	(	PUNCT
cana-3960	145	2	ii	ii	NOUN
cana-3960	145	3	)	)	PUNCT
cana-3960	145	4	.	.	PUNCT
cana-3960	146	1	suppose	suppose	VERB
cana-3960	146	2	that	that	SCONJ
cana-3960	146	3	(	(	PUNCT
cana-3960	146	4	2.2	2.2	NUM
cana-3960	146	5	)	)	PUNCT
cana-3960	146	6	is	be	AUX
cana-3960	146	7	controllable	controllable	ADJ
cana-3960	146	8	on	on	ADP
cana-3960	146	9	[	[	X
cana-3960	146	10	𝑡0	𝑡0	NOUN
cana-3960	146	11	,	,	PUNCT
cana-3960	146	12	𝑡𝑓]𝕋.	𝑡𝑓]𝕋.	PRON
cana-3960	146	13	we	we	PRON
cana-3960	146	14	have	have	VERB
cana-3960	146	15	to	to	PART
cana-3960	146	16	show	show	VERB
cana-3960	146	17	that	that	SCONJ
cana-3960	146	18	𝑟𝑎𝑛𝑘	𝑟𝑎𝑛𝑘	PROPN
cana-3960	146	19	{	{	PUNCT
cana-3960	146	20	𝐻1	𝐻1	PROPN
cana-3960	146	21	,	,	PUNCT
cana-3960	146	22	…	…	PUNCT
cana-3960	146	23	,	,	PUNCT
cana-3960	146	24	𝐻𝑙	𝐻𝑙	ADJ
cana-3960	146	25	}	}	PUNCT
cana-3960	146	26	=	=	SYM
cana-3960	146	27	𝑛2	𝑛2	NOUN
cana-3960	146	28	assume	assume	VERB
cana-3960	146	29	that	that	SCONJ
cana-3960	146	30	𝑟𝑎𝑛𝑘	𝑟𝑎𝑛𝑘	PROPN
cana-3960	146	31	{	{	PUNCT
cana-3960	146	32	𝐻1	𝐻1	PROPN
cana-3960	146	33	,	,	PUNCT
cana-3960	146	34	…	…	PUNCT
cana-3960	146	35	,	,	PUNCT
cana-3960	146	36	𝐻𝑙	𝐻𝑙	NOUN
cana-3960	146	37	}	}	PUNCT
cana-3960	146	38	<	<	X
cana-3960	146	39	𝑛	𝑛	PRON
cana-3960	146	40	2	2	NUM
cana-3960	146	41	communications	communication	NOUN
cana-3960	146	42	on	on	ADP
cana-3960	146	43	applied	apply	VERB
cana-3960	146	44	nonlinear	nonlinear	ADJ
cana-3960	146	45	analysis	analysis	NOUN
cana-3960	146	46	issn	issn	NOUN
cana-3960	146	47	:	:	PUNCT
cana-3960	146	48	1074	1074	NUM
cana-3960	146	49	-	-	PUNCT
cana-3960	146	50	133x	133x	NUM
cana-3960	146	51	vol	vol	NOUN
cana-3960	146	52	32	32	NUM
cana-3960	147	1	no	no	NOUN
cana-3960	147	2	.	.	PUNCT
cana-3960	148	1	9s	9s	NUM
cana-3960	148	2	(	(	PUNCT
cana-3960	148	3	2025	2025	NUM
cana-3960	148	4	)	)	PUNCT
cana-3960	148	5	499	499	NUM
cana-3960	149	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3960	149	2	then	then	ADV
cana-3960	149	3	,	,	PUNCT
cana-3960	149	4	there	there	PRON
cana-3960	149	5	exists	exist	VERB
cana-3960	149	6	a	a	DET
cana-3960	149	7	non	non	ADJ
cana-3960	149	8	-	-	ADJ
cana-3960	149	9	zero	zero	NUM
cana-3960	149	10	𝑧𝛼	𝑧𝛼	ADP
cana-3960	149	11	≠	≠	PROPN
cana-3960	149	12	0	0	NUM
cana-3960	149	13	∈	∈	PROPN
cana-3960	150	1	ℝ𝑛	ℝ𝑛	ADP
cana-3960	150	2	2	2	NUM
cana-3960	150	3	such	such	ADJ
cana-3960	150	4	that	that	DET
cana-3960	150	5	𝑧𝛼	𝑧𝛼	ADP
cana-3960	150	6	𝑇𝐻𝑗(𝑡𝑗−1	𝑇𝐻𝑗(𝑡𝑗−1	PROPN
cana-3960	150	7	,	,	PUNCT
cana-3960	150	8	𝑡𝑗	𝑡𝑗	PROPN
cana-3960	150	9	,	,	PUNCT
cana-3960	150	10	𝑡𝑓)𝑧𝛼	𝑡𝑓)𝑧𝛼	PROPN
cana-3960	150	11	=	=	SYM
cana-3960	150	12	0	0	NUM
cana-3960	150	13	,	,	PUNCT
cana-3960	150	14	𝑗	𝑗	NOUN
cana-3960	150	15	=	=	SYM
cana-3960	150	16	1,2	1,2	NUM
cana-3960	150	17	,	,	PUNCT
cana-3960	150	18	…	…	PUNCT
cana-3960	150	19	,	,	PUNCT
cana-3960	150	20	𝑙.	𝑙.	NOUN
cana-3960	150	21	for	for	ADP
cana-3960	150	22	j=1	j=1	PROPN
cana-3960	150	23	𝑧𝛼	𝑧𝛼	X
cana-3960	150	24	𝑇𝐻1𝑧𝛼	𝑇𝐻1𝑧𝛼	PROPN
cana-3960	150	25	=	=	SYM
cana-3960	150	26	∫	∫	PROPN
cana-3960	150	27	𝑧𝛼	𝑧𝛼	X
cana-3960	150	28	𝑇𝜓𝐺1	𝑇𝜓𝐺1	PRON
cana-3960	150	29	𝑇	𝑇	PROPN
cana-3960	150	30	(	(	PUNCT
cana-3960	150	31	𝜏	𝜏	NOUN
cana-3960	150	32	,	,	PUNCT
cana-3960	150	33	𝑡𝑓	𝑡𝑓	X
cana-3960	150	34	)	)	PUNCT
cana-3960	150	35	𝑡𝑓	𝑡𝑓	VERB
cana-3960	150	36	𝑡0	𝑡0	PROPN
cana-3960	150	37	𝐴1(𝜏)𝐴1	𝐴1(𝜏)𝐴1	PROPN
cana-3960	150	38	𝑇(𝜏)𝜓𝐺1(𝜏	𝑇(𝜏)𝜓𝐺1(𝜏	PROPN
cana-3960	150	39	,	,	PUNCT
cana-3960	150	40	𝑡𝑓)𝑧𝛼∆𝜏	𝑡𝑓)𝑧𝛼∆𝜏	ADJ
cana-3960	150	41	,	,	PUNCT
cana-3960	150	42	as	as	SCONJ
cana-3960	150	43	𝑧𝛼	𝑧𝛼	X
cana-3960	150	44	𝑇𝜓𝐺1	𝑇𝜓𝐺1	PRON
cana-3960	150	45	𝑇	𝑇	PROPN
cana-3960	150	46	(	(	PUNCT
cana-3960	150	47	𝑡	𝑡	NOUN
cana-3960	150	48	,	,	PUNCT
cana-3960	150	49	𝑡𝑓)𝐴1(𝑡	𝑡𝑓)𝐴1(𝑡	NOUN
cana-3960	150	50	)	)	PUNCT
cana-3960	150	51	is	be	AUX
cana-3960	150	52	rd	rd	NOUN
cana-3960	150	53	-	-	ADJ
cana-3960	150	54	continuous	continuous	ADJ
cana-3960	150	55	functions	function	NOUN
cana-3960	151	1	so	so	SCONJ
cana-3960	151	2	‖𝑧𝛼	‖𝑧𝛼	PROPN
cana-3960	151	3	𝑇𝜓𝐺1	𝑇𝜓𝐺1	DET
cana-3960	151	4	𝑇	𝑇	PROPN
cana-3960	151	5	(	(	PUNCT
cana-3960	151	6	𝜏	𝜏	NOUN
cana-3960	151	7	,	,	PUNCT
cana-3960	151	8	𝑡𝑓)𝐴1(𝑡)‖	𝑡𝑓)𝐴1(𝑡)‖	ADJ
cana-3960	151	9	2	2	NUM
cana-3960	151	10	=	=	SYM
cana-3960	151	11	0	0	NUM
cana-3960	151	12	.	.	NOUN
cana-3960	151	13	which	which	PRON
cana-3960	151	14	according	accord	VERB
cana-3960	151	15	to	to	ADP
cana-3960	151	16	𝐴1	𝐴1	PROPN
cana-3960	151	17	𝑇(𝜏)𝜓𝐺1	𝑇(𝜏)𝜓𝐺1	VERB
cana-3960	151	18	𝑇	𝑇	PROPN
cana-3960	151	19	(	(	PUNCT
cana-3960	151	20	𝑡	𝑡	PROPN
cana-3960	151	21	,	,	PUNCT
cana-3960	151	22	𝑡𝑓)𝑧𝛼	𝑡𝑓)𝑧𝛼	X
cana-3960	151	23	=	=	SYM
cana-3960	151	24	0	0	NUM
cana-3960	151	25	,	,	PUNCT
cana-3960	151	26	𝑡	𝑡	PROPN
cana-3960	151	27	∈	∈	PROPN
cana-3960	152	1	[	[	X
cana-3960	152	2	𝑡0	𝑡0	NOUN
cana-3960	152	3	,	,	PUNCT
cana-3960	152	4	𝑡1]𝕋.	𝑡1]𝕋.	PROPN
cana-3960	152	5	(	(	PUNCT
cana-3960	152	6	3.6	3.6	NUM
cana-3960	152	7	)	)	PUNCT
cana-3960	152	8	for	for	ADP
cana-3960	152	9	𝑘	𝑘	NOUN
cana-3960	152	10	=	=	SYM
cana-3960	152	11	2,3	2,3	NUM
cana-3960	152	12	,	,	PUNCT
cana-3960	152	13	⋯	⋯	PROPN
cana-3960	152	14	,	,	PUNCT
cana-3960	152	15	𝑙	𝑙	PROPN
cana-3960	152	16	−	−	PROPN
cana-3960	152	17	1	1	NUM
cana-3960	152	18	.	.	PUNCT
cana-3960	153	1	𝑧𝛼	𝑧𝛼	X
cana-3960	153	2	𝑇𝐻𝑘𝑧𝛼	𝑇𝐻𝑘𝑧𝛼	PROPN
cana-3960	153	3	=	=	SYM
cana-3960	153	4	∫	∫	PROPN
cana-3960	154	1	𝑧𝛼	𝑧𝛼	ADP
cana-3960	154	2	𝑇𝜓𝐺𝑘	𝑇𝜓𝐺𝑘	PROPN
cana-3960	154	3	𝑇	𝑇	PROPN
cana-3960	154	4	(	(	PUNCT
cana-3960	154	5	𝜏	𝜏	NOUN
cana-3960	154	6	,	,	PUNCT
cana-3960	154	7	𝑡𝑓	𝑡𝑓	X
cana-3960	154	8	)	)	PUNCT
cana-3960	154	9	𝑡𝑓	𝑡𝑓	PROPN
cana-3960	154	10	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	154	11	𝐴𝑘(𝜏)𝐴𝑘	𝐴𝑘(𝜏)𝐴𝑘	ADJ
cana-3960	154	12	𝑇(𝜏)𝜓𝐺𝑘(𝜏	𝑇(𝜏)𝜓𝐺𝑘(𝜏	NOUN
cana-3960	154	13	,	,	PUNCT
cana-3960	154	14	𝑡𝑓)𝑧𝛼∆𝜏	𝑡𝑓)𝑧𝛼∆𝜏	NOUN
cana-3960	154	15	=	=	SYM
cana-3960	154	16	0	0	NUM
cana-3960	154	17	,	,	PUNCT
cana-3960	154	18	𝑧𝛼	𝑧𝛼	CCONJ
cana-3960	154	19	𝑇𝜓𝐺𝑘	𝑇𝜓𝐺𝑘	PROPN
cana-3960	154	20	𝑇	𝑇	PROPN
cana-3960	154	21	(	(	PUNCT
cana-3960	154	22	𝑡	𝑡	PROPN
cana-3960	154	23	,	,	PUNCT
cana-3960	154	24	𝑡𝑓)𝐴𝑘(𝑡)𝐴𝑘	𝑡𝑓)𝐴𝑘(𝑡)𝐴𝑘	PROPN
cana-3960	154	25	𝑇(𝑡)𝜓𝐺𝑘(𝑡	𝑇(𝑡)𝜓𝐺𝑘(𝑡	PROPN
cana-3960	154	26	,	,	PUNCT
cana-3960	154	27	𝑡𝑓)𝑧𝛼	𝑡𝑓)𝑧𝛼	SYM
cana-3960	154	28	=	=	SYM
cana-3960	154	29	‖𝑧𝛼	‖𝑧𝛼	PROPN
cana-3960	154	30	𝑇𝜓𝐺𝑘	𝑇𝜓𝐺𝑘	PROPN
cana-3960	154	31	𝑇	𝑇	PROPN
cana-3960	154	32	(	(	PUNCT
cana-3960	154	33	𝑡	𝑡	PROPN
cana-3960	154	34	,	,	PUNCT
cana-3960	154	35	𝑡𝑓)𝐴𝑘(𝑡)‖	𝑡𝑓)𝐴𝑘(𝑡)‖	ADJ
cana-3960	154	36	2	2	NUM
cana-3960	154	37	𝐴𝑘	𝐴𝑘	PROPN
cana-3960	154	38	𝑇(𝑡)𝜓𝐺𝑘	𝑇(𝑡)𝜓𝐺𝑘	PROPN
cana-3960	154	39	𝑇	𝑇	PROPN
cana-3960	154	40	(	(	PUNCT
cana-3960	154	41	𝑡	𝑡	PROPN
cana-3960	154	42	,	,	PUNCT
cana-3960	154	43	𝑡𝑓)𝑧𝛼	𝑡𝑓)𝑧𝛼	X
cana-3960	154	44	=	=	SYM
cana-3960	154	45	0	0	NUM
cana-3960	154	46	,	,	PUNCT
cana-3960	154	47	𝑡	𝑡	PROPN
cana-3960	154	48	∈	∈	PROPN
cana-3960	155	1	[	[	X
cana-3960	155	2	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	155	3	,	,	PUNCT
cana-3960	155	4	𝑡𝑘)𝕋	𝑡𝑘)𝕋	NOUN
cana-3960	155	5	,	,	PUNCT
cana-3960	155	6	(	(	PUNCT
cana-3960	155	7	3.7	3.7	NUM
cana-3960	155	8	)	)	PUNCT
cana-3960	155	9	similarly	similarly	ADV
cana-3960	155	10	,	,	PUNCT
cana-3960	155	11	𝐴𝑙	𝐴𝑙	PROPN
cana-3960	155	12	𝑇(𝑡)𝜓𝐺𝑙	𝑇(𝑡)𝜓𝐺𝑙	PROPN
cana-3960	155	13	𝑇	𝑇	PROPN
cana-3960	155	14	(	(	PUNCT
cana-3960	155	15	𝑡	𝑡	PROPN
cana-3960	155	16	,	,	PUNCT
cana-3960	155	17	𝑡𝑓)𝑧𝛼	𝑡𝑓)𝑧𝛼	X
cana-3960	155	18	=	=	SYM
cana-3960	155	19	0	0	NUM
cana-3960	155	20	,	,	PUNCT
cana-3960	155	21	𝑡	𝑡	PROPN
cana-3960	155	22	∈	∈	PROPN
cana-3960	156	1	[	[	X
cana-3960	156	2	𝑡𝑙−1	𝑡𝑙−1	X
cana-3960	156	3	,	,	PUNCT
cana-3960	156	4	𝑡𝑙)𝕋	𝑡𝑙)𝕋	ADP
cana-3960	156	5	,	,	PUNCT
cana-3960	156	6	(	(	PUNCT
cana-3960	156	7	3.8	3.8	NUM
cana-3960	156	8	)	)	PUNCT
cana-3960	156	9	however	however	ADV
cana-3960	156	10	,	,	PUNCT
cana-3960	156	11	the	the	DET
cana-3960	156	12	impulsive	impulsive	ADJ
cana-3960	156	13	system	system	NOUN
cana-3960	156	14	(	(	PUNCT
cana-3960	156	15	2.2	2.2	NUM
cana-3960	156	16	)	)	PUNCT
cana-3960	156	17	is	be	AUX
cana-3960	156	18	controllability	controllability	NOUN
cana-3960	156	19	on	on	ADP
cana-3960	156	20	[	[	X
cana-3960	156	21	𝑡0	𝑡0	NOUN
cana-3960	156	22	,	,	PUNCT
cana-3960	156	23	𝑡1]𝕋	𝑡1]𝕋	PROPN
cana-3960	156	24	,	,	PUNCT
cana-3960	156	25	and	and	CCONJ
cana-3960	156	26	so	so	ADV
cana-3960	156	27	choosing	choose	VERB
cana-3960	156	28	𝑧0	𝑧0	PROPN
cana-3960	156	29	=	=	SYM
cana-3960	157	1	𝑧𝛼	𝑧𝛼	NUM
cana-3960	157	2	,	,	PUNCT
cana-3960	157	3	there	there	PRON
cana-3960	157	4	exists	exist	VERB
cana-3960	157	5	a	a	DET
cana-3960	157	6	piecewise	piecewise	NOUN
cana-3960	157	7	rd	rd	NOUN
cana-3960	157	8	-	-	ADJ
cana-3960	157	9	continuous	continuous	ADJ
cana-3960	157	10	control	control	NOUN
cana-3960	157	11	function	function	NOUN
cana-3960	157	12	�	�	PROPN
cana-3960	157	13	̂	̂	NOUN
cana-3960	157	14	�	�	NOUN
cana-3960	157	15	(𝑡	(𝑡	NOUN
cana-3960	157	16	)	)	PUNCT
cana-3960	157	17	such	such	ADJ
cana-3960	157	18	that	that	DET
cana-3960	157	19	0	0	NUM
cana-3960	157	20	=	=	SYM
cana-3960	157	21	𝑧(𝑡𝑓	𝑧(𝑡𝑓	NUM
cana-3960	157	22	)	)	PUNCT
cana-3960	157	23	=	=	SYM
cana-3960	157	24	𝜓𝐺1	𝜓𝐺1	NOUN
cana-3960	157	25	𝑇	𝑇	PROPN
cana-3960	157	26	(	(	PUNCT
cana-3960	157	27	𝑡0	𝑡0	PROPN
cana-3960	157	28	,	,	PUNCT
cana-3960	157	29	𝑡𝑓)𝑧𝛼	𝑡𝑓)𝑧𝛼	PROPN
cana-3960	157	30	+	+	NUM
cana-3960	157	31	∫	∫	PROPN
cana-3960	157	32	𝜓𝐺1	𝜓𝐺1	PROPN
cana-3960	157	33	𝑇	𝑇	PROPN
cana-3960	157	34	(	(	PUNCT
cana-3960	157	35	𝜏	𝜏	NOUN
cana-3960	157	36	,	,	PUNCT
cana-3960	157	37	𝑡𝑓	𝑡𝑓	X
cana-3960	157	38	)	)	PUNCT
cana-3960	157	39	𝑡𝑓	𝑡𝑓	VERB
cana-3960	157	40	𝑡0	𝑡0	PROPN
cana-3960	157	41	𝐴1(𝜏)	𝐴1(𝜏)	PROPN
cana-3960	157	42	�	�	PROPN
cana-3960	157	43	̂	̂	NOUN
cana-3960	157	44	�	�	NOUN
cana-3960	157	45	(𝜏)∆𝜏	(𝜏)∆𝜏	PROPN
cana-3960	157	46	;	;	PUNCT
cana-3960	157	47	𝑙	𝑙	SYM
cana-3960	157	48	=	=	SYM
cana-3960	157	49	1	1	NUM
cana-3960	157	50	(	(	PUNCT
cana-3960	157	51	3.9	3.9	NUM
cana-3960	157	52	)	)	PUNCT
cana-3960	157	53	multiply	multiply	NOUN
cana-3960	157	54	through	through	ADP
cana-3960	157	55	by	by	ADP
cana-3960	157	56	𝑧𝛼	𝑧𝛼	X
cana-3960	157	57	𝑇	𝑇	PROPN
cana-3960	157	58	in	in	ADP
cana-3960	157	59	(	(	PUNCT
cana-3960	157	60	3.9	3.9	NUM
cana-3960	157	61	)	)	PUNCT
cana-3960	157	62	and	and	CCONJ
cana-3960	157	63	by	by	ADP
cana-3960	157	64	using	use	VERB
cana-3960	157	65	the	the	DET
cana-3960	157	66	transpose	transpose	NOUN
cana-3960	157	67	of	of	ADP
cana-3960	157	68	the	the	DET
cana-3960	157	69	equations	equation	NOUN
cana-3960	157	70	(	(	PUNCT
cana-3960	157	71	3.6	3.6	NUM
cana-3960	157	72	)	)	PUNCT
cana-3960	157	73	,	,	PUNCT
cana-3960	157	74	we	we	PRON
cana-3960	157	75	have	have	VERB
cana-3960	157	76	𝜓𝐺1	𝜓𝐺1	NOUN
cana-3960	157	77	𝑇	𝑇	PROPN
cana-3960	157	78	(	(	PUNCT
cana-3960	157	79	𝑡0	𝑡0	PROPN
cana-3960	157	80	,	,	PUNCT
cana-3960	157	81	𝑡𝑓)𝑧𝛼	𝑡𝑓)𝑧𝛼	PROPN
cana-3960	157	82	𝑇𝑧𝛼	𝑇𝑧𝛼	PROPN
cana-3960	157	83	=	=	SYM
cana-3960	157	84	0	0	PROPN
cana-3960	157	85	.	.	PUNCT
cana-3960	158	1	(	(	PUNCT
cana-3960	158	2	3.10	3.10	NUM
cana-3960	158	3	)	)	PUNCT
cana-3960	158	4	similarly	similarly	ADV
cana-3960	158	5	,	,	PUNCT
cana-3960	158	6	communications	communication	NOUN
cana-3960	158	7	on	on	ADP
cana-3960	158	8	applied	apply	VERB
cana-3960	158	9	nonlinear	nonlinear	ADJ
cana-3960	158	10	analysis	analysis	NOUN
cana-3960	158	11	issn	issn	NOUN
cana-3960	158	12	:	:	PUNCT
cana-3960	158	13	1074	1074	NUM
cana-3960	158	14	-	-	PUNCT
cana-3960	158	15	133x	133x	NUM
cana-3960	158	16	vol	vol	NOUN
cana-3960	158	17	32	32	NUM
cana-3960	158	18	no	no	NOUN
cana-3960	158	19	.	.	PUNCT
cana-3960	159	1	9s	9s	NUM
cana-3960	159	2	(	(	PUNCT
cana-3960	159	3	2025	2025	NUM
cana-3960	159	4	)	)	PUNCT
cana-3960	159	5	500	500	NUM
cana-3960	159	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3960	159	7	𝑧(𝑡𝑓	𝑧(𝑡𝑓	NUM
cana-3960	159	8	)	)	PUNCT
cana-3960	159	9	=	=	SYM
cana-3960	159	10	𝜓𝐺𝑙	𝜓𝐺𝑙	PROPN
cana-3960	159	11	𝑇	𝑇	PROPN
cana-3960	159	12	(	(	PUNCT
cana-3960	159	13	𝑡𝑙−1	𝑡𝑙−1	X
cana-3960	159	14	,	,	PUNCT
cana-3960	159	15	𝑡𝑓){∏	𝑡𝑓){∏	PROPN
cana-3960	159	16	[	[	X
cana-3960	159	17	𝐼𝑛⊗𝑅𝑗	𝐼𝑛⊗𝑅𝑗	X
cana-3960	159	18	]	]	X
cana-3960	159	19	1	1	NUM
cana-3960	159	20	𝑗=𝑙−1	𝑗=𝑙−1	PROPN
cana-3960	159	21	∏	∏	PROPN
cana-3960	159	22	𝜓𝐺𝑗	𝜓𝐺𝑗	NOUN
cana-3960	159	23	𝑇	𝑇	PROPN
cana-3960	159	24	(	(	PUNCT
cana-3960	159	25	𝑡𝑗−1	𝑡𝑗−1	PROPN
cana-3960	159	26	,	,	PUNCT
cana-3960	159	27	𝑡𝑗	𝑡𝑗	PROPN
cana-3960	159	28	)	)	PUNCT
cana-3960	159	29	1	1	NUM
cana-3960	159	30	𝑗=𝑙−1	𝑗=𝑙−1	PROPN
cana-3960	159	31	𝑧𝛼	𝑧𝛼	ADP
cana-3960	159	32	+	+	ADJ
cana-3960	159	33	∑	∑	PROPN
cana-3960	159	34	(	(	PUNCT
cana-3960	159	35	∏	∏	X
cana-3960	159	36	[	[	X
cana-3960	159	37	𝐼𝐼	𝐼𝐼	PROPN
cana-3960	159	38	1	1	NUM
cana-3960	159	39	𝐼=𝐼−1	𝐼=𝐼−1	PROPN
cana-3960	159	40	𝐼−2	𝐼−2	PROPN
cana-3960	159	41	𝐼=1	𝐼=1	PROPN
cana-3960	159	42	⊗𝑅𝑗	⊗𝑅𝑗	PROPN
cana-3960	159	43	]	]	X
cana-3960	159	44	∏	∏	PROPN
cana-3960	159	45	𝜓𝐺𝑖	𝜓𝐺𝑖	ADJ
cana-3960	159	46	𝑇	𝑇	PROPN
cana-3960	159	47	(	(	PUNCT
cana-3960	159	48	𝑡𝑖−1	𝑡𝑖−1	PROPN
cana-3960	159	49	,	,	PUNCT
cana-3960	159	50	𝑡𝑖	𝑡𝑖	NOUN
cana-3960	159	51	)	)	PUNCT
cana-3960	159	52	𝑗+1	𝑗+1	X
cana-3960	160	1	𝑖=𝑙−1	𝑖=𝑙−1	PUNCT
cana-3960	160	2	∫	∫	PROPN
cana-3960	161	1	𝜓𝐺𝑗	𝜓𝐺𝑗	NOUN
cana-3960	161	2	𝑇	𝑇	PROPN
cana-3960	161	3	(	(	PUNCT
cana-3960	161	4	𝜏	𝜏	PROPN
cana-3960	161	5	,	,	PUNCT
cana-3960	161	6	𝑡𝑗	𝑡𝑗	NOUN
cana-3960	161	7	)	)	PUNCT
cana-3960	161	8	𝑡𝑗	𝑡𝑗	PROPN
cana-3960	161	9	𝑡𝑗−1	𝑡𝑗−1	PROPN
cana-3960	161	10	𝐴𝑗(𝜏)𝑈(𝜏)∆𝜏	𝐴𝑗(𝜏)𝑈(𝜏)∆𝜏	PROPN
cana-3960	161	11	)	)	PUNCT
cana-3960	161	12	+	+	CCONJ
cana-3960	162	1	[	[	X
cana-3960	162	2	𝐼𝑛⊗𝑅𝑙−1	𝐼𝑛⊗𝑅𝑙−1	X
cana-3960	162	3	]	]	X
cana-3960	162	4	∫	∫	PROPN
cana-3960	162	5	𝜓𝐺𝑙−1	𝜓𝐺𝑙−1	X
cana-3960	162	6	𝑇	𝑇	PROPN
cana-3960	162	7	(	(	PUNCT
cana-3960	162	8	𝜏	𝜏	NOUN
cana-3960	162	9	,	,	PUNCT
cana-3960	162	10	𝑡𝑙−1	𝑡𝑙−1	ADJ
cana-3960	162	11	)	)	PUNCT
cana-3960	162	12	𝑡𝑙−1	𝑡𝑙−1	PROPN
cana-3960	162	13	𝑡𝑙−2	𝑡𝑙−2	PROPN
cana-3960	162	14	𝐴𝑙−1(𝜏)	𝐴𝑙−1(𝜏)	PROPN
cana-3960	162	15	�	�	PROPN
cana-3960	162	16	̂	̂	NOUN
cana-3960	162	17	�	�	NOUN
cana-3960	162	18	(𝜏)∆𝜏	(𝜏)∆𝜏	SYM
cana-3960	162	19	}	}	PUNCT
cana-3960	162	20	+	+	NUM
cana-3960	162	21	∫	∫	PROPN
cana-3960	162	22	𝜓𝐺𝑙	𝜓𝐺𝑙	PROPN
cana-3960	162	23	𝑇	𝑇	PROPN
cana-3960	162	24	(	(	PUNCT
cana-3960	162	25	𝜏	𝜏	NOUN
cana-3960	162	26	,	,	PUNCT
cana-3960	162	27	𝑡𝑓	𝑡𝑓	X
cana-3960	162	28	)	)	PUNCT
cana-3960	162	29	𝑡𝑓	𝑡𝑓	VERB
cana-3960	162	30	𝑡𝑙−1	𝑡𝑙−1	ADV
cana-3960	162	31	𝐴𝑙(𝜏)	𝐴𝑙(𝜏)	X
cana-3960	162	32	�	�	PROPN
cana-3960	162	33	̂	̂	VERB
cana-3960	162	34	�	�	NOUN
cana-3960	162	35	(𝜏)∆𝜏	(𝜏)∆𝜏	PROPN
cana-3960	162	36	,	,	PUNCT
cana-3960	162	37	𝑙	𝑙	X
cana-3960	162	38	=	=	SYM
cana-3960	162	39	2,3	2,3	NUM
cana-3960	162	40	,	,	PUNCT
cana-3960	162	41	…	…	PUNCT
cana-3960	162	42	(	(	PUNCT
cana-3960	162	43	3.11	3.11	NUM
cana-3960	162	44	)	)	PUNCT
cana-3960	162	45	multiply	multiply	ADV
cana-3960	162	46	by	by	ADP
cana-3960	162	47	𝜓𝐺1	𝜓𝐺1	NOUN
cana-3960	162	48	𝑇	𝑇	PROPN
cana-3960	162	49	(	(	PUNCT
cana-3960	162	50	𝑡1	𝑡1	NOUN
cana-3960	162	51	,	,	PUNCT
cana-3960	162	52	𝑡2)𝜓𝐺2	𝑡2)𝜓𝐺2	NOUN
cana-3960	162	53	𝑇	𝑇	PROPN
cana-3960	162	54	(	(	PUNCT
cana-3960	162	55	𝑡2	𝑡2	PROPN
cana-3960	162	56	,	,	PUNCT
cana-3960	162	57	𝑡3)	𝑡3)	NUM
cana-3960	162	58	…	…	SYM
cana-3960	162	59	𝜓𝐺𝑘	𝜓𝐺𝑘	ADJ
cana-3960	162	60	𝑇	𝑇	PROPN
cana-3960	162	61	(	(	PUNCT
cana-3960	162	62	𝑡𝑙−1	𝑡𝑙−1	X
cana-3960	162	63	,	,	PUNCT
cana-3960	162	64	𝑡𝑓	𝑡𝑓	ADV
cana-3960	162	65	)	)	PUNCT
cana-3960	162	66	and	and	CCONJ
cana-3960	162	67	𝑧𝛼	𝑧𝛼	ADP
cana-3960	162	68	𝑇	𝑇	PROPN
cana-3960	162	69	in	in	ADP
cana-3960	162	70	the	the	DET
cana-3960	162	71	equation	equation	NOUN
cana-3960	162	72	(	(	PUNCT
cana-3960	162	73	3.11	3.11	NUM
cana-3960	162	74	)	)	PUNCT
cana-3960	162	75	,	,	PUNCT
cana-3960	162	76	using	use	VERB
cana-3960	162	77	equations	equation	NOUN
cana-3960	162	78	(	(	PUNCT
cana-3960	162	79	3.7	3.7	NUM
cana-3960	162	80	)	)	PUNCT
cana-3960	162	81	,	,	PUNCT
cana-3960	162	82	and	and	CCONJ
cana-3960	162	83	(	(	PUNCT
cana-3960	162	84	3.8	3.8	NUM
cana-3960	162	85	)	)	PUNCT
cana-3960	162	86	,	,	PUNCT
cana-3960	162	87	we	we	PRON
cana-3960	162	88	have	have	VERB
cana-3960	162	89	∏[𝐼𝑛⊗𝑅𝑗]𝑧𝛼	∏[𝐼𝑛⊗𝑅𝑗]𝑧𝛼	ADJ
cana-3960	162	90	𝑇𝑧𝛼	𝑇𝑧𝛼	PROPN
cana-3960	162	91	=	=	SYM
cana-3960	162	92	0	0	NUM
cana-3960	162	93	,	,	PUNCT
cana-3960	162	94	𝑙	𝑙	X
cana-3960	162	95	𝑖=2	𝑖=2	PROPN
cana-3960	162	96	(	(	PUNCT
cana-3960	162	97	3.12	3.12	NUM
cana-3960	162	98	)	)	PUNCT
cana-3960	162	99	from	from	ADP
cana-3960	162	100	equations	equation	NOUN
cana-3960	162	101	(	(	PUNCT
cana-3960	162	102	3.10	3.10	NUM
cana-3960	162	103	)	)	PUNCT
cana-3960	162	104	and	and	CCONJ
cana-3960	162	105	(	(	PUNCT
cana-3960	162	106	3.12	3.12	NUM
cana-3960	162	107	)	)	PUNCT
cana-3960	162	108	,	,	PUNCT
cana-3960	162	109	according	accord	VERB
cana-3960	162	110	to	to	ADP
cana-3960	162	111	that	that	PRON
cana-3960	162	112	𝑧𝛼	𝑧𝛼	VERB
cana-3960	162	113	𝑇𝑧𝛼	𝑇𝑧𝛼	PROPN
cana-3960	162	114	=	=	SYM
cana-3960	162	115	0	0	PROPN
cana-3960	162	116	.	.	PUNCT
cana-3960	163	1	this	this	PRON
cana-3960	163	2	contradicts	contradict	VERB
cana-3960	163	3	𝑧𝛼	𝑧𝛼	X
cana-3960	163	4	≠	≠	PROPN
cana-3960	163	5	0	0	NUM
cana-3960	164	1	and	and	CCONJ
cana-3960	164	2	so	so	ADV
cana-3960	164	3	,	,	PUNCT
cana-3960	164	4	we	we	PRON
cana-3960	164	5	conclude	conclude	VERB
cana-3960	164	6	that	that	SCONJ
cana-3960	164	7	𝑟𝑎𝑛𝑘	𝑟𝑎𝑛𝑘	PROPN
cana-3960	164	8	{	{	PUNCT
cana-3960	164	9	𝐻1	𝐻1	PROPN
cana-3960	164	10	,	,	PUNCT
cana-3960	164	11	…	…	PUNCT
cana-3960	164	12	,	,	PUNCT
cana-3960	164	13	𝐻𝑙	𝐻𝑙	NOUN
cana-3960	164	14	}	}	PUNCT
cana-3960	164	15	=	=	SYM
cana-3960	164	16	𝑛	𝑛	DET
cana-3960	164	17	2	2	NUM
cana-3960	164	18	theorem	theorem	VERB
cana-3960	164	19	3.2	3.2	NUM
cana-3960	164	20	.	.	PUNCT
cana-3960	164	21	suppose	suppose	VERB
cana-3960	164	22	that	that	SCONJ
cana-3960	164	23	(	(	PUNCT
cana-3960	164	24	𝐼𝑛⊗𝑅𝑗	𝐼𝑛⊗𝑅𝑗	NOUN
cana-3960	164	25	)	)	PUNCT
cana-3960	164	26	≠	≠	PROPN
cana-3960	164	27	−1	−1	NOUN
cana-3960	164	28	,	,	PUNCT
cana-3960	164	29	𝑗	𝑗	NOUN
cana-3960	164	30	=	=	SYM
cana-3960	164	31	1,2	1,2	NUM
cana-3960	164	32	,	,	PUNCT
cana-3960	164	33	…	…	PUNCT
cana-3960	164	34	,	,	PUNCT
cana-3960	164	35	𝑘	𝑘	X
cana-3960	164	36	and	and	CCONJ
cana-3960	164	37	𝐺𝑘(𝑡	𝐺𝑘(𝑡	VERB
cana-3960	164	38	)	)	PUNCT
cana-3960	164	39	=	=	SYM
cana-3960	165	1	𝐺𝑘	𝐺𝑘	NOUN
cana-3960	165	2	,	,	PUNCT
cana-3960	165	3	𝐴𝑘(𝑡	𝐴𝑘(𝑡	VERB
cana-3960	165	4	)	)	PUNCT
cana-3960	165	5	=	=	PUNCT
cana-3960	166	1	𝐴𝑘	𝐴𝑘	PROPN
cana-3960	166	2	are	be	AUX
cana-3960	166	3	constant	constant	ADJ
cana-3960	166	4	matrices	matrix	NOUN
cana-3960	166	5	.	.	PUNCT
cana-3960	167	1	then	then	ADV
cana-3960	167	2	,	,	PUNCT
cana-3960	167	3	the	the	DET
cana-3960	167	4	system	system	NOUN
cana-3960	167	5	(	(	PUNCT
cana-3960	167	6	2.2	2.2	NUM
cana-3960	167	7	)	)	PUNCT
cana-3960	167	8	is	be	AUX
cana-3960	167	9	a	a	DET
cana-3960	167	10	controllable	controllable	ADJ
cana-3960	167	11	on	on	ADP
cana-3960	167	12	[	[	X
cana-3960	167	13	𝑡0	𝑡0	NOUN
cana-3960	167	14	,	,	PUNCT
cana-3960	167	15	𝑡𝑓]𝕋	𝑡𝑓]𝕋	PROPN
cana-3960	167	16	(	(	PUNCT
cana-3960	167	17	𝑡	𝑡	PROPN
cana-3960	167	18	∈	∈	PROPN
cana-3960	167	19	[	[	X
cana-3960	167	20	𝑡𝑙−1	𝑡𝑙−1	X
cana-3960	167	21	,	,	PUNCT
cana-3960	167	22	𝑡𝑙)𝕋	𝑡𝑙)𝕋	NOUN
cana-3960	167	23	)	)	PUNCT
cana-3960	167	24	,	,	PUNCT
cana-3960	167	25	if	if	SCONJ
cana-3960	167	26	and	and	CCONJ
cana-3960	167	27	only	only	ADV
cana-3960	167	28	if	if	SCONJ
cana-3960	167	29	𝑟𝑎𝑛𝑘	𝑟𝑎𝑛𝑘	PROPN
cana-3960	167	30	{	{	PUNCT
cana-3960	167	31	𝑀1	𝑀1	PROPN
cana-3960	167	32	,	,	PUNCT
cana-3960	167	33	𝑀2	𝑀2	PROPN
cana-3960	167	34	,	,	PUNCT
cana-3960	167	35	…	…	PUNCT
cana-3960	167	36	,	,	PUNCT
cana-3960	167	37	𝑀𝑙	𝑀𝑙	VERB
cana-3960	167	38	}	}	PUNCT
cana-3960	167	39	=	=	SYM
cana-3960	167	40	𝑛2	𝑛2	NOUN
cana-3960	167	41	(	(	PUNCT
cana-3960	167	42	3.13	3.13	NUM
cana-3960	167	43	)	)	PUNCT
cana-3960	167	44	since	since	SCONJ
cana-3960	167	45	,	,	PUNCT
cana-3960	167	46	𝑀𝑗	𝑀𝑗	PROPN
cana-3960	167	47	=	=	PUNCT
cana-3960	168	1	[	[	X
cana-3960	168	2	𝐴𝑗	𝐴𝑗	PROPN
cana-3960	168	3	𝑇	𝑇	PROPN
cana-3960	168	4	𝐴𝑗	𝐴𝑗	PROPN
cana-3960	168	5	𝑇𝐺𝑗	𝑇𝐺𝑗	NOUN
cana-3960	168	6	…	…	PUNCT
cana-3960	169	1	𝐴𝑗	𝐴𝑗	ADP
cana-3960	169	2	𝑇𝐺𝑗	𝑇𝐺𝑗	VERB
cana-3960	169	3	𝑛2−1	𝑛2−1	ADP
cana-3960	169	4	]	]	PUNCT
cana-3960	169	5	,	,	PUNCT
cana-3960	169	6	𝑗	𝑗	NOUN
cana-3960	169	7	=	=	SYM
cana-3960	169	8	1,2	1,2	NUM
cana-3960	169	9	,	,	PUNCT
cana-3960	169	10	…	…	PUNCT
cana-3960	169	11	,	,	PUNCT
cana-3960	169	12	𝑙.	𝑙.	ADJ
cana-3960	169	13	proof	proof	NOUN
cana-3960	169	14	:	:	PUNCT
cana-3960	169	15	suppose	suppose	VERB
cana-3960	169	16	that	that	SCONJ
cana-3960	169	17	the	the	DET
cana-3960	169	18	system	system	NOUN
cana-3960	169	19	(	(	PUNCT
cana-3960	169	20	2.2	2.2	NUM
cana-3960	169	21	)	)	PUNCT
cana-3960	169	22	is	be	AUX
cana-3960	169	23	a	a	DET
cana-3960	169	24	controllability	controllability	NOUN
cana-3960	169	25	on	on	ADP
cana-3960	169	26	[	[	X
cana-3960	169	27	𝑡0	𝑡0	NOUN
cana-3960	169	28	,	,	PUNCT
cana-3960	169	29	𝑡𝑓]𝕋.	𝑡𝑓]𝕋.	ADJ
cana-3960	169	30	if	if	SCONJ
cana-3960	169	31	the	the	DET
cana-3960	169	32	rank	rank	NOUN
cana-3960	169	33	condition	condition	NOUN
cana-3960	169	34	(	(	PUNCT
cana-3960	169	35	3.13	3.13	NUM
cana-3960	169	36	)	)	PUNCT
cana-3960	169	37	does	do	AUX
cana-3960	169	38	not	not	PART
cana-3960	169	39	hold	hold	VERB
cana-3960	169	40	,	,	PUNCT
cana-3960	169	41	if	if	SCONJ
cana-3960	169	42	there	there	PRON
cana-3960	169	43	exist	exist	VERB
cana-3960	169	44	𝑧𝛼	𝑧𝛼	X
cana-3960	169	45	∈	∈	PUNCT
cana-3960	170	1	ℝ𝑛	ℝ𝑛	ADP
cana-3960	170	2	2	2	NUM
cana-3960	170	3	with	with	ADP
cana-3960	170	4	𝑧𝛼	𝑧𝛼	ADP
cana-3960	170	5	≠	≠	PROPN
cana-3960	170	6	0	0	NUM
cana-3960	170	7	,	,	PUNCT
cana-3960	170	8	such	such	ADJ
cana-3960	170	9	that	that	DET
cana-3960	170	10	𝐴𝑗𝐺𝑗	𝐴𝑗𝐺𝑗	PROPN
cana-3960	170	11	𝑖𝑧𝛼	𝑖𝑧𝛼	NOUN
cana-3960	170	12	=	=	NOUN
cana-3960	170	13	0	0	PROPN
cana-3960	170	14	.	.	PUNCT
cana-3960	171	1	(	(	PUNCT
cana-3960	171	2	3.14	3.14	NUM
cana-3960	171	3	)	)	PUNCT
cana-3960	171	4	for	for	ADP
cana-3960	171	5	,	,	PUNCT
cana-3960	171	6	j	j	PROPN
cana-3960	171	7	=	=	SYM
cana-3960	171	8	1	1	NUM
cana-3960	171	9	,	,	PUNCT
cana-3960	171	10	…	…	PUNCT
cana-3960	171	11	,	,	PUNCT
cana-3960	172	1	k	k	X
cana-3960	172	2	,	,	PUNCT
cana-3960	172	3	i	i	NOUN
cana-3960	172	4	=	=	NOUN
cana-3960	172	5	0,1	0,1	NUM
cana-3960	172	6	,	,	PUNCT
cana-3960	172	7	…	…	PUNCT
cana-3960	172	8	,	,	PUNCT
cana-3960	172	9	𝑛2	𝑛2	NOUN
cana-3960	172	10	−	−	PROPN
cana-3960	172	11	1	1	NUM
cana-3960	172	12	we	we	PRON
cana-3960	172	13	consider	consider	VERB
cana-3960	172	14	𝐻1(𝑡0	𝐻1(𝑡0	NOUN
cana-3960	172	15	,	,	PUNCT
cana-3960	172	16	𝑡𝑓	𝑡𝑓	ADV
cana-3960	172	17	,	,	PUNCT
cana-3960	172	18	𝑡𝑓)𝑧𝛼	𝑡𝑓)𝑧𝛼	PROPN
cana-3960	172	19	=	=	SYM
cana-3960	172	20	∫	∫	PROPN
cana-3960	172	21	𝑒𝐺1	𝑒𝐺1	PROPN
cana-3960	172	22	𝑇	𝑇	PROPN
cana-3960	172	23	(	(	PUNCT
cana-3960	172	24	𝜏	𝜏	NOUN
cana-3960	172	25	,	,	PUNCT
cana-3960	172	26	𝑡𝑓	𝑡𝑓	X
cana-3960	172	27	)	)	PUNCT
cana-3960	172	28	𝑡𝑓	𝑡𝑓	VERB
cana-3960	172	29	𝑡0	𝑡0	PROPN
cana-3960	172	30	𝐴1𝐴1	𝐴1𝐴1	PROPN
cana-3960	172	31	𝑇𝑒𝐺1(𝜏	𝑇𝑒𝐺1(𝜏	PROPN
cana-3960	172	32	,	,	PUNCT
cana-3960	172	33	𝑡𝑓)𝑧𝛼∆𝜏.	𝑡𝑓)𝑧𝛼∆𝜏.	PUNCT
cana-3960	172	34	from	from	ADP
cana-3960	172	35	equation	equation	NOUN
cana-3960	172	36	(	(	PUNCT
cana-3960	172	37	3.14	3.14	NUM
cana-3960	172	38	)	)	PUNCT
cana-3960	172	39	and	and	CCONJ
cana-3960	172	40	using	use	VERB
cana-3960	172	41	proposition	proposition	NOUN
cana-3960	172	42	2.1	2.1	NUM
cana-3960	172	43	,	,	PUNCT
cana-3960	172	44	we	we	PRON
cana-3960	172	45	have	have	VERB
cana-3960	172	46	communications	communication	NOUN
cana-3960	172	47	on	on	ADP
cana-3960	172	48	applied	apply	VERB
cana-3960	172	49	nonlinear	nonlinear	ADJ
cana-3960	172	50	analysis	analysis	NOUN
cana-3960	172	51	issn	issn	NOUN
cana-3960	172	52	:	:	PUNCT
cana-3960	172	53	1074	1074	NUM
cana-3960	172	54	-	-	PUNCT
cana-3960	172	55	133x	133x	NUM
cana-3960	172	56	vol	vol	NOUN
cana-3960	172	57	32	32	NUM
cana-3960	173	1	no	no	NOUN
cana-3960	173	2	.	.	PUNCT
cana-3960	174	1	9s	9s	NUM
cana-3960	174	2	(	(	PUNCT
cana-3960	174	3	2025	2025	NUM
cana-3960	174	4	)	)	PUNCT
cana-3960	174	5	501	501	NUM
cana-3960	174	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3960	174	7	𝐻1(𝑡0	𝐻1(𝑡0	PROPN
cana-3960	174	8	,	,	PUNCT
cana-3960	174	9	𝑡𝑓	𝑡𝑓	ADV
cana-3960	174	10	,	,	PUNCT
cana-3960	174	11	𝑡𝑓)𝑧𝛼	𝑡𝑓)𝑧𝛼	PROPN
cana-3960	174	12	=	=	SYM
cana-3960	174	13	∫	∫	PROPN
cana-3960	174	14	𝑒𝐺1	𝑒𝐺1	PROPN
cana-3960	174	15	𝑇	𝑇	PROPN
cana-3960	174	16	(	(	PUNCT
cana-3960	174	17	𝜏	𝜏	NOUN
cana-3960	174	18	,	,	PUNCT
cana-3960	174	19	𝑡𝑓	𝑡𝑓	X
cana-3960	174	20	)	)	PUNCT
cana-3960	174	21	𝑡𝑓	𝑡𝑓	VERB
cana-3960	174	22	𝑡0	𝑡0	PROPN
cana-3960	174	23	𝐴1𝐴1	𝐴1𝐴1	PROPN
cana-3960	174	24	𝑇	𝑇	PROPN
cana-3960	174	25	∑	∑	PUNCT
cana-3960	174	26	𝜒1𝑖(𝜏	𝜒1𝑖(𝜏	PROPN
cana-3960	174	27	,	,	PUNCT
cana-3960	174	28	𝑡𝑓	𝑡𝑓	NOUN
cana-3960	174	29	)	)	PUNCT
cana-3960	174	30	𝑛2−1	𝑛2−1	ADP
cana-3960	174	31	𝑖=0	𝑖=0	PROPN
cana-3960	174	32	𝐺1	𝐺1	NOUN
cana-3960	174	33	𝑖𝑧𝛼∆𝜏	𝑖𝑧𝛼∆𝜏	PUNCT
cana-3960	174	34	,	,	PUNCT
cana-3960	174	35	=	=	SYM
cana-3960	174	36	∫	∫	PROPN
cana-3960	174	37	𝑒𝐺1	𝑒𝐺1	PROPN
cana-3960	174	38	𝑇	𝑇	PROPN
cana-3960	174	39	(	(	PUNCT
cana-3960	174	40	𝜏	𝜏	NOUN
cana-3960	174	41	,	,	PUNCT
cana-3960	174	42	𝑡𝑓	𝑡𝑓	X
cana-3960	174	43	)	)	PUNCT
cana-3960	174	44	𝑡𝑓	𝑡𝑓	VERB
cana-3960	174	45	𝑡0	𝑡0	PROPN
cana-3960	174	46	𝐴1	𝐴1	PROPN
cana-3960	174	47	∑	∑	PROPN
cana-3960	174	48	𝜒1𝑖(𝜏	𝜒1𝑖(𝜏	PROPN
cana-3960	174	49	,	,	PUNCT
cana-3960	174	50	𝑡𝑓	𝑡𝑓	NOUN
cana-3960	174	51	)	)	PUNCT
cana-3960	174	52	𝑛2−1	𝑛2−1	SYM
cana-3960	174	53	𝑖=0	𝑖=0	PROPN
cana-3960	175	1	𝐴1	𝐴1	PROPN
cana-3960	175	2	𝑇𝐺1	𝑇𝐺1	PROPN
cana-3960	175	3	𝑖	𝑖	SYM
cana-3960	175	4	𝑧𝛼∆𝜏	𝑧𝛼∆𝜏	PROPN
cana-3960	175	5	=	=	NOUN
cana-3960	175	6	0	0	X
cana-3960	175	7	.	.	PUNCT
cana-3960	176	1	by	by	ADP
cana-3960	176	2	again	again	ADV
cana-3960	176	3	equation	equation	NOUN
cana-3960	176	4	(	(	PUNCT
cana-3960	176	5	3.14	3.14	NUM
cana-3960	176	6	)	)	PUNCT
cana-3960	176	7	and	and	CCONJ
cana-3960	176	8	using	use	VERB
cana-3960	176	9	proposition	proposition	NOUN
cana-3960	176	10	2.1	2.1	NUM
cana-3960	176	11	,	,	PUNCT
cana-3960	176	12	according	accord	VERB
cana-3960	176	13	to	to	ADP
cana-3960	176	14	𝐻𝑘(𝑡𝑘−1	𝐻𝑘(𝑡𝑘−1	PROPN
cana-3960	176	15	,	,	PUNCT
cana-3960	176	16	𝑡𝑘	𝑡𝑘	ADV
cana-3960	176	17	,	,	PUNCT
cana-3960	176	18	𝑡𝑓)𝑧𝛼	𝑡𝑓)𝑧𝛼	PROPN
cana-3960	176	19	=	=	SYM
cana-3960	176	20	∫	∫	PROPN
cana-3960	176	21	𝑒𝐺𝑘	𝑒𝐺𝑘	PROPN
cana-3960	176	22	𝑇	𝑇	PROPN
cana-3960	176	23	(	(	PUNCT
cana-3960	176	24	𝜏	𝜏	NOUN
cana-3960	176	25	,	,	PUNCT
cana-3960	176	26	𝑡𝑓	𝑡𝑓	X
cana-3960	176	27	)	)	PUNCT
cana-3960	176	28	𝑡𝑓	𝑡𝑓	VERB
cana-3960	176	29	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	176	30	𝐴𝑘𝐴𝑘	𝐴𝑘𝐴𝑘	PROPN
cana-3960	176	31	𝑇𝑒𝐺𝑘(𝜏	𝑇𝑒𝐺𝑘(𝜏	PROPN
cana-3960	176	32	,	,	PUNCT
cana-3960	176	33	𝑡𝑓)𝑧𝛼∆𝜏	𝑡𝑓)𝑧𝛼∆𝜏	X
cana-3960	176	34	=	=	SYM
cana-3960	176	35	∫	∫	PROPN
cana-3960	176	36	𝑒𝐺𝑘	𝑒𝐺𝑘	NOUN
cana-3960	176	37	𝑇	𝑇	PROPN
cana-3960	176	38	(	(	PUNCT
cana-3960	176	39	𝜏	𝜏	NOUN
cana-3960	176	40	,	,	PUNCT
cana-3960	176	41	𝑡𝑓	𝑡𝑓	X
cana-3960	176	42	)	)	PUNCT
cana-3960	176	43	𝑡𝑓	𝑡𝑓	VERB
cana-3960	176	44	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	176	45	𝐴𝑘	𝐴𝑘	PROPN
cana-3960	176	46	∑	∑	PROPN
cana-3960	176	47	𝜒1𝑖(𝜏	𝜒1𝑖(𝜏	PROPN
cana-3960	176	48	,	,	PUNCT
cana-3960	176	49	𝑡𝑓	𝑡𝑓	NOUN
cana-3960	176	50	)	)	PUNCT
cana-3960	176	51	𝑛2−1	𝑛2−1	SYM
cana-3960	176	52	𝑖=0	𝑖=0	PUNCT
cana-3960	177	1	𝐴𝑘	𝐴𝑘	PROPN
cana-3960	177	2	𝑇𝐺𝑘	𝑇𝐺𝑘	VERB
cana-3960	177	3	𝑖	𝑖	PRON
cana-3960	177	4	𝑧𝛼∆𝜏	𝑧𝛼∆𝜏	X
cana-3960	177	5	=	=	NOUN
cana-3960	177	6	0	0	X
cana-3960	177	7	.	.	PUNCT
cana-3960	178	1	for	for	ADP
cana-3960	178	2	2	2	NUM
cana-3960	178	3	≤	≤	NOUN
cana-3960	178	4	𝑘	𝑘	DET
cana-3960	178	5	≤	≤	NOUN
cana-3960	178	6	𝑙	𝑙	DET
cana-3960	178	7	−	−	PROPN
cana-3960	178	8	1	1	NUM
cana-3960	178	9	,	,	PUNCT
cana-3960	178	10	similarly	similarly	ADV
cana-3960	178	11	,	,	PUNCT
cana-3960	178	12	𝐻𝑙(𝑡𝑙−1	𝐻𝑙(𝑡𝑙−1	NOUN
cana-3960	178	13	,	,	PUNCT
cana-3960	178	14	𝑡𝑙	𝑡𝑙	NOUN
cana-3960	178	15	,	,	PUNCT
cana-3960	178	16	𝑡𝑓)𝑧𝛼	𝑡𝑓)𝑧𝛼	X
cana-3960	178	17	=	=	SYM
cana-3960	178	18	0	0	NUM
cana-3960	178	19	,	,	PUNCT
cana-3960	178	20	according	accord	VERB
cana-3960	178	21	to	to	ADP
cana-3960	178	22	𝑟𝑎𝑛𝑘	𝑟𝑎𝑛𝑘	PROPN
cana-3960	178	23	{	{	PUNCT
cana-3960	178	24	𝑀1	𝑀1	PROPN
cana-3960	178	25	,	,	PUNCT
cana-3960	178	26	…	…	PUNCT
cana-3960	178	27	,	,	PUNCT
cana-3960	178	28	𝑀𝑙	𝑀𝑙	VERB
cana-3960	178	29	}	}	PUNCT
cana-3960	178	30	<	<	X
cana-3960	178	31	𝑛	𝑛	DET
cana-3960	178	32	2	2	NUM
cana-3960	178	33	hence	hence	ADV
cana-3960	178	34	,	,	PUNCT
cana-3960	178	35	it	it	PRON
cana-3960	178	36	is	be	AUX
cana-3960	178	37	contradicting	contradict	VERB
cana-3960	178	38	the	the	DET
cana-3960	178	39	conclusion	conclusion	NOUN
cana-3960	178	40	(	(	PUNCT
cana-3960	178	41	ii	ii	NOUN
cana-3960	178	42	)	)	PUNCT
cana-3960	178	43	of	of	ADP
cana-3960	178	44	theorem	theorem	NOUN
cana-3960	178	45	(	(	PUNCT
cana-3960	178	46	3.1	3.1	NUM
cana-3960	178	47	)	)	PUNCT
cana-3960	178	48	and	and	CCONJ
cana-3960	178	49	thus	thus	ADV
cana-3960	178	50	,	,	PUNCT
cana-3960	178	51	we	we	PRON
cana-3960	178	52	can	can	AUX
cana-3960	178	53	conclude	conclude	VERB
cana-3960	178	54	that	that	SCONJ
cana-3960	178	55	the	the	DET
cana-3960	178	56	condition	condition	NOUN
cana-3960	178	57	(	(	PUNCT
cana-3960	178	58	3.13	3.13	NUM
cana-3960	178	59	)	)	PUNCT
cana-3960	178	60	is	be	AUX
cana-3960	178	61	true	true	ADJ
cana-3960	178	62	.	.	PUNCT
cana-3960	179	1	conversely	conversely	ADV
cana-3960	179	2	,	,	PUNCT
cana-3960	179	3	assume	assume	VERB
cana-3960	179	4	that	that	SCONJ
cana-3960	179	5	the	the	DET
cana-3960	179	6	condition	condition	NOUN
cana-3960	179	7	(	(	PUNCT
cana-3960	179	8	3.13	3.13	NUM
cana-3960	179	9	)	)	PUNCT
cana-3960	179	10	is	be	AUX
cana-3960	179	11	satisfied	satisfied	ADJ
cana-3960	179	12	.	.	PUNCT
cana-3960	180	1	if	if	SCONJ
cana-3960	180	2	the	the	DET
cana-3960	180	3	impulsive	impulsive	ADJ
cana-3960	180	4	system	system	NOUN
cana-3960	180	5	(	(	PUNCT
cana-3960	180	6	2.2	2.2	NUM
cana-3960	180	7	)	)	PUNCT
cana-3960	180	8	is	be	AUX
cana-3960	180	9	not	not	PART
cana-3960	180	10	controllable	controllable	ADJ
cana-3960	180	11	on	on	ADP
cana-3960	180	12	𝑡	𝑡	PROPN
cana-3960	180	13	∈	∈	PROPN
cana-3960	180	14	[	[	X
cana-3960	180	15	𝑡0	𝑡0	NOUN
cana-3960	180	16	,	,	PUNCT
cana-3960	180	17	𝑡𝑓]𝕋	𝑡𝑓]𝕋	PROPN
cana-3960	180	18	(	(	PUNCT
cana-3960	180	19	𝑡	𝑡	PROPN
cana-3960	180	20	∈	∈	PROPN
cana-3960	181	1	[	[	X
cana-3960	181	2	𝑡𝑙−1	𝑡𝑙−1	X
cana-3960	181	3	,	,	PUNCT
cana-3960	181	4	𝑡𝑙)𝕋	𝑡𝑙)𝕋	NOUN
cana-3960	181	5	)	)	PUNCT
cana-3960	181	6	,	,	PUNCT
cana-3960	181	7	then	then	ADV
cana-3960	181	8	it	it	PRON
cana-3960	181	9	follows	follow	VERB
cana-3960	181	10	that	that	SCONJ
cana-3960	181	11	from	from	ADP
cana-3960	181	12	the	the	DET
cana-3960	181	13	conclusion	conclusion	NOUN
cana-3960	181	14	(	(	PUNCT
cana-3960	181	15	i	i	NOUN
cana-3960	181	16	)	)	PUNCT
cana-3960	181	17	of	of	ADP
cana-3960	181	18	theorem	theorem	NOUN
cana-3960	181	19	3.1	3.1	NUM
cana-3960	181	20	.	.	NUM
cana-3960	181	21	,	,	PUNCT
cana-3960	181	22	that	that	SCONJ
cana-3960	181	23	the	the	DET
cana-3960	181	24	matrices	matrix	NOUN
cana-3960	181	25	𝐻1(𝑡0	𝐻1(𝑡0	VERB
cana-3960	181	26	,	,	PUNCT
cana-3960	181	27	𝑡𝑓	𝑡𝑓	INTJ
cana-3960	181	28	,	,	PUNCT
cana-3960	181	29	𝑡𝑓	𝑡𝑓	NOUN
cana-3960	181	30	)	)	PUNCT
cana-3960	181	31	,	,	PUNCT
cana-3960	181	32	𝐻𝑘(𝑡𝑘−1	𝐻𝑘(𝑡𝑘−1	PROPN
cana-3960	181	33	,	,	PUNCT
cana-3960	181	34	𝑡𝑘	𝑡𝑘	ADV
cana-3960	181	35	,	,	PUNCT
cana-3960	181	36	𝑡𝑓	𝑡𝑓	ADV
cana-3960	181	37	)	)	PUNCT
cana-3960	181	38	and	and	CCONJ
cana-3960	181	39	𝐻𝑙(𝑡𝑙−1	𝐻𝑙(𝑡𝑙−1	NOUN
cana-3960	181	40	,	,	PUNCT
cana-3960	181	41	𝑡𝑙	𝑡𝑙	NOUN
cana-3960	181	42	,	,	PUNCT
cana-3960	181	43	𝑡𝑓	𝑡𝑓	X
cana-3960	181	44	)	)	PUNCT
cana-3960	181	45	are	be	AUX
cana-3960	181	46	not	not	PART
cana-3960	181	47	invertible	invertible	ADJ
cana-3960	181	48	.	.	PUNCT
cana-3960	182	1	if	if	SCONJ
cana-3960	182	2	there	there	PRON
cana-3960	182	3	exist	exist	VERB
cana-3960	182	4	𝑧𝛼	𝑧𝛼	X
cana-3960	182	5	∈	∈	PUNCT
cana-3960	183	1	ℝ𝑛	ℝ𝑛	ADP
cana-3960	183	2	2	2	NUM
cana-3960	183	3	with	with	ADP
cana-3960	183	4	𝑧𝛼	𝑧𝛼	ADP
cana-3960	183	5	≠	≠	PROPN
cana-3960	183	6	0	0	NUM
cana-3960	183	7	,	,	PUNCT
cana-3960	183	8	such	such	ADJ
cana-3960	183	9	that	that	PRON
cana-3960	183	10	𝑧𝛼	𝑧𝛼	ADP
cana-3960	183	11	𝑇𝐻1(𝑡0	𝑇𝐻1(𝑡0	PROPN
cana-3960	183	12	,	,	PUNCT
cana-3960	183	13	𝑡𝑓	𝑡𝑓	ADV
cana-3960	183	14	,	,	PUNCT
cana-3960	183	15	𝑡𝑓)𝑧𝛼	𝑡𝑓)𝑧𝛼	PROPN
cana-3960	183	16	=	=	SYM
cana-3960	183	17	∫	∫	PROPN
cana-3960	183	18	𝑧𝛼	𝑧𝛼	ADP
cana-3960	183	19	𝑇𝑒𝐺1	𝑇𝑒𝐺1	PROPN
cana-3960	183	20	𝑇	𝑇	PROPN
cana-3960	183	21	(	(	PUNCT
cana-3960	183	22	𝜏	𝜏	NOUN
cana-3960	183	23	,	,	PUNCT
cana-3960	183	24	𝑡𝑓	𝑡𝑓	X
cana-3960	183	25	)	)	PUNCT
cana-3960	183	26	𝑡𝑓	𝑡𝑓	VERB
cana-3960	183	27	𝑡0	𝑡0	PROPN
cana-3960	183	28	𝐴1𝐴1	𝐴1𝐴1	PROPN
cana-3960	183	29	𝑇𝑒𝐺1(𝜏	𝑇𝑒𝐺1(𝜏	PROPN
cana-3960	183	30	,	,	PUNCT
cana-3960	183	31	𝑡𝑓)𝑧𝛼∆𝜏	𝑡𝑓)𝑧𝛼∆𝜏	NOUN
cana-3960	183	32	=	=	SYM
cana-3960	183	33	0	0	NUM
cana-3960	183	34	,	,	PUNCT
cana-3960	183	35	𝑧𝛼	𝑧𝛼	CCONJ
cana-3960	183	36	𝑇𝐻𝑘(𝑡𝑘−1	𝑇𝐻𝑘(𝑡𝑘−1	PROPN
cana-3960	183	37	,	,	PUNCT
cana-3960	183	38	𝑡𝑘	𝑡𝑘	ADV
cana-3960	183	39	,	,	PUNCT
cana-3960	183	40	𝑡𝑓)𝑧𝛼	𝑡𝑓)𝑧𝛼	PROPN
cana-3960	183	41	=	=	SYM
cana-3960	183	42	∫	∫	PROPN
cana-3960	183	43	𝑧𝛼	𝑧𝛼	NUM
cana-3960	183	44	𝑇𝑒𝐺𝑘	𝑇𝑒𝐺𝑘	PROPN
cana-3960	183	45	𝑇	𝑇	PROPN
cana-3960	183	46	(	(	PUNCT
cana-3960	183	47	𝜏	𝜏	NOUN
cana-3960	183	48	,	,	PUNCT
cana-3960	183	49	𝑡𝑓	𝑡𝑓	X
cana-3960	183	50	)	)	PUNCT
cana-3960	183	51	𝑡𝑓	𝑡𝑓	VERB
cana-3960	183	52	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	183	53	𝐴𝑘𝐴𝑘	𝐴𝑘𝐴𝑘	PROPN
cana-3960	183	54	𝑇𝑒𝐺𝑘(𝜏	𝑇𝑒𝐺𝑘(𝜏	PROPN
cana-3960	183	55	,	,	PUNCT
cana-3960	183	56	𝑡𝑓)𝑧𝛼∆𝜏	𝑡𝑓)𝑧𝛼∆𝜏	NOUN
cana-3960	183	57	=	=	SYM
cana-3960	183	58	0	0	NUM
cana-3960	183	59	,	,	PUNCT
cana-3960	183	60	2	2	NUM
cana-3960	183	61	≤	≤	NOUN
cana-3960	183	62	𝑘	𝑘	DET
cana-3960	183	63	≤	≤	NUM
cana-3960	183	64	𝑙	𝑙	PRON
cana-3960	183	65	−	−	NUM
cana-3960	183	66	1	1	NUM
cana-3960	183	67	𝑧𝛼	𝑧𝛼	ADP
cana-3960	183	68	𝑇𝐻𝑙(𝑡𝑙−1	𝑇𝐻𝑙(𝑡𝑙−1	PROPN
cana-3960	183	69	,	,	PUNCT
cana-3960	183	70	𝑡𝑙	𝑡𝑙	NOUN
cana-3960	183	71	,	,	PUNCT
cana-3960	183	72	𝑡𝑓)𝑧𝛼	𝑡𝑓)𝑧𝛼	X
cana-3960	183	73	=	=	SYM
cana-3960	183	74	∫	∫	PROPN
cana-3960	183	75	𝑧𝛼	𝑧𝛼	X
cana-3960	183	76	𝑇𝑒𝐺𝑙	𝑇𝑒𝐺𝑙	PROPN
cana-3960	183	77	𝑇	𝑇	PROPN
cana-3960	183	78	(	(	PUNCT
cana-3960	183	79	𝜏	𝜏	NOUN
cana-3960	183	80	,	,	PUNCT
cana-3960	183	81	𝑡𝑓	𝑡𝑓	X
cana-3960	183	82	)	)	PUNCT
cana-3960	183	83	𝑡𝑓	𝑡𝑓	VERB
cana-3960	184	1	𝑡𝑙−1	𝑡𝑙−1	PROPN
cana-3960	184	2	𝐴𝑙𝐴𝑙	𝐴𝑙𝐴𝑙	PROPN
cana-3960	184	3	𝑇𝑒𝐺𝑙(𝜏	𝑇𝑒𝐺𝑙(𝜏	PROPN
cana-3960	184	4	,	,	PUNCT
cana-3960	184	5	𝑡𝑓)𝑧𝛼∆𝜏	𝑡𝑓)𝑧𝛼∆𝜏	NOUN
cana-3960	184	6	=	=	SYM
cana-3960	184	7	0	0	NUM
cana-3960	184	8	,	,	PUNCT
cana-3960	184	9	exactly	exactly	ADV
cana-3960	184	10	same	same	ADJ
cana-3960	184	11	as	as	ADP
cana-3960	184	12	in	in	ADP
cana-3960	184	13	proof	proof	NOUN
cana-3960	184	14	of	of	ADP
cana-3960	184	15	theorem	theorem	ADJ
cana-3960	184	16	3.1	3.1	NUM
cana-3960	184	17	.	.	NUM
cana-3960	184	18	,	,	PUNCT
cana-3960	184	19	according	accord	VERB
cana-3960	184	20	to	to	ADP
cana-3960	184	21	𝐴1	𝐴1	PROPN
cana-3960	184	22	𝑇𝑒𝐺1(𝑡	𝑇𝑒𝐺1(𝑡	PROPN
cana-3960	184	23	,	,	PUNCT
cana-3960	184	24	𝑡𝑓)𝑧𝛼	𝑡𝑓)𝑧𝛼	X
cana-3960	184	25	=	=	SYM
cana-3960	184	26	0	0	PROPN
cana-3960	184	27	.	.	PUNCT
cana-3960	185	1	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
cana-3960	185	2	𝑡	𝑡	PROPN
cana-3960	185	3	∈	∈	PROPN
cana-3960	186	1	[	[	X
cana-3960	186	2	𝑡0	𝑡0	NOUN
cana-3960	186	3	,	,	PUNCT
cana-3960	186	4	𝑡𝑓]𝕋	𝑡𝑓]𝕋	PROPN
cana-3960	186	5	(	(	PUNCT
cana-3960	186	6	3.15	3.15	NUM
cana-3960	186	7	)	)	PUNCT
cana-3960	187	1	𝐴𝑘	𝐴𝑘	PROPN
cana-3960	187	2	𝑇𝑒𝐺𝑘(𝑡	𝑇𝑒𝐺𝑘(𝑡	ADJ
cana-3960	187	3	,	,	PUNCT
cana-3960	187	4	𝑡𝑓)𝑧𝛼	𝑡𝑓)𝑧𝛼	PROPN
cana-3960	187	5	=	=	SYM
cana-3960	187	6	0	0	PROPN
cana-3960	187	7	.	.	PUNCT
cana-3960	188	1	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
cana-3960	188	2	𝑡	𝑡	PROPN
cana-3960	188	3	∈	∈	PROPN
cana-3960	189	1	[	[	X
cana-3960	189	2	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	189	3	,	,	PUNCT
cana-3960	189	4	𝑡𝑘)𝕋.	𝑡𝑘)𝕋.	NOUN
cana-3960	189	5	(	(	PUNCT
cana-3960	189	6	3.16	3.16	NUM
cana-3960	189	7	)	)	PUNCT
cana-3960	189	8	where	where	SCONJ
cana-3960	189	9	2	2	NUM
cana-3960	189	10	≤	≤	NOUN
cana-3960	189	11	𝑘	𝑘	PRON
cana-3960	189	12	≤	≤	NOUN
cana-3960	189	13	𝑙	𝑙	PRON
cana-3960	189	14	−	−	PROPN
cana-3960	189	15	1	1	NUM
cana-3960	189	16	,	,	PUNCT
cana-3960	189	17	and	and	CCONJ
cana-3960	189	18	𝐴𝑙	𝐴𝑙	PROPN
cana-3960	189	19	𝑇𝑒𝐺𝑙(𝑡	𝑇𝑒𝐺𝑙(𝑡	PROPN
cana-3960	189	20	,	,	PUNCT
cana-3960	189	21	𝑡𝑓)𝑧𝛼	𝑡𝑓)𝑧𝛼	X
cana-3960	189	22	=	=	SYM
cana-3960	189	23	0	0	PROPN
cana-3960	189	24	.	.	PUNCT
cana-3960	190	1	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
cana-3960	190	2	𝑡	𝑡	PROPN
cana-3960	190	3	∈	∈	PROPN
cana-3960	191	1	[	[	X
cana-3960	191	2	𝑡𝑙−1	𝑡𝑙−1	X
cana-3960	191	3	,	,	PUNCT
cana-3960	191	4	𝑡𝑙)𝕋	𝑡𝑙)𝕋	X
cana-3960	191	5	(	(	PUNCT
cana-3960	191	6	3.17	3.17	NUM
cana-3960	191	7	)	)	PUNCT
cana-3960	191	8	differentiating	differentiate	VERB
cana-3960	191	9	equations	equation	NOUN
cana-3960	191	10	(	(	PUNCT
cana-3960	191	11	3.13	3.13	NUM
cana-3960	191	12	)	)	PUNCT
cana-3960	191	13	,	,	PUNCT
cana-3960	191	14	(	(	PUNCT
cana-3960	191	15	3.14	3.14	NUM
cana-3960	191	16	)	)	PUNCT
cana-3960	191	17	and	and	CCONJ
cana-3960	191	18	(	(	PUNCT
cana-3960	191	19	3.15	3.15	NUM
cana-3960	191	20	)	)	PUNCT
cana-3960	191	21	𝑖𝑡ℎ	𝑖𝑡ℎ	NOUN
cana-3960	191	22	times	time	NOUN
cana-3960	191	23	,	,	PUNCT
cana-3960	191	24	where	where	SCONJ
cana-3960	191	25	(	(	PUNCT
cana-3960	191	26	0	0	NUM
cana-3960	191	27	≤	≤	NUM
cana-3960	191	28	𝑖	𝑖	SYM
cana-3960	191	29	≤	≤	NOUN
cana-3960	191	30	𝑛2	𝑛2	NOUN
cana-3960	191	31	−	−	PROPN
cana-3960	191	32	1	1	NUM
cana-3960	191	33	)	)	PUNCT
cana-3960	191	34	,	,	PUNCT
cana-3960	191	35	we	we	PRON
cana-3960	191	36	obtain	obtain	VERB
cana-3960	191	37	communications	communication	NOUN
cana-3960	191	38	on	on	ADP
cana-3960	191	39	applied	apply	VERB
cana-3960	191	40	nonlinear	nonlinear	ADJ
cana-3960	191	41	analysis	analysis	NOUN
cana-3960	191	42	issn	issn	NOUN
cana-3960	191	43	:	:	PUNCT
cana-3960	191	44	1074	1074	NUM
cana-3960	191	45	-	-	PUNCT
cana-3960	191	46	133x	133x	NUM
cana-3960	191	47	vol	vol	NOUN
cana-3960	191	48	32	32	NUM
cana-3960	191	49	no	no	NOUN
cana-3960	191	50	.	.	PUNCT
cana-3960	192	1	9s	9s	NUM
cana-3960	192	2	(	(	PUNCT
cana-3960	192	3	2025	2025	NUM
cana-3960	192	4	)	)	PUNCT
cana-3960	193	1	502	502	NUM
cana-3960	193	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-3960	193	3	𝐴1	𝐴1	PROPN
cana-3960	194	1	𝑇𝐺1	𝑇𝐺1	PROPN
cana-3960	194	2	𝑖𝑒𝐺1(𝜏	𝑖𝑒𝐺1(𝜏	PROPN
cana-3960	194	3	,	,	PUNCT
cana-3960	194	4	𝑡𝑓)𝑧𝛼	𝑡𝑓)𝑧𝛼	X
cana-3960	194	5	=	=	SYM
cana-3960	194	6	0	0	PROPN
cana-3960	194	7	.	.	PUNCT
cana-3960	195	1	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
cana-3960	195	2	𝑡	𝑡	PROPN
cana-3960	195	3	∈	∈	PROPN
cana-3960	196	1	[	[	X
cana-3960	196	2	𝑡0	𝑡0	NOUN
cana-3960	196	3	,	,	PUNCT
cana-3960	196	4	𝑡1]𝕋.	𝑡1]𝕋.	PROPN
cana-3960	196	5	(	(	PUNCT
cana-3960	196	6	3.18	3.18	NUM
cana-3960	196	7	)	)	PUNCT
cana-3960	197	1	𝐴𝑘	𝐴𝑘	PROPN
cana-3960	197	2	𝑇𝐺𝑘	𝑇𝐺𝑘	VERB
cana-3960	197	3	𝑖𝑒𝐺𝑘(𝜏	𝑖𝑒𝐺𝑘(𝜏	NOUN
cana-3960	197	4	,	,	PUNCT
cana-3960	197	5	𝑡𝑓)𝑧𝛼	𝑡𝑓)𝑧𝛼	X
cana-3960	197	6	=	=	SYM
cana-3960	197	7	0	0	PROPN
cana-3960	197	8	.	.	PUNCT
cana-3960	198	1	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
cana-3960	198	2	𝑡	𝑡	PROPN
cana-3960	198	3	∈	∈	PROPN
cana-3960	199	1	[	[	X
cana-3960	199	2	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	199	3	,	,	PUNCT
cana-3960	199	4	𝑡𝑘)𝕋.	𝑡𝑘)𝕋.	NOUN
cana-3960	199	5	(	(	PUNCT
cana-3960	199	6	3.19	3.19	NUM
cana-3960	199	7	)	)	PUNCT
cana-3960	199	8	where	where	SCONJ
cana-3960	199	9	2	2	NUM
cana-3960	199	10	≤	≤	NOUN
cana-3960	199	11	𝑘	𝑘	PRON
cana-3960	199	12	≤	≤	NOUN
cana-3960	199	13	𝑙	𝑙	PRON
cana-3960	199	14	−	−	PROPN
cana-3960	199	15	1	1	NUM
cana-3960	199	16	,	,	PUNCT
cana-3960	199	17	and	and	CCONJ
cana-3960	199	18	𝐴𝑙	𝐴𝑙	PROPN
cana-3960	199	19	𝑇𝐺𝑙	𝑇𝐺𝑙	PROPN
cana-3960	199	20	𝑖𝑒𝐺𝑙(𝜏	𝑖𝑒𝐺𝑙(𝜏	NOUN
cana-3960	199	21	,	,	PUNCT
cana-3960	199	22	𝑡𝑓)𝑧𝛼	𝑡𝑓)𝑧𝛼	PROPN
cana-3960	199	23	=	=	SYM
cana-3960	199	24	0	0	PROPN
cana-3960	199	25	.	.	PUNCT
cana-3960	200	1	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
cana-3960	200	2	𝑡	𝑡	PROPN
cana-3960	200	3	∈	∈	PROPN
cana-3960	201	1	[	[	X
cana-3960	201	2	𝑡𝑙−1	𝑡𝑙−1	X
cana-3960	201	3	,	,	PUNCT
cana-3960	201	4	𝑡𝑙)𝕋.	𝑡𝑙)𝕋.	NOUN
cana-3960	201	5	(	(	PUNCT
cana-3960	201	6	3.20	3.20	NUM
cana-3960	201	7	)	)	PUNCT
cana-3960	201	8	if	if	SCONJ
cana-3960	201	9	we	we	PRON
cana-3960	201	10	take	take	VERB
cana-3960	201	11	𝑡	𝑡	NOUN
cana-3960	201	12	=	=	PUNCT
cana-3960	201	13	𝑡𝑓	𝑡𝑓	NOUN
cana-3960	201	14	in	in	ADP
cana-3960	201	15	equations	equation	NOUN
cana-3960	201	16	(	(	PUNCT
cana-3960	201	17	3.18	3.18	NUM
cana-3960	201	18	)	)	PUNCT
cana-3960	201	19	,	,	PUNCT
cana-3960	201	20	(	(	PUNCT
cana-3960	201	21	3.19	3.19	NUM
cana-3960	201	22	)	)	PUNCT
cana-3960	201	23	and	and	CCONJ
cana-3960	201	24	(	(	PUNCT
cana-3960	201	25	3.20	3.20	NUM
cana-3960	201	26	)	)	PUNCT
cana-3960	201	27	,	,	PUNCT
cana-3960	201	28	then	then	ADV
cana-3960	201	29	it	it	PRON
cana-3960	201	30	follows	follow	VERB
cana-3960	201	31	that	that	SCONJ
cana-3960	201	32	𝐴𝑗	𝐴𝑗	PROPN
cana-3960	201	33	𝑇𝐺𝑗	𝑇𝐺𝑗	VERB
cana-3960	201	34	𝑖𝑧𝛼	𝑖𝑧𝛼	NOUN
cana-3960	202	1	=	=	SYM
cana-3960	202	2	0	0	PROPN
cana-3960	202	3	,	,	PUNCT
cana-3960	202	4	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-3960	202	5	𝑗	𝑗	X
cana-3960	202	6	=	=	SYM
cana-3960	202	7	1	1	NUM
cana-3960	202	8	,	,	PUNCT
cana-3960	202	9	…	…	PUNCT
cana-3960	203	1	,	,	PUNCT
cana-3960	203	2	𝑘	𝑘	NOUN
cana-3960	203	3	and	and	CCONJ
cana-3960	203	4	𝑖	𝑖	NOUN
cana-3960	203	5	=	=	SYM
cana-3960	203	6	0,1	0,1	NUM
cana-3960	203	7	,	,	PUNCT
cana-3960	203	8	…	…	PUNCT
cana-3960	203	9	,	,	PUNCT
cana-3960	203	10	𝑛2	𝑛2	NOUN
cana-3960	203	11	−	−	PROPN
cana-3960	203	12	1	1	NUM
cana-3960	203	13	.	.	NOUN
cana-3960	203	14	which	which	PRON
cana-3960	203	15	implies	imply	VERB
cana-3960	203	16	that	that	SCONJ
cana-3960	203	17	the	the	DET
cana-3960	203	18	rank	rank	NOUN
cana-3960	203	19	condition	condition	NOUN
cana-3960	203	20	(	(	PUNCT
cana-3960	203	21	3.11	3.11	NUM
cana-3960	203	22	)	)	PUNCT
cana-3960	203	23	fails	fail	VERB
cana-3960	203	24	,	,	PUNCT
cana-3960	203	25	which	which	PRON
cana-3960	203	26	gives	give	VERB
cana-3960	203	27	contradiction	contradiction	NOUN
cana-3960	203	28	.	.	PUNCT
cana-3960	204	1	so	so	ADV
cana-3960	204	2	,	,	PUNCT
cana-3960	204	3	the	the	DET
cana-3960	204	4	impulsive	impulsive	ADJ
cana-3960	204	5	system	system	NOUN
cana-3960	204	6	(	(	PUNCT
cana-3960	204	7	2.1	2.1	NUM
cana-3960	204	8	)	)	PUNCT
cana-3960	204	9	is	be	AUX
cana-3960	204	10	controllable	controllable	ADJ
cana-3960	204	11	on	on	ADP
cana-3960	204	12	𝑡	𝑡	PROPN
cana-3960	204	13	∈	∈	PROPN
cana-3960	204	14	[	[	X
cana-3960	204	15	𝑡0	𝑡0	NOUN
cana-3960	204	16	,	,	PUNCT
cana-3960	204	17	𝑡𝑓]𝕋	𝑡𝑓]𝕋	PROPN
cana-3960	204	18	(	(	PUNCT
cana-3960	204	19	𝑡𝑓	𝑡𝑓	ADV
cana-3960	204	20	∈	∈	PROPN
cana-3960	205	1	[	[	X
cana-3960	205	2	𝑡𝑙−1	𝑡𝑙−1	X
cana-3960	205	3	,	,	PUNCT
cana-3960	205	4	𝑡𝑙)𝕋	𝑡𝑙)𝕋	NOUN
cana-3960	205	5	)	)	PUNCT
cana-3960	205	6	.	.	PUNCT
cana-3960	206	1	so	so	ADV
cana-3960	206	2	that	that	SCONJ
cana-3960	206	3	the	the	DET
cana-3960	206	4	system	system	NOUN
cana-3960	206	5	(	(	PUNCT
cana-3960	206	6	2.2	2.2	NUM
cana-3960	206	7	)	)	PUNCT
cana-3960	206	8	is	be	AUX
cana-3960	206	9	controllable	controllable	ADJ
cana-3960	206	10	by	by	ADP
cana-3960	206	11	theorem	theorem	NOUN
cana-3960	206	12	3.2	3.2	NUM
cana-3960	206	13	.	.	PUNCT
cana-3960	207	1	4	4	NUM
cana-3960	207	2	.	.	X
cana-3960	207	3	complete	complete	ADJ
cana-3960	207	4	observability	observability	NOUN
cana-3960	207	5	in	in	ADP
cana-3960	207	6	this	this	DET
cana-3960	207	7	section	section	NOUN
cana-3960	208	1	,	,	PUNCT
cana-3960	208	2	we	we	PRON
cana-3960	208	3	present	present	VERB
cana-3960	208	4	the	the	DET
cana-3960	208	5	observability	observability	NOUN
cana-3960	208	6	in	in	ADP
cana-3960	208	7	time	time	NOUN
cana-3960	208	8	variant	variant	NOUN
cana-3960	208	9	and	and	CCONJ
cana-3960	208	10	time	time	NOUN
cana-3960	208	11	invariant	invariant	PROPN
cana-3960	208	12	adjoint	adjoint	PROPN
cana-3960	208	13	dynamic	dynamic	ADJ
cana-3960	208	14	system	system	NOUN
cana-3960	208	15	(	(	PUNCT
cana-3960	208	16	2.3	2.3	NUM
cana-3960	208	17	)	)	PUNCT
cana-3960	208	18	on	on	ADP
cana-3960	208	19	time	time	NOUN
cana-3960	208	20	scales	scale	NOUN
cana-3960	208	21	.	.	PUNCT
cana-3960	209	1	definition	definition	NOUN
cana-3960	209	2	4.1	4.1	NUM
cana-3960	209	3	.	.	PUNCT
cana-3960	210	1	the	the	DET
cana-3960	210	2	system	system	NOUN
cana-3960	210	3	(	(	PUNCT
cana-3960	210	4	2.2	2.2	NUM
cana-3960	210	5	)	)	PUNCT
cana-3960	210	6	is	be	AUX
cana-3960	210	7	said	say	VERB
cana-3960	210	8	to	to	PART
cana-3960	210	9	be	be	AUX
cana-3960	210	10	completely	completely	ADV
cana-3960	210	11	observability	observability	ADJ
cana-3960	210	12	on	on	ADP
cana-3960	210	13	[	[	X
cana-3960	210	14	𝑡0	𝑡0	NOUN
cana-3960	210	15	,	,	PUNCT
cana-3960	210	16	𝑡𝑓]𝕋	𝑡𝑓]𝕋	PROPN
cana-3960	210	17	(	(	PUNCT
cana-3960	210	18	𝑡𝑓	𝑡𝑓	NOUN
cana-3960	210	19	>	>	X
cana-3960	210	20	𝑡0	𝑡0	PROPN
cana-3960	210	21	)	)	PUNCT
cana-3960	210	22	if	if	SCONJ
cana-3960	210	23	any	any	DET
cana-3960	210	24	initial	initial	ADJ
cana-3960	210	25	state	state	NOUN
cana-3960	210	26	𝑧(𝑡0	𝑧(𝑡0	NUM
cana-3960	210	27	)	)	PUNCT
cana-3960	210	28	=	=	NOUN
cana-3960	210	29	𝑧0	𝑧0	PROPN
cana-3960	210	30	∈	∈	PROPN
cana-3960	211	1	ℝ𝑛	ℝ𝑛	X
cana-3960	211	2	2	2	NUM
cana-3960	211	3	is	be	AUX
cana-3960	211	4	uniquely	uniquely	ADV
cana-3960	211	5	determined	determine	VERB
cana-3960	211	6	by	by	ADP
cana-3960	211	7	the	the	DET
cana-3960	211	8	corresponding	correspond	VERB
cana-3960	211	9	system	system	NOUN
cana-3960	211	10	input	input	NOUN
cana-3960	211	11	�	�	PROPN
cana-3960	211	12	̂	̂	NOUN
cana-3960	211	13	�	�	NOUN
cana-3960	211	14	(𝑡	(𝑡	NOUN
cana-3960	211	15	)	)	PUNCT
cana-3960	211	16	and	and	CCONJ
cana-3960	211	17	the	the	DET
cana-3960	211	18	system	system	NOUN
cana-3960	211	19	output	output	NOUN
cana-3960	211	20	y(t	y(t	PROPN
cana-3960	211	21	)	)	PUNCT
cana-3960	211	22	for	for	ADP
cana-3960	211	23	[	[	X
cana-3960	211	24	𝑡0	𝑡0	NOUN
cana-3960	211	25	,	,	PUNCT
cana-3960	211	26	𝑡𝑓]𝕋.	𝑡𝑓]𝕋.	PRON
cana-3960	211	27	theorem	theorem	ADJ
cana-3960	211	28	4.1	4.1	NUM
cana-3960	211	29	.	.	PUNCT
cana-3960	211	30	suppose	suppose	VERB
cana-3960	211	31	that	that	SCONJ
cana-3960	211	32	[	[	X
cana-3960	211	33	𝐼𝑛⊗𝑅𝑗	𝐼𝑛⊗𝑅𝑗	X
cana-3960	211	34	]	]	X
cana-3960	211	35	≥	≥	X
cana-3960	211	36	0	0	NUM
cana-3960	211	37	,	,	PUNCT
cana-3960	211	38	𝑗	𝑗	NOUN
cana-3960	211	39	=	=	SYM
cana-3960	211	40	1,2	1,2	NUM
cana-3960	211	41	,	,	PUNCT
cana-3960	211	42	…	…	PUNCT
cana-3960	211	43	,	,	PUNCT
cana-3960	211	44	𝑙.	𝑙.	ADV
cana-3960	211	45	then	then	ADV
cana-3960	211	46	,	,	PUNCT
cana-3960	211	47	the	the	DET
cana-3960	211	48	impulsive	impulsive	ADJ
cana-3960	211	49	system	system	NOUN
cana-3960	211	50	(	(	PUNCT
cana-3960	211	51	2.2	2.2	NUM
cana-3960	211	52	)	)	PUNCT
cana-3960	211	53	is	be	AUX
cana-3960	211	54	observable	observable	ADJ
cana-3960	211	55	on	on	ADP
cana-3960	211	56	𝑡	𝑡	PROPN
cana-3960	211	57	∈	∈	PROPN
cana-3960	211	58	[	[	X
cana-3960	211	59	𝑡0	𝑡0	NOUN
cana-3960	211	60	,	,	PUNCT
cana-3960	211	61	𝑡𝑓]𝕋	𝑡𝑓]𝕋	PROPN
cana-3960	211	62	(	(	PUNCT
cana-3960	211	63	𝑡𝑓	𝑡𝑓	ADV
cana-3960	211	64	∈	∈	PROPN
cana-3960	212	1	[	[	X
cana-3960	212	2	𝑡𝑙−1	𝑡𝑙−1	X
cana-3960	212	3	,	,	PUNCT
cana-3960	212	4	𝑡𝑙)𝕋	𝑡𝑙)𝕋	NOUN
cana-3960	212	5	)	)	PUNCT
cana-3960	212	6	if	if	SCONJ
cana-3960	212	7	and	and	CCONJ
cana-3960	212	8	only	only	ADV
cana-3960	212	9	if	if	SCONJ
cana-3960	212	10	the	the	DET
cana-3960	212	11	matrix	matrix	NOUN
cana-3960	212	12	𝑊(𝑡0	𝑊(𝑡0	NOUN
cana-3960	212	13	,	,	PUNCT
cana-3960	212	14	𝑡𝑓	𝑡𝑓	NOUN
cana-3960	212	15	)	)	PUNCT
cana-3960	212	16	≔	≔	NOUN
cana-3960	212	17	𝑊(𝑡0	𝑊(𝑡0	PROPN
cana-3960	212	18	,	,	PUNCT
cana-3960	212	19	𝑡0	𝑡0	PROPN
cana-3960	212	20	,	,	PUNCT
cana-3960	212	21	𝑡1	𝑡1	NOUN
cana-3960	212	22	)	)	PUNCT
cana-3960	213	1	+	+	NOUN
cana-3960	213	2	∑∏[𝐼𝑛⊗𝑅𝑖]𝑊(𝑡0	∑∏[𝐼𝑛⊗𝑅𝑖]𝑊(𝑡0	NUM
cana-3960	213	3	,	,	PUNCT
cana-3960	213	4	𝑡𝑗−1	𝑡𝑗−1	PROPN
cana-3960	213	5	,	,	PUNCT
cana-3960	213	6	𝑡𝑗	𝑡𝑗	NOUN
cana-3960	213	7	)	)	PUNCT
cana-3960	213	8	+	+	NUM
cana-3960	213	9	𝑗	𝑗	PROPN
cana-3960	213	10	𝑖=1	𝑖=1	PROPN
cana-3960	213	11	𝑙−1	𝑙−1	PROPN
cana-3960	213	12	𝑗=2	𝑗=2	PROPN
cana-3960	213	13	∏[𝐼𝑛⊗𝑅𝑖]𝑊(𝑡0	∏[𝐼𝑛⊗𝑅𝑖]𝑊(𝑡0	NOUN
cana-3960	213	14	,	,	PUNCT
cana-3960	213	15	𝑡𝑙−1	𝑡𝑙−1	X
cana-3960	213	16	,	,	PUNCT
cana-3960	213	17	𝑡𝑓	𝑡𝑓	NOUN
cana-3960	213	18	)	)	PUNCT
cana-3960	213	19	𝑙	𝑙	X
cana-3960	213	20	𝑖=1	𝑖=1	PROPN
cana-3960	213	21	is	be	AUX
cana-3960	213	22	invertible	invertible	ADJ
cana-3960	213	23	,	,	PUNCT
cana-3960	213	24	where	where	SCONJ
cana-3960	213	25	𝑊(𝑡0	𝑊(𝑡0	PROPN
cana-3960	213	26	,	,	PUNCT
cana-3960	213	27	𝑡0	𝑡0	PROPN
cana-3960	213	28	,	,	PUNCT
cana-3960	213	29	𝑡1	𝑡1	NOUN
cana-3960	213	30	)	)	PUNCT
cana-3960	213	31	≔	≔	NOUN
cana-3960	213	32	∫	∫	PROPN
cana-3960	213	33	𝜓𝐺1(𝑡0	𝜓𝐺1(𝑡0	PROPN
cana-3960	213	34	,	,	PUNCT
cana-3960	213	35	𝜏)(i⊗𝐶1	𝜏)(i⊗𝐶1	NUM
cana-3960	213	36	)	)	PUNCT
cana-3960	213	37	𝑇(𝜏)(i⊗𝐶1)(𝜏)𝜓𝐺1	𝑇(𝜏)(i⊗𝐶1)(𝜏)𝜓𝐺1	VERB
cana-3960	213	38	𝑇	𝑇	PROPN
cana-3960	213	39	(	(	PUNCT
cana-3960	213	40	𝑡0	𝑡0	PROPN
cana-3960	213	41	,	,	PUNCT
cana-3960	213	42	𝜏)∆𝜏	𝜏)∆𝜏	VERB
cana-3960	213	43	𝑡1	𝑡1	NOUN
cana-3960	213	44	𝑡0	𝑡0	NOUN
cana-3960	213	45	,	,	PUNCT
cana-3960	213	46	𝑊(𝑡0	𝑊(𝑡0	PROPN
cana-3960	213	47	,	,	PUNCT
cana-3960	213	48	𝑡𝑗−1	𝑡𝑗−1	PROPN
cana-3960	213	49	,	,	PUNCT
cana-3960	213	50	𝑡𝑗	𝑡𝑗	NOUN
cana-3960	213	51	)	)	PUNCT
cana-3960	213	52	≔	≔	NOUN
cana-3960	213	53	∫	∫	PROPN
cana-3960	213	54	ω𝑗(𝑡0	ω𝑗(𝑡0	PROPN
cana-3960	213	55	,	,	PUNCT
cana-3960	213	56	𝜏)(i⊗𝐶𝑗	𝜏)(i⊗𝐶𝑗	PROPN
cana-3960	213	57	)	)	PUNCT
cana-3960	213	58	𝑇	𝑇	PROPN
cana-3960	213	59	(	(	PUNCT
cana-3960	213	60	𝜏)(i⊗𝐶𝑗)(𝜏)ω𝑗	𝜏)(i⊗𝐶𝑗)(𝜏)ω𝑗	PROPN
cana-3960	213	61	𝑇(𝑡0	𝑇(𝑡0	NUM
cana-3960	213	62	,	,	PUNCT
cana-3960	213	63	𝜏)∆𝜏	𝜏)∆𝜏	VERB
cana-3960	213	64	𝑡𝑗	𝑡𝑗	PROPN
cana-3960	213	65	𝑡𝑗−1	𝑡𝑗−1	PROPN
cana-3960	213	66	,	,	PUNCT
cana-3960	213	67	𝑗	𝑗	NOUN
cana-3960	213	68	=	=	SYM
cana-3960	213	69	2	2	NUM
cana-3960	213	70	,	,	PUNCT
cana-3960	213	71	…	…	PUNCT
cana-3960	213	72	,	,	PUNCT
cana-3960	213	73	𝑙	𝑙	X
cana-3960	213	74	−	−	PROPN
cana-3960	213	75	1	1	NUM
cana-3960	213	76	,	,	PUNCT
cana-3960	213	77	and	and	CCONJ
cana-3960	213	78	𝑊(𝑡0	𝑊(𝑡0	PROPN
cana-3960	213	79	,	,	PUNCT
cana-3960	213	80	𝑡𝑙−1	𝑡𝑙−1	X
cana-3960	213	81	,	,	PUNCT
cana-3960	213	82	𝑡𝑓	𝑡𝑓	ADV
cana-3960	213	83	)	)	PUNCT
cana-3960	213	84	≔	≔	VERB
cana-3960	213	85	∫	∫	PROPN
cana-3960	213	86	ω𝑙(𝑡0	ω𝑙(𝑡0	ADP
cana-3960	213	87	,	,	PUNCT
cana-3960	213	88	𝜏	𝜏	NOUN
cana-3960	213	89	,	,	PUNCT
cana-3960	213	90	)	)	PUNCT
cana-3960	213	91	(	(	PUNCT
cana-3960	213	92	i⊗𝐶𝑙	i⊗𝐶𝑙	NOUN
cana-3960	213	93	)	)	PUNCT
cana-3960	213	94	𝑇(𝜏)(i⊗𝐶𝑙)(𝜏)ω𝑙	𝑇(𝜏)(i⊗𝐶𝑙)(𝜏)ω𝑙	CCONJ
cana-3960	214	1	𝑇(𝑡0	𝑇(𝑡0	ADJ
cana-3960	214	2	,	,	PUNCT
cana-3960	214	3	𝜏)∆𝜏	𝜏)∆𝜏	NOUN
cana-3960	214	4	𝑡𝑓	𝑡𝑓	X
cana-3960	214	5	𝑡𝑙−1	𝑡𝑙−1	ADV
cana-3960	214	6	,	,	PUNCT
cana-3960	214	7	with	with	ADP
cana-3960	214	8	ω𝑗	ω𝑗	ADP
cana-3960	214	9	𝑇(𝑡0	𝑇(𝑡0	NUM
cana-3960	214	10	,	,	PUNCT
cana-3960	214	11	𝜏	𝜏	NOUN
cana-3960	214	12	)	)	PUNCT
cana-3960	214	13	=	=	PUNCT
cana-3960	215	1	𝜓𝐺𝑗	𝜓𝐺𝑗	PROPN
cana-3960	215	2	𝑇	𝑇	PROPN
cana-3960	215	3	(	(	PUNCT
cana-3960	215	4	𝑡𝑗−1	𝑡𝑗−1	PROPN
cana-3960	215	5	,	,	PUNCT
cana-3960	215	6	𝜏)𝜓𝐺𝑗−1	𝜏)𝜓𝐺𝑗−1	ADP
cana-3960	215	7	𝑇	𝑇	PROPN
cana-3960	215	8	(	(	PUNCT
cana-3960	215	9	𝑡𝑗−2	𝑡𝑗−2	PROPN
cana-3960	215	10	,	,	PUNCT
cana-3960	215	11	𝑡𝑗−1)	𝑡𝑗−1)	PRON
cana-3960	215	12	…	…	SYM
cana-3960	215	13	𝜓𝐺𝑗	𝜓𝐺𝑗	NOUN
cana-3960	215	14	𝑇	𝑇	PROPN
cana-3960	215	15	(	(	PUNCT
cana-3960	215	16	𝑡0	𝑡0	PROPN
cana-3960	215	17	,	,	PUNCT
cana-3960	215	18	𝑡1	𝑡1	NOUN
cana-3960	215	19	)	)	PUNCT
cana-3960	215	20	,	,	PUNCT
cana-3960	215	21	𝑗	𝑗	NOUN
cana-3960	215	22	=	=	SYM
cana-3960	215	23	1	1	NUM
cana-3960	215	24	,	,	PUNCT
cana-3960	215	25	…	…	PUNCT
cana-3960	215	26	,	,	PUNCT
cana-3960	215	27	𝑘	𝑘	DET
cana-3960	215	28	proof	proof	NOUN
cana-3960	215	29	:	:	PUNCT
cana-3960	215	30	assume	assume	VERB
cana-3960	215	31	that	that	SCONJ
cana-3960	215	32	the	the	DET
cana-3960	215	33	matrix	matrix	NOUN
cana-3960	215	34	𝑊(𝑡0	𝑊(𝑡0	NOUN
cana-3960	215	35	,	,	PUNCT
cana-3960	215	36	𝑡𝑓)𝕋	𝑡𝑓)𝕋	PROPN
cana-3960	215	37	is	be	AUX
cana-3960	215	38	invertible	invertible	ADJ
cana-3960	215	39	.	.	PUNCT
cana-3960	216	1	from	from	ADP
cana-3960	216	2	the	the	DET
cana-3960	216	3	system	system	NOUN
cana-3960	216	4	(	(	PUNCT
cana-3960	216	5	2.2	2.2	NUM
cana-3960	216	6	)	)	PUNCT
cana-3960	216	7	and	and	CCONJ
cana-3960	216	8	the	the	DET
cana-3960	216	9	equation	equation	NOUN
cana-3960	216	10	(	(	PUNCT
cana-3960	216	11	3.1	3.1	NUM
cana-3960	216	12	)	)	PUNCT
cana-3960	216	13	,	,	PUNCT
cana-3960	216	14	we	we	PRON
cana-3960	216	15	have	have	VERB
cana-3960	216	16	for	for	ADP
cana-3960	216	17	𝑡	𝑡	PROPN
cana-3960	216	18	∈	∈	PROPN
cana-3960	217	1	[	[	X
cana-3960	217	2	𝑡0	𝑡0	NOUN
cana-3960	217	3	,	,	PUNCT
cana-3960	217	4	𝑡𝑓]𝕋	𝑡𝑓]𝕋	PROPN
cana-3960	217	5	communications	communication	NOUN
cana-3960	217	6	on	on	ADP
cana-3960	217	7	applied	apply	VERB
cana-3960	217	8	nonlinear	nonlinear	ADJ
cana-3960	217	9	analysis	analysis	NOUN
cana-3960	217	10	issn	issn	NOUN
cana-3960	217	11	:	:	PUNCT
cana-3960	217	12	1074	1074	NUM
cana-3960	217	13	-	-	PUNCT
cana-3960	217	14	133x	133x	NUM
cana-3960	217	15	vol	vol	NOUN
cana-3960	217	16	32	32	NUM
cana-3960	217	17	no	no	NOUN
cana-3960	217	18	.	.	PUNCT
cana-3960	218	1	9s	9s	NUM
cana-3960	218	2	(	(	PUNCT
cana-3960	218	3	2025	2025	NUM
cana-3960	218	4	)	)	PUNCT
cana-3960	218	5	503	503	NUM
cana-3960	218	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3960	218	7	𝑦(𝑡	𝑦(𝑡	NUM
cana-3960	218	8	)	)	PUNCT
cana-3960	218	9	=	=	PUNCT
cana-3960	218	10	(	(	PUNCT
cana-3960	218	11	i⊗𝐶1)(𝑡)𝜓𝐺1	i⊗𝐶1)(𝑡)𝜓𝐺1	NOUN
cana-3960	218	12	𝑇	𝑇	PROPN
cana-3960	218	13	(	(	PUNCT
cana-3960	218	14	𝑡0	𝑡0	PROPN
cana-3960	218	15	,	,	PUNCT
cana-3960	218	16	𝜏)𝑧0	𝜏)𝑧0	PROPN
cana-3960	218	17	+	+	CCONJ
cana-3960	218	18	(	(	PUNCT
cana-3960	218	19	i⊗𝐶1)(𝑡	i⊗𝐶1)(𝑡	PROPN
cana-3960	218	20	)	)	PUNCT
cana-3960	218	21	∫	∫	PROPN
cana-3960	218	22	𝜓𝐺1	𝜓𝐺1	PROPN
cana-3960	218	23	𝑇	𝑇	PROPN
cana-3960	218	24	(	(	PUNCT
cana-3960	218	25	𝜏	𝜏	PROPN
cana-3960	218	26	,	,	PUNCT
cana-3960	218	27	𝑡)𝐴1(𝜏)𝑈(𝑡)∆𝜏	𝑡)𝐴1(𝜏)𝑈(𝑡)∆𝜏	PUNCT
cana-3960	218	28	𝑡1	𝑡1	NOUN
cana-3960	218	29	𝑡0	𝑡0	NOUN
cana-3960	218	30	+	+	CCONJ
cana-3960	218	31	(	(	PUNCT
cana-3960	218	32	i⊗𝐷1)	i⊗𝐷1)	PROPN
cana-3960	218	33	�	�	PROPN
cana-3960	218	34	̂	̂	NOUN
cana-3960	218	35	�	�	NOUN
cana-3960	218	36	(𝑡	(𝑡	NOUN
cana-3960	218	37	)	)	PUNCT
cana-3960	218	38	,	,	PUNCT
cana-3960	218	39	(	(	PUNCT
cana-3960	218	40	4.1	4.1	NUM
cana-3960	218	41	)	)	PUNCT
cana-3960	218	42	and	and	CCONJ
cana-3960	218	43	for	for	ADP
cana-3960	218	44	𝑡	𝑡	PROPN
cana-3960	218	45	∈	∈	PROPN
cana-3960	218	46	(	(	PUNCT
cana-3960	218	47	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	218	48	,	,	PUNCT
cana-3960	218	49	𝑡𝑘]𝕋	𝑡𝑘]𝕋	NOUN
cana-3960	218	50	,	,	PUNCT
cana-3960	218	51	𝑘	𝑘	X
cana-3960	218	52	=	=	NOUN
cana-3960	218	53	2,3	2,3	NUM
cana-3960	218	54	,	,	PUNCT
cana-3960	218	55	…	…	PUNCT
cana-3960	218	56	𝑙.	𝑙.	NOUN
cana-3960	218	57	𝑦(𝑡	𝑦(𝑡	NUM
cana-3960	218	58	)	)	PUNCT
cana-3960	218	59	=	=	PUNCT
cana-3960	218	60	(	(	PUNCT
cana-3960	218	61	i⊗𝐶𝑘)(𝑡)𝜓𝐺𝑘	i⊗𝐶𝑘)(𝑡)𝜓𝐺𝑘	PROPN
cana-3960	218	62	𝑇	𝑇	PROPN
cana-3960	218	63	(	(	PUNCT
cana-3960	218	64	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	218	65	,	,	PUNCT
cana-3960	218	66	𝑡	𝑡	NOUN
cana-3960	218	67	)	)	PUNCT
cana-3960	218	68	{	{	PUNCT
cana-3960	219	1	∏	∏	PROPN
cana-3960	219	2	[	[	X
cana-3960	219	3	𝐼𝑛⊗𝑅𝑗	𝐼𝑛⊗𝑅𝑗	X
cana-3960	219	4	]	]	X
cana-3960	219	5	1	1	NUM
cana-3960	219	6	𝑗=𝑘−1	𝑗=𝑘−1	PROPN
cana-3960	219	7	∏	∏	PROPN
cana-3960	219	8	𝜓𝐺𝑗	𝜓𝐺𝑗	NOUN
cana-3960	219	9	𝑇	𝑇	PROPN
cana-3960	219	10	(	(	PUNCT
cana-3960	219	11	𝑡𝑗−1	𝑡𝑗−1	PROPN
cana-3960	219	12	,	,	PUNCT
cana-3960	219	13	𝑡𝑗	𝑡𝑗	PROPN
cana-3960	219	14	)	)	PUNCT
cana-3960	219	15	1	1	NUM
cana-3960	219	16	𝑗=𝑘−1	𝑗=𝑘−1	PROPN
cana-3960	219	17	𝑧0(i⊗𝐶𝑘)(𝑡	𝑧0(i⊗𝐶𝑘)(𝑡	PROPN
cana-3960	219	18	)	)	PUNCT
cana-3960	220	1	+	+	NOUN
cana-3960	220	2	∑(∏	∑(∏	PROPN
cana-3960	220	3	[	[	X
cana-3960	220	4	𝐼𝑛⊗𝑅𝑖	𝐼𝑛⊗𝑅𝑖	X
cana-3960	220	5	]	]	X
cana-3960	220	6	𝑘	𝑘	X
cana-3960	220	7	𝑖=𝑘−1	𝑖=𝑘−1	PUNCT
cana-3960	220	8	∏	∏	PROPN
cana-3960	220	9	𝜓𝐺𝑖	𝜓𝐺𝑖	ADJ
cana-3960	220	10	𝑇	𝑇	PROPN
cana-3960	220	11	(	(	PUNCT
cana-3960	220	12	𝑡𝑖−1	𝑡𝑖−1	PROPN
cana-3960	220	13	,	,	PUNCT
cana-3960	220	14	𝑡𝑖	𝑡𝑖	NOUN
cana-3960	220	15	)	)	PUNCT
cana-3960	220	16	𝑘+1	𝑘+1	NUM
cana-3960	220	17	𝑖=𝑘−1	𝑖=𝑘−1	SYM
cana-3960	220	18	∫	∫	PROPN
cana-3960	221	1	𝜓𝐺𝑗	𝜓𝐺𝑗	NOUN
cana-3960	221	2	𝑇	𝑇	PROPN
cana-3960	221	3	(	(	PUNCT
cana-3960	221	4	𝜏	𝜏	PROPN
cana-3960	221	5	,	,	PUNCT
cana-3960	221	6	𝑡𝑗	𝑡𝑗	NOUN
cana-3960	221	7	)	)	PUNCT
cana-3960	221	8	𝑡𝑗	𝑡𝑗	PROPN
cana-3960	221	9	𝑡𝑗−1	𝑡𝑗−1	PROPN
cana-3960	221	10	𝐴𝑗(𝜏)	𝐴𝑗(𝜏)	NOUN
cana-3960	221	11	�	�	PROPN
cana-3960	221	12	̂	̂	NOUN
cana-3960	221	13	�	�	NOUN
cana-3960	221	14	(𝜏)∆𝜏	(𝜏)∆𝜏	NOUN
cana-3960	221	15	)	)	PUNCT
cana-3960	221	16	𝑘−2	𝑘−2	PROPN
cana-3960	221	17	𝑗=1	𝑗=1	PROPN
cana-3960	221	18	(	(	PUNCT
cana-3960	221	19	i⊗𝐶𝑘)(𝑡	i⊗𝐶𝑘)(𝑡	PROPN
cana-3960	221	20	)	)	PUNCT
cana-3960	221	21	+	+	PROPN
cana-3960	221	22	[	[	X
cana-3960	221	23	𝐼𝑛⊗𝑅𝑙−1	𝐼𝑛⊗𝑅𝑙−1	X
cana-3960	221	24	]	]	X
cana-3960	221	25	∫	∫	PROPN
cana-3960	221	26	𝜓𝐺𝑘−1	𝜓𝐺𝑘−1	VERB
cana-3960	221	27	𝑇	𝑇	PROPN
cana-3960	221	28	(	(	PUNCT
cana-3960	221	29	𝜏	𝜏	PROPN
cana-3960	221	30	,	,	PUNCT
cana-3960	221	31	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	221	32	)	)	PUNCT
cana-3960	221	33	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	221	34	𝑡𝑘−2	𝑡𝑘−2	PROPN
cana-3960	221	35	𝐴𝑘−1(𝜏)	𝐴𝑘−1(𝜏)	NOUN
cana-3960	221	36	�	�	PROPN
cana-3960	221	37	̂	̂	NOUN
cana-3960	221	38	�	�	NOUN
cana-3960	221	39	(𝜏)∆𝜏}(i⊗𝐶𝑘)(𝑡	(𝜏)∆𝜏}(i⊗𝐶𝑘)(𝑡	NOUN
cana-3960	221	40	)	)	PUNCT
cana-3960	222	1	+	+	NUM
cana-3960	222	2	∫	∫	PROPN
cana-3960	222	3	𝜓𝐺𝑘	𝜓𝐺𝑘	PROPN
cana-3960	222	4	𝑇	𝑇	PROPN
cana-3960	222	5	(	(	PUNCT
cana-3960	222	6	𝜏	𝜏	PROPN
cana-3960	222	7	,	,	PUNCT
cana-3960	222	8	𝑡	𝑡	NOUN
cana-3960	222	9	)	)	PUNCT
cana-3960	222	10	𝑡	𝑡	PROPN
cana-3960	222	11	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	222	12	𝐴𝑘(𝜏)	𝐴𝑘(𝜏)	NOUN
cana-3960	222	13	�	�	PROPN
cana-3960	222	14	̂	̂	VERB
cana-3960	222	15	�	�	NOUN
cana-3960	222	16	(𝜏)∆𝜏	(𝜏)∆𝜏	SYM
cana-3960	222	17	+	+	CCONJ
cana-3960	222	18	(	(	PUNCT
cana-3960	222	19	i⊗𝐷𝑘)	i⊗𝐷𝑘)	PROPN
cana-3960	222	20	�	�	PROPN
cana-3960	222	21	̂	̂	NUM
cana-3960	222	22	�	�	NOUN
cana-3960	222	23	(𝑡	(𝑡	NOUN
cana-3960	222	24	)	)	PUNCT
cana-3960	222	25	(	(	PUNCT
cana-3960	222	26	4.2	4.2	NUM
cana-3960	222	27	)	)	PUNCT
cana-3960	222	28	from	from	ADP
cana-3960	222	29	the	the	DET
cana-3960	222	30	definition	definition	NOUN
cana-3960	222	31	4.1	4.1	NUM
cana-3960	222	32	.	.	PUNCT
cana-3960	222	33	,	,	PUNCT
cana-3960	222	34	that	that	SCONJ
cana-3960	222	35	the	the	DET
cana-3960	222	36	observability	observability	NOUN
cana-3960	222	37	of	of	ADP
cana-3960	222	38	the	the	DET
cana-3960	222	39	system	system	NOUN
cana-3960	222	40	(	(	PUNCT
cana-3960	222	41	2.2	2.2	NUM
cana-3960	222	42	)	)	PUNCT
cana-3960	222	43	is	be	AUX
cana-3960	222	44	𝑦(𝑡	𝑦(𝑡	NUM
cana-3960	222	45	)	)	PUNCT
cana-3960	222	46	=	=	PRON
cana-3960	222	47	{	{	PUNCT
cana-3960	222	48	(	(	PUNCT
cana-3960	222	49	i⊗𝐶𝑘)(𝑡)𝜓𝐺𝑘	i⊗𝐶𝑘)(𝑡)𝜓𝐺𝑘	PROPN
cana-3960	222	50	𝑇	𝑇	PROPN
cana-3960	222	51	(	(	PUNCT
cana-3960	222	52	𝑡0	𝑡0	PROPN
cana-3960	222	53	,	,	PUNCT
cana-3960	222	54	𝑡)𝑧0	𝑡)𝑧0	PROPN
cana-3960	222	55	,	,	PUNCT
cana-3960	222	56	𝑡	𝑡	PROPN
cana-3960	222	57	∈	∈	PROPN
cana-3960	222	58	[	[	X
cana-3960	222	59	𝑡0	𝑡0	NOUN
cana-3960	222	60	,	,	PUNCT
cana-3960	222	61	𝑡1]𝕋	𝑡1]𝕋	PROPN
cana-3960	222	62	∏	∏	PROPN
cana-3960	223	1	[	[	X
cana-3960	223	2	𝐼𝑛⊗𝑅𝑗](i⊗𝐶𝑘)𝜓𝐺𝑘	𝐼𝑛⊗𝑅𝑗](i⊗𝐶𝑘)𝜓𝐺𝑘	PROPN
cana-3960	223	3	𝑇	𝑇	PROPN
cana-3960	223	4	(	(	PUNCT
cana-3960	223	5	𝑡0	𝑡0	PROPN
cana-3960	223	6	,	,	PUNCT
cana-3960	223	7	𝑡	𝑡	PROPN
cana-3960	223	8	)	)	PUNCT
cana-3960	223	9	1	1	NUM
cana-3960	223	10	𝑗=𝑘−1	𝑗=𝑘−1	PROPN
cana-3960	223	11	𝑧0	𝑧0	PROPN
cana-3960	223	12	,	,	PUNCT
cana-3960	223	13	𝑡	𝑡	PROPN
cana-3960	223	14	∈	∈	PROPN
cana-3960	223	15	(	(	PUNCT
cana-3960	223	16	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	223	17	,	,	PUNCT
cana-3960	223	18	𝑡𝑘]𝕋	𝑡𝑘]𝕋	NOUN
cana-3960	223	19	,	,	PUNCT
cana-3960	223	20	𝑘	𝑘	X
cana-3960	223	21	=	=	NOUN
cana-3960	223	22	2,3	2,3	NUM
cana-3960	223	23	,	,	PUNCT
cana-3960	223	24	…	…	PUNCT
cana-3960	223	25	,	,	PUNCT
cana-3960	223	26	𝑙	𝑙	X
cana-3960	223	27	(	(	PUNCT
cana-3960	223	28	4.3	4.3	NUM
cana-3960	223	29	)	)	PUNCT
cana-3960	223	30	as	as	ADP
cana-3960	223	31	�	�	PROPN
cana-3960	223	32	̂	̂	X
cana-3960	223	33	�	�	NOUN
cana-3960	223	34	(𝑡	(𝑡	NOUN
cana-3960	223	35	)	)	PUNCT
cana-3960	223	36	=	=	SYM
cana-3960	223	37	0	0	X
cana-3960	223	38	.	.	PUNCT
cana-3960	223	39	now	now	ADV
cana-3960	223	40	multiply	multiply	VERB
cana-3960	223	41	by	by	ADP
cana-3960	223	42	ω𝑘(𝑡0	ω𝑘(𝑡0	NUM
cana-3960	223	43	,	,	PUNCT
cana-3960	223	44	𝑡)(i⊗𝐶𝑘	𝑡)(i⊗𝐶𝑘	NUM
cana-3960	223	45	)	)	PUNCT
cana-3960	223	46	𝑇(𝑡	𝑇(𝑡	NOUN
cana-3960	223	47	)	)	PUNCT
cana-3960	223	48	to	to	ADP
cana-3960	223	49	both	both	DET
cana-3960	223	50	sides	side	NOUN
cana-3960	223	51	of	of	ADP
cana-3960	223	52	the	the	DET
cana-3960	223	53	equation	equation	NOUN
cana-3960	223	54	(	(	PUNCT
cana-3960	223	55	4.3	4.3	NUM
cana-3960	223	56	)	)	PUNCT
cana-3960	223	57	and	and	CCONJ
cana-3960	223	58	integrating	integrate	VERB
cana-3960	223	59	with	with	ADP
cana-3960	223	60	respect	respect	NOUN
cana-3960	223	61	to	to	ADP
cana-3960	223	62	𝑡0	𝑡0	PROPN
cana-3960	223	63	𝑡𝑜	𝑡𝑜	PROPN
cana-3960	223	64	𝑡𝑓	𝑡𝑓	ADV
cana-3960	223	65	,	,	PUNCT
cana-3960	223	66	we	we	PRON
cana-3960	223	67	get	get	VERB
cana-3960	223	68	∫	∫	PROPN
cana-3960	223	69	ω𝐼(𝐼0,𝐼	ω𝐼(𝐼0,𝐼	NOUN
cana-3960	223	70	,	,	PUNCT
cana-3960	223	71	)	)	PUNCT
cana-3960	223	72	(	(	PUNCT
cana-3960	223	73	i⊗	i⊗	PROPN
cana-3960	223	74	𝐼𝐼	𝐼𝐼	PROPN
cana-3960	223	75	)	)	PUNCT
cana-3960	223	76	𝐼(𝐼)𝐼(𝐼)∆𝐼	𝐼(𝐼)𝐼(𝐼)∆𝐼	PROPN
cana-3960	223	77	𝐼𝐼	𝐼𝐼	PROPN
cana-3960	223	78	𝐼0	𝐼0	NOUN
cana-3960	223	79	=	=	PUNCT
cana-3960	223	80	[	[	PUNCT
cana-3960	223	81	∫	∫	X
cana-3960	223	82	𝐼𝐼1	𝐼𝐼1	PROPN
cana-3960	223	83	(	(	PUNCT
cana-3960	223	84	𝐼0,𝐼	𝐼0,𝐼	PROPN
cana-3960	223	85	,	,	PUNCT
cana-3960	223	86	)	)	PUNCT
cana-3960	223	87	(	(	PUNCT
cana-3960	223	88	i⊗𝐼1	i⊗𝐼1	X
cana-3960	223	89	)	)	PUNCT
cana-3960	223	90	𝐼(𝐼)(i⊗𝐼1)(𝐼)𝐼𝐼1	𝐼(𝐼)(i⊗𝐼1)(𝐼)𝐼𝐼1	PUNCT
cana-3960	224	1	𝐼	𝐼	PROPN
cana-3960	224	2	(	(	PUNCT
cana-3960	224	3	𝐼0,𝐼)∆𝐼	𝐼0,𝐼)∆𝐼	PROPN
cana-3960	224	4	𝐼1	𝐼1	NOUN
cana-3960	224	5	𝐼0	𝐼0	PROPN
cana-3960	224	6	+	+	SYM
cana-3960	224	7	∑∏[𝐼𝐼⊗𝐼𝐼	∑∏[𝐼𝐼⊗𝐼𝐼	VERB
cana-3960	224	8	]	]	PUNCT
cana-3960	224	9	𝐼	𝐼	PROPN
cana-3960	224	10	𝐼=𝐼	𝐼=𝐼	NOUN
cana-3960	224	11	∫	∫	PROPN
cana-3960	224	12	ω𝐼(𝐼0,𝐼	ω𝐼(𝐼0,𝐼	NOUN
cana-3960	224	13	,	,	PUNCT
cana-3960	224	14	)	)	PUNCT
cana-3960	224	15	(	(	PUNCT
cana-3960	224	16	i⊗𝐼𝐼	i⊗𝐼𝐼	NOUN
cana-3960	224	17	)	)	PUNCT
cana-3960	224	18	𝐼(𝐼)(i⊗𝐼𝐼)(𝐼)ω𝐼	𝐼(𝐼)(i⊗𝐼𝐼)(𝐼)ω𝐼	PROPN
cana-3960	224	19	𝐼(𝐼0,𝐼)∆𝐼	𝐼(𝐼0,𝐼)∆𝐼	PUNCT
cana-3960	224	20	𝐼𝐼	𝐼𝐼	PROPN
cana-3960	224	21	𝐼𝐼−1	𝐼𝐼−1	NOUN
cana-3960	224	22	𝐼−1	𝐼−1	PROPN
cana-3960	224	23	𝐼=2	𝐼=2	PUNCT
cana-3960	225	1	+	+	NOUN
cana-3960	225	2	∏[𝐼𝑛⊗𝑅𝑖	∏[𝐼𝑛⊗𝑅𝑖	X
cana-3960	225	3	]	]	X
cana-3960	225	4	𝑘	𝑘	X
cana-3960	225	5	𝑖=1	𝑖=1	PROPN
cana-3960	225	6	∫	∫	PROPN
cana-3960	225	7	ω𝑙(𝑡0	ω𝑙(𝑡0	NUM
cana-3960	225	8	,	,	PUNCT
cana-3960	225	9	𝜏	𝜏	NOUN
cana-3960	225	10	,	,	PUNCT
cana-3960	225	11	)	)	PUNCT
cana-3960	225	12	(	(	PUNCT
cana-3960	225	13	i⊗𝐶𝑙	i⊗𝐶𝑙	NOUN
cana-3960	225	14	)	)	PUNCT
cana-3960	225	15	𝑇(𝜏)(i⊗𝐶𝑙)(𝜏)ω𝑙	𝑇(𝜏)(i⊗𝐶𝑙)(𝜏)ω𝑙	CCONJ
cana-3960	226	1	𝑇(𝑡0	𝑇(𝑡0	ADJ
cana-3960	226	2	,	,	PUNCT
cana-3960	226	3	𝜏)∆𝜏	𝜏)∆𝜏	NOUN
cana-3960	226	4	𝑡𝑓	𝑡𝑓	X
cana-3960	226	5	𝑡𝑙−1	𝑡𝑙−1	X
cana-3960	226	6	]	]	PUNCT
cana-3960	226	7	𝑧0	𝑧0	PROPN
cana-3960	226	8	and	and	CCONJ
cana-3960	226	9	so	so	ADV
cana-3960	226	10	,	,	PUNCT
cana-3960	226	11	communications	communication	NOUN
cana-3960	226	12	on	on	ADP
cana-3960	226	13	applied	apply	VERB
cana-3960	226	14	nonlinear	nonlinear	ADJ
cana-3960	226	15	analysis	analysis	NOUN
cana-3960	226	16	issn	issn	NOUN
cana-3960	226	17	:	:	PUNCT
cana-3960	226	18	1074	1074	NUM
cana-3960	226	19	-	-	PUNCT
cana-3960	226	20	133x	133x	NUM
cana-3960	226	21	vol	vol	NOUN
cana-3960	226	22	32	32	NUM
cana-3960	227	1	no	no	NOUN
cana-3960	227	2	.	.	PUNCT
cana-3960	228	1	9s	9s	NUM
cana-3960	228	2	(	(	PUNCT
cana-3960	228	3	2025	2025	NUM
cana-3960	228	4	)	)	PUNCT
cana-3960	228	5	504	504	NUM
cana-3960	228	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3960	228	7	∫	∫	PROPN
cana-3960	228	8	ω𝑘(𝑡0	ω𝑘(𝑡0	NUM
cana-3960	228	9	,	,	PUNCT
cana-3960	228	10	𝜏	𝜏	NOUN
cana-3960	228	11	,	,	PUNCT
cana-3960	228	12	)	)	PUNCT
cana-3960	228	13	(	(	PUNCT
cana-3960	228	14	i⊗𝐶𝑘	i⊗𝐶𝑘	NOUN
cana-3960	228	15	)	)	PUNCT
cana-3960	228	16	𝑇(𝜏)𝑦(𝜏)∆𝜏	𝑇(𝜏)𝑦(𝜏)∆𝜏	NOUN
cana-3960	228	17	𝑡𝑓	𝑡𝑓	NOUN
cana-3960	228	18	𝑡0	𝑡0	NOUN
cana-3960	228	19	=	=	SYM
cana-3960	228	20	𝑊(𝑡0	𝑊(𝑡0	PROPN
cana-3960	228	21	,	,	PUNCT
cana-3960	228	22	𝑡𝑓)𝑧0	𝑡𝑓)𝑧0	NOUN
cana-3960	228	23	.	.	PUNCT
cana-3960	229	1	(	(	PUNCT
cana-3960	229	2	4.4	4.4	NUM
cana-3960	229	3	)	)	PUNCT
cana-3960	229	4	obviously	obviously	ADV
cana-3960	229	5	,	,	PUNCT
cana-3960	229	6	the	the	DET
cana-3960	229	7	left	leave	VERB
cana-3960	229	8	-	-	PUNCT
cana-3960	229	9	hand	hand	NOUN
cana-3960	229	10	side	side	NOUN
cana-3960	229	11	of	of	ADP
cana-3960	229	12	equation	equation	NOUN
cana-3960	229	13	(	(	PUNCT
cana-3960	229	14	4.4	4.4	NUM
cana-3960	229	15	)	)	PUNCT
cana-3960	229	16	depends	depend	VERB
cana-3960	229	17	on	on	ADP
cana-3960	229	18	𝑦(𝑡	𝑦(𝑡	NUM
cana-3960	229	19	)	)	PUNCT
cana-3960	229	20	,	,	PUNCT
cana-3960	229	21	𝑡	𝑡	PROPN
cana-3960	229	22	∈	∈	PROPN
cana-3960	229	23	[	[	X
cana-3960	229	24	𝑡0	𝑡0	NOUN
cana-3960	229	25	,	,	PUNCT
cana-3960	229	26	𝑡𝑓]𝕋	𝑡𝑓]𝕋	PROPN
cana-3960	229	27	.	.	PUNCT
cana-3960	230	1	since	since	SCONJ
cana-3960	230	2	the	the	DET
cana-3960	230	3	matrix	matrix	NOUN
cana-3960	230	4	𝑊(𝑡0	𝑊(𝑡0	NOUN
cana-3960	230	5	,	,	PUNCT
cana-3960	230	6	𝑡𝑓	𝑡𝑓	X
cana-3960	230	7	)	)	PUNCT
cana-3960	230	8	is	be	AUX
cana-3960	230	9	invertible	invertible	ADJ
cana-3960	230	10	,	,	PUNCT
cana-3960	230	11	then	then	ADV
cana-3960	230	12	from	from	ADP
cana-3960	230	13	linear	linear	ADJ
cana-3960	230	14	algebraic	algebraic	ADJ
cana-3960	230	15	equations	equation	NOUN
cana-3960	230	16	(	(	PUNCT
cana-3960	230	17	4.4	4.4	NUM
cana-3960	230	18	)	)	PUNCT
cana-3960	230	19	we	we	PRON
cana-3960	230	20	deduce	deduce	VERB
cana-3960	230	21	that	that	PRON
cana-3960	230	22	𝑧(𝑡0	𝑧(𝑡0	X
cana-3960	230	23	)	)	PUNCT
cana-3960	231	1	=	=	NOUN
cana-3960	231	2	𝑧0	𝑧0	PROPN
cana-3960	231	3	is	be	AUX
cana-3960	231	4	a	a	PRON
cana-3960	231	5	uniquely	uniquely	ADV
cana-3960	231	6	determined	determine	VERB
cana-3960	231	7	by	by	ADP
cana-3960	231	8	the	the	DET
cana-3960	231	9	corresponding	correspond	VERB
cana-3960	231	10	system	system	NOUN
cana-3960	231	11	output	output	NOUN
cana-3960	231	12	𝑦(𝑡	𝑦(𝑡	NUM
cana-3960	231	13	)	)	PUNCT
cana-3960	231	14	,	,	PUNCT
cana-3960	231	15	𝑡	𝑡	PROPN
cana-3960	231	16	∈	∈	PROPN
cana-3960	231	17	[	[	X
cana-3960	231	18	𝑡0	𝑡0	NOUN
cana-3960	231	19	,	,	PUNCT
cana-3960	231	20	𝑡𝑓]𝕋.	𝑡𝑓]𝕋.	PRON
cana-3960	231	21	conversely	conversely	ADV
cana-3960	231	22	,	,	PUNCT
cana-3960	231	23	assume	assume	VERB
cana-3960	231	24	that	that	SCONJ
cana-3960	231	25	the	the	DET
cana-3960	231	26	matrix	matrix	NOUN
cana-3960	231	27	𝑊(𝑡0	𝑊(𝑡0	NOUN
cana-3960	231	28	,	,	PUNCT
cana-3960	231	29	𝑡𝑓	𝑡𝑓	ADV
cana-3960	231	30	)	)	PUNCT
cana-3960	231	31	is	be	AUX
cana-3960	231	32	not	not	PART
cana-3960	231	33	invertible	invertible	ADJ
cana-3960	231	34	,	,	PUNCT
cana-3960	231	35	then	then	ADV
cana-3960	231	36	there	there	PRON
cana-3960	231	37	exists	exist	VERB
cana-3960	231	38	a	a	DET
cana-3960	231	39	nonzero	nonzero	NOUN
cana-3960	231	40	𝑧𝛼	𝑧𝛼	ADP
cana-3960	231	41	∈	∈	PROPN
cana-3960	232	1	ℝ𝑛	ℝ𝑛	ADP
cana-3960	232	2	2	2	NUM
cana-3960	232	3	,	,	PUNCT
cana-3960	232	4	such	such	ADJ
cana-3960	232	5	that	that	PRON
cana-3960	232	6	𝑧𝛼	𝑧𝛼	ADP
cana-3960	232	7	𝑇𝑊(𝑡0	𝑇𝑊(𝑡0	PROPN
cana-3960	232	8	,	,	PUNCT
cana-3960	232	9	𝑡𝑓)𝑧𝛼	𝑡𝑓)𝑧𝛼	X
cana-3960	232	10	=	=	SYM
cana-3960	232	11	0	0	PROPN
cana-3960	232	12	.	.	PUNCT
cana-3960	233	1	since	since	SCONJ
cana-3960	233	2	,	,	PUNCT
cana-3960	233	3	[	[	X
cana-3960	233	4	𝐼𝑛⊗𝑅𝑗	𝐼𝑛⊗𝑅𝑗	X
cana-3960	233	5	]	]	X
cana-3960	233	6	≥	≥	X
cana-3960	233	7	0	0	NUM
cana-3960	233	8	,	,	PUNCT
cana-3960	233	9	𝑗	𝑗	NOUN
cana-3960	233	10	=	=	SYM
cana-3960	233	11	1,2	1,2	NUM
cana-3960	233	12	,	,	PUNCT
cana-3960	233	13	…	…	PUNCT
cana-3960	233	14	,	,	PUNCT
cana-3960	233	15	𝑙	𝑙	X
cana-3960	233	16	,	,	PUNCT
cana-3960	233	17	𝑊(𝑡0	𝑊(𝑡0	PROPN
cana-3960	233	18	,	,	PUNCT
cana-3960	233	19	𝑡0	𝑡0	PROPN
cana-3960	233	20	,	,	PUNCT
cana-3960	233	21	𝑡1	𝑡1	NOUN
cana-3960	233	22	)	)	PUNCT
cana-3960	233	23	,	,	PUNCT
cana-3960	233	24	𝑊(𝑡0	𝑊(𝑡0	PROPN
cana-3960	233	25	,	,	PUNCT
cana-3960	233	26	𝑡𝑗−1	𝑡𝑗−1	PROPN
cana-3960	233	27	,	,	PUNCT
cana-3960	233	28	𝑡𝑗	𝑡𝑗	PROPN
cana-3960	233	29	)	)	PUNCT
cana-3960	233	30	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-3960	233	31	𝑗	𝑗	NOUN
cana-3960	233	32	=	=	SYM
cana-3960	233	33	2,3	2,3	NUM
cana-3960	233	34	,	,	PUNCT
cana-3960	233	35	…	…	PUNCT
cana-3960	233	36	𝑙	𝑙	X
cana-3960	233	37	−	−	PROPN
cana-3960	233	38	1	1	NUM
cana-3960	233	39	and	and	CCONJ
cana-3960	233	40	𝑊(𝑡0	𝑊(𝑡0	PROPN
cana-3960	233	41	,	,	PUNCT
cana-3960	233	42	𝑡𝑙−1	𝑡𝑙−1	ADV
cana-3960	233	43	,	,	PUNCT
cana-3960	233	44	𝑡𝑓	𝑡𝑓	ADV
cana-3960	233	45	)	)	PUNCT
cana-3960	233	46	are	be	AUX
cana-3960	233	47	positive	positive	ADJ
cana-3960	233	48	semidefinite	semidefinite	NOUN
cana-3960	233	49	matrices	matrix	NOUN
cana-3960	233	50	,	,	PUNCT
cana-3960	233	51	we	we	PRON
cana-3960	233	52	get	get	VERB
cana-3960	233	53	𝑧𝛼	𝑧𝛼	ADP
cana-3960	233	54	𝑇𝑊(𝑡0	𝑇𝑊(𝑡0	NOUN
cana-3960	233	55	,	,	PUNCT
cana-3960	233	56	𝑡0	𝑡0	PROPN
cana-3960	233	57	,	,	PUNCT
cana-3960	233	58	𝑡1)𝑧𝛼	𝑡1)𝑧𝛼	PROPN
cana-3960	233	59	=	=	SYM
cana-3960	233	60	0	0	NUM
cana-3960	233	61	.	.	PUNCT
cana-3960	234	1	𝑧𝛼	𝑧𝛼	X
cana-3960	234	2	𝑇𝑊(𝑡0	𝑇𝑊(𝑡0	PROPN
cana-3960	234	3	,	,	PUNCT
cana-3960	234	4	𝑡𝑗−1	𝑡𝑗−1	PROPN
cana-3960	234	5	,	,	PUNCT
cana-3960	234	6	𝑡𝑗)𝑧𝛼	𝑡𝑗)𝑧𝛼	PUNCT
cana-3960	234	7	=	=	SYM
cana-3960	234	8	0	0	PROPN
cana-3960	234	9	.	.	PUNCT
cana-3960	235	1	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
cana-3960	235	2	𝑗	𝑗	X
cana-3960	235	3	=	=	ADJ
cana-3960	235	4	2	2	NUM
cana-3960	235	5	,	,	PUNCT
cana-3960	235	6	…	…	PUNCT
cana-3960	235	7	𝑙	𝑙	X
cana-3960	236	1	−	−	NUM
cana-3960	236	2	1	1	NUM
cana-3960	236	3	(	(	PUNCT
cana-3960	236	4	4.5	4.5	NUM
cana-3960	236	5	)	)	PUNCT
cana-3960	236	6	𝑧𝛼	𝑧𝛼	ADP
cana-3960	236	7	𝑇𝑊(𝑡0	𝑇𝑊(𝑡0	PROPN
cana-3960	236	8	,	,	PUNCT
cana-3960	236	9	𝑡𝑙−1	𝑡𝑙−1	X
cana-3960	236	10	,	,	PUNCT
cana-3960	236	11	𝑡𝑓)𝑧𝛼	𝑡𝑓)𝑧𝛼	PROPN
cana-3960	236	12	=	=	SYM
cana-3960	236	13	0	0	X
cana-3960	236	14	.	.	PUNCT
cana-3960	237	1	we	we	PRON
cana-3960	237	2	choose	choose	VERB
cana-3960	237	3	𝑧0	𝑧0	PROPN
cana-3960	237	4	=	=	SYM
cana-3960	237	5	𝑧𝛼.	𝑧𝛼.	NOUN
cana-3960	237	6	thus	thus	ADV
cana-3960	237	7	,	,	PUNCT
cana-3960	237	8	from	from	ADP
cana-3960	237	9	equations	equation	NOUN
cana-3960	237	10	(	(	PUNCT
cana-3960	237	11	4.3	4.3	NUM
cana-3960	237	12	)	)	PUNCT
cana-3960	237	13	and	and	CCONJ
cana-3960	237	14	(	(	PUNCT
cana-3960	237	15	4.5	4.5	NUM
cana-3960	237	16	)	)	PUNCT
cana-3960	237	17	,	,	PUNCT
cana-3960	237	18	according	accord	VERB
cana-3960	237	19	to	to	ADP
cana-3960	237	20	∫	∫	PROPN
cana-3960	237	21	𝑦𝑇(𝜏)𝑦(𝜏)∆𝜏	𝑦𝑇(𝜏)𝑦(𝜏)∆𝜏	NOUN
cana-3960	237	22	𝑡𝑓	𝑡𝑓	NOUN
cana-3960	237	23	𝑡0	𝑡0	PROPN
cana-3960	237	24	=	=	SYM
cana-3960	237	25	∫	∫	PROPN
cana-3960	237	26	𝑧𝛼	𝑧𝛼	ADP
cana-3960	237	27	𝑇𝜓𝐺1(𝑡0	𝑇𝜓𝐺1(𝑡0	NOUN
cana-3960	237	28	,	,	PUNCT
cana-3960	237	29	𝜏	𝜏	NOUN
cana-3960	237	30	,	,	PUNCT
cana-3960	237	31	)	)	PUNCT
cana-3960	237	32	(	(	PUNCT
cana-3960	237	33	i⊗𝐶1	i⊗𝐶1	X
cana-3960	237	34	)	)	PUNCT
cana-3960	237	35	𝑇(𝜏)(i⊗𝐶1)(𝜏)𝜓𝐺1	𝑇(𝜏)(i⊗𝐶1)(𝜏)𝜓𝐺1	PROPN
cana-3960	237	36	𝑇	𝑇	PROPN
cana-3960	237	37	(	(	PUNCT
cana-3960	237	38	𝑡0	𝑡0	PROPN
cana-3960	237	39	,	,	PUNCT
cana-3960	237	40	𝜏)𝑧𝛼∆𝜏	𝜏)𝑧𝛼∆𝜏	NOUN
cana-3960	237	41	𝑡1	𝑡1	PROPN
cana-3960	237	42	𝑡0	𝑡0	PROPN
cana-3960	238	1	+	+	CCONJ
cana-3960	238	2	∑[∏[𝐼𝑛⊗𝑅𝑖	∑[∏[𝐼𝑛⊗𝑅𝑖	ADJ
cana-3960	238	3	]	]	X
cana-3960	238	4	𝑗	𝑗	X
cana-3960	238	5	𝑖=1	𝑖=1	PUNCT
cana-3960	238	6	]	]	PUNCT
cana-3960	238	7	2	2	NUM
cana-3960	238	8	𝑙−1	𝑙−1	NOUN
cana-3960	238	9	𝑗=2	𝑗=2	PROPN
cana-3960	238	10	∫	∫	PROPN
cana-3960	239	1	𝑧𝛼	𝑧𝛼	X
cana-3960	239	2	𝑇ω	𝑇ω	PROPN
cana-3960	239	3	𝑗	𝑗	PROPN
cana-3960	239	4	(	(	PUNCT
cana-3960	239	5	𝑡0	𝑡0	PROPN
cana-3960	239	6	,	,	PUNCT
cana-3960	239	7	𝜏	𝜏	NOUN
cana-3960	239	8	,	,	PUNCT
cana-3960	239	9	)	)	PUNCT
cana-3960	239	10	(	(	PUNCT
cana-3960	239	11	i⊗𝐶𝑗	i⊗𝐶𝑗	NOUN
cana-3960	239	12	)	)	PUNCT
cana-3960	239	13	𝑇	𝑇	PROPN
cana-3960	239	14	(	(	PUNCT
cana-3960	239	15	𝜏)(i⊗𝐶𝑗)(𝜏)ω𝑗	𝜏)(i⊗𝐶𝑗)(𝜏)ω𝑗	PROPN
cana-3960	239	16	𝑇(𝑡0	𝑇(𝑡0	NUM
cana-3960	239	17	,	,	PUNCT
cana-3960	239	18	𝜏)𝑧𝛼∆𝜏	𝜏)𝑧𝛼∆𝜏	VERB
cana-3960	239	19	𝑡𝑗	𝑡𝑗	PROPN
cana-3960	239	20	𝑡𝑗−1	𝑡𝑗−1	PROPN
cana-3960	239	21	+	+	PROPN
cana-3960	239	22	[	[	X
cana-3960	239	23	∏[𝐼𝑛⊗𝑅𝑖	∏[𝐼𝑛⊗𝑅𝑖	X
cana-3960	239	24	]	]	X
cana-3960	239	25	𝑙	𝑙	X
cana-3960	239	26	𝑖=1	𝑖=1	PUNCT
cana-3960	239	27	]	]	PUNCT
cana-3960	239	28	2	2	NUM
cana-3960	239	29	∫	∫	NOUN
cana-3960	239	30	𝑧𝛼	𝑧𝛼	ADP
cana-3960	239	31	𝑇ω𝑙(𝑡0	𝑇ω𝑙(𝑡0	PROPN
cana-3960	239	32	,	,	PUNCT
cana-3960	239	33	𝜏	𝜏	NOUN
cana-3960	239	34	,	,	PUNCT
cana-3960	239	35	)	)	PUNCT
cana-3960	239	36	(	(	PUNCT
cana-3960	239	37	i⊗𝐶𝑙	i⊗𝐶𝑙	NOUN
cana-3960	239	38	)	)	PUNCT
cana-3960	239	39	𝑇(𝜏)(i⊗𝐶𝑙)(𝜏)ω𝑙	𝑇(𝜏)(i⊗𝐶𝑙)(𝜏)ω𝑙	CCONJ
cana-3960	240	1	𝑇(𝑡0	𝑇(𝑡0	NUM
cana-3960	240	2	,	,	PUNCT
cana-3960	240	3	𝜏)𝑧𝛼∆𝜏	𝜏)𝑧𝛼∆𝜏	NOUN
cana-3960	240	4	𝑡𝑓	𝑡𝑓	INTJ
cana-3960	240	5	𝑡𝑙−1	𝑡𝑙−1	PROPN
cana-3960	240	6	.	.	PUNCT
cana-3960	240	7	implies	imply	VERB
cana-3960	240	8	∫‖𝑦(𝜏)‖2∆𝜏	∫‖𝑦(𝜏)‖2∆𝜏	PROPN
cana-3960	240	9	𝑡𝑓	𝑡𝑓	NOUN
cana-3960	240	10	𝑡0	𝑡0	PROPN
cana-3960	240	11	=	=	SYM
cana-3960	240	12	0	0	PROPN
cana-3960	240	13	.	.	PUNCT
cana-3960	241	1	according	accord	VERB
cana-3960	241	2	to	to	ADP
cana-3960	241	3	0	0	NUM
cana-3960	241	4	=	=	SYM
cana-3960	241	5	𝑦(𝑡	𝑦(𝑡	PROPN
cana-3960	241	6	)	)	PUNCT
cana-3960	241	7	=	=	PRON
cana-3960	241	8	{	{	PUNCT
cana-3960	241	9	(	(	PUNCT
cana-3960	241	10	i⊗𝐶1)(𝑡)𝜓𝐺1	i⊗𝐶1)(𝑡)𝜓𝐺1	NOUN
cana-3960	241	11	𝑇	𝑇	PROPN
cana-3960	241	12	(	(	PUNCT
cana-3960	241	13	𝑡0	𝑡0	PROPN
cana-3960	241	14	,	,	PUNCT
cana-3960	241	15	𝑡)𝑧0	𝑡)𝑧0	PROPN
cana-3960	241	16	,	,	PUNCT
cana-3960	241	17	𝑡	𝑡	PROPN
cana-3960	241	18	∈	∈	PROPN
cana-3960	242	1	[	[	X
cana-3960	242	2	𝑡0	𝑡0	NOUN
cana-3960	242	3	,	,	PUNCT
cana-3960	242	4	𝑡1]𝕋	𝑡1]𝕋	NUM
cana-3960	242	5	∏[𝐼𝑛⊗𝑅𝑖](i⊗𝐶𝑘)ω𝑘	∏[𝐼𝑛⊗𝑅𝑖](i⊗𝐶𝑘)ω𝑘	PROPN
cana-3960	242	6	𝑇(𝑡0	𝑇(𝑡0	PROPN
cana-3960	242	7	,	,	PUNCT
cana-3960	242	8	𝑡	𝑡	PROPN
cana-3960	242	9	)	)	PUNCT
cana-3960	242	10	𝑘	𝑘	PRON
cana-3960	242	11	𝑖=1	𝑖=1	PROPN
cana-3960	242	12	𝑧0	𝑧0	PROPN
cana-3960	242	13	,	,	PUNCT
cana-3960	242	14	𝑡	𝑡	PROPN
cana-3960	242	15	∈	∈	PROPN
cana-3960	242	16	(	(	PUNCT
cana-3960	242	17	𝑡𝑘−1	𝑡𝑘−1	PROPN
cana-3960	242	18	,	,	PUNCT
cana-3960	242	19	𝑡𝑘]𝕋	𝑡𝑘]𝕋	NOUN
cana-3960	242	20	,	,	PUNCT
cana-3960	242	21	𝑘	𝑘	X
cana-3960	242	22	=	=	SYM
cana-3960	242	23	2	2	NUM
cana-3960	242	24	,	,	PUNCT
cana-3960	242	25	…	…	PUNCT
cana-3960	242	26	,	,	PUNCT
cana-3960	242	27	𝑙	𝑙	PRON
cana-3960	242	28	−	−	PROPN
cana-3960	242	29	1	1	NUM
cana-3960	242	30	,	,	PUNCT
cana-3960	242	31	∏[𝐼𝑛⊗𝑅𝑖](i⊗𝐶𝑙)ω𝑙	∏[𝐼𝑛⊗𝑅𝑖](i⊗𝐶𝑙)ω𝑙	ADV
cana-3960	242	32	𝑇(𝑡0	𝑇(𝑡0	NUM
cana-3960	242	33	,	,	PUNCT
cana-3960	242	34	𝑡	𝑡	PROPN
cana-3960	242	35	)	)	PUNCT
cana-3960	242	36	𝑙	𝑙	PRON
cana-3960	242	37	𝑖=1	𝑖=1	PROPN
cana-3960	242	38	𝑧0	𝑧0	PROPN
cana-3960	242	39	,	,	PUNCT
cana-3960	242	40	𝑡	𝑡	PROPN
cana-3960	242	41	∈	∈	PROPN
cana-3960	242	42	(	(	PUNCT
cana-3960	242	43	𝑡𝑙−1	𝑡𝑙−1	X
cana-3960	242	44	,	,	PUNCT
cana-3960	242	45	𝑡𝑙]𝕋.	𝑡𝑙]𝕋.	NUM
cana-3960	242	46	communications	communication	NOUN
cana-3960	242	47	on	on	ADP
cana-3960	242	48	applied	apply	VERB
cana-3960	242	49	nonlinear	nonlinear	ADJ
cana-3960	242	50	analysis	analysis	NOUN
cana-3960	242	51	issn	issn	NOUN
cana-3960	242	52	:	:	PUNCT
cana-3960	242	53	1074	1074	NUM
cana-3960	242	54	-	-	PUNCT
cana-3960	242	55	133x	133x	NUM
cana-3960	242	56	vol	vol	NOUN
cana-3960	242	57	32	32	NUM
cana-3960	242	58	no	no	NOUN
cana-3960	242	59	.	.	PUNCT
cana-3960	243	1	9s	9s	NUM
cana-3960	243	2	(	(	PUNCT
cana-3960	243	3	2025	2025	NUM
cana-3960	243	4	)	)	PUNCT
cana-3960	243	5	505	505	NUM
cana-3960	243	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3960	243	7	the	the	DET
cana-3960	243	8	last	last	ADJ
cana-3960	243	9	equality	equality	NOUN
cana-3960	243	10	implies	imply	VERB
cana-3960	243	11	,	,	PUNCT
cana-3960	243	12	by	by	ADP
cana-3960	243	13	definition	definition	NOUN
cana-3960	243	14	4.1	4.1	NUM
cana-3960	243	15	.	.	PUNCT
cana-3960	243	16	,	,	PUNCT
cana-3960	243	17	that	that	SCONJ
cana-3960	243	18	the	the	DET
cana-3960	243	19	system	system	NOUN
cana-3960	243	20	(	(	PUNCT
cana-3960	243	21	2.2	2.2	NUM
cana-3960	243	22	)	)	PUNCT
cana-3960	243	23	is	be	AUX
cana-3960	243	24	not	not	PART
cana-3960	243	25	observable	observable	ADJ
cana-3960	243	26	on	on	ADP
cana-3960	243	27	𝑡	𝑡	PROPN
cana-3960	243	28	∈	∈	PROPN
cana-3960	243	29	[	[	X
cana-3960	243	30	𝑡0	𝑡0	NOUN
cana-3960	243	31	,	,	PUNCT
cana-3960	243	32	𝑡𝑓]𝕋	𝑡𝑓]𝕋	PROPN
cana-3960	243	33	(	(	PUNCT
cana-3960	243	34	𝑡𝑓	𝑡𝑓	ADV
cana-3960	243	35	∈	∈	PROPN
cana-3960	244	1	[	[	X
cana-3960	244	2	𝑡𝑙−1	𝑡𝑙−1	X
cana-3960	244	3	,	,	PUNCT
cana-3960	244	4	𝑡𝑙)𝕋	𝑡𝑙)𝕋	NOUN
cana-3960	244	5	)	)	PUNCT
cana-3960	244	6	.	.	PUNCT
cana-3960	245	1	theorem	theorem	VERB
cana-3960	245	2	4.2	4.2	NUM
cana-3960	245	3	.	.	PUNCT
cana-3960	246	1	assume	assume	VERB
cana-3960	246	2	that	that	SCONJ
cana-3960	246	3	[	[	X
cana-3960	246	4	𝐼𝑛⊗𝑅𝑗	𝐼𝑛⊗𝑅𝑗	X
cana-3960	246	5	]	]	X
cana-3960	246	6	≥	≥	X
cana-3960	246	7	0	0	NUM
cana-3960	246	8	,	,	PUNCT
cana-3960	246	9	𝑗	𝑗	NOUN
cana-3960	246	10	=	=	SYM
cana-3960	246	11	1,2	1,2	NUM
cana-3960	246	12	,	,	PUNCT
cana-3960	246	13	…	…	PUNCT
cana-3960	246	14	,	,	PUNCT
cana-3960	246	15	𝑙	𝑙	NOUN
cana-3960	246	16	and	and	CCONJ
cana-3960	246	17	𝐺𝑘(𝑡	𝐺𝑘(𝑡	PROPN
cana-3960	246	18	)	)	PUNCT
cana-3960	246	19	=	=	SYM
cana-3960	247	1	𝐺𝑘	𝐺𝑘	NOUN
cana-3960	247	2	,	,	PUNCT
cana-3960	247	3	(	(	PUNCT
cana-3960	247	4	i⊗𝐶𝑘)(𝑡	i⊗𝐶𝑘)(𝑡	PROPN
cana-3960	247	5	)	)	PUNCT
cana-3960	247	6	=	=	SYM
cana-3960	247	7	(	(	PUNCT
cana-3960	247	8	i⊗𝐶𝑘	i⊗𝐶𝑘	NOUN
cana-3960	247	9	)	)	PUNCT
cana-3960	247	10	are	be	AUX
cana-3960	247	11	constant	constant	ADJ
cana-3960	247	12	matrices	matrix	NOUN
cana-3960	247	13	.	.	PUNCT
cana-3960	248	1	then	then	ADV
cana-3960	248	2	,	,	PUNCT
cana-3960	248	3	the	the	DET
cana-3960	248	4	system	system	NOUN
cana-3960	248	5	(	(	PUNCT
cana-3960	248	6	2.2	2.2	NUM
cana-3960	248	7	)	)	PUNCT
cana-3960	248	8	is	be	AUX
cana-3960	248	9	observable	observable	ADJ
cana-3960	248	10	on	on	ADP
cana-3960	248	11	𝑡	𝑡	PROPN
cana-3960	248	12	∈	∈	PROPN
cana-3960	248	13	[	[	X
cana-3960	248	14	𝑡0	𝑡0	NOUN
cana-3960	248	15	,	,	PUNCT
cana-3960	248	16	𝑡𝑓]𝕋	𝑡𝑓]𝕋	PROPN
cana-3960	248	17	(	(	PUNCT
cana-3960	248	18	𝑡	𝑡	PROPN
cana-3960	248	19	∈	∈	PROPN
cana-3960	248	20	[	[	X
cana-3960	248	21	𝑡𝑙−1	𝑡𝑙−1	X
cana-3960	248	22	,	,	PUNCT
cana-3960	248	23	𝑡𝑙)𝕋	𝑡𝑙)𝕋	NOUN
cana-3960	248	24	)	)	PUNCT
cana-3960	248	25	,	,	PUNCT
cana-3960	248	26	if	if	SCONJ
cana-3960	248	27	and	and	CCONJ
cana-3960	248	28	only	only	ADV
cana-3960	248	29	if	if	SCONJ
cana-3960	248	30	𝑟𝑎𝑛𝑘	𝑟𝑎𝑛𝑘	PROPN
cana-3960	248	31	(	(	PUNCT
cana-3960	248	32	𝑆	𝑆	PROPN
cana-3960	248	33	)	)	PUNCT
cana-3960	248	34	=	=	NOUN
cana-3960	248	35	𝑛2	𝑛2	NOUN
cana-3960	248	36	.	.	PUNCT
cana-3960	249	1	let	let	VERB
cana-3960	249	2	us	we	PRON
cana-3960	249	3	define	define	VERB
cana-3960	249	4	the	the	DET
cana-3960	249	5	following	follow	VERB
cana-3960	249	6	matrix	matrix	NOUN
cana-3960	249	7	𝑆	𝑆	PROPN
cana-3960	250	1	=	=	PRON
cana-3960	250	2	[	[	PUNCT
cana-3960	250	3	(	(	PUNCT
cana-3960	250	4	i⊗𝐺1𝑗	i⊗𝐺1𝑗	PROPN
cana-3960	250	5	)	)	PUNCT
cana-3960	250	6	(	(	PUNCT
cana-3960	250	7	i⊗𝐺1𝑘)𝑃𝑗	i⊗𝐺1𝑘)𝑃𝑗	PROPN
cana-3960	250	8	𝑇	𝑇	PROPN
cana-3960	250	9	⋮	⋮	NOUN
cana-3960	250	10	(	(	PUNCT
cana-3960	250	11	i⊗𝐺𝑘)(𝑃𝑗	i⊗𝐺𝑘)(𝑃𝑗	ADJ
cana-3960	250	12	𝑇	𝑇	PROPN
cana-3960	250	13	)	)	PUNCT
cana-3960	250	14	𝑛2−1	𝑛2−1	NOUN
cana-3960	250	15	]	]	PUNCT
cana-3960	250	16	(	(	PUNCT
cana-3960	250	17	4.6	4.6	NUM
cana-3960	250	18	)	)	PUNCT
cana-3960	250	19	proof	proof	NOUN
cana-3960	250	20	:	:	PUNCT
cana-3960	250	21	assume	assume	VERB
cana-3960	250	22	that	that	SCONJ
cana-3960	250	23	r𝑎𝑛𝑘	r𝑎𝑛𝑘	PROPN
cana-3960	250	24	(	(	PUNCT
cana-3960	250	25	𝑆	𝑆	PROPN
cana-3960	250	26	)	)	PUNCT
cana-3960	250	27	=	=	SYM
cana-3960	250	28	𝑛2	𝑛2	NOUN
cana-3960	250	29	.	.	PUNCT
cana-3960	251	1	and	and	CCONJ
cana-3960	251	2	we	we	PRON
cana-3960	251	3	aim	aim	VERB
cana-3960	251	4	to	to	PART
cana-3960	251	5	show	show	VERB
cana-3960	251	6	that	that	SCONJ
cana-3960	251	7	the	the	DET
cana-3960	251	8	system	system	NOUN
cana-3960	251	9	(	(	PUNCT
cana-3960	251	10	2.2	2.2	NUM
cana-3960	251	11	)	)	PUNCT
cana-3960	251	12	is	be	AUX
cana-3960	251	13	observability	observability	NOUN
cana-3960	251	14	on	on	ADP
cana-3960	251	15	𝑡	𝑡	PROPN
cana-3960	251	16	∈	∈	PROPN
cana-3960	251	17	[	[	X
cana-3960	251	18	𝑡0	𝑡0	NOUN
cana-3960	251	19	,	,	PUNCT
cana-3960	251	20	𝑡𝑓]𝕋	𝑡𝑓]𝕋	PROPN
cana-3960	251	21	(	(	PUNCT
cana-3960	251	22	𝑡	𝑡	PROPN
cana-3960	251	23	∈	∈	PROPN
cana-3960	252	1	[	[	X
cana-3960	252	2	𝑡𝑙−1	𝑡𝑙−1	X
cana-3960	252	3	,	,	PUNCT
cana-3960	252	4	𝑡𝑙)𝕋	𝑡𝑙)𝕋	NOUN
cana-3960	252	5	)	)	PUNCT
cana-3960	252	6	.	.	PUNCT
cana-3960	253	1	if	if	SCONJ
cana-3960	253	2	otherwise	otherwise	ADV
cana-3960	253	3	,	,	PUNCT
cana-3960	253	4	namely	namely	ADV
cana-3960	253	5	the	the	DET
cana-3960	253	6	system	system	NOUN
cana-3960	253	7	(	(	PUNCT
cana-3960	253	8	2.2	2.2	NUM
cana-3960	253	9	)	)	PUNCT
cana-3960	253	10	is	be	AUX
cana-3960	253	11	not	not	PART
cana-3960	253	12	observability	observability	NOUN
cana-3960	253	13	then	then	ADV
cana-3960	253	14	by	by	ADP
cana-3960	253	15	theorem	theorem	NOUN
cana-3960	253	16	4.1	4.1	NUM
cana-3960	253	17	.	.	PUNCT
cana-3960	253	18	,	,	PUNCT
cana-3960	253	19	according	accord	VERB
cana-3960	253	20	to	to	ADP
cana-3960	253	21	the	the	DET
cana-3960	253	22	matrix	matrix	NOUN
cana-3960	253	23	𝑊(𝑡0	𝑊(𝑡0	NOUN
cana-3960	253	24	,	,	PUNCT
cana-3960	253	25	𝑡𝑓	𝑡𝑓	ADV
cana-3960	253	26	)	)	PUNCT
cana-3960	253	27	is	be	AUX
cana-3960	253	28	not	not	PART
cana-3960	253	29	invertible	invertible	ADJ
cana-3960	253	30	,	,	PUNCT
cana-3960	253	31	which	which	PRON
cana-3960	253	32	leads	lead	VERB
cana-3960	253	33	to	to	ADP
cana-3960	253	34	that	that	SCONJ
cana-3960	253	35	there	there	PRON
cana-3960	253	36	exists	exist	VERB
cana-3960	253	37	a	a	DET
cana-3960	253	38	nonzero	nonzero	PROPN
cana-3960	253	39	vector	vector	NOUN
cana-3960	253	40	𝑧𝛼	𝑧𝛼	ADP
cana-3960	253	41	≠	≠	PROPN
cana-3960	253	42	0	0	NUM
cana-3960	253	43	.	.	PUNCT
cana-3960	254	1	then	then	ADV
cana-3960	254	2	by	by	ADP
cana-3960	254	3	using	use	VERB
cana-3960	254	4	theorem	theorem	NOUN
cana-3960	254	5	4.1	4.1	NUM
cana-3960	254	6	.	.	PUNCT
cana-3960	254	7	,	,	PUNCT
cana-3960	254	8	we	we	PRON
cana-3960	254	9	have	have	VERB
cana-3960	254	10	𝑧𝛼	𝑧𝛼	NUM
cana-3960	254	11	𝑇𝑊(𝑡0	𝑇𝑊(𝑡0	PRON
cana-3960	254	12	,	,	PUNCT
cana-3960	254	13	𝑡0	𝑡0	PROPN
cana-3960	254	14	,	,	PUNCT
cana-3960	254	15	𝑡1)𝑧𝛼	𝑡1)𝑧𝛼	PROPN
cana-3960	254	16	=	=	SYM
cana-3960	254	17	∫	∫	PROPN
cana-3960	255	1	𝑧𝛼	𝑧𝛼	X
cana-3960	255	2	𝑇𝑒𝐺1(𝑡0	𝑇𝑒𝐺1(𝑡0	PROPN
cana-3960	255	3	,	,	PUNCT
cana-3960	255	4	𝜏)(i⊗𝐶1)(i⊗𝐶1	𝜏)(i⊗𝐶1)(i⊗𝐶1	ADJ
cana-3960	255	5	)	)	PUNCT
cana-3960	255	6	𝑇𝑒𝐺1	𝑇𝑒𝐺1	NOUN
cana-3960	255	7	𝑇	𝑇	PROPN
cana-3960	255	8	(	(	PUNCT
cana-3960	255	9	𝑡0	𝑡0	PROPN
cana-3960	255	10	,	,	PUNCT
cana-3960	255	11	𝜏)𝑧𝛼∆𝜏	𝜏)𝑧𝛼∆𝜏	NOUN
cana-3960	255	12	𝑡1	𝑡1	NOUN
cana-3960	255	13	𝑡0	𝑡0	NOUN
cana-3960	255	14	=	=	SYM
cana-3960	255	15	∫[(i	∫[(i	PROPN
cana-3960	255	16	⊗𝐶1)𝑒𝐺1	⊗𝐶1)𝑒𝐺1	PROPN
cana-3960	255	17	𝑇	𝑇	PROPN
cana-3960	255	18	(	(	PUNCT
cana-3960	255	19	𝑡0	𝑡0	PROPN
cana-3960	255	20	,	,	PUNCT
cana-3960	255	21	𝜏)𝑧𝛼	𝜏)𝑧𝛼	PROPN
cana-3960	255	22	]	]	X
cana-3960	255	23	𝑇	𝑇	PROPN
cana-3960	255	24	[	[	X
cana-3960	255	25	(	(	PUNCT
cana-3960	255	26	i⊗𝐶1)𝑒𝐺1	i⊗𝐶1)𝑒𝐺1	PROPN
cana-3960	255	27	𝑇	𝑇	PROPN
cana-3960	255	28	(	(	PUNCT
cana-3960	255	29	𝑡0	𝑡0	PROPN
cana-3960	255	30	,	,	PUNCT
cana-3960	255	31	𝜏)𝑧𝛼	𝜏)𝑧𝛼	PROPN
cana-3960	255	32	]	]	PUNCT
cana-3960	255	33	𝑇	𝑇	PROPN
cana-3960	255	34	∆𝜏	∆𝜏	PROPN
cana-3960	255	35	𝑡1	𝑡1	NOUN
cana-3960	255	36	𝑡0	𝑡0	NOUN
cana-3960	255	37	,	,	PUNCT
cana-3960	255	38	similarly	similarly	ADV
cana-3960	255	39	,	,	PUNCT
cana-3960	255	40	(	(	PUNCT
cana-3960	255	41	i⊗𝐶𝑗)ω𝑗	i⊗𝐶𝑗)ω𝑗	PROPN
cana-3960	255	42	𝑇(𝑡0	𝑇(𝑡0	PROPN
cana-3960	255	43	,	,	PUNCT
cana-3960	255	44	𝑡)𝑧𝛼	𝑡)𝑧𝛼	PROPN
cana-3960	255	45	=	=	SYM
cana-3960	255	46	0	0	NUM
cana-3960	255	47	,	,	PUNCT
cana-3960	255	48	𝑗	𝑗	NOUN
cana-3960	255	49	=	=	SYM
cana-3960	255	50	1	1	NUM
cana-3960	255	51	,	,	PUNCT
cana-3960	255	52	…	…	PUNCT
cana-3960	255	53	,	,	PUNCT
cana-3960	255	54	𝑙	𝑙	X
cana-3960	255	55	−	−	PROPN
cana-3960	255	56	1	1	NUM
cana-3960	255	57	,	,	PUNCT
cana-3960	255	58	(	(	PUNCT
cana-3960	255	59	4.8	4.8	NUM
cana-3960	255	60	)	)	PUNCT
cana-3960	255	61	and	and	CCONJ
cana-3960	255	62	(	(	PUNCT
cana-3960	255	63	i⊗𝐶𝑙)ω𝑙	i⊗𝐶𝑙)ω𝑙	ADV
cana-3960	255	64	𝑇(𝑡0	𝑇(𝑡0	ADJ
cana-3960	255	65	,	,	PUNCT
cana-3960	255	66	𝑡)𝑧𝛼	𝑡)𝑧𝛼	PROPN
cana-3960	255	67	=	=	SYM
cana-3960	255	68	0	0	NUM
cana-3960	255	69	,	,	PUNCT
cana-3960	255	70	(	(	PUNCT
cana-3960	255	71	4.9	4.9	NUM
cana-3960	255	72	)	)	PUNCT
cana-3960	255	73	where	where	SCONJ
cana-3960	255	74	ω𝑗	ω𝑗	ADP
cana-3960	255	75	𝑇(𝑡0	𝑇(𝑡0	NUM
cana-3960	255	76	,	,	PUNCT
cana-3960	255	77	𝑡	𝑡	PROPN
cana-3960	255	78	)	)	PUNCT
cana-3960	255	79	=	=	PUNCT
cana-3960	256	1	𝑒𝐺𝑗	𝑒𝐺𝑗	NUM
cana-3960	256	2	𝑇	𝑇	PROPN
cana-3960	256	3	(	(	PUNCT
cana-3960	256	4	𝑡𝑗−1	𝑡𝑗−1	PROPN
cana-3960	256	5	,	,	PUNCT
cana-3960	256	6	𝑡)𝑒𝐺𝑗−1	𝑡)𝑒𝐺𝑗−1	NOUN
cana-3960	256	7	𝑇	𝑇	PROPN
cana-3960	256	8	(	(	PUNCT
cana-3960	256	9	𝑡𝑗−2	𝑡𝑗−2	PROPN
cana-3960	256	10	,	,	PUNCT
cana-3960	256	11	𝑡𝑗−1	𝑡𝑗−1	PROPN
cana-3960	256	12	)	)	PUNCT
cana-3960	256	13	…	…	PUNCT
cana-3960	256	14	𝑒𝐺𝑗	𝑒𝐺𝑗	PROPN
cana-3960	256	15	𝑇	𝑇	PROPN
cana-3960	256	16	(	(	PUNCT
cana-3960	256	17	𝑡0	𝑡0	PROPN
cana-3960	256	18	,	,	PUNCT
cana-3960	256	19	𝑡1	𝑡1	NOUN
cana-3960	256	20	)	)	PUNCT
cana-3960	256	21	.	.	PUNCT
cana-3960	257	1	obviously	obviously	ADV
cana-3960	257	2	,	,	PUNCT
cana-3960	257	3	at	at	ADP
cana-3960	257	4	𝑡	𝑡	PROPN
cana-3960	257	5	=	=	SYM
cana-3960	257	6	𝑡0	𝑡0	PROPN
cana-3960	257	7	,	,	PUNCT
cana-3960	257	8	we	we	PRON
cana-3960	257	9	obtain	obtain	VERB
cana-3960	257	10	(	(	PUNCT
cana-3960	257	11	i⊗𝐶𝑗)𝑧𝛼	i⊗𝐶𝑗)𝑧𝛼	PROPN
cana-3960	257	12	=	=	SYM
cana-3960	257	13	0	0	PROPN
cana-3960	257	14	,	,	PUNCT
cana-3960	257	15	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-3960	257	16	𝑗	𝑗	X
cana-3960	257	17	=	=	SYM
cana-3960	257	18	1	1	NUM
cana-3960	257	19	,	,	PUNCT
cana-3960	257	20	…	…	PUNCT
cana-3960	257	21	,	,	PUNCT
cana-3960	257	22	𝑙	𝑙	X
cana-3960	257	23	−	−	PROPN
cana-3960	257	24	1	1	NUM
cana-3960	257	25	,	,	PUNCT
cana-3960	257	26	and	and	CCONJ
cana-3960	257	27	differentiating	differentiate	VERB
cana-3960	257	28	the	the	DET
cana-3960	257	29	equations	equation	NOUN
cana-3960	257	30	(	(	PUNCT
cana-3960	257	31	4.7	4.7	NUM
cana-3960	257	32	)	)	PUNCT
cana-3960	257	33	,	,	PUNCT
cana-3960	257	34	(	(	PUNCT
cana-3960	257	35	4.8	4.8	NUM
cana-3960	257	36	)	)	PUNCT
cana-3960	257	37	and	and	CCONJ
cana-3960	257	38	(	(	PUNCT
cana-3960	257	39	4.9	4.9	NUM
cana-3960	257	40	)	)	PUNCT
cana-3960	257	41	𝑛2	𝑛2	NOUN
cana-3960	257	42	−	−	PROPN
cana-3960	257	43	1	1	NUM
cana-3960	257	44	times	time	NOUN
cana-3960	257	45	and	and	CCONJ
cana-3960	257	46	evaluating	evaluate	VERB
cana-3960	257	47	the	the	DET
cana-3960	257	48	results	result	NOUN
cana-3960	257	49	at	at	ADP
cana-3960	257	50	𝑡	𝑡	PROPN
cana-3960	257	51	=	=	PROPN
cana-3960	257	52	𝑡0	𝑡0	PROPN
cana-3960	257	53	gives	give	VERB
cana-3960	257	54	(	(	PUNCT
cana-3960	257	55	i⊗𝐶𝑗)𝐺𝑗	i⊗𝐶𝑗)𝐺𝑗	PROPN
cana-3960	257	56	𝑖𝑧𝛼	𝑖𝑧𝛼	NOUN
cana-3960	257	57	=	=	SYM
cana-3960	257	58	0	0	NUM
cana-3960	257	59	,	,	PUNCT
cana-3960	257	60	𝑖	𝑖	NOUN
cana-3960	257	61	=	=	SYM
cana-3960	257	62	0,1	0,1	NUM
cana-3960	257	63	,	,	PUNCT
cana-3960	257	64	…	…	PUNCT
cana-3960	257	65	,	,	PUNCT
cana-3960	257	66	𝑛2	𝑛2	NOUN
cana-3960	257	67	−	−	PROPN
cana-3960	257	68	1	1	NUM
cana-3960	257	69	,	,	PUNCT
cana-3960	257	70	𝑗	𝑗	NOUN
cana-3960	257	71	=	=	SYM
cana-3960	257	72	1,2	1,2	NUM
cana-3960	257	73	,	,	PUNCT
cana-3960	257	74	…	…	PUNCT
cana-3960	257	75	,	,	PUNCT
cana-3960	257	76	𝑙	𝑙	X
cana-3960	257	77	(	(	PUNCT
cana-3960	257	78	4.10	4.10	NUM
cana-3960	257	79	)	)	PUNCT
cana-3960	257	80	therefore	therefore	ADV
cana-3960	257	81	,	,	PUNCT
cana-3960	257	82	by	by	ADP
cana-3960	257	83	the	the	DET
cana-3960	257	84	equations	equation	NOUN
cana-3960	257	85	(	(	PUNCT
cana-3960	257	86	4.6	4.6	NUM
cana-3960	257	87	)	)	PUNCT
cana-3960	257	88	and	and	CCONJ
cana-3960	257	89	(	(	PUNCT
cana-3960	257	90	4.9	4.9	NUM
cana-3960	257	91	)	)	PUNCT
cana-3960	257	92	we	we	PRON
cana-3960	257	93	have	have	VERB
cana-3960	257	94	𝑆𝑧𝛼	𝑆𝑧𝛼	PROPN
cana-3960	257	95	=	=	SYM
cana-3960	257	96	0	0	NUM
cana-3960	257	97	,	,	PUNCT
cana-3960	257	98	and	and	CCONJ
cana-3960	257	99	furthermore	furthermore	ADV
cana-3960	257	100	,	,	PUNCT
cana-3960	257	101	𝑧𝛼	𝑧𝛼	CCONJ
cana-3960	257	102	≠	≠	PROPN
cana-3960	257	103	0	0	NUM
cana-3960	257	104	implies	imply	VERB
cana-3960	257	105	that	that	SCONJ
cana-3960	257	106	𝑟𝑎𝑛𝑘	𝑟𝑎𝑛𝑘	PROPN
cana-3960	257	107	(	(	PUNCT
cana-3960	257	108	𝑆	𝑆	PROPN
cana-3960	257	109	)	)	PUNCT
cana-3960	257	110	<	<	X
cana-3960	258	1	𝑛2	𝑛2	NOUN
cana-3960	258	2	which	which	PRON
cana-3960	258	3	leads	lead	VERB
cana-3960	258	4	to	to	ADP
cana-3960	258	5	a	a	DET
cana-3960	258	6	contradiction	contradiction	NOUN
cana-3960	258	7	with	with	ADP
cana-3960	258	8	the	the	DET
cana-3960	258	9	assumptions	assumption	NOUN
cana-3960	258	10	that	that	PRON
cana-3960	258	11	𝑟𝑎𝑛𝑘	𝑟𝑎𝑛𝑘	PROPN
cana-3960	258	12	(	(	PUNCT
cana-3960	258	13	𝑆	𝑆	PROPN
cana-3960	258	14	)	)	PUNCT
cana-3960	258	15	=	=	NOUN
cana-3960	258	16	𝑛2	𝑛2	NOUN
cana-3960	258	17	.	.	PUNCT
cana-3960	259	1	conversely	conversely	ADV
cana-3960	259	2	,	,	PUNCT
cana-3960	259	3	we	we	PRON
cana-3960	259	4	assume	assume	VERB
cana-3960	259	5	that	that	SCONJ
cana-3960	259	6	r𝑎𝑛𝑘	r𝑎𝑛𝑘	PROPN
cana-3960	259	7	(	(	PUNCT
cana-3960	259	8	𝑆	𝑆	PROPN
cana-3960	259	9	)	)	PUNCT
cana-3960	259	10	<	<	X
cana-3960	259	11	𝑛2	𝑛2	NOUN
cana-3960	259	12	.	.	PUNCT
cana-3960	260	1	thus	thus	ADV
cana-3960	260	2	,	,	PUNCT
cana-3960	260	3	there	there	PRON
cana-3960	260	4	exists	exist	VERB
cana-3960	260	5	𝑧𝛼	𝑧𝛼	ADP
cana-3960	260	6	≠	≠	PROPN
cana-3960	260	7	0	0	NUM
cana-3960	260	8	such	such	ADJ
cana-3960	260	9	that	that	DET
cana-3960	260	10	𝑆𝑧𝛼	𝑆𝑧𝛼	PROPN
cana-3960	260	11	=	=	SYM
cana-3960	260	12	0	0	PROPN
cana-3960	260	13	,	,	PUNCT
cana-3960	260	14	which	which	PRON
cana-3960	260	15	leads	lead	VERB
cana-3960	260	16	to	to	ADP
cana-3960	260	17	the	the	DET
cana-3960	260	18	equation	equation	NOUN
cana-3960	260	19	(	(	PUNCT
cana-3960	260	20	4.10	4.10	NUM
cana-3960	260	21	)	)	PUNCT
cana-3960	260	22	.	.	PUNCT
cana-3960	261	1	from	from	ADP
cana-3960	261	2	equation	equation	NOUN
cana-3960	261	3	(	(	PUNCT
cana-3960	261	4	4.10	4.10	NUM
cana-3960	261	5	)	)	PUNCT
cana-3960	261	6	and	and	CCONJ
cana-3960	261	7	using	use	VERB
cana-3960	261	8	proposition	proposition	NOUN
cana-3960	261	9	2.1	2.1	NUM
cana-3960	261	10	.	.	PUNCT
cana-3960	262	1	we	we	PRON
cana-3960	262	2	obtain	obtain	VERB
cana-3960	262	3	𝑊(𝑡0	𝑊(𝑡0	PROPN
cana-3960	262	4	,	,	PUNCT
cana-3960	262	5	𝑡0	𝑡0	PROPN
cana-3960	262	6	,	,	PUNCT
cana-3960	262	7	𝑡1)𝑧𝛼	𝑡1)𝑧𝛼	PROPN
cana-3960	262	8	=	=	SYM
cana-3960	262	9	∫	∫	PROPN
cana-3960	262	10	∑	∑	PROPN
cana-3960	262	11	𝜒1𝑖	𝜒1𝑖	PROPN
cana-3960	262	12	𝑛2−1	𝑛2−1	PRON
cana-3960	262	13	𝑖=0	𝑖=0	PROPN
cana-3960	262	14	(	(	PUNCT
cana-3960	262	15	𝑡0	𝑡0	PROPN
cana-3960	262	16	,	,	PUNCT
cana-3960	262	17	𝜏)𝑒𝐺1(𝑡0	𝜏)𝑒𝐺1(𝑡0	PROPN
cana-3960	262	18	,	,	PUNCT
cana-3960	262	19	𝜏)(i⊗𝐶1	𝜏)(i⊗𝐶1	PROPN
cana-3960	262	20	)	)	PUNCT
cana-3960	262	21	𝑇(i⊗𝐶1)𝑒𝐺1	𝑇(i⊗𝐶1)𝑒𝐺1	NUM
cana-3960	262	22	𝑇	𝑇	PROPN
cana-3960	262	23	(	(	PUNCT
cana-3960	262	24	𝑡0	𝑡0	PROPN
cana-3960	262	25	,	,	PUNCT
cana-3960	262	26	𝜏)𝑧𝛼∆𝜏	𝜏)𝑧𝛼∆𝜏	NOUN
cana-3960	262	27	𝑡1	𝑡1	PROPN
cana-3960	262	28	𝑡0	𝑡0	PROPN
cana-3960	262	29	communications	communication	NOUN
cana-3960	262	30	on	on	ADP
cana-3960	262	31	applied	apply	VERB
cana-3960	262	32	nonlinear	nonlinear	ADJ
cana-3960	262	33	analysis	analysis	NOUN
cana-3960	262	34	issn	issn	NOUN
cana-3960	262	35	:	:	PUNCT
cana-3960	262	36	1074	1074	NUM
cana-3960	262	37	-	-	PUNCT
cana-3960	262	38	133x	133x	NUM
cana-3960	262	39	vol	vol	NOUN
cana-3960	262	40	32	32	NUM
cana-3960	263	1	no	no	NOUN
cana-3960	263	2	.	.	PUNCT
cana-3960	264	1	9s	9s	NUM
cana-3960	264	2	(	(	PUNCT
cana-3960	264	3	2025	2025	NUM
cana-3960	264	4	)	)	PUNCT
cana-3960	264	5	506	506	NUM
cana-3960	264	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3960	264	7	=	=	SYM
cana-3960	264	8	∫	∫	PROPN
cana-3960	264	9	∑	∑	PROPN
cana-3960	264	10	𝜒1𝑖	𝜒1𝑖	PROPN
cana-3960	264	11	𝑛2−1	𝑛2−1	PRON
cana-3960	264	12	𝑖=0	𝑖=0	PROPN
cana-3960	264	13	(	(	PUNCT
cana-3960	264	14	𝑡0	𝑡0	PROPN
cana-3960	264	15	,	,	PUNCT
cana-3960	264	16	𝜏)𝑒𝐺1(𝑡0	𝜏)𝑒𝐺1(𝑡0	PROPN
cana-3960	264	17	,	,	PUNCT
cana-3960	264	18	𝜏)(i⊗𝐶1	𝜏)(i⊗𝐶1	PROPN
cana-3960	264	19	)	)	PUNCT
cana-3960	264	20	𝑇(i⊗𝐶1)𝑒𝐺1	𝑇(i⊗𝐶1)𝑒𝐺1	NUM
cana-3960	265	1	𝑇	𝑇	PROPN
cana-3960	265	2	(	(	PUNCT
cana-3960	265	3	𝑡0	𝑡0	PROPN
cana-3960	265	4	,	,	PUNCT
cana-3960	265	5	𝜏)𝑧𝛼∆𝜏	𝜏)𝑧𝛼∆𝜏	NOUN
cana-3960	265	6	𝑡1	𝑡1	NOUN
cana-3960	265	7	𝑡0	𝑡0	PROPN
cana-3960	265	8	=	=	SYM
cana-3960	265	9	0	0	NUM
cana-3960	265	10	,	,	PUNCT
cana-3960	265	11	similarly	similarly	ADV
cana-3960	265	12	,	,	PUNCT
cana-3960	265	13	for	for	ADP
cana-3960	265	14	𝑗	𝑗	NOUN
cana-3960	265	15	=	=	SYM
cana-3960	265	16	1,2	1,2	NUM
cana-3960	265	17	,	,	PUNCT
cana-3960	265	18	…	…	PUNCT
cana-3960	265	19	,	,	PUNCT
cana-3960	265	20	𝑙	𝑙	X
cana-3960	265	21	−	−	PROPN
cana-3960	265	22	1	1	NUM
cana-3960	265	23	𝑊(𝑡0	𝑊(𝑡0	PROPN
cana-3960	265	24	,	,	PUNCT
cana-3960	265	25	𝑡𝑗−1	𝑡𝑗−1	PROPN
cana-3960	265	26	,	,	PUNCT
cana-3960	265	27	𝑡𝑗)𝑧𝛼	𝑡𝑗)𝑧𝛼	PUNCT
cana-3960	265	28	=	=	SYM
cana-3960	265	29	0	0	NUM
cana-3960	265	30	,	,	PUNCT
cana-3960	265	31	and	and	CCONJ
cana-3960	265	32	𝑊(𝑡0	𝑊(𝑡0	PROPN
cana-3960	265	33	,	,	PUNCT
cana-3960	265	34	𝑡𝑙−1	𝑡𝑙−1	ADV
cana-3960	265	35	,	,	PUNCT
cana-3960	265	36	𝑡𝑙)𝑧𝛼	𝑡𝑙)𝑧𝛼	PUNCT
cana-3960	265	37	=	=	SYM
cana-3960	265	38	0	0	X
cana-3960	265	39	.	.	PUNCT
cana-3960	266	1	the	the	DET
cana-3960	266	2	equation	equation	NOUN
cana-3960	266	3	(	(	PUNCT
cana-3960	266	4	4.10	4.10	NUM
cana-3960	266	5	)	)	PUNCT
cana-3960	266	6	yields	yield	VERB
cana-3960	266	7	𝑊(𝑡0	𝑊(𝑡0	PROPN
cana-3960	266	8	,	,	PUNCT
cana-3960	266	9	𝑡𝑓)𝑧𝛼	𝑡𝑓)𝑧𝛼	PROPN
cana-3960	266	10	=	=	SYM
cana-3960	266	11	0	0	PROPN
cana-3960	266	12	.	.	PUNCT
cana-3960	266	13	since	since	SCONJ
cana-3960	266	14	𝑧𝛼	𝑧𝛼	ADP
cana-3960	266	15	≠	≠	PROPN
cana-3960	266	16	0	0	NUM
cana-3960	266	17	,	,	PUNCT
cana-3960	266	18	the	the	DET
cana-3960	266	19	matrix	matrix	NOUN
cana-3960	266	20	𝑊(𝑡0	𝑊(𝑡0	NOUN
cana-3960	266	21	,	,	PUNCT
cana-3960	266	22	𝑡𝑓	𝑡𝑓	ADV
cana-3960	266	23	)	)	PUNCT
cana-3960	266	24	is	be	AUX
cana-3960	266	25	not	not	PART
cana-3960	266	26	invertible	invertible	ADJ
cana-3960	266	27	.	.	PUNCT
cana-3960	267	1	hence	hence	ADV
cana-3960	267	2	the	the	DET
cana-3960	267	3	system	system	NOUN
cana-3960	267	4	(	(	PUNCT
cana-3960	267	5	2.2	2.2	NUM
cana-3960	267	6	)	)	PUNCT
cana-3960	267	7	is	be	AUX
cana-3960	267	8	not	not	PART
cana-3960	267	9	observable	observable	ADJ
cana-3960	267	10	,	,	PUNCT
cana-3960	267	11	and	and	CCONJ
cana-3960	267	12	it	it	PRON
cana-3960	267	13	is	be	AUX
cana-3960	267	14	contradicting	contradict	VERB
cana-3960	267	15	with	with	ADP
cana-3960	267	16	the	the	DET
cana-3960	267	17	assumption	assumption	NOUN
cana-3960	267	18	of	of	ADP
cana-3960	267	19	observability	observability	NOUN
cana-3960	267	20	.	.	PUNCT
cana-3960	268	1	references	reference	NOUN
cana-3960	268	2	.	.	PUNCT
cana-3960	269	1	[	[	X
cana-3960	269	2	1	1	X
cana-3960	269	3	]	]	X
cana-3960	269	4	agarwal	agarwal	PROPN
cana-3960	269	5	r.	r.	PROPN
cana-3960	269	6	p	p	PROPN
cana-3960	269	7	,	,	PUNCT
cana-3960	269	8	bohner	bohner	ADJ
cana-3960	269	9	m.	m.	NOUN
cana-3960	269	10	,	,	PUNCT
cana-3960	269	11	regan	regan	PROPN
cana-3960	269	12	d.o	d.o	PROPN
cana-3960	269	13	and	and	CCONJ
cana-3960	269	14	peterson	peterson	PROPN
cana-3960	269	15	a.	a.	PROPN
cana-3960	269	16	dynamic	dynamic	PROPN
cana-3960	269	17	equations	equation	NOUN
cana-3960	269	18	on	on	ADP
cana-3960	269	19	time	time	NOUN
cana-3960	269	20	scales	scale	NOUN
cana-3960	269	21	.	.	PUNCT
cana-3960	270	1	a	a	DET
cana-3960	270	2	survey	survey	NOUN
cana-3960	270	3	,	,	PUNCT
cana-3960	270	4	j	j	PROPN
cana-3960	270	5	comput	comput	NOUN
cana-3960	270	6	.	.	PUNCT
cana-3960	271	1	appl	appl	PROPN
cana-3960	271	2	.	.	PROPN
cana-3960	271	3	math	math	PROPN
cana-3960	271	4	,	,	PUNCT
cana-3960	271	5	no.4	no.4	PROPN
cana-3960	271	6	,	,	PUNCT
cana-3960	271	7	(	(	PUNCT
cana-3960	271	8	2002	2002	NUM
cana-3960	271	9	)	)	PUNCT
cana-3960	271	10	,	,	PUNCT
cana-3960	271	11	1	1	NUM
cana-3960	271	12	-	-	SYM
cana-3960	271	13	26	26	NUM
cana-3960	271	14	.	.	PUNCT
cana-3960	272	1	[	[	X
cana-3960	272	2	2	2	NUM
cana-3960	272	3	]	]	X
cana-3960	272	4	alexander	alexander	NOUN
cana-3960	272	5	g	g	PROPN
cana-3960	272	6	,	,	PUNCT
cana-3960	272	7	kronecker	kronecker	NOUN
cana-3960	272	8	products	product	NOUN
cana-3960	272	9	and	and	CCONJ
cana-3960	272	10	matrix	matrix	NOUN
cana-3960	272	11	calculus	calculus	NOUN
cana-3960	272	12	;	;	PUNCT
cana-3960	272	13	with	with	ADP
cana-3960	272	14	applications	application	NOUN
cana-3960	272	15	,	,	PUNCT
cana-3960	272	16	ellis	ellis	PROPN
cana-3960	272	17	hordwood	hordwood	PROPN
cana-3960	272	18	ltd	ltd	PROPN
cana-3960	272	19	.	.	PROPN
cana-3960	272	20	,	,	PUNCT
cana-3960	272	21	england	england	PROPN
cana-3960	272	22	,	,	PUNCT
cana-3960	272	23	(	(	PUNCT
cana-3960	272	24	1981	1981	NUM
cana-3960	272	25	)	)	PUNCT
cana-3960	272	26	.	.	PUNCT
cana-3960	273	1	[	[	X
cana-3960	273	2	3	3	X
cana-3960	273	3	]	]	X
cana-3960	273	4	appa	appa	PROPN
cana-3960	273	5	rao	rao	PROPN
cana-3960	273	6	b	b	PROPN
cana-3960	273	7	v	v	PROPN
cana-3960	273	8	and	and	CCONJ
cana-3960	273	9	prasad	prasad	PROPN
cana-3960	273	10	kasnv	kasnv	PROPN
cana-3960	273	11	,	,	PUNCT
cana-3960	273	12	controllability	controllability	NOUN
cana-3960	273	13	and	and	CCONJ
cana-3960	273	14	observability	observability	NOUN
cana-3960	273	15	of	of	ADP
cana-3960	273	16	sylvester	sylvester	ADJ
cana-3960	273	17	matrix	matrix	NOUN
cana-3960	273	18	dynamical	dynamical	ADJ
cana-3960	273	19	systems	system	NOUN
cana-3960	273	20	on	on	ADP
cana-3960	273	21	time	time	NOUN
cana-3960	273	22	scales	scale	NOUN
cana-3960	273	23	kyungpook	kyungpook	PROPN
cana-3960	273	24	math.j	math.j	PROPN
cana-3960	273	25	.	.	PUNCT
cana-3960	274	1	56(2016	56(2016	PROPN
cana-3960	274	2	)	)	PUNCT
cana-3960	274	3	,	,	PUNCT
cana-3960	274	4	529	529	NUM
cana-3960	274	5	-	-	SYM
cana-3960	274	6	539	539	NUM
cana-3960	274	7	.	.	PUNCT
cana-3960	275	1	[	[	X
cana-3960	275	2	4	4	X
cana-3960	275	3	]	]	X
cana-3960	275	4	appa	appa	PROPN
cana-3960	275	5	rao	rao	PROPN
cana-3960	275	6	b	b	PROPN
cana-3960	275	7	v	v	PROPN
cana-3960	275	8	and	and	CCONJ
cana-3960	275	9	prasad	prasad	PROPN
cana-3960	275	10	kasnv	kasnv	PROPN
cana-3960	275	11	,	,	PUNCT
cana-3960	275	12	existence	existence	NOUN
cana-3960	275	13	of	of	ADP
cana-3960	275	14	psi	psi	NOUN
cana-3960	275	15	-	-	PUNCT
cana-3960	275	16	bounded	bound	VERB
cana-3960	275	17	solutions	solution	NOUN
cana-3960	275	18	for	for	ADP
cana-3960	275	19	sylvester	sylvester	ADJ
cana-3960	275	20	matrix	matrix	NOUN
cana-3960	275	21	dynamical	dynamical	ADJ
cana-3960	275	22	systems	system	NOUN
cana-3960	275	23	on	on	ADP
cana-3960	275	24	time	time	NOUN
cana-3960	275	25	scales	scale	NOUN
cana-3960	275	26	,	,	PUNCT
cana-3960	275	27	filomat	filomat	NOUN
cana-3960	275	28	,	,	PUNCT
cana-3960	275	29	32(12	32(12	NUM
cana-3960	275	30	)	)	PUNCT
cana-3960	275	31	(	(	PUNCT
cana-3960	275	32	2018	2018	NUM
cana-3960	275	33	)	)	PUNCT
cana-3960	275	34	,	,	PUNCT
cana-3960	275	35	4209	4209	NUM
cana-3960	275	36	-	-	SYM
cana-3960	275	37	4219	4219	NUM
cana-3960	275	38	.	.	PUNCT
cana-3960	276	1	[	[	X
cana-3960	276	2	5	5	NUM
cana-3960	276	3	]	]	X
cana-3960	276	4	atici	atici	PROPN
cana-3960	276	5	f.m	f.m	PROPN
cana-3960	276	6	,	,	PUNCT
cana-3960	276	7	biles	biles	PROPN
cana-3960	276	8	d.c	d.c	PROPN
cana-3960	276	9	,	,	PUNCT
cana-3960	276	10	first	first	ADJ
cana-3960	276	11	and	and	CCONJ
cana-3960	276	12	second	second	ADJ
cana-3960	276	13	order	order	NOUN
cana-3960	276	14	dynamic	dynamic	ADJ
cana-3960	276	15	equations	equation	NOUN
cana-3960	276	16	with	with	ADP
cana-3960	276	17	impulse	impulse	ADJ
cana-3960	276	18	,	,	PUNCT
cana-3960	276	19	adv.difference	adv.difference	NOUN
cana-3960	277	1	equ.2(2005),119	equ.2(2005),119	PROPN
cana-3960	277	2	-	-	PUNCT
cana-3960	277	3	132	132	NUM
cana-3960	277	4	.	.	PUNCT
cana-3960	278	1	[	[	X
cana-3960	278	2	6	6	NUM
cana-3960	278	3	]	]	X
cana-3960	278	4	bhoner	bhoner	NOUN
cana-3960	278	5	m.	m.	NOUN
cana-3960	278	6	and	and	CCONJ
cana-3960	278	7	peterson	peterson	PROPN
cana-3960	278	8	a	a	DET
cana-3960	278	9	,	,	PUNCT
cana-3960	278	10	dynamic	dynamic	ADJ
cana-3960	278	11	equations	equation	NOUN
cana-3960	278	12	on	on	ADP
cana-3960	278	13	time	time	NOUN
cana-3960	278	14	scales	scale	NOUN
cana-3960	278	15	,	,	PUNCT
cana-3960	278	16	birkhauser	birkhauser	PROPN
cana-3960	278	17	,	,	PUNCT
cana-3960	278	18	boston	boston	PROPN
cana-3960	278	19	,	,	PUNCT
cana-3960	278	20	(	(	PUNCT
cana-3960	278	21	2001	2001	NUM
cana-3960	278	22	)	)	PUNCT
cana-3960	278	23	.	.	PUNCT
cana-3960	279	1	[	[	X
cana-3960	279	2	7	7	NUM
cana-3960	279	3	]	]	X
cana-3960	279	4	bhoner	bhoner	NOUN
cana-3960	279	5	m.	m.	NOUN
cana-3960	279	6	and	and	CCONJ
cana-3960	279	7	peterson	peterson	PROPN
cana-3960	279	8	a	a	PROPN
cana-3960	279	9	,	,	PUNCT
cana-3960	279	10	advances	advance	NOUN
cana-3960	279	11	in	in	ADP
cana-3960	279	12	dynamic	dynamic	ADJ
cana-3960	279	13	equations	equation	NOUN
cana-3960	279	14	on	on	ADP
cana-3960	279	15	time	time	NOUN
cana-3960	279	16	scales	scale	NOUN
cana-3960	279	17	,	,	PUNCT
cana-3960	279	18	birkhauser	birkhauser	PROPN
cana-3960	279	19	,	,	PUNCT
cana-3960	279	20	boston	boston	PROPN
cana-3960	279	21	,	,	PUNCT
cana-3960	279	22	(	(	PUNCT
cana-3960	279	23	2003	2003	NUM
cana-3960	279	24	)	)	PUNCT
cana-3960	279	25	.	.	PUNCT
cana-3960	280	1	[	[	X
cana-3960	280	2	8	8	NUM
cana-3960	280	3	]	]	PUNCT
cana-3960	280	4	cevikel	cevikel	PROPN
cana-3960	280	5	ac	ac	PROPN
cana-3960	280	6	,	,	PUNCT
cana-3960	280	7	bekir	bekir	VERB
cana-3960	280	8	a	a	PRON
cana-3960	280	9	,	,	PUNCT
cana-3960	280	10	abu	abu	PROPN
cana-3960	280	11	arqub	arqub	NOUN
cana-3960	280	12	o	o	PROPN
cana-3960	280	13	and	and	CCONJ
cana-3960	280	14	abukhaled	abukhale	VERB
cana-3960	280	15	m	m	PROPN
cana-3960	280	16	(	(	PUNCT
cana-3960	280	17	2022	2022	NUM
cana-3960	280	18	)	)	PUNCT
cana-3960	280	19	solitary	solitary	ADJ
cana-3960	280	20	wave	wave	NOUN
cana-3960	280	21	solutions	solution	NOUN
cana-3960	280	22	of	of	ADP
cana-3960	280	23	fitzhugh	fitzhugh	PROPN
cana-3960	280	24	–	–	PUNCT
cana-3960	280	25	nagumo	nagumo	ADJ
cana-3960	280	26	-	-	PUNCT
cana-3960	280	27	type	type	NOUN
cana-3960	280	28	equations	equation	NOUN
cana-3960	280	29	with	with	ADP
cana-3960	280	30	conformable	conformable	ADJ
cana-3960	280	31	derivatives	derivative	NOUN
cana-3960	280	32	.	.	PUNCT
cana-3960	281	1	front	front	ADJ
cana-3960	281	2	.	.	PUNCT
cana-3960	282	1	phys	phy	NOUN
cana-3960	282	2	.	.	PUNCT
cana-3960	283	1	10:1028668	10:1028668	X
cana-3960	283	2	.	.	PUNCT
cana-3960	283	3	doi	doi	NOUN
cana-3960	283	4	:	:	PUNCT
cana-3960	283	5	10.3389	10.3389	NUM
cana-3960	283	6	/	/	SYM
cana-3960	283	7	fphy.2022.1028668	fphy.2022.1028668	NOUN
cana-3960	283	8	[	[	X
cana-3960	283	9	9	9	NUM
cana-3960	283	10	]	]	PUNCT
cana-3960	283	11	dacunha	dacunha	VERB
cana-3960	283	12	j.j	j.j	PROPN
cana-3960	283	13	.	.	PROPN
cana-3960	283	14	,	,	PUNCT
cana-3960	283	15	transition	transition	NOUN
cana-3960	283	16	matrix	matrix	NOUN
cana-3960	283	17	and	and	CCONJ
cana-3960	283	18	generalized	generalized	ADJ
cana-3960	283	19	matrix	matrix	NOUN
cana-3960	283	20	exponential	exponential	NOUN
cana-3960	283	21	via	via	ADP
cana-3960	283	22	the	the	DET
cana-3960	283	23	peano	peano	PROPN
cana-3960	283	24	-	-	PUNCT
cana-3960	283	25	baker	baker	PROPN
cana-3960	283	26	series	series	PROPN
cana-3960	283	27	,	,	PUNCT
cana-3960	283	28	j.	j.	PROPN
cana-3960	283	29	differ	differ	VERB
cana-3960	283	30	.	.	PUNCT
cana-3960	284	1	equ	equ	PROPN
cana-3960	284	2	.	.	PUNCT
cana-3960	284	3	appl.11	appl.11	PROPN
cana-3960	284	4	,	,	PUNCT
cana-3960	284	5	(	(	PUNCT
cana-3960	284	6	15	15	NUM
cana-3960	284	7	)	)	PUNCT
cana-3960	284	8	,	,	PUNCT
cana-3960	284	9	(	(	PUNCT
cana-3960	284	10	2005),1245	2005),1245	NUM
cana-3960	284	11	-	-	SYM
cana-3960	284	12	1264	1264	NUM
cana-3960	284	13	.	.	PUNCT
cana-3960	285	1	[	[	X
cana-3960	285	2	10	10	NUM
cana-3960	285	3	]	]	X
cana-3960	285	4	m	m	NOUN
cana-3960	285	5	benchohra	benchohra	NOUN
cana-3960	285	6	,	,	PUNCT
cana-3960	285	7	j	j	PROPN
cana-3960	285	8	henderson	henderson	PROPN
cana-3960	285	9	,	,	PUNCT
cana-3960	285	10	and	and	CCONJ
cana-3960	285	11	sk	sk	ADP
cana-3960	285	12	ntouyas	ntouyas	NOUN
cana-3960	285	13	,	,	PUNCT
cana-3960	285	14	impulsive	impulsive	ADJ
cana-3960	285	15	differential	differential	ADJ
cana-3960	285	16	equations	equation	NOUN
cana-3960	285	17	and	and	CCONJ
cana-3960	285	18	inclusions	inclusion	NOUN
cana-3960	285	19	,	,	PUNCT
cana-3960	285	20	vol	vol	NOUN
cana-3960	285	21	.	.	PROPN
cana-3960	285	22	2	2	NUM
cana-3960	285	23	,	,	PUNCT
cana-3960	285	24	hindawi	hindawi	ADJ
cana-3960	285	25	publishing	publishing	NOUN
cana-3960	285	26	corporation	corporation	NOUN
cana-3960	285	27	,	,	PUNCT
cana-3960	285	28	new	new	PROPN
cana-3960	285	29	york	york	PROPN
cana-3960	285	30	,	,	PUNCT
cana-3960	285	31	2006	2006	NUM
cana-3960	285	32	[	[	X
cana-3960	285	33	11	11	NUM
cana-3960	285	34	]	]	X
cana-3960	285	35	davis	davis	PROPN
cana-3960	285	36	jhon	jhon	PROPN
cana-3960	285	37	m	m	PROPN
cana-3960	285	38	,	,	PUNCT
cana-3960	285	39	gravagre	gravagre	PROPN
cana-3960	285	40	ian	ian	PROPN
cana-3960	285	41	a	a	PROPN
cana-3960	285	42	,	,	PUNCT
cana-3960	285	43	jackson	jackson	PROPN
cana-3960	285	44	billy	billy	PROPN
cana-3960	285	45	j	j	PROPN
cana-3960	285	46	,	,	PUNCT
cana-3960	285	47	and	and	CCONJ
cana-3960	285	48	marks	mark	VERB
cana-3960	285	49	robert	robert	PROPN
cana-3960	285	50	j	j	PROPN
cana-3960	285	51	,	,	PUNCT
cana-3960	285	52	controllability	controllability	NOUN
cana-3960	285	53	,	,	PUNCT
cana-3960	285	54	observability	observability	NOUN
cana-3960	285	55	,	,	PUNCT
cana-3960	285	56	realizability	realizability	NOUN
cana-3960	285	57	and	and	CCONJ
cana-3960	285	58	stability	stability	NOUN
cana-3960	285	59	of	of	ADP
cana-3960	285	60	dynamic	dynamic	ADJ
cana-3960	285	61	linear	linear	NOUN
cana-3960	285	62	systems	system	NOUN
cana-3960	285	63	,	,	PUNCT
cana-3960	285	64	electronic	electronic	ADJ
cana-3960	285	65	journal	journal	NOUN
cana-3960	285	66	of	of	ADP
cana-3960	285	67	differential	differential	ADJ
cana-3960	285	68	equations	equation	NOUN
cana-3960	285	69	vol	vol	VERB
cana-3960	285	70	2009	2009	NUM
cana-3960	285	71	no.37	no.37	NOUN
cana-3960	285	72	,	,	PUNCT
cana-3960	285	73	(	(	PUNCT
cana-3960	285	74	2009	2009	NUM
cana-3960	285	75	)	)	PUNCT
cana-3960	285	76	,	,	PUNCT
cana-3960	285	77	1	1	NUM
cana-3960	285	78	-	-	SYM
cana-3960	285	79	32	32	NUM
cana-3960	285	80	.	.	PUNCT
cana-3960	286	1	[	[	X
cana-3960	286	2	12	12	NUM
cana-3960	286	3	]	]	PUNCT
cana-3960	286	4	dashkovskiy	dashkovskiy	NOUN
cana-3960	286	5	,	,	PUNCT
cana-3960	286	6	s	s	NOUN
cana-3960	286	7	,	,	PUNCT
cana-3960	286	8	slynko	slynko	ADJ
cana-3960	286	9	,	,	PUNCT
cana-3960	286	10	v	v	ADP
cana-3960	286	11	stability	stability	NOUN
cana-3960	286	12	conditions	condition	NOUN
cana-3960	286	13	for	for	ADP
cana-3960	286	14	impulsive	impulsive	ADJ
cana-3960	286	15	dynamical	dynamical	ADJ
cana-3960	286	16	systems	system	NOUN
cana-3960	286	17	,	,	PUNCT
cana-3960	286	18	mathematics	mathematic	NOUN
cana-3960	286	19	of	of	ADP
cana-3960	286	20	control	control	NOUN
cana-3960	286	21	,	,	PUNCT
cana-3960	286	22	signals	signal	NOUN
cana-3960	286	23	,	,	PUNCT
cana-3960	286	24	and	and	CCONJ
cana-3960	286	25	systems	system	NOUN
cana-3960	286	26	,	,	PUNCT
cana-3960	286	27	2021	2021	NUM
cana-3960	286	28	,	,	PUNCT
cana-3960	286	29	https://doi.org/10.1007/s00498-021-00305-y	https://doi.org/10.1007/s00498-021-00305-y	X
cana-3960	286	30	[	[	X
cana-3960	286	31	13	13	NUM
cana-3960	286	32	]	]	X
cana-3960	286	33	fausett	fausett	PROPN
cana-3960	286	34	l.	l.	PROPN
cana-3960	286	35	v.	v.	PROPN
cana-3960	286	36	and	and	CCONJ
cana-3960	286	37	murty	murty	PROPN
cana-3960	286	38	k.n	k.n	PROPN
cana-3960	286	39	.	.	PROPN
cana-3960	286	40	controllability	controllability	PROPN
cana-3960	286	41	,	,	PUNCT
cana-3960	286	42	observability	observability	NOUN
cana-3960	286	43	,	,	PUNCT
cana-3960	286	44	and	and	CCONJ
cana-3960	286	45	realizability	realizability	NOUN
cana-3960	286	46	criteria	criterion	NOUN
cana-3960	286	47	on	on	ADP
cana-3960	286	48	time	time	NOUN
cana-3960	286	49	scale	scale	NOUN
cana-3960	286	50	dynamical	dynamical	ADJ
cana-3960	286	51	systems	system	NOUN
cana-3960	286	52	,	,	PUNCT
cana-3960	286	53	nonlinear	nonlinear	ADJ
cana-3960	286	54	stud.11	stud.11	ADJ
cana-3960	286	55	(	(	PUNCT
cana-3960	286	56	2004	2004	NUM
cana-3960	286	57	)	)	PUNCT
cana-3960	286	58	,	,	PUNCT
cana-3960	286	59	627–638	627–638	NUM
cana-3960	286	60	.	.	PUNCT
cana-3960	287	1	[	[	X
cana-3960	287	2	14	14	NUM
cana-3960	287	3	]	]	X
cana-3960	287	4	hilger	hilger	NOUN
cana-3960	287	5	s	s	PROPN
cana-3960	287	6	,	,	PUNCT
cana-3960	287	7	analysis	analysis	NOUN
cana-3960	287	8	on	on	ADP
cana-3960	287	9	measure	measure	NOUN
cana-3960	287	10	chains	chain	NOUN
cana-3960	287	11	a	a	DET
cana-3960	287	12	unified	unified	ADJ
cana-3960	287	13	approach	approach	NOUN
cana-3960	287	14	to	to	ADP
cana-3960	287	15	continuous	continuous	ADJ
cana-3960	287	16	and	and	CCONJ
cana-3960	287	17	discrete	discrete	ADJ
cana-3960	287	18	calculus	calculus	NOUN
cana-3960	287	19	,	,	PUNCT
cana-3960	287	20	results	result	VERB
cana-3960	287	21	math	math	NOUN
cana-3960	287	22	.	.	PUNCT
cana-3960	288	1	18	18	NUM
cana-3960	288	2	,	,	PUNCT
cana-3960	288	3	(	(	PUNCT
cana-3960	288	4	1990	1990	NUM
cana-3960	288	5	)	)	PUNCT
cana-3960	288	6	,	,	PUNCT
cana-3960	288	7	18–56	18–56	NUM
cana-3960	288	8	.	.	PUNCT
cana-3960	289	1	[	[	X
cana-3960	289	2	15	15	NUM
cana-3960	289	3	]	]	X
cana-3960	289	4	hilger	hilger	NOUN
cana-3960	289	5	s.	s.	PROPN
cana-3960	289	6	ein	ein	AUX
cana-3960	289	7	ma𝛽kettenkalkul	ma𝛽kettenkalkul	PROPN
cana-3960	289	8	mit	mit	PROPN
cana-3960	289	9	anwendung	anwendung	PROPN
cana-3960	289	10	auf	auf	PROPN
cana-3960	289	11	zentrmsmannigfaltingkeiten	zentrmsmannigfaltingkeiten	VERB
cana-3960	289	12	.	.	PUNCT
cana-3960	290	1	phd	phd	NOUN
cana-3960	290	2	thesis	thesis	NOUN
cana-3960	290	3	,	,	PUNCT
cana-3960	290	4	universi	universi	ADJ
cana-3960	290	5	.	.	PUNCT
cana-3960	291	1	wurzburg	wurzburg	PROPN
cana-3960	291	2	;	;	PUNCT
cana-3960	291	3	1988	1988	NUM
cana-3960	291	4	.	.	PUNCT
cana-3960	292	1	[	[	X
cana-3960	292	2	16	16	NUM
cana-3960	292	3	]	]	X
cana-3960	292	4	ionescu	ionescu	PROPN
cana-3960	292	5	,	,	PUNCT
cana-3960	292	6	adela	adela	PROPN
cana-3960	292	7	.	.	PUNCT
cana-3960	293	1	(	(	PUNCT
cana-3960	293	2	2021	2021	NUM
cana-3960	293	3	)	)	PUNCT
cana-3960	294	1	,	,	PUNCT
cana-3960	294	2	qualitative	qualitative	VERB
cana-3960	294	3	analysis	analysis	NOUN
cana-3960	294	4	for	for	ADP
cana-3960	294	5	controllable	controllable	ADJ
cana-3960	294	6	dynamical	dynamical	ADJ
cana-3960	294	7	systems	system	NOUN
cana-3960	294	8	:	:	PUNCT
cana-3960	294	9	stability	stability	NOUN
cana-3960	294	10	with	with	ADP
cana-3960	294	11	control	control	NOUN
cana-3960	294	12	lyapunov	lyapunov	NOUN
cana-3960	294	13	functions	function	NOUN
cana-3960	294	14	”	"	PUNCT
cana-3960	294	15	.	.	PUNCT
cana-3960	295	1	advances	advance	NOUN
cana-3960	295	2	in	in	ADP
cana-3960	295	3	dynamical	dynamical	ADJ
cana-3960	295	4	systems	system	NOUN
cana-3960	295	5	theory	theory	NOUN
cana-3960	295	6	,	,	PUNCT
cana-3960	295	7	models	model	NOUN
cana-3960	295	8	,	,	PUNCT
cana-3960	295	9	algorithms	algorithm	NOUN
cana-3960	295	10	and	and	CCONJ
cana-3960	295	11	applications	application	NOUN
cana-3960	295	12	,	,	PUNCT
cana-3960	295	13	edited	edit	VERB
cana-3960	295	14	by	by	ADP
cana-3960	295	15	bruno	bruno	PROPN
cana-3960	295	16	carpentieri	carpentieri	PROPN
cana-3960	295	17	,	,	PUNCT
cana-3960	295	18	intechopen	intechopen	ADJ
cana-3960	295	19	.	.	PUNCT
cana-3960	296	1	10.5772	10.5772	NUM
cana-3960	296	2	/	/	SYM
cana-3960	296	3	intechopen.96872	intechopen.96872	PROPN
cana-3960	296	4	.	.	PUNCT
cana-3960	297	1	[	[	X
cana-3960	297	2	17	17	NUM
cana-3960	297	3	]	]	X
cana-3960	297	4	liu	liu	PROPN
cana-3960	297	5	h	h	PROPN
cana-3960	297	6	,	,	PUNCT
cana-3960	297	7	xiang	xiang	PROPN
cana-3960	297	8	x	x	PROPN
cana-3960	297	9	,	,	PUNCT
cana-3960	297	10	a	a	DET
cana-3960	297	11	class	class	NOUN
cana-3960	297	12	of	of	ADP
cana-3960	297	13	the	the	DET
cana-3960	297	14	first	first	ADJ
cana-3960	297	15	order	order	NOUN
cana-3960	297	16	impulsive	impulsive	ADJ
cana-3960	297	17	dynamic	dynamic	ADJ
cana-3960	297	18	equations	equation	NOUN
cana-3960	297	19	on	on	ADP
cana-3960	297	20	time	time	NOUN
cana-3960	297	21	scales	scale	NOUN
cana-3960	297	22	,	,	PUNCT
cana-3960	297	23	nonlinear	nonlinear	ADJ
cana-3960	297	24	analysis	analysis	NOUN
cana-3960	297	25	,	,	PUNCT
cana-3960	297	26	69	69	NUM
cana-3960	297	27	(	(	PUNCT
cana-3960	297	28	2008	2008	NUM
cana-3960	297	29	)	)	PUNCT
cana-3960	297	30	,	,	PUNCT
cana-3960	297	31	2803	2803	NUM
cana-3960	297	32	-	-	SYM
cana-3960	297	33	2811	2811	NUM
cana-3960	297	34	.	.	PUNCT
cana-3960	298	1	[	[	X
cana-3960	298	2	18	18	NUM
cana-3960	298	3	]	]	X
cana-3960	298	4	maayah	maayah	PROPN
cana-3960	298	5	,	,	PUNCT
cana-3960	298	6	b.	b.	PROPN
cana-3960	298	7	,	,	PUNCT
cana-3960	298	8	abu	abu	PROPN
cana-3960	298	9	arqub	arqub	PROPN
cana-3960	298	10	,	,	PUNCT
cana-3960	298	11	o.	o.	PROPN
cana-3960	298	12	,	,	PUNCT
cana-3960	298	13	alnabulsi	alnabulsi	NOUN
cana-3960	298	14	,	,	PUNCT
cana-3960	298	15	s.	s.	PROPN
cana-3960	298	16	,	,	PUNCT
cana-3960	298	17	&	&	CCONJ
cana-3960	298	18	alsulami	alsulami	PROPN
cana-3960	298	19	,	,	PUNCT
cana-3960	298	20	h.	h.	PROPN
cana-3960	298	21	(	(	PUNCT
cana-3960	298	22	2022	2022	NUM
cana-3960	298	23	)	)	PUNCT
cana-3960	298	24	.	.	PUNCT
cana-3960	299	1	numerical	numerical	ADJ
cana-3960	299	2	solutions	solution	NOUN
cana-3960	299	3	and	and	CCONJ
cana-3960	299	4	geometric	geometric	ADJ
cana-3960	299	5	attractors	attractor	NOUN
cana-3960	299	6	of	of	ADP
cana-3960	299	7	a	a	DET
cana-3960	299	8	fractional	fractional	ADJ
cana-3960	299	9	model	model	NOUN
cana-3960	299	10	of	of	ADP
cana-3960	299	11	the	the	DET
cana-3960	299	12	cancer	cancer	NOUN
cana-3960	299	13	-	-	PUNCT
cana-3960	299	14	immune	immune	NOUN
cana-3960	299	15	based	base	VERB
cana-3960	299	16	on	on	ADP
cana-3960	299	17	the	the	DET
cana-3960	299	18	atangana	atangana	PROPN
cana-3960	299	19	-	-	PUNCT
cana-3960	299	20	baleanu	baleanu	PROPN
cana-3960	299	21	-	-	PUNCT
cana-3960	299	22	caputo	caputo	PROPN
cana-3960	299	23	derivative	derivative	NOUN
cana-3960	299	24	and	and	CCONJ
cana-3960	299	25	the	the	DET
cana-3960	299	26	reproducing	reproduce	VERB
cana-3960	299	27	kernel	kernel	NOUN
cana-3960	299	28	scheme	scheme	NOUN
cana-3960	299	29	.	.	PUNCT
cana-3960	300	1	chinese	chinese	ADJ
cana-3960	300	2	journal	journal	PROPN
cana-3960	300	3	of	of	ADP
cana-3960	300	4	physics	physics	PROPN
cana-3960	300	5	,	,	PUNCT
cana-3960	300	6	80	80	NUM
cana-3960	300	7	,	,	PUNCT
cana-3960	300	8	463	463	NUM
cana-3960	300	9	-	-	SYM
cana-3960	300	10	483	483	NUM
cana-3960	300	11	.	.	PUNCT
cana-3960	301	1	[	[	X
cana-3960	301	2	19	19	NUM
cana-3960	301	3	]	]	X
cana-3960	301	4	mohammad	mohammad	PROPN
cana-3960	301	5	abdel	abdel	PROPN
cana-3960	301	6	aal	aal	PROPN
cana-3960	301	7	,	,	PUNCT
cana-3960	301	8	smina	smina	ADJ
cana-3960	301	9	djennadi	djennadi	NOUN
cana-3960	301	10	,	,	PUNCT
cana-3960	301	11	omar	omar	PROPN
cana-3960	301	12	abu	abu	PROPN
cana-3960	301	13	arqub	arqub	PROPN
cana-3960	301	14	,	,	PUNCT
cana-3960	301	15	hamed	hamed	PROPN
cana-3960	301	16	alsulami	alsulami	PROPN
cana-3960	301	17	,	,	PUNCT
cana-3960	301	18	"	"	PUNCT
cana-3960	301	19	on	on	ADP
cana-3960	301	20	the	the	DET
cana-3960	301	21	recovery	recovery	NOUN
cana-3960	301	22	of	of	ADP
cana-3960	301	23	a	a	DET
cana-3960	301	24	conformable	conformable	ADJ
cana-3960	301	25	time	time	NOUN
cana-3960	301	26	-	-	PUNCT
cana-3960	301	27	dependent	dependent	ADJ
cana-3960	301	28	inverse	inverse	NOUN
cana-3960	301	29	coefficient	coefficient	NOUN
cana-3960	301	30	problem	problem	NOUN
cana-3960	301	31	for	for	ADP
cana-3960	301	32	diffusion	diffusion	NOUN
cana-3960	301	33	equation	equation	NOUN
cana-3960	301	34	of	of	ADP
cana-3960	301	35	periodic	periodic	ADJ
cana-3960	301	36	constraints	constraint	NOUN
cana-3960	301	37	type	type	NOUN
cana-3960	301	38	and	and	CCONJ
cana-3960	301	39	integral	integral	ADJ
cana-3960	301	40	overposed	overposed	ADJ
cana-3960	301	41	data	datum	NOUN
cana-3960	301	42	"	"	PUNCT
cana-3960	301	43	,	,	PUNCT
cana-3960	301	44	mathematical	mathematical	ADJ
cana-3960	301	45	problems	problem	NOUN
cana-3960	301	46	in	in	ADP
cana-3960	301	47	engineering	engineering	NOUN
cana-3960	301	48	,	,	PUNCT
cana-3960	301	49	vol	vol	NOUN
cana-3960	301	50	.	.	NOUN
cana-3960	301	51	2022	2022	NUM
cana-3960	301	52	,	,	PUNCT
cana-3960	301	53	article	article	NOUN
cana-3960	301	54	i	i	PROPN
cana-3960	301	55	d	d	PROPN
cana-3960	301	56	5104725	5104725	NUM
cana-3960	301	57	,	,	PUNCT
cana-3960	301	58	12	12	NUM
cana-3960	301	59	pages	page	NOUN
cana-3960	301	60	,	,	PUNCT
cana-3960	301	61	2022	2022	NUM
cana-3960	301	62	.	.	PUNCT
cana-3960	302	1	https://doi.org/10.1155/2022/5104725	https://doi.org/10.1155/2022/5104725	PROPN
cana-3960	303	1	[	[	X
cana-3960	303	2	20	20	NUM
cana-3960	303	3	]	]	X
cana-3960	303	4	murty	murty	NOUN
cana-3960	303	5	m.s.n	m.s.n	NOUN
cana-3960	303	6	.	.	PUNCT
cana-3960	304	1	kumar	kumar	PROPN
cana-3960	304	2	g.s	g.s	PROPN
cana-3960	304	3	,	,	PUNCT
cana-3960	304	4	appa	appa	PROPN
cana-3960	304	5	rao	rao	PROPN
cana-3960	304	6	b	b	PROPN
cana-3960	304	7	v	v	PROPN
cana-3960	304	8	and	and	CCONJ
cana-3960	304	9	prasad	prasad	PROPN
cana-3960	304	10	kasnv	kasnv	PROPN
cana-3960	304	11	,	,	PUNCT
cana-3960	304	12	on	on	ADP
cana-3960	304	13	controllability	controllability	NOUN
cana-3960	304	14	of	of	ADP
cana-3960	304	15	fuzzy	fuzzy	ADJ
cana-3960	304	16	dynamical	dynamical	ADJ
cana-3960	304	17	matrix	matrix	NOUN
cana-3960	304	18	lyapunov	lyapunov	NOUN
cana-3960	304	19	systems	system	NOUN
cana-3960	304	20	,	,	PUNCT
cana-3960	304	21	analele	analele	ADP
cana-3960	304	22	university	university	PROPN
cana-3960	304	23	,	,	PUNCT
cana-3960	304	24	di	di	X
cana-3960	304	25	vest	vest	PROPN
cana-3960	304	26	timsora	timsora	PROPN
cana-3960	304	27	,	,	PUNCT
cana-3960	304	28	seria	seria	PROPN
cana-3960	304	29	mathematica	mathematica	PROPN
cana-3960	304	30	informatica	informatica	PROPN
cana-3960	304	31	li	li	PROPN
cana-3960	304	32	,	,	PUNCT
cana-3960	304	33	2(13	2(13	NUM
cana-3960	304	34	)	)	PUNCT
cana-3960	304	35	,	,	PUNCT
cana-3960	304	36	73	73	NUM
cana-3960	304	37	-	-	SYM
cana-3960	304	38	86	86	NUM
cana-3960	304	39	.	.	PUNCT
cana-3960	305	1	communications	communication	NOUN
cana-3960	305	2	on	on	ADP
cana-3960	305	3	applied	apply	VERB
cana-3960	305	4	nonlinear	nonlinear	ADJ
cana-3960	305	5	analysis	analysis	NOUN
cana-3960	305	6	issn	issn	NOUN
cana-3960	305	7	:	:	PUNCT
cana-3960	305	8	1074	1074	NUM
cana-3960	305	9	-	-	PUNCT
cana-3960	305	10	133x	133x	NUM
cana-3960	305	11	vol	vol	NOUN
cana-3960	305	12	32	32	NUM
cana-3960	305	13	no	no	NOUN
cana-3960	305	14	.	.	PUNCT
cana-3960	306	1	9s	9s	NUM
cana-3960	306	2	(	(	PUNCT
cana-3960	306	3	2025	2025	NUM
cana-3960	306	4	)	)	PUNCT
cana-3960	306	5	507	507	NUM
cana-3960	306	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3960	307	1	[	[	X
cana-3960	307	2	21	21	NUM
cana-3960	307	3	]	]	X
cana-3960	307	4	murty	murty	NOUN
cana-3960	307	5	m.s.n	m.s.n	NOUN
cana-3960	307	6	.	.	PUNCT
cana-3960	307	7	,	,	PUNCT
cana-3960	307	8	appa	appa	PROPN
cana-3960	307	9	rao	rao	PROPN
cana-3960	307	10	b.	b.	PROPN
cana-3960	307	11	v	v	PROPN
cana-3960	307	12	,	,	PUNCT
cana-3960	307	13	controllability	controllability	NOUN
cana-3960	307	14	and	and	CCONJ
cana-3960	307	15	observability	observability	NOUN
cana-3960	307	16	of	of	ADP
cana-3960	307	17	matrix	matrix	NOUN
cana-3960	307	18	lyapunov	lyapunov	NOUN
cana-3960	307	19	systems	system	NOUN
cana-3960	307	20	,	,	PUNCT
cana-3960	307	21	ranchi	ranchi	PROPN
cana-3960	307	22	univ.math	univ.math	PROPN
cana-3960	307	23	.	.	PUNCT
cana-3960	307	24	journal	journal	PROPN
cana-3960	307	25	vol.no.32	vol.no.32	PROPN
cana-3960	307	26	,	,	PUNCT
cana-3960	307	27	(	(	PUNCT
cana-3960	307	28	2005),55	2005),55	NUM
cana-3960	307	29	-	-	SYM
cana-3960	307	30	65	65	NUM
cana-3960	307	31	.	.	PUNCT
cana-3960	308	1	[	[	X
cana-3960	308	2	22	22	NUM
cana-3960	308	3	]	]	X
cana-3960	308	4	murty	murty	NOUN
cana-3960	308	5	m.s.n	m.s.n	NOUN
cana-3960	308	6	.	.	PUNCT
cana-3960	308	7	,	,	PUNCT
cana-3960	308	8	appa	appa	PROPN
cana-3960	308	9	rao	rao	PROPN
cana-3960	308	10	b.	b.	PROPN
cana-3960	308	11	v	v	PROPN
cana-3960	308	12	and	and	CCONJ
cana-3960	308	13	suresh	suresh	PROPN
cana-3960	308	14	kumar	kumar	PROPN
cana-3960	308	15	g.	g.	PROPN
cana-3960	308	16	,	,	PUNCT
cana-3960	308	17	controllability	controllability	NOUN
cana-3960	308	18	,	,	PUNCT
cana-3960	308	19	observability	observability	NOUN
cana-3960	308	20	and	and	CCONJ
cana-3960	308	21	realizability	realizability	NOUN
cana-3960	308	22	of	of	ADP
cana-3960	308	23	matrix	matrix	NOUN
cana-3960	308	24	lyapunov	lyapunov	NOUN
cana-3960	308	25	systems	system	NOUN
cana-3960	308	26	,	,	PUNCT
cana-3960	308	27	bull.korean	bull.korean	PROPN
cana-3960	308	28	math.soc	math.soc	PROPN
cana-3960	308	29	.	.	PROPN
cana-3960	308	30	,43	,43	NOUN
cana-3960	308	31	,	,	PUNCT
cana-3960	308	32	no.1	no.1	NUM
cana-3960	308	33	,	,	PUNCT
cana-3960	308	34	(	(	PUNCT
cana-3960	308	35	2006	2006	NUM
cana-3960	308	36	)	)	PUNCT
cana-3960	308	37	,	,	PUNCT
cana-3960	308	38	149	149	NUM
cana-3960	308	39	-	-	SYM
cana-3960	308	40	159	159	NUM
cana-3960	308	41	.	.	PUNCT
cana-3960	309	1	[	[	X
cana-3960	309	2	23	23	NUM
cana-3960	309	3	]	]	X
cana-3960	309	4	sreenivasulu	sreenivasulu	PROPN
cana-3960	309	5	,	,	PUNCT
cana-3960	309	6	a	a	PRON
cana-3960	309	7	,	,	PUNCT
cana-3960	309	8	appa	appa	PROPN
cana-3960	309	9	rao	rao	PROPN
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cana-3960	309	13	,	,	PUNCT
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cana-3960	309	16	observability	observability	NOUN
cana-3960	309	17	for	for	ADP
cana-3960	309	18	matrix	matrix	NOUN
cana-3960	309	19	sylvester	sylvester	NOUN
cana-3960	309	20	impulsive	impulsive	ADJ
cana-3960	309	21	non	non	ADJ
cana-3960	309	22	-	-	ADJ
cana-3960	309	23	linear	linear	ADJ
cana-3960	309	24	dynamic	dynamic	ADJ
cana-3960	309	25	system	system	NOUN
cana-3960	309	26	with	with	ADP
cana-3960	309	27	delta	delta	NOUN
cana-3960	309	28	settings	setting	NOUN
cana-3960	309	29	,	,	PUNCT
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cana-3960	309	32	in	in	ADP
cana-3960	309	33	aip	aip	PROPN
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cana-3960	309	35	proceedings	proceeding	NOUN
cana-3960	309	36	,	,	PUNCT
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cana-3960	309	38	.	.	PUNCT
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cana-3960	310	2	24	24	NUM
cana-3960	310	3	]	]	PUNCT
cana-3960	310	4	xie	xie	PROPN
cana-3960	310	5	g.m	g.m	PROPN
cana-3960	310	6	,	,	PUNCT
cana-3960	310	7	wang	wang	PROPN
cana-3960	310	8	l.	l.	PROPN
cana-3960	310	9	,	,	PUNCT
cana-3960	310	10	controllability	controllability	NOUN
cana-3960	310	11	and	and	CCONJ
cana-3960	310	12	observability	observability	NOUN
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cana-3960	310	14	a	a	DET
cana-3960	310	15	class	class	NOUN
cana-3960	310	16	of	of	ADP
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cana-3960	310	18	impulsive	impulsive	ADJ
cana-3960	310	19	systems	system	NOUN
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cana-3960	310	21	.	.	PUNCT
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cana-3960	311	2	-	-	PUNCT
cana-3960	311	3	355	355	NUM
cana-3960	311	4	.	.	PUNCT
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cana-3960	312	2	25	25	NUM
cana-3960	312	3	]	]	X
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cana-3960	312	5	s.	s.	PROPN
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cana-3960	312	11	and	and	CCONJ
cana-3960	312	12	observability	observability	NOUN
cana-3960	312	13	of	of	ADP
cana-3960	312	14	a	a	DET
cana-3960	312	15	class	class	NOUN
cana-3960	312	16	of	of	ADP
cana-3960	312	17	time	time	NOUN
cana-3960	312	18	varying	vary	VERB
cana-3960	312	19	impulsive	impulsive	ADJ
cana-3960	312	20	systems	system	NOUN
cana-3960	312	21	,	,	PUNCT
cana-3960	312	22	nonlinear	nonlinear	ADJ
cana-3960	312	23	analysis	analysis	NOUN
cana-3960	312	24	:	:	PUNCT
cana-3960	312	25	real	real	ADJ
cana-3960	312	26	world	world	NOUN
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cana-3960	312	30	)	)	PUNCT
cana-3960	312	31	,	,	PUNCT
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cana-3960	312	33	-	-	SYM
cana-3960	312	34	1380	1380	NUM
cana-3960	312	35	.	.	PUNCT
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cana-3960	313	2	26	26	NUM
cana-3960	313	3	]	]	X
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cana-3960	313	8	.	.	PROPN
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cana-3960	313	10	and	and	CCONJ
cana-3960	313	11	l.	l.	PROPN
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cana-3960	313	17	hybrid	hybrid	ADJ
cana-3960	313	18	dynamical	dynamical	ADJ
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cana-3960	313	20	,	,	PUNCT
cana-3960	313	21	ieee	ieee	NOUN
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cana-3960	313	23	.	.	PUNCT
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cana-3960	314	2	.	.	PUNCT
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cana-3960	315	4	)	)	PUNCT
cana-3960	315	5	,	,	PUNCT
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cana-3960	315	7	,	,	PUNCT
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cana-3960	315	9	-	-	SYM
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cana-3960	315	11	.	.	PUNCT
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cana-3960	316	2	27	27	NUM
cana-3960	316	3	]	]	PUNCT
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cana-3960	316	9	.	.	PUNCT
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cana-3960	317	3	x.	x.	PROPN
cana-3960	317	4	h.	h.	PROPN
cana-3960	317	5	xu	xu	PROPN
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cana-3960	317	10	a	a	DET
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cana-3960	317	13	hybrid	hybrid	ADJ
cana-3960	317	14	dynamic	dynamic	ADJ
cana-3960	317	15	system	system	NOUN
cana-3960	317	16	,	,	PUNCT
cana-3960	317	17	automatica	automatica	PROPN
cana-3960	317	18	,	,	PUNCT
cana-3960	317	19	36(200	36(200	NUM
cana-3960	317	20	)	)	PUNCT
cana-3960	317	21	,	,	PUNCT
cana-3960	317	22	297	297	NUM
cana-3960	317	23	-	-	SYM
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cana-3960	317	25	.	.	PUNCT
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cana-3960	318	2	28	28	NUM
cana-3960	318	3	]	]	X
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cana-3960	318	5	,	,	PUNCT
cana-3960	318	6	marco	marco	PROPN
cana-3960	318	7	;	;	PUNCT
cana-3960	318	8	kumar	kumar	PROPN
cana-3960	318	9	,	,	PUNCT
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cana-3960	318	11	,	,	PUNCT
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cana-3960	318	13	c	c	NOUN
cana-3960	318	14	-	-	PUNCT
cana-3960	318	15	almost	almost	ADV
cana-3960	318	16	periodic	periodic	ADJ
cana-3960	318	17	type	type	NOUN
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cana-3960	318	19	in	in	ADP
cana-3960	318	20	rn	rn	PROPN
cana-3960	318	21	.	.	PROPN
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cana-3960	318	23	mathematicum	mathematicum	PROPN
cana-3960	318	24	,	,	PUNCT
cana-3960	318	25	58	58	NUM
cana-3960	318	26	(	(	PUNCT
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cana-3960	318	28	)	)	PUNCT
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cana-3960	318	34	.	.	PUNCT
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cana-3960	319	2	-	-	SYM
cana-3960	319	3	104	104	NUM
cana-3960	319	4	doi	doi	NOUN
cana-3960	319	5	:	:	PUNCT
cana-3960	319	6	10.5817	10.5817	NUM
cana-3960	319	7	/	/	SYM
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cana-3960	319	9	-	-	PUNCT
cana-3960	319	10	2	2	NUM
cana-3960	319	11	-	-	PUNCT
cana-3960	319	12	85	85	NUM
cana-3960	319	13	.	.	PUNCT
cana-3960	320	1	[	[	X
cana-3960	320	2	29	29	NUM
cana-3960	320	3	]	]	X
cana-3960	320	4	kumar	kumar	PROPN
cana-3960	320	5	,	,	PUNCT
cana-3960	320	6	vipin	vipin	PROPN
cana-3960	320	7	and	and	CCONJ
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cana-3960	320	10	muslim	muslim	PROPN
cana-3960	320	11	.	.	PUNCT
cana-3960	320	12	existence	existence	NOUN
cana-3960	320	13	and	and	CCONJ
cana-3960	320	14	stability	stability	NOUN
cana-3960	320	15	results	result	NOUN
cana-3960	320	16	of	of	ADP
cana-3960	320	17	nonlinear	nonlinear	ADJ
cana-3960	320	18	fractional	fractional	ADJ
cana-3960	320	19	differential	differential	ADJ
cana-3960	320	20	equations	equation	NOUN
cana-3960	320	21	with	with	ADP
cana-3960	320	22	nonlinear	nonlinear	ADJ
cana-3960	320	23	integral	integral	ADJ
cana-3960	320	24	boundary	boundary	ADJ
cana-3960	320	25	condition	condition	NOUN
cana-3960	320	26	on	on	ADP
cana-3960	320	27	time	time	NOUN
cana-3960	320	28	scales	scale	NOUN
cana-3960	320	29	,	,	PUNCT
cana-3960	320	30	applications	application	NOUN
cana-3960	320	31	and	and	CCONJ
cana-3960	320	32	applied	apply	VERB
cana-3960	320	33	mathematics	mathematic	NOUN
cana-3960	320	34	:	:	PUNCT
cana-3960	320	35	an	an	DET
cana-3960	320	36	international	international	ADJ
cana-3960	320	37	journal	journal	NOUN
cana-3960	320	38	(	(	PUNCT
cana-3960	320	39	aam	aam	PROPN
cana-3960	320	40	)	)	PUNCT
cana-3960	320	41	,	,	PUNCT
cana-3960	320	42	15	15	NUM
cana-3960	320	43	,	,	PUNCT
cana-3960	320	44	iss	iss	PROPN
cana-3960	320	45	.	.	PROPN
cana-3960	320	46	3	3	NUM
cana-3960	320	47	,	,	PUNCT
cana-3960	320	48	(	(	PUNCT
cana-3960	320	49	2020	2020	NUM
cana-3960	320	50	)	)	PUNCT
cana-3960	320	51	.	.	PUNCT
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cana-3960	321	2	.	.	PUNCT
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cana-3960	322	2	30	30	NUM
cana-3960	322	3	]	]	X
cana-3960	322	4	vipin	vipin	PROPN
cana-3960	322	5	kumar	kumar	PROPN
cana-3960	322	6	,	,	PUNCT
cana-3960	322	7	muslim	muslim	PROPN
cana-3960	322	8	malik	malik	PROPN
cana-3960	322	9	,	,	PUNCT
cana-3960	322	10	mohamed	mohamed	PROPN
cana-3960	322	11	djemai	djemai	PROPN
cana-3960	322	12	,	,	PUNCT
cana-3960	322	13	results	result	NOUN
cana-3960	322	14	on	on	ADP
cana-3960	322	15	abstract	abstract	ADJ
cana-3960	322	16	integro	integro	ADJ
cana-3960	322	17	hybrid	hybrid	ADJ
cana-3960	322	18	evolution	evolution	NOUN
cana-3960	322	19	system	system	NOUN
cana-3960	322	20	with	with	ADP
cana-3960	322	21	impulses	impulse	NOUN
cana-3960	322	22	on	on	ADP
cana-3960	322	23	time	time	NOUN
cana-3960	322	24	scales	scale	NOUN
cana-3960	322	25	,	,	PUNCT
cana-3960	322	26	nonlinear	nonlinear	ADJ
cana-3960	322	27	analysis	analysis	NOUN
cana-3960	322	28	:	:	PUNCT
cana-3960	322	29	hybrid	hybrid	ADJ
cana-3960	322	30	systems	system	NOUN
cana-3960	322	31	,	,	PUNCT
cana-3960	322	32	39	39	NUM
cana-3960	322	33	,	,	PUNCT
cana-3960	322	34	(	(	PUNCT
cana-3960	322	35	2021	2021	NUM
cana-3960	322	36	)	)	PUNCT
cana-3960	322	37	.	.	PUNCT
cana-3960	323	1	[	[	X
cana-3960	323	2	31	31	NUM
cana-3960	323	3	]	]	PUNCT
cana-3960	323	4	vipin	vipin	PROPN
cana-3960	323	5	kumar	kumar	PROPN
cana-3960	323	6	,	,	PUNCT
cana-3960	323	7	mohamed	mohamed	PROPN
cana-3960	323	8	djemai	djemai	PROPN
cana-3960	323	9	,	,	PUNCT
cana-3960	323	10	existence	existence	NOUN
cana-3960	323	11	,	,	PUNCT
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cana-3960	323	13	and	and	CCONJ
cana-3960	323	14	controllability	controllability	NOUN
cana-3960	323	15	of	of	ADP
cana-3960	323	16	piece	piece	NOUN
cana-3960	323	17	wise	wise	ADJ
cana-3960	323	18	impulsive	impulsive	ADJ
cana-3960	323	19	dynamic	dynamic	ADJ
cana-3960	323	20	systems	system	NOUN
cana-3960	323	21	on	on	ADP
cana-3960	323	22	arbitrary	arbitrary	ADJ
cana-3960	323	23	time	time	NOUN
cana-3960	323	24	domain	domain	NOUN
cana-3960	323	25	,	,	PUNCT
cana-3960	323	26	applied	apply	VERB
cana-3960	323	27	mathematical	mathematical	ADJ
cana-3960	323	28	modelling,117	modelling,117	PROPN
cana-3960	323	29	,	,	PUNCT
cana-3960	323	30	(	(	PUNCT
cana-3960	323	31	2023	2023	NUM
cana-3960	323	32	)	)	PUNCT
cana-3960	323	33	,	,	PUNCT
cana-3960	323	34	pages	page	NOUN
cana-3960	323	35	529	529	NUM
cana-3960	323	36	-	-	SYM
cana-3960	323	37	548	548	NUM
cana-3960	323	38	.	.	PUNCT
cana-3960	324	1	https://doi.org/10.1016/j.apm.2022.12.027	https://doi.org/10.1016/j.apm.2022.12.027	PROPN
cana-3960	324	2	.	.	PUNCT
cana-3960	325	1	[	[	X
cana-3960	325	2	32	32	NUM
cana-3960	325	3	]	]	SYM
cana-3960	325	4	sweis	sweis	PROPN
cana-3960	325	5	,	,	PUNCT
cana-3960	325	6	h.	h.	PROPN
cana-3960	325	7	,	,	PUNCT
cana-3960	325	8	arqub	arqub	NOUN
cana-3960	325	9	,	,	PUNCT
cana-3960	325	10	o.	o.	NOUN
cana-3960	325	11	a.	a.	NOUN
cana-3960	325	12	,	,	PUNCT
cana-3960	325	13	and	and	CCONJ
cana-3960	325	14	shawagfeh	shawagfeh	NOUN
cana-3960	325	15	,	,	PUNCT
cana-3960	325	16	n.	n.	NOUN
cana-3960	325	17	,	,	PUNCT
cana-3960	325	18	fractional	fractional	ADJ
cana-3960	325	19	delay	delay	NOUN
cana-3960	325	20	integrodifferential	integrodifferential	ADJ
cana-3960	325	21	equations	equation	NOUN
cana-3960	325	22	of	of	ADP
cana-3960	325	23	nonsingular	nonsingular	ADJ
cana-3960	325	24	kernels	kernel	NOUN
cana-3960	325	25	:	:	PUNCT
cana-3960	325	26	existence	existence	NOUN
cana-3960	325	27	,	,	PUNCT
cana-3960	325	28	uniqueness	uniqueness	NOUN
cana-3960	325	29	,	,	PUNCT
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cana-3960	325	32	solutions	solution	NOUN
cana-3960	325	33	using	use	VERB
cana-3960	325	34	galerkin	galerkin	ADJ
cana-3960	325	35	algorithm	algorithm	NOUN
cana-3960	325	36	based	base	VERB
cana-3960	325	37	on	on	ADP
cana-3960	325	38	shifted	shift	VERB
cana-3960	325	39	legendre	legendre	PROPN
cana-3960	325	40	polynomials	polynomial	NOUN
cana-3960	325	41	,	,	PUNCT
cana-3960	325	42	<	<	X
cana-3960	325	43	i	i	PROPN
cana-3960	325	44	>	>	SYM
cana-3960	325	45	international	international	ADJ
cana-3960	325	46	journal	journal	NOUN
cana-3960	325	47	of	of	ADP
cana-3960	325	48	modern	modern	ADJ
cana-3960	325	49	physics	physics	PROPN
cana-3960	325	50	c</i	c</i	PROPN
cana-3960	325	51	>	>	PROPN
cana-3960	325	52	,	,	PUNCT
cana-3960	325	53	vol	vol	NOUN
cana-3960	325	54	.	.	PROPN
cana-3960	326	1	34	34	NUM
cana-3960	326	2	,	,	PUNCT
cana-3960	326	3	no	no	INTJ
cana-3960	326	4	.	.	NOUN
cana-3960	326	5	4	4	NUM
cana-3960	326	6	,	,	PUNCT
cana-3960	326	7	2023	2023	NUM
cana-3960	326	8	.	.	PUNCT
cana-3960	327	1	doi:10.1142	doi:10.1142	NOUN
cana-3960	327	2	/	/	SYM
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