id sid tid token lemma pos cana-2410 1 1 communications communication NOUN cana-2410 1 2 on on ADP cana-2410 1 3 applied apply VERB cana-2410 1 4 nonlinear nonlinear ADJ cana-2410 1 5 analysis analysis NOUN cana-2410 1 6 issn issn NOUN cana-2410 1 7 : : PUNCT cana-2410 1 8 1074 1074 NUM cana-2410 1 9 - - PUNCT cana-2410 1 10 133x 133x NUM cana-2410 1 11 vol vol NOUN cana-2410 1 12 32 32 NUM cana-2410 1 13 no no NOUN cana-2410 1 14 . . PUNCT cana-2410 2 1 2s 2s NUM cana-2410 2 2 ( ( PUNCT cana-2410 2 3 2025 2025 NUM cana-2410 2 4 ) ) PUNCT cana-2410 2 5 350 350 NUM cana-2410 2 6 https://internationalpubls.com https://internationalpubls.com X cana-2410 2 7 controllability controllability NOUN cana-2410 2 8 for for ADP cana-2410 2 9 volterra volterra PROPN cana-2410 2 10 integro integro PROPN cana-2410 2 11 - - PUNCT cana-2410 2 12 dynamic dynamic ADJ cana-2410 2 13 sylvester sylvester NOUN cana-2410 2 14 matrix matrix NOUN cana-2410 2 15 systems system NOUN cana-2410 2 16 with with ADP cana-2410 2 17 impulse impulse NOUN cana-2410 2 18 on on ADP cana-2410 2 19 time time NOUN cana-2410 2 20 scales scale VERB cana-2410 2 21 a a DET cana-2410 2 22 sreenivasulu1 sreenivasulu1 NOUN cana-2410 2 23 * * NOUN cana-2410 2 24 , , PUNCT cana-2410 2 25 b b PROPN cana-2410 2 26 v v ADP cana-2410 2 27 appa appa PROPN cana-2410 2 28 rao1 rao1 PROPN cana-2410 2 29 , , PUNCT cana-2410 2 30 durga durga VERB cana-2410 2 31 prasad prasad PROPN cana-2410 2 32 ravutla2 ravutla2 PROPN cana-2410 2 33 and and CCONJ cana-2410 2 34 gudala gudala PROPN cana-2410 2 35 balaji balaji PROPN cana-2410 2 36 prakash3 prakash3 PROPN cana-2410 2 37 1department 1department NUM cana-2410 2 38 of of ADP cana-2410 2 39 engineering engineering NOUN cana-2410 2 40 mathematics mathematic NOUN cana-2410 2 41 , , PUNCT cana-2410 2 42 koneru koneru PROPN cana-2410 2 43 lakshmaiah lakshmaiah PROPN cana-2410 2 44 education education PROPN cana-2410 2 45 foundation foundation PROPN cana-2410 2 46 , , PUNCT cana-2410 2 47 vaddeswaram vaddeswaram NOUN cana-2410 2 48 , , PUNCT cana-2410 2 49 guntur-522502 guntur-522502 NOUN cana-2410 2 50 , , PUNCT cana-2410 2 51 andhra andhra PROPN cana-2410 2 52 pradesh pradesh PROPN cana-2410 2 53 , , PUNCT cana-2410 2 54 india india PROPN cana-2410 2 55 . . PUNCT cana-2410 3 1 2department 2department NUM cana-2410 3 2 of of ADP cana-2410 3 3 human human NOUN cana-2410 3 4 and and CCONJ cana-2410 3 5 science science NOUN cana-2410 3 6 , , PUNCT cana-2410 3 7 kg kg PROPN cana-2410 3 8 reddy reddy PROPN cana-2410 3 9 college college PROPN cana-2410 3 10 of of ADP cana-2410 3 11 engineering engineering NOUN cana-2410 3 12 and and CCONJ cana-2410 3 13 technology technology NOUN cana-2410 3 14 , , PUNCT cana-2410 3 15 moinabadh moinabadh NOUN cana-2410 3 16 mandal mandal NOUN cana-2410 3 17 , , PUNCT cana-2410 3 18 r.r r.r PROPN cana-2410 3 19 district district PROPN cana-2410 3 20 -501504 -501504 PROPN cana-2410 3 21 , , PUNCT cana-2410 3 22 telangana telangana PROPN cana-2410 3 23 state state PROPN cana-2410 3 24 , , PUNCT cana-2410 3 25 india india PROPN cana-2410 3 26 . . PUNCT cana-2410 4 1 3department 3department NUM cana-2410 4 2 of of ADP cana-2410 4 3 mathematics mathematic NOUN cana-2410 4 4 , , PUNCT cana-2410 4 5 aditya aditya PROPN cana-2410 4 6 university university PROPN cana-2410 4 7 , , PUNCT cana-2410 4 8 surampalem surampalem NOUN cana-2410 4 9 , , PUNCT cana-2410 4 10 andhra andhra PROPN cana-2410 4 11 pradesh pradesh PROPN cana-2410 4 12 , , PUNCT cana-2410 4 13 pin pin PROPN cana-2410 4 14 code code PROPN cana-2410 4 15 533437 533437 NUM cana-2410 4 16 , , PUNCT cana-2410 4 17 india india PROPN cana-2410 4 18 * * PUNCT cana-2410 4 19 corresponding correspond VERB cana-2410 4 20 author author NOUN cana-2410 4 21 . . PUNCT cana-2410 5 1 e e X cana-2410 5 2 - - NOUN cana-2410 5 3 mail mail NOUN cana-2410 5 4 : : PUNCT cana-2410 5 5 asreenivasulu@kluniversity.in asreenivasulu@kluniversity.in PROPN cana-2410 5 6 contributing contribute VERB cana-2410 5 7 author author NOUN cana-2410 5 8 : : PUNCT cana-2410 5 9 bvardr2010@kluniversity.in bvardr2010@kluniversity.in PROPN cana-2410 5 10 , , PUNCT cana-2410 5 11 durgaprasadravutla@kgr.ac.in durgaprasadravutla@kgr.ac.in ADV cana-2410 5 12 , , PUNCT cana-2410 5 13 and and CCONJ cana-2410 5 14 balajiprakashgudala@gmail.com balajiprakashgudala@gmail.com NOUN cana-2410 5 15 article article NOUN cana-2410 5 16 history history NOUN cana-2410 5 17 : : PUNCT cana-2410 5 18 received receive VERB cana-2410 5 19 : : PUNCT cana-2410 5 20 16 16 NUM cana-2410 5 21 - - SYM cana-2410 5 22 09 09 NUM cana-2410 5 23 - - PUNCT cana-2410 5 24 2024 2024 NUM cana-2410 5 25 revised revise VERB cana-2410 5 26 : : PUNCT cana-2410 5 27 23 23 NUM cana-2410 5 28 - - SYM cana-2410 5 29 10 10 NUM cana-2410 5 30 - - PUNCT cana-2410 5 31 2024 2024 NUM cana-2410 5 32 accepted accept VERB cana-2410 5 33 : : PUNCT cana-2410 5 34 04 04 NUM cana-2410 5 35 - - SYM cana-2410 5 36 11 11 NUM cana-2410 5 37 - - PUNCT cana-2410 5 38 2024 2024 NUM cana-2410 5 39 abstract abstract NOUN cana-2410 5 40 : : PUNCT cana-2410 5 41 this this DET cana-2410 5 42 article article NOUN cana-2410 5 43 presents present VERB cana-2410 5 44 the the DET cana-2410 5 45 complete complete ADJ cana-2410 5 46 controllability controllability NOUN cana-2410 5 47 for for ADP cana-2410 5 48 a a DET cana-2410 5 49 volterra volterra NOUN cana-2410 5 50 integro integro ADJ cana-2410 5 51 - - PUNCT cana-2410 5 52 dynamic dynamic ADJ cana-2410 5 53 sylvester sylvester NOUN cana-2410 5 54 matrix matrix NOUN cana-2410 5 55 system system NOUN cana-2410 5 56 with with ADP cana-2410 5 57 time time NOUN cana-2410 5 58 scale scale NOUN cana-2410 5 59 impulses impulse NOUN cana-2410 5 60 in in ADP cana-2410 5 61 a a DET cana-2410 5 62 finite finite ADJ cana-2410 5 63 - - ADJ cana-2410 5 64 dimensional dimensional ADJ cana-2410 5 65 space space NOUN cana-2410 5 66 rn rn NOUN cana-2410 5 67 . . PUNCT cana-2410 6 1 we we PRON cana-2410 6 2 utilized utilize VERB cana-2410 6 3 the the DET cana-2410 6 4 banach banach ADV cana-2410 6 5 fixed fix VERB cana-2410 6 6 point point NOUN cana-2410 6 7 theorem theorem ADJ cana-2410 6 8 and and CCONJ cana-2410 6 9 nonlinear nonlinear ADJ cana-2410 6 10 functional functional ADJ cana-2410 6 11 analysis analysis NOUN cana-2410 6 12 to to PART cana-2410 6 13 determine determine VERB cana-2410 6 14 the the DET cana-2410 6 15 existence existence NOUN cana-2410 6 16 of of ADP cana-2410 6 17 a a DET cana-2410 6 18 unique unique ADJ cana-2410 6 19 solution solution NOUN cana-2410 6 20 for for ADP cana-2410 6 21 the the DET cana-2410 6 22 system system NOUN cana-2410 6 23 , , PUNCT cana-2410 6 24 and and CCONJ cana-2410 6 25 conducted conduct VERB cana-2410 6 26 an an DET cana-2410 6 27 analysis analysis NOUN cana-2410 6 28 of of ADP cana-2410 6 29 complete complete ADJ cana-2410 6 30 controllability controllability NOUN cana-2410 6 31 using use VERB cana-2410 6 32 the the DET cana-2410 6 33 gramian gramian ADJ cana-2410 6 34 matrix matrix NOUN cana-2410 6 35 and and CCONJ cana-2410 6 36 various various ADJ cana-2410 6 37 parameter parameter NOUN cana-2410 6 38 changes change NOUN cana-2410 6 39 . . PUNCT cana-2410 7 1 we we PRON cana-2410 7 2 provided provide VERB cana-2410 7 3 a a DET cana-2410 7 4 numerical numerical ADJ cana-2410 7 5 example example NOUN cana-2410 7 6 using use VERB cana-2410 7 7 simulation simulation NOUN cana-2410 7 8 to to PART cana-2410 7 9 demonstrate demonstrate VERB cana-2410 7 10 the the DET cana-2410 7 11 application application NOUN cana-2410 7 12 of of ADP cana-2410 7 13 these these DET cana-2410 7 14 conclusions conclusion NOUN cana-2410 7 15 for for ADP cana-2410 7 16 two two NUM cana-2410 7 17 different different ADJ cana-2410 7 18 time time NOUN cana-2410 7 19 scales scale NOUN cana-2410 7 20 , , PUNCT cana-2410 7 21 t t PROPN cana-2410 7 22 = = SYM cana-2410 7 23 r r NOUN cana-2410 7 24 and and CCONJ cana-2410 7 25 t t NOUN cana-2410 7 26 = = NOUN cana-2410 7 27 p_1,1 p_1,1 NOUN cana-2410 7 28 . . PUNCT cana-2410 8 1 keywords keyword NOUN cana-2410 8 2 : : PUNCT cana-2410 8 3 controllability controllability NOUN cana-2410 8 4 , , PUNCT cana-2410 8 5 impulses impulse NOUN cana-2410 8 6 , , PUNCT cana-2410 8 7 volterra volterra PROPN cana-2410 8 8 integro integro PROPN cana-2410 8 9 - - PUNCT cana-2410 8 10 dynamic dynamic ADJ cana-2410 8 11 system system NOUN cana-2410 8 12 , , PUNCT cana-2410 8 13 time time NOUN cana-2410 8 14 scale scale NOUN cana-2410 8 15 . . PUNCT cana-2410 9 1 ams am NOUN cana-2410 9 2 subject subject ADJ cana-2410 9 3 classification classification NOUN cana-2410 9 4 : : PUNCT cana-2410 9 5 34a37 34a37 NUM cana-2410 9 6 , , PUNCT cana-2410 9 7 93b05 93b05 NUM cana-2410 9 8 , , PUNCT cana-2410 9 9 34h05 34h05 NUM cana-2410 9 10 , , PUNCT cana-2410 9 11 34n05 34n05 NUM cana-2410 9 12 . . PUNCT cana-2410 10 1 1 1 X cana-2410 10 2 . . X cana-2410 10 3 introduction introduction NOUN cana-2410 10 4 there there PRON cana-2410 10 5 are be VERB cana-2410 10 6 numerous numerous ADJ cana-2410 10 7 health health NOUN cana-2410 10 8 issues issue NOUN cana-2410 10 9 that that PRON cana-2410 10 10 exhibit exhibit VERB cana-2410 10 11 abrupt abrupt ADJ cana-2410 10 12 shifts shift NOUN cana-2410 10 13 in in ADP cana-2410 10 14 their their PRON cana-2410 10 15 states state NOUN cana-2410 10 16 . . PUNCT cana-2410 11 1 we we PRON cana-2410 11 2 refer refer VERB cana-2410 11 3 to to ADP cana-2410 11 4 these these DET cana-2410 11 5 abrupt abrupt ADJ cana-2410 11 6 alterations alteration NOUN cana-2410 11 7 as as ADP cana-2410 11 8 impulsive impulsive ADJ cana-2410 11 9 impacts impact NOUN cana-2410 11 10 within within ADP cana-2410 11 11 the the DET cana-2410 11 12 system system NOUN cana-2410 11 13 . . PUNCT cana-2410 12 1 impulsive impulsive ADJ cana-2410 12 2 differential differential ADJ cana-2410 12 3 equations equation NOUN cana-2410 12 4 are be AUX cana-2410 12 5 those those PRON cana-2410 12 6 that that PRON cana-2410 12 7 incorporate incorporate VERB cana-2410 12 8 the the DET cana-2410 12 9 impact impact NOUN cana-2410 12 10 of of ADP cana-2410 12 11 impulses impulse NOUN cana-2410 12 12 . . PUNCT cana-2410 13 1 these these PRON cana-2410 13 2 have have VERB cana-2410 13 3 substantial substantial ADJ cana-2410 13 4 applications application NOUN cana-2410 13 5 in in ADP cana-2410 13 6 several several ADJ cana-2410 13 7 real real ADJ cana-2410 13 8 - - PUNCT cana-2410 13 9 world world NOUN cana-2410 13 10 situations situation NOUN cana-2410 13 11 , , PUNCT cana-2410 13 12 specifically specifically ADV cana-2410 13 13 in in ADP cana-2410 13 14 mechanical mechanical ADJ cana-2410 13 15 systems system NOUN cana-2410 13 16 involving involve VERB cana-2410 13 17 impact impact NOUN cana-2410 13 18 , , PUNCT cana-2410 13 19 biological biological ADJ cana-2410 13 20 systems system NOUN cana-2410 13 21 like like ADP cana-2410 13 22 heartbeats heartbeat NOUN cana-2410 13 23 and and CCONJ cana-2410 13 24 population population NOUN cana-2410 13 25 dynamics dynamic NOUN cana-2410 13 26 , , PUNCT cana-2410 13 27 blood blood NOUN cana-2410 13 28 flow flow NOUN cana-2410 13 29 , , PUNCT cana-2410 13 30 ecology ecology NOUN cana-2410 13 31 , , PUNCT cana-2410 13 32 medicine medicine NOUN cana-2410 13 33 , , PUNCT cana-2410 13 34 control control NOUN cana-2410 13 35 theory theory NOUN cana-2410 13 36 , , PUNCT cana-2410 13 37 and and CCONJ cana-2410 13 38 more more ADJ cana-2410 13 39 . . PUNCT cana-2410 14 1 within within ADP cana-2410 14 2 the the DET cana-2410 14 3 current current ADJ cana-2410 14 4 body body NOUN cana-2410 14 5 of of ADP cana-2410 14 6 knowledge knowledge NOUN cana-2410 14 7 , , PUNCT cana-2410 14 8 there there PRON cana-2410 14 9 are be VERB cana-2410 14 10 two two NUM cana-2410 14 11 distinct distinct ADJ cana-2410 14 12 categories category NOUN cana-2410 14 13 of of ADP cana-2410 14 14 impulsive impulsive ADJ cana-2410 14 15 systems system NOUN cana-2410 14 16 . . PUNCT cana-2410 15 1 there there PRON cana-2410 15 2 are be VERB cana-2410 15 3 two two NUM cana-2410 15 4 types type NOUN cana-2410 15 5 of of ADP cana-2410 15 6 systems system NOUN cana-2410 15 7 : : PUNCT cana-2410 15 8 impulsive impulsive ADJ cana-2410 15 9 and and CCONJ cana-2410 15 10 non non ADJ cana-2410 15 11 - - ADJ cana-2410 15 12 impulsive impulsive ADJ cana-2410 15 13 . . PUNCT cana-2410 16 1 in in ADP cana-2410 16 2 the the DET cana-2410 16 3 impulsive impulsive ADJ cana-2410 16 4 system system NOUN cana-2410 16 5 , , PUNCT cana-2410 16 6 the the DET cana-2410 16 7 period period NOUN cana-2410 16 8 of of ADP cana-2410 16 9 abrupt abrupt ADJ cana-2410 16 10 changes change NOUN cana-2410 16 11 is be AUX cana-2410 16 12 significantly significantly ADV cana-2410 16 13 shorter short ADJ cana-2410 16 14 compared compare VERB cana-2410 16 15 to to ADP cana-2410 16 16 the the DET cana-2410 16 17 overall overall ADJ cana-2410 16 18 duration duration NOUN cana-2410 16 19 of of ADP cana-2410 16 20 an an DET cana-2410 16 21 evolutionary evolutionary ADJ cana-2410 16 22 process process NOUN cana-2410 16 23 , , PUNCT cana-2410 16 24 such such ADJ cana-2410 16 25 as as ADP cana-2410 16 26 shocks shock NOUN cana-2410 16 27 , , PUNCT cana-2410 16 28 natural natural ADJ cana-2410 16 29 disasters disaster NOUN cana-2410 16 30 , , PUNCT cana-2410 16 31 and and CCONJ cana-2410 16 32 non non ADJ cana-2410 16 33 - - ADJ cana-2410 16 34 impulsive impulsive ADJ cana-2410 16 35 events event NOUN cana-2410 16 36 . . PUNCT cana-2410 17 1 the the DET cana-2410 17 2 duration duration NOUN cana-2410 17 3 of of ADP cana-2410 17 4 these these DET cana-2410 17 5 modifications modification NOUN cana-2410 17 6 persists persist VERB cana-2410 17 7 throughout throughout ADP cana-2410 17 8 a a DET cana-2410 17 9 limited limited ADJ cana-2410 17 10 time time NOUN cana-2410 17 11 span span NOUN cana-2410 17 12 . . PUNCT cana-2410 18 1 insulin insulin NOUN cana-2410 18 2 administration administration NOUN cana-2410 18 3 into into ADP cana-2410 18 4 the the DET cana-2410 18 5 bloodstream bloodstream NOUN cana-2410 18 6 is be AUX cana-2410 18 7 an an DET cana-2410 18 8 important important ADJ cana-2410 18 9 application application NOUN cana-2410 18 10 of of ADP cana-2410 18 11 non non NOUN cana-2410 18 12 - - NOUN cana-2410 18 13 impulses impulse NOUN cana-2410 18 14 is be AUX cana-2410 18 15 the the DET cana-2410 18 16 administration administration NOUN cana-2410 18 17 of of ADP cana-2410 18 18 insulin insulin NOUN cana-2410 18 19 into into ADP cana-2410 18 20 the the DET cana-2410 18 21 bloodstream bloodstream NOUN cana-2410 18 22 . . PUNCT cana-2410 19 1 this this PRON cana-2410 19 2 involves involve VERB cana-2410 19 3 a a DET cana-2410 19 4 sudden sudden ADJ cana-2410 19 5 shift shift NOUN cana-2410 19 6 followed follow VERB cana-2410 19 7 by by ADP cana-2410 19 8 a a DET cana-2410 19 9 gradual gradual ADJ cana-2410 19 10 absorption absorption NOUN cana-2410 19 11 process process NOUN cana-2410 19 12 , , PUNCT cana-2410 19 13 with with ADP cana-2410 19 14 the the DET cana-2410 19 15 insulin insulin NOUN cana-2410 19 16 remaining remain VERB cana-2410 19 17 active active ADJ cana-2410 19 18 for for ADP cana-2410 19 19 a a DET cana-2410 19 20 specific specific ADJ cana-2410 19 21 period period NOUN cana-2410 19 22 of of ADP cana-2410 19 23 time time NOUN cana-2410 19 24 . . PUNCT cana-2410 20 1 some some DET cana-2410 20 2 references reference NOUN cana-2410 20 3 [ [ X cana-2410 20 4 1 1 NUM cana-2410 20 5 , , PUNCT cana-2410 20 6 10 10 NUM cana-2410 20 7 , , PUNCT cana-2410 20 8 14 14 NUM cana-2410 20 9 ] ] PUNCT cana-2410 20 10 . . PUNCT cana-2410 21 1 kalman kalman PROPN cana-2410 21 2 presented present VERB cana-2410 21 3 the the DET cana-2410 21 4 idea idea NOUN cana-2410 21 5 of of ADP cana-2410 21 6 controllability controllability NOUN cana-2410 21 7 and and CCONJ cana-2410 21 8 observability observability NOUN cana-2410 21 9 in in ADP cana-2410 21 10 the the DET cana-2410 21 11 year year NOUN cana-2410 21 12 1960 1960 NUM cana-2410 21 13 , , PUNCT cana-2410 21 14 and and CCONJ cana-2410 21 15 it it PRON cana-2410 21 16 quickly quickly ADV cana-2410 21 17 became become VERB cana-2410 21 18 a a DET cana-2410 21 19 subject subject NOUN cana-2410 21 20 of of ADP cana-2410 21 21 examination examination NOUN cana-2410 21 22 for for ADP cana-2410 21 23 a a DET cana-2410 21 24 significant significant ADJ cana-2410 21 25 number number NOUN cana-2410 21 26 of of ADP cana-2410 21 27 researchers researcher NOUN cana-2410 21 28 immediately immediately ADV cana-2410 21 29 after after ADP cana-2410 21 30 its its PRON cana-2410 21 31 introduction introduction NOUN cana-2410 21 32 . . PUNCT cana-2410 22 1 in in ADP cana-2410 22 2 a a DET cana-2410 22 3 general general ADJ cana-2410 22 4 sense sense NOUN cana-2410 22 5 , , PUNCT cana-2410 22 6 controllability controllability NOUN cana-2410 22 7 refers refer VERB cana-2410 22 8 to to ADP cana-2410 22 9 the the DET cana-2410 22 10 ability ability NOUN cana-2410 22 11 of of ADP cana-2410 22 12 a a DET cana-2410 22 13 control control NOUN cana-2410 22 14 dynamical dynamical ADJ cana-2410 22 15 system system NOUN cana-2410 22 16 to to PART cana-2410 22 17 guide guide VERB cana-2410 22 18 itself itself PRON cana-2410 22 19 from from ADP cana-2410 22 20 mailto:asreenivasulu@kluniversity.in mailto:asreenivasulu@kluniversity.in PROPN cana-2410 22 21 mailto:bvardr2010@kluniversity.in mailto:bvardr2010@kluniversity.in PROPN cana-2410 22 22 communications communication NOUN cana-2410 22 23 on on ADP cana-2410 22 24 applied apply VERB cana-2410 22 25 nonlinear nonlinear ADJ cana-2410 22 26 analysis analysis NOUN cana-2410 22 27 issn issn NOUN cana-2410 22 28 : : PUNCT cana-2410 22 29 1074 1074 NUM cana-2410 22 30 - - PUNCT cana-2410 22 31 133x 133x NUM cana-2410 22 32 vol vol NOUN cana-2410 22 33 32 32 NUM cana-2410 22 34 no no NOUN cana-2410 22 35 . . PUNCT cana-2410 23 1 2s 2s NUM cana-2410 23 2 ( ( PUNCT cana-2410 23 3 2025 2025 NUM cana-2410 23 4 ) ) PUNCT cana-2410 23 5 351 351 NUM cana-2410 23 6 https://internationalpubls.com https://internationalpubls.com X cana-2410 23 7 a a DET cana-2410 23 8 state state NOUN cana-2410 23 9 initial initial NOUN cana-2410 23 10 to to ADP cana-2410 23 11 the the DET cana-2410 23 12 intended intend VERB cana-2410 23 13 final final NOUN cana-2410 23 14 by by ADP cana-2410 23 15 making make VERB cana-2410 23 16 use use NOUN cana-2410 23 17 of of ADP cana-2410 23 18 a a DET cana-2410 23 19 control control NOUN cana-2410 23 20 that that PRON cana-2410 23 21 is be AUX cana-2410 23 22 accessible accessible ADJ cana-2410 23 23 within within ADP cana-2410 23 24 the the DET cana-2410 23 25 system system NOUN cana-2410 23 26 . . PUNCT cana-2410 24 1 recently recently ADV cana-2410 24 2 , , PUNCT cana-2410 24 3 numerous numerous ADJ cana-2410 24 4 authors author NOUN cana-2410 24 5 have have AUX cana-2410 24 6 published publish VERB cana-2410 24 7 their their PRON cana-2410 24 8 research research NOUN cana-2410 24 9 articles article NOUN cana-2410 24 10 [ [ X cana-2410 24 11 3 3 NUM cana-2410 24 12 , , PUNCT cana-2410 24 13 8 8 NUM cana-2410 24 14 , , PUNCT cana-2410 24 15 9 9 NUM cana-2410 24 16 , , PUNCT cana-2410 24 17 11 11 NUM cana-2410 24 18 - - SYM cana-2410 24 19 13 13 NUM cana-2410 24 20 , , PUNCT cana-2410 24 21 15 15 NUM cana-2410 24 22 - - SYM cana-2410 24 23 24 24 NUM cana-2410 24 24 ] ] PUNCT cana-2410 24 25 . . PUNCT cana-2410 25 1 we we PRON cana-2410 25 2 design design VERB cana-2410 25 3 with with ADP cana-2410 25 4 nonlinear nonlinear ADJ cana-2410 25 5 time time NOUN cana-2410 25 6 - - PUNCT cana-2410 25 7 varying vary VERB cana-2410 25 8 complete complete ADJ cana-2410 25 9 controllability controllability PROPN cana-2410 25 10 volterra volterra PROPN cana-2410 25 11 integro integro PROPN cana-2410 25 12 - - PUNCT cana-2410 25 13 dynamic dynamic ADJ cana-2410 25 14 sylvester sylvester NOUN cana-2410 25 15 matrix matrix NOUN cana-2410 25 16 system system NOUN cana-2410 25 17 with with ADP cana-2410 25 18 an an DET cana-2410 25 19 impulse impulse ADJ cana-2410 25 20 control control NOUN cana-2410 25 21 system system NOUN cana-2410 25 22 in in ADP cana-2410 25 23 rn rn PROPN cana-2410 25 24 π‘‹βˆ†(𝑑 π‘‹βˆ†(𝑑 PROPN cana-2410 25 25 ) ) PUNCT cana-2410 26 1 = = SYM cana-2410 26 2 𝐴(𝑑)𝑋(𝑑 𝐴(𝑑)𝑋(𝑑 NOUN cana-2410 26 3 ) ) PUNCT cana-2410 26 4 + + CCONJ cana-2410 26 5 𝑋(𝑑)𝐡(𝑑 𝑋(𝑑)𝐡(𝑑 NOUN cana-2410 26 6 ) ) PUNCT cana-2410 27 1 + + NUM cana-2410 27 2 ∫ ∫ PROPN cana-2410 27 3 ( ( PUNCT cana-2410 27 4 𝐾1(𝑑 𝐾1(𝑑 PROPN cana-2410 27 5 , , PUNCT cana-2410 27 6 𝑠)𝑋(𝑠 𝑠)𝑋(𝑠 NOUN cana-2410 27 7 ) ) PUNCT cana-2410 27 8 + + CCONJ cana-2410 27 9 𝑋(𝑠)𝐾2(𝑑 𝑋(𝑠)𝐾2(𝑑 NOUN cana-2410 27 10 , , PUNCT cana-2410 27 11 𝑠))βˆ†π‘  𝑠))βˆ†π‘  PROPN cana-2410 27 12 𝑑 𝑑 PROPN cana-2410 27 13 𝑑0 𝑑0 NOUN cana-2410 27 14 + + CCONJ cana-2410 27 15 𝐢(𝑑)π‘ˆ(𝑑 𝐢(𝑑)π‘ˆ(𝑑 NOUN cana-2410 27 16 ) ) PUNCT cana-2410 27 17 + + CCONJ cana-2410 27 18 𝐹(𝑑 𝐹(𝑑 NUM cana-2410 27 19 , , PUNCT cana-2410 27 20 𝑋(𝑑 𝑋(𝑑 NOUN cana-2410 27 21 ) ) PUNCT cana-2410 27 22 ) ) PUNCT cana-2410 27 23 ( ( PUNCT cana-2410 27 24 1.1 1.1 NUM cana-2410 27 25 ) ) PUNCT cana-2410 27 26 𝑋(𝑑 𝑋(𝑑 PROPN cana-2410 27 27 ) ) PUNCT cana-2410 27 28 = = PUNCT cana-2410 27 29 ( ( PUNCT cana-2410 27 30 1 1 NUM cana-2410 27 31 + + NUM cana-2410 27 32 𝐷𝑗)𝑋(𝑑𝑗 𝐷𝑗)𝑋(𝑑𝑗 NUM cana-2410 27 33 βˆ’ βˆ’ NUM cana-2410 27 34 ) ) PUNCT cana-2410 27 35 , , PUNCT cana-2410 27 36 𝑗 𝑗 NOUN cana-2410 27 37 = = SYM cana-2410 27 38 1,2 1,2 NUM cana-2410 27 39 , , PUNCT cana-2410 27 40 … … PUNCT cana-2410 27 41 . . PUNCT cana-2410 27 42 , , PUNCT cana-2410 27 43 ( ( PUNCT cana-2410 27 44 1.2 1.2 NUM cana-2410 27 45 ) ) PUNCT cana-2410 27 46 where where SCONJ cana-2410 27 47 a(t a(t NOUN cana-2410 27 48 ) ) PUNCT cana-2410 27 49 , , PUNCT cana-2410 27 50 b(t b(t PROPN cana-2410 27 51 ) ) PUNCT cana-2410 27 52 , , PUNCT cana-2410 27 53 c(t c(t PROPN cana-2410 27 54 ) ) PUNCT cana-2410 27 55 , , PUNCT cana-2410 27 56 𝐾1(𝑑 𝐾1(𝑑 PROPN cana-2410 27 57 ) ) PUNCT cana-2410 27 58 and and CCONJ cana-2410 27 59 𝐾2(𝑑 𝐾2(𝑑 PROPN cana-2410 27 60 ) ) PUNCT cana-2410 27 61 are be AUX cana-2410 27 62 rd rd NOUN cana-2410 27 63 - - ADJ cana-2410 27 64 continuous continuous ADJ cana-2410 27 65 matrices matrix NOUN cana-2410 27 66 orders order NOUN cana-2410 27 67 𝑛 𝑛 DET cana-2410 27 68 Γ— Γ— NOUN cana-2410 27 69 𝑛. 𝑛. NOUN cana-2410 27 70 𝐹 𝐹 PROPN cana-2410 27 71 : : PUNCT cana-2410 27 72 𝐼 𝐼 ADP cana-2410 27 73 Γ— Γ— NOUN cana-2410 28 1 ℝ𝑛 ℝ𝑛 INTJ cana-2410 28 2 β†’ β†’ PUNCT cana-2410 28 3 ℝ𝑛 ℝ𝑛 PROPN cana-2410 28 4 is be AUX cana-2410 28 5 rd rd NOUN cana-2410 28 6 - - NOUN cana-2410 28 7 continuous continuous ADJ cana-2410 28 8 on on ADP cana-2410 28 9 𝕋0 𝕋0 NOUN cana-2410 28 10 . . PUNCT cana-2410 29 1 𝐷𝑗 𝐷𝑗 PROPN cana-2410 29 2 ∈ ∈ PROPN cana-2410 29 3 𝑀𝑛×𝑛(ℝ 𝑀𝑛×𝑛(ℝ PROPN cana-2410 29 4 ) ) PUNCT cana-2410 29 5 , , PUNCT cana-2410 29 6 𝑋(𝑑 𝑋(𝑑 NOUN cana-2410 29 7 ) ) PUNCT cana-2410 29 8 ∈ ∈ PROPN cana-2410 30 1 ℝ𝑛 ℝ𝑛 PROPN cana-2410 30 2 is be AUX cana-2410 30 3 state state NOUN cana-2410 30 4 variable variable NOUN cana-2410 30 5 . . PUNCT cana-2410 31 1 π‘ˆ(𝑑 π‘ˆ(𝑑 NUM cana-2410 31 2 ) ) PUNCT cana-2410 31 3 ∈ ∈ PROPN cana-2410 32 1 β„π‘š β„π‘š ADP cana-2410 32 2 is be AUX cana-2410 32 3 the the DET cana-2410 32 4 control control NOUN cana-2410 32 5 input input NOUN cana-2410 32 6 . . PUNCT cana-2410 33 1 time time NOUN cana-2410 33 2 scale scale NOUN cana-2410 33 3 theory theory NOUN cana-2410 33 4 incorporates incorporate VERB cana-2410 33 5 both both CCONJ cana-2410 33 6 discrete discrete ADJ cana-2410 33 7 and and CCONJ cana-2410 33 8 continuous continuous ADJ cana-2410 33 9 theories theory NOUN cana-2410 33 10 , , PUNCT cana-2410 33 11 as as ADV cana-2410 33 12 well well ADV cana-2410 33 13 as as ADP cana-2410 33 14 a a DET cana-2410 33 15 hybrid hybrid NOUN cana-2410 33 16 of of ADP cana-2410 33 17 the the DET cana-2410 33 18 two two NUM cana-2410 33 19 . . PUNCT cana-2410 34 1 thus thus ADV cana-2410 34 2 , , PUNCT cana-2410 34 3 in in ADP cana-2410 34 4 contrast contrast NOUN cana-2410 34 5 to to ADP cana-2410 34 6 previous previous ADJ cana-2410 34 7 findings finding NOUN cana-2410 34 8 in in ADP cana-2410 34 9 the the DET cana-2410 34 10 literature literature NOUN cana-2410 34 11 , , PUNCT cana-2410 34 12 our our PRON cana-2410 34 13 findings finding NOUN cana-2410 34 14 are be AUX cana-2410 34 15 more more ADV cana-2410 34 16 applicable applicable ADJ cana-2410 34 17 to to ADP cana-2410 34 18 a a DET cana-2410 34 19 wider wide ADJ cana-2410 34 20 range range NOUN cana-2410 34 21 of of ADP cana-2410 34 22 situations situation NOUN cana-2410 34 23 . . PUNCT cana-2410 35 1 in in ADP cana-2410 35 2 this this DET cana-2410 35 3 paper paper NOUN cana-2410 35 4 , , PUNCT cana-2410 35 5 the the DET cana-2410 35 6 following follow VERB cana-2410 35 7 structure structure NOUN cana-2410 35 8 is be AUX cana-2410 35 9 used use VERB cana-2410 35 10 : : PUNCT cana-2410 35 11 we we PRON cana-2410 35 12 lay lay VERB cana-2410 35 13 the the DET cana-2410 35 14 groundwork groundwork NOUN cana-2410 35 15 , , PUNCT cana-2410 35 16 provide provide VERB cana-2410 35 17 some some DET cana-2410 35 18 definitions definition NOUN cana-2410 35 19 , , PUNCT cana-2410 35 20 state state VERB cana-2410 35 21 some some DET cana-2410 35 22 key key ADJ cana-2410 35 23 lemmas lemma NOUN cana-2410 35 24 and and CCONJ cana-2410 35 25 theorems theorem NOUN cana-2410 35 26 in in ADP cana-2410 35 27 section section NOUN cana-2410 35 28 2 2 NUM cana-2410 35 29 . . PUNCT cana-2410 35 30 section section NOUN cana-2410 35 31 3 3 NUM cana-2410 35 32 presents present VERB cana-2410 35 33 the the DET cana-2410 35 34 results result NOUN cana-2410 35 35 for for ADP cana-2410 35 36 complete complete ADJ cana-2410 35 37 controllability controllability NOUN cana-2410 35 38 with with ADP cana-2410 35 39 gramian gramian ADJ cana-2410 35 40 matrix matrix NOUN cana-2410 35 41 . . PUNCT cana-2410 36 1 2 2 X cana-2410 36 2 . . NUM cana-2410 36 3 preliminaries preliminary NOUN cana-2410 36 4 stefan stefan PROPN cana-2410 36 5 hilger hilger PROPN cana-2410 36 6 ’s ’s PART cana-2410 36 7 1988 1988 NUM cana-2410 36 8 doctoral doctoral ADJ cana-2410 36 9 thesis thesis NOUN cana-2410 36 10 was be AUX cana-2410 36 11 the the DET cana-2410 36 12 first first ADJ cana-2410 36 13 to to PART cana-2410 36 14 present present VERB cana-2410 36 15 the the DET cana-2410 36 16 time time NOUN cana-2410 36 17 scales scale VERB cana-2410 36 18 calculus calculus NOUN cana-2410 36 19 . . PUNCT cana-2410 37 1 he he PRON cana-2410 37 2 is be AUX cana-2410 37 3 bringing bring VERB cana-2410 37 4 together together ADV cana-2410 37 5 the the DET cana-2410 37 6 system system NOUN cana-2410 37 7 ’s ’s PART cana-2410 37 8 discrete discrete ADJ cana-2410 37 9 and and CCONJ cana-2410 37 10 continuous continuous ADJ cana-2410 37 11 analysis analysis NOUN cana-2410 37 12 . . PUNCT cana-2410 38 1 a a DET cana-2410 38 2 time time NOUN cana-2410 38 3 scale scale NOUN cana-2410 38 4 𝕋 𝕋 NOUN cana-2410 38 5 is be AUX cana-2410 38 6 defined define VERB cana-2410 38 7 as as ADP cana-2410 38 8 a a DET cana-2410 38 9 non non ADJ cana-2410 38 10 - - ADJ cana-2410 38 11 empty empty ADJ cana-2410 38 12 closed closed ADJ cana-2410 38 13 subset subset NOUN cana-2410 38 14 of of ADP cana-2410 38 15 ℝ. ℝ. PROPN cana-2410 38 16 if if SCONJ cana-2410 38 17 max max PROPN cana-2410 38 18 𝕋 𝕋 PROPN cana-2410 38 19 exists exist VERB cana-2410 38 20 , , PUNCT cana-2410 38 21 we we PRON cana-2410 38 22 define define VERB cana-2410 38 23 π•‹π‘˜ π•‹π‘˜ PROPN cana-2410 38 24 = = SYM cana-2410 38 25 𝕋{π‘šπ‘Žπ‘₯𝕋 𝕋{π‘šπ‘Žπ‘₯𝕋 PROPN cana-2410 38 26 } } PUNCT cana-2410 38 27 . . PUNCT cana-2410 39 1 but but CCONJ cana-2410 39 2 if if SCONJ cana-2410 39 3 that that PRON cana-2410 39 4 is be AUX cana-2410 39 5 not not PART cana-2410 39 6 the the DET cana-2410 39 7 case case NOUN cana-2410 39 8 , , PUNCT cana-2410 39 9 π•‹π‘˜ π•‹π‘˜ PROPN cana-2410 39 10 = = SYM cana-2410 39 11 𝕋. 𝕋. NOUN cana-2410 39 12 according according NOUN cana-2410 39 13 , , PUNCT cana-2410 39 14 we we PRON cana-2410 39 15 define define VERB cana-2410 39 16 ( ( PUNCT cana-2410 39 17 π‘Ž π‘Ž NOUN cana-2410 39 18 , , PUNCT cana-2410 39 19 𝑏)𝕋 𝑏)𝕋 ADV cana-2410 39 20 , , PUNCT cana-2410 39 21 [ [ X cana-2410 39 22 π‘Ž π‘Ž X cana-2410 39 23 , , PUNCT cana-2410 39 24 𝑏)𝕋 𝑏)𝕋 NOUN cana-2410 39 25 , , PUNCT cana-2410 39 26 ( ( PUNCT cana-2410 39 27 π‘Ž π‘Ž X cana-2410 39 28 , , PUNCT cana-2410 39 29 𝑏]𝕋 𝑏]𝕋 NOUN cana-2410 39 30 and and CCONJ cana-2410 39 31 so so ADV cana-2410 39 32 on on ADV cana-2410 39 33 as as ADP cana-2410 39 34 a a DET cana-2410 39 35 time time NOUN cana-2410 39 36 scale scale NOUN cana-2410 39 37 interval interval NOUN cana-2410 39 38 , , PUNCT cana-2410 39 39 where where SCONJ cana-2410 39 40 [ [ X cana-2410 39 41 π‘Ž π‘Ž NOUN cana-2410 39 42 , , PUNCT cana-2410 39 43 𝑏]𝕋 𝑏]𝕋 NOUN cana-2410 39 44 = = SYM cana-2410 39 45 { { PUNCT cana-2410 39 46 𝑑 𝑑 NOUN cana-2410 39 47 ∈ ∈ PROPN cana-2410 39 48 𝕋 𝕋 PROPN cana-2410 39 49 : : PUNCT cana-2410 39 50 π‘Ž π‘Ž PROPN cana-2410 39 51 ≀ ≀ NUM cana-2410 39 52 𝑑 𝑑 X cana-2410 39 53 ≀ ≀ NOUN cana-2410 39 54 𝑏 𝑏 NOUN cana-2410 39 55 } } PUNCT cana-2410 39 56 . . PUNCT cana-2410 40 1 with with ADP cana-2410 40 2 the the DET cana-2410 40 3 substitution substitution NOUN cana-2410 40 4 𝑠𝑒𝑝𝕋 𝑠𝑒𝑝𝕋 NOUN cana-2410 40 5 for for ADP cana-2410 40 6 inf{βˆ… inf{βˆ… PROPN cana-2410 40 7 } } PUNCT cana-2410 40 8 , , PUNCT cana-2410 40 9 the the DET cana-2410 40 10 forward forward ADJ cana-2410 40 11 jump jump NOUN cana-2410 40 12 operator operator NOUN cana-2410 40 13 𝜎 𝜎 NOUN cana-2410 40 14 : : PUNCT cana-2410 40 15 π•‹π‘˜ π•‹π‘˜ PROPN cana-2410 40 16 β†’ β†’ SYM cana-2410 40 17 𝕋 𝕋 PROPN cana-2410 40 18 is be AUX cana-2410 40 19 defined define VERB cana-2410 40 20 as as ADP cana-2410 40 21 𝜎(𝑑 𝜎(𝑑 NUM cana-2410 40 22 ) ) PUNCT cana-2410 41 1 = = SYM cana-2410 41 2 𝑖𝑛𝑓 𝑖𝑛𝑓 PROPN cana-2410 41 3 { { PUNCT cana-2410 41 4 𝑠 𝑠 PROPN cana-2410 41 5 ∈ ∈ PROPN cana-2410 41 6 𝕋 𝕋 PROPN cana-2410 41 7 : : PUNCT cana-2410 41 8 𝑠 𝑠 PROPN cana-2410 41 9 > > PUNCT cana-2410 41 10 𝑑 𝑑 PROPN cana-2410 41 11 } } PUNCT cana-2410 41 12 ∈ ∈ NOUN cana-2410 41 13 𝕋. 𝕋. NOUN cana-2410 41 14 the the DET cana-2410 41 15 operator operator NOUN cana-2410 41 16 𝜌 𝜌 ADP cana-2410 41 17 : : PUNCT cana-2410 41 18 π•‹π‘˜ π•‹π‘˜ PROPN cana-2410 41 19 β†’ β†’ SYM cana-2410 41 20 𝕋 𝕋 PROPN cana-2410 41 21 , , PUNCT cana-2410 41 22 which which PRON cana-2410 41 23 is be AUX cana-2410 41 24 defined define VERB cana-2410 41 25 as as ADP cana-2410 41 26 𝜌(𝑑 𝜌(𝑑 NOUN cana-2410 41 27 ) ) PUNCT cana-2410 41 28 = = SYM cana-2410 41 29 sup sup NOUN cana-2410 41 30 { { PUNCT cana-2410 41 31 𝑠 𝑠 PRON cana-2410 41 32 ∈ ∈ PROPN cana-2410 41 33 𝕋 𝕋 PROPN cana-2410 41 34 : : PUNCT cana-2410 41 35 𝑠 𝑠 PROPN cana-2410 41 36 > > PUNCT cana-2410 41 37 𝑑 𝑑 PROPN cana-2410 41 38 } } PUNCT cana-2410 41 39 ∈ ∈ PROPN cana-2410 41 40 𝕋 𝕋 PROPN cana-2410 41 41 , , PUNCT cana-2410 41 42 can can AUX cana-2410 41 43 be be AUX cana-2410 41 44 expanded expand VERB cana-2410 41 45 with with ADP cana-2410 41 46 the the DET cana-2410 41 47 substitution substitution NOUN cana-2410 41 48 sup{βˆ… sup{βˆ… NOUN cana-2410 41 49 } } PUNCT cana-2410 42 1 = = SYM cana-2410 42 2 inf𝕋. inf𝕋. NOUN cana-2410 42 3 at at ADP cana-2410 42 4 last last ADJ cana-2410 42 5 , , PUNCT cana-2410 42 6 for for ADP cana-2410 42 7 𝑑 𝑑 PROPN cana-2410 42 8 ∈ ∈ PROPN cana-2410 42 9 𝕋 𝕋 PROPN cana-2410 42 10 , , PUNCT cana-2410 42 11 the the DET cana-2410 42 12 graininess graininess NOUN cana-2410 42 13 function function VERB cana-2410 42 14 πœ‡(𝑑 πœ‡(𝑑 NOUN cana-2410 42 15 ) ) PUNCT cana-2410 42 16 follows follow VERB cana-2410 42 17 the the DET cana-2410 42 18 equation equation NOUN cana-2410 42 19 𝜎(𝑑 𝜎(𝑑 NUM cana-2410 42 20 ) ) PUNCT cana-2410 42 21 βˆ’ βˆ’ PROPN cana-2410 42 22 𝑑. 𝑑. NOUN cana-2410 42 23 when when SCONJ cana-2410 42 24 𝑑 𝑑 PROPN cana-2410 42 25 = = SYM cana-2410 42 26 𝑠𝑒𝑝𝕋 𝑠𝑒𝑝𝕋 NOUN cana-2410 42 27 , , PUNCT cana-2410 42 28 choose choose VERB cana-2410 42 29 𝜏 𝜏 PRON cana-2410 42 30 such such ADJ cana-2410 42 31 that that DET cana-2410 42 32 mapping mapping NOUN cana-2410 42 33 x x PUNCT cana-2410 42 34 from from ADP cana-2410 42 35 𝕋 𝕋 PRON cana-2410 42 36 to to ADP cana-2410 42 37 ℝ ℝ PROPN cana-2410 42 38 is be AUX cana-2410 42 39 not not PART cana-2410 42 40 left leave VERB cana-2410 42 41 scattered scatter VERB cana-2410 42 42 . . PUNCT cana-2410 43 1 if if SCONJ cana-2410 43 2 νœ€ νœ€ PRON cana-2410 43 3 > > X cana-2410 43 4 0 0 NUM cana-2410 43 5 , , PUNCT cana-2410 43 6 then then ADV cana-2410 43 7 the the DET cana-2410 43 8 generalized generalized ADJ cana-2410 43 9 delta delta NOUN cana-2410 43 10 derivative derivative NOUN cana-2410 43 11 of of ADP cana-2410 43 12 x(t x(t PROPN cana-2410 43 13 ) ) PUNCT cana-2410 43 14 , , PUNCT cana-2410 43 15 denoted denote VERB cana-2410 43 16 as as ADP cana-2410 43 17 π‘₯βˆ†(t π‘₯βˆ†(t NOUN cana-2410 43 18 ) ) PUNCT cana-2410 43 19 , , PUNCT cana-2410 43 20 is be AUX cana-2410 43 21 of of ADP cana-2410 43 22 the the DET cana-2410 43 23 form form NOUN cana-2410 43 24 that that PRON cana-2410 43 25 . . PUNCT cana-2410 44 1 given give VERB cana-2410 44 2 that that SCONJ cana-2410 44 3 u(t u(t NOUN cana-2410 44 4 ) ) PUNCT cana-2410 44 5 is be AUX cana-2410 44 6 a a DET cana-2410 44 7 neighbourhood neighbourhood NOUN cana-2410 44 8 , , PUNCT cana-2410 44 9 it it PRON cana-2410 44 10 follows follow VERB cana-2410 44 11 that that DET cana-2410 44 12 |[π‘₯(𝜎(𝑑 |[π‘₯(𝜎(𝑑 NOUN cana-2410 44 13 ) ) PUNCT cana-2410 44 14 βˆ’ βˆ’ PROPN cana-2410 44 15 π‘₯(𝑠 π‘₯(𝑠 NOUN cana-2410 44 16 ) ) PUNCT cana-2410 44 17 ] ] PUNCT cana-2410 45 1 βˆ’ βˆ’ PROPN cana-2410 45 2 π‘₯βˆ†(𝑑)[𝜎(𝑑 π‘₯βˆ†(𝑑)[𝜎(𝑑 NOUN cana-2410 45 3 ) ) PUNCT cana-2410 45 4 βˆ’ βˆ’ PROPN cana-2410 46 1 𝑠]| 𝑠]| PROPN cana-2410 46 2 ≀ ≀ ADV cana-2410 46 3 νœ€|𝜎(𝑑 νœ€|𝜎(𝑑 NOUN cana-2410 46 4 ) ) PUNCT cana-2410 46 5 βˆ’ βˆ’ PROPN cana-2410 47 1 𝑠| 𝑠| PROPN cana-2410 47 2 , , PUNCT cana-2410 47 3 for for ADP cana-2410 47 4 𝑠 𝑠 PROPN cana-2410 47 5 ∈ ∈ PROPN cana-2410 47 6 π‘ˆ. π‘ˆ. PROPN cana-2410 47 7 the the DET cana-2410 47 8 process process NOUN cana-2410 47 9 of of ADP cana-2410 47 10 mapping mapping NOUN cana-2410 47 11 x x PUNCT cana-2410 47 12 from from ADP cana-2410 47 13 𝕋 𝕋 PRON cana-2410 47 14 to to ADP cana-2410 47 15 ℝ ℝ PROPN cana-2410 47 16 is be AUX cana-2410 47 17 known know VERB cana-2410 47 18 as as ADP cana-2410 47 19 the the DET cana-2410 47 20 generalized generalized ADJ cana-2410 47 21 delta delta NOUN cana-2410 47 22 derivative derivative NOUN cana-2410 47 23 on on ADP cana-2410 47 24 time time NOUN cana-2410 47 25 scales scale NOUN cana-2410 47 26 calculus calculus NOUN cana-2410 47 27 , , PUNCT cana-2410 47 28 where where SCONJ cana-2410 47 29 x x PROPN cana-2410 47 30 is be AUX cana-2410 47 31 delta delta NOUN cana-2410 47 32 derivative derivative NOUN cana-2410 47 33 for for ADP cana-2410 47 34 every every DET cana-2410 47 35 𝑑 𝑑 PROPN cana-2410 47 36 ∈ ∈ PROPN cana-2410 47 37 𝕋. 𝕋. NOUN cana-2410 47 38 the the DET cana-2410 47 39 right right ADJ cana-2410 47 40 dense dense ADJ cana-2410 47 41 points point NOUN cana-2410 47 42 in in ADP cana-2410 47 43 𝕋 𝕋 NOUN cana-2410 47 44 are be AUX cana-2410 47 45 considered consider VERB cana-2410 47 46 to to PART cana-2410 47 47 represent represent VERB cana-2410 47 48 the the DET cana-2410 47 49 origins origin NOUN cana-2410 47 50 of of ADP cana-2410 47 51 rd rd NOUN cana-2410 47 52 - - ADJ cana-2410 47 53 continuous continuous ADJ cana-2410 47 54 m m VERB cana-2410 47 55 mapping mapping NOUN cana-2410 47 56 from from ADP cana-2410 47 57 𝕋 𝕋 PRON cana-2410 47 58 to to ADP cana-2410 47 59 ℝ ℝ PROPN cana-2410 47 60 , , PUNCT cana-2410 47 61 whereas whereas SCONJ cana-2410 47 62 the the DET cana-2410 47 63 left left ADJ cana-2410 47 64 dense dense ADJ cana-2410 47 65 points point NOUN cana-2410 47 66 in in ADP cana-2410 47 67 𝕋 𝕋 NOUN cana-2410 47 68 are be AUX cana-2410 47 69 the the DET cana-2410 47 70 locations location NOUN cana-2410 47 71 of of ADP cana-2410 47 72 its its PRON cana-2410 47 73 finite finite NOUN cana-2410 47 74 left leave VERB cana-2410 47 75 sided sided ADJ cana-2410 47 76 limits limit NOUN cana-2410 47 77 . . PUNCT cana-2410 48 1 the the DET cana-2410 48 2 set set NOUN cana-2410 48 3 of of ADP cana-2410 48 4 rd rd NOUN cana-2410 48 5 - - ADJ cana-2410 48 6 continuous continuous ADJ cana-2410 48 7 functions function NOUN cana-2410 48 8 m m VERB cana-2410 48 9 is be AUX cana-2410 48 10 denoted denote VERB cana-2410 48 11 by by ADP cana-2410 48 12 πΆπ‘Ÿπ‘‘ πΆπ‘Ÿπ‘‘ PROPN cana-2410 48 13 = = PUNCT cana-2410 48 14 πΆπ‘Ÿπ‘‘(𝕋 πΆπ‘Ÿπ‘‘(𝕋 X cana-2410 48 15 ) ) PUNCT cana-2410 48 16 = = PUNCT cana-2410 48 17 πΆπ‘Ÿπ‘‘(𝕋,ℝ πΆπ‘Ÿπ‘‘(𝕋,ℝ NOUN cana-2410 48 18 ) ) PUNCT cana-2410 48 19 . . PUNCT cana-2410 49 1 assuming assume VERB cana-2410 49 2 π‘€βˆ†(𝜏 π‘€βˆ†(𝜏 NOUN cana-2410 49 3 ) ) PUNCT cana-2410 49 4 = = SYM cana-2410 49 5 𝑀(𝜏 𝑀(𝜏 PROPN cana-2410 49 6 ) ) PUNCT cana-2410 49 7 for for ADP cana-2410 49 8 every every DET cana-2410 49 9 𝜏 𝜏 PROPN cana-2410 49 10 ∈ ∈ PROPN cana-2410 49 11 π•‹π‘˜ π•‹π‘˜ PROPN cana-2410 49 12 , , PUNCT cana-2410 49 13 the the DET cana-2410 49 14 mapping mapping NOUN cana-2410 49 15 from from ADP cana-2410 49 16 π•‹π‘˜ π•‹π‘˜ PROPN cana-2410 49 17 to to ADP cana-2410 49 18 ℝ ℝ PROPN cana-2410 49 19 is be AUX cana-2410 49 20 referred refer VERB cana-2410 49 21 to to ADP cana-2410 49 22 as as ADP cana-2410 49 23 the the DET cana-2410 49 24 antiderivative antiderivative NOUN cana-2410 49 25 of of ADP cana-2410 49 26 m m NOUN cana-2410 49 27 from from ADP cana-2410 49 28 π•‹π‘˜ π•‹π‘˜ PROPN cana-2410 49 29 to to AUX cana-2410 49 30 ℝ. ℝ. PROPN cana-2410 49 31 we we PRON cana-2410 49 32 continue continue VERB cana-2410 49 33 by by ADP cana-2410 49 34 creating create VERB cana-2410 49 35 the the DET cana-2410 49 36 integral integral ADJ cana-2410 49 37 ∫ ∫ NOUN cana-2410 49 38 π‘š(𝑑)βˆ†π‘‘ π‘š(𝑑)βˆ†π‘‘ X cana-2410 49 39 = = SYM cana-2410 49 40 𝑀(𝑏 𝑀(𝑏 PROPN cana-2410 49 41 ) ) PUNCT cana-2410 49 42 βˆ’ βˆ’ PROPN cana-2410 49 43 𝑀(π‘Ž 𝑀(π‘Ž PROPN cana-2410 49 44 ) ) PUNCT cana-2410 49 45 . . PUNCT cana-2410 50 1 𝑏 𝑏 PRON cana-2410 50 2 π‘Ž π‘Ž DET cana-2410 50 3 definition definition NOUN cana-2410 50 4 2.1.[5 2.1.[5 NUM cana-2410 50 5 ] ] PUNCT cana-2410 50 6 : : PUNCT cana-2410 50 7 the the DET cana-2410 50 8 function function NOUN cana-2410 50 9 m(t m(t PROPN cana-2410 50 10 ) ) PUNCT cana-2410 50 11 that that PRON cana-2410 50 12 maps map VERB cana-2410 50 13 from from ADP cana-2410 50 14 𝕋 𝕋 PRON cana-2410 50 15 to to ADP cana-2410 50 16 ℝ ℝ PROPN cana-2410 50 17 is be AUX cana-2410 50 18 regressive regressive ADJ cana-2410 50 19 is be AUX cana-2410 50 20 defined define VERB cana-2410 50 21 as1 as1 NOUN cana-2410 50 22 + + CCONJ cana-2410 50 23 πœ‡(𝑑)(𝑑 πœ‡(𝑑)(𝑑 PROPN cana-2410 50 24 ) ) PUNCT cana-2410 50 25 β‰  β‰  PROPN cana-2410 50 26 0 0 NUM cana-2410 50 27 βˆ€ βˆ€ NOUN cana-2410 50 28 𝑑 𝑑 NOUN cana-2410 50 29 ∈ ∈ NOUN cana-2410 50 30 𝕋. 𝕋. NOUN cana-2410 50 31 the the DET cana-2410 50 32 right right ADV cana-2410 50 33 dense dense ADJ cana-2410 50 34 continuous continuous ADJ cana-2410 50 35 function function NOUN cana-2410 50 36 β„› β„› NOUN cana-2410 50 37 = = SYM cana-2410 50 38 β„›(𝑑 β„›(𝑑 NUM cana-2410 50 39 ) ) PUNCT cana-2410 50 40 = = SYM cana-2410 51 1 β„›(𝕋,ℝ β„›(𝕋,ℝ NOUN cana-2410 51 2 ) ) PUNCT cana-2410 51 3 is be AUX cana-2410 51 4 the the DET cana-2410 51 5 sum sum NOUN cana-2410 51 6 of of ADP cana-2410 51 7 all all DET cana-2410 51 8 regressive regressive ADJ cana-2410 51 9 communications communication NOUN cana-2410 51 10 on on ADP cana-2410 51 11 applied apply VERB cana-2410 51 12 nonlinear nonlinear ADJ cana-2410 51 13 analysis analysis NOUN cana-2410 51 14 issn issn NOUN cana-2410 51 15 : : PUNCT cana-2410 51 16 1074 1074 NUM cana-2410 51 17 - - PUNCT cana-2410 51 18 133x 133x NUM cana-2410 51 19 vol vol NOUN cana-2410 51 20 32 32 NUM cana-2410 51 21 no no NOUN cana-2410 51 22 . . PUNCT cana-2410 52 1 2s 2s NUM cana-2410 52 2 ( ( PUNCT cana-2410 52 3 2025 2025 NUM cana-2410 52 4 ) ) PUNCT cana-2410 52 5 352 352 NUM cana-2410 52 6 https://internationalpubls.com https://internationalpubls.com X cana-2410 52 7 functions function NOUN cana-2410 52 8 . . PUNCT cana-2410 53 1 likewise likewise ADV cana-2410 53 2 , , PUNCT cana-2410 53 3 β„›+ β„›+ NOUN cana-2410 53 4 = = SYM cana-2410 53 5 β„›+(𝕋,ℝ β„›+(𝕋,ℝ X cana-2410 53 6 ) ) PUNCT cana-2410 53 7 = = SYM cana-2410 53 8 { { PUNCT cana-2410 53 9 𝑀 𝑀 PROPN cana-2410 53 10 ∈ ∈ PROPN cana-2410 53 11 β„› β„› PROPN cana-2410 53 12 : : PUNCT cana-2410 53 13 1 1 NUM cana-2410 53 14 + + NUM cana-2410 53 15 πœ‡(𝑑)𝑀(𝑑 πœ‡(𝑑)𝑀(𝑑 NOUN cana-2410 53 16 ) ) PUNCT cana-2410 53 17 > > X cana-2410 53 18 0 0 NUM cana-2410 53 19 , , PUNCT cana-2410 53 20 βˆ€ βˆ€ NUM cana-2410 53 21 𝑑 𝑑 NOUN cana-2410 53 22 ∈ ∈ PROPN cana-2410 53 23 𝕋 𝕋 PROPN cana-2410 53 24 } } PUNCT cana-2410 53 25 denotes denote NOUN cana-2410 53 26 all all DET cana-2410 53 27 positively positively ADV cana-2410 53 28 regressive regressive ADJ cana-2410 53 29 function function NOUN cana-2410 53 30 . . PUNCT cana-2410 54 1 lemma lemma PROPN cana-2410 54 2 2.1.[6 2.1.[6 NUM cana-2410 54 3 ] ] X cana-2410 54 4 : : PUNCT cana-2410 54 5 when when SCONJ cana-2410 54 6 𝑀,𝑁 𝑀,𝑁 NOUN cana-2410 54 7 ∈ ∈ NOUN cana-2410 54 8 β„› β„› PROPN cana-2410 54 9 matrices matrix NOUN cana-2410 54 10 on on ADP cana-2410 54 11 𝕋 𝕋 PROPN cana-2410 54 12 , , PUNCT cana-2410 54 13 thus thus ADV cana-2410 54 14 i. i. PROPN cana-2410 54 15 𝑒𝑀 𝑒𝑀 PROPN cana-2410 54 16 βˆ’1(𝜏 βˆ’1(𝜏 PROPN cana-2410 54 17 , , PUNCT cana-2410 54 18 𝑠 𝑠 NOUN cana-2410 54 19 ) ) PUNCT cana-2410 54 20 ≑ ≑ PROPN cana-2410 54 21 π‘’βŠ–π‘€ π‘’βŠ–π‘€ PROPN cana-2410 54 22 βˆ— βˆ— NOUN cana-2410 54 23 ( ( PUNCT cana-2410 54 24 𝜏 𝜏 NOUN cana-2410 54 25 , , PUNCT cana-2410 54 26 𝑠 𝑠 NOUN cana-2410 54 27 ) ) PUNCT cana-2410 54 28 ; ; PUNCT cana-2410 54 29 ii ii X cana-2410 54 30 . . PUNCT cana-2410 55 1 𝑒0(𝜏 𝑒0(𝜏 PROPN cana-2410 55 2 , , PUNCT cana-2410 55 3 𝑠 𝑠 NOUN cana-2410 55 4 ) ) PUNCT cana-2410 55 5 ≑ ≑ PROPN cana-2410 55 6 𝐼 𝐼 PROPN cana-2410 55 7 and and CCONJ cana-2410 55 8 𝑒0(𝜏 𝑒0(𝜏 PROPN cana-2410 55 9 , , PUNCT cana-2410 55 10 𝜏 𝜏 NOUN cana-2410 55 11 ) ) PUNCT cana-2410 55 12 ≑ ≑ PROPN cana-2410 55 13 𝐼 𝐼 PROPN cana-2410 55 14 ; ; PUNCT cana-2410 55 15 iii iii X cana-2410 55 16 . . PUNCT cana-2410 56 1 𝑒𝑀(𝜎(𝜏 𝑒𝑀(𝜎(𝜏 X cana-2410 56 2 ) ) PUNCT cana-2410 56 3 , , PUNCT cana-2410 56 4 𝑠 𝑠 X cana-2410 56 5 ) ) PUNCT cana-2410 56 6 ≑ ≑ PROPN cana-2410 56 7 ( ( PUNCT cana-2410 56 8 𝐼 𝐼 PROPN cana-2410 56 9 + + CCONJ cana-2410 56 10 πœ‡(𝜏)𝑀(𝜏))𝑒𝑀(𝜏 πœ‡(𝜏)𝑀(𝜏))𝑒𝑀(𝜏 PROPN cana-2410 56 11 , , PUNCT cana-2410 56 12 𝑠 𝑠 PROPN cana-2410 56 13 ) ) PUNCT cana-2410 56 14 ; ; PUNCT cana-2410 57 1 iv iv X cana-2410 57 2 . . PUNCT cana-2410 57 3 𝑒𝑀(𝜏 𝑒𝑀(𝜏 PROPN cana-2410 57 4 , , PUNCT cana-2410 57 5 𝑠 𝑠 PROPN cana-2410 57 6 ) ) PUNCT cana-2410 57 7 = = PUNCT cana-2410 57 8 𝑒𝑀 𝑒𝑀 PROPN cana-2410 57 9 βˆ’1(𝑠 βˆ’1(𝑠 NOUN cana-2410 57 10 , , PUNCT cana-2410 57 11 𝜏 𝜏 NOUN cana-2410 57 12 ) ) PUNCT cana-2410 57 13 = = SYM cana-2410 57 14 π‘’βŠ–π‘€βˆ— π‘’βŠ–π‘€βˆ— PROPN cana-2410 57 15 βˆ— βˆ— NOUN cana-2410 57 16 ( ( PUNCT cana-2410 57 17 𝑠 𝑠 PROPN cana-2410 57 18 , , PUNCT cana-2410 57 19 𝜏 𝜏 NOUN cana-2410 57 20 ) ) PUNCT cana-2410 57 21 ; ; PUNCT cana-2410 58 1 v. v. PROPN cana-2410 58 2 𝑒𝑀(𝜏 𝑒𝑀(𝜏 PROPN cana-2410 58 3 , , PUNCT cana-2410 58 4 𝑠)𝑒𝑁(𝜏 𝑠)𝑒𝑁(𝜏 PROPN cana-2410 58 5 , , PUNCT cana-2410 58 6 𝑠 𝑠 NOUN cana-2410 58 7 ) ) PUNCT cana-2410 58 8 = = SYM cana-2410 59 1 π‘’π‘€βŠ•π‘(𝜏 π‘’π‘€βŠ•π‘(𝜏 PROPN cana-2410 59 2 , , PUNCT cana-2410 59 3 𝑠 𝑠 PROPN cana-2410 59 4 ) ) PUNCT cana-2410 59 5 ; ; PUNCT cana-2410 59 6 vi vi X cana-2410 59 7 . . PROPN cana-2410 59 8 𝑒𝑀(𝜏 𝑒𝑀(𝜏 PROPN cana-2410 59 9 , , PUNCT cana-2410 59 10 𝑠)𝑒𝑀(𝑠 𝑠)𝑒𝑀(𝑠 PROPN cana-2410 59 11 , , PUNCT cana-2410 59 12 π‘Ÿ π‘Ÿ NOUN cana-2410 59 13 ) ) PUNCT cana-2410 59 14 = = SYM cana-2410 59 15 𝑒𝑀(𝜏 𝑒𝑀(𝜏 PROPN cana-2410 59 16 , , PUNCT cana-2410 59 17 π‘Ÿ π‘Ÿ NOUN cana-2410 59 18 ) ) PUNCT cana-2410 59 19 ; ; PUNCT cana-2410 59 20 lemma lemma PROPN cana-2410 59 21 2.2.[4 2.2.[4 NUM cana-2410 59 22 ] ] PRON cana-2410 59 23 : : PUNCT cana-2410 59 24 consider consider VERB cana-2410 59 25 a a DET cana-2410 59 26 matrix matrix NOUN cana-2410 59 27 m m NOUN cana-2410 59 28 of of ADP cana-2410 59 29 size size NOUN cana-2410 59 30 𝑛 𝑛 DET cana-2410 59 31 Γ— Γ— NOUN cana-2410 59 32 𝑛 𝑛 PROPN cana-2410 59 33 on on ADP cana-2410 59 34 a a DET cana-2410 59 35 time time NOUN cana-2410 59 36 scale scale NOUN cana-2410 59 37 . . PUNCT cana-2410 60 1 assume assume VERB cana-2410 60 2 that that SCONJ cana-2410 60 3 the the DET cana-2410 60 4 mapping mapping NOUN cana-2410 60 5 f f NOUN cana-2410 60 6 from from ADP cana-2410 60 7 𝕋 𝕋 PRON cana-2410 60 8 to to ADP cana-2410 60 9 ℝ𝑛 ℝ𝑛 PROPN cana-2410 60 10 is be AUX cana-2410 60 11 continuous continuous ADJ cana-2410 60 12 and and CCONJ cana-2410 60 13 right right ADV cana-2410 60 14 dense dense ADJ cana-2410 60 15 . . PUNCT cana-2410 61 1 given give VERB cana-2410 61 2 that that DET cana-2410 61 3 𝑑0 𝑑0 PROPN cana-2410 61 4 belongs belong VERB cana-2410 61 5 to to ADP cana-2410 61 6 the the DET cana-2410 61 7 set set NOUN cana-2410 61 8 𝕋 𝕋 NOUN cana-2410 61 9 and and CCONJ cana-2410 61 10 𝑝0 𝑝0 NOUN cana-2410 61 11 belongs belong VERB cana-2410 61 12 to to ADP cana-2410 61 13 ℝ𝑛 ℝ𝑛 PROPN cana-2410 61 14 , , PUNCT cana-2410 61 15 this this PRON cana-2410 61 16 implies imply VERB cana-2410 61 17 the the DET cana-2410 61 18 initial initial ADJ cana-2410 61 19 value value NOUN cana-2410 61 20 problem problem NOUN cana-2410 61 21 ( ( PUNCT cana-2410 61 22 ivp ivp NOUN cana-2410 61 23 ) ) PUNCT cana-2410 61 24 . . PUNCT cana-2410 62 1 π‘βˆ†(𝑑 π‘βˆ†(𝑑 X cana-2410 62 2 ) ) PUNCT cana-2410 62 3 = = SYM cana-2410 62 4 𝑀(𝑑)𝑝(𝑑 𝑀(𝑑)𝑝(𝑑 NOUN cana-2410 62 5 ) ) PUNCT cana-2410 63 1 + + CCONJ cana-2410 64 1 𝑙(𝑑 𝑙(𝑑 NOUN cana-2410 64 2 ) ) PUNCT cana-2410 64 3 , , PUNCT cana-2410 64 4 𝑝(𝑑0 𝑝(𝑑0 ADJ cana-2410 64 5 ) ) PUNCT cana-2410 64 6 = = NOUN cana-2410 64 7 𝑝0 𝑝0 PROPN cana-2410 64 8 , , PUNCT cana-2410 64 9 having have VERB cana-2410 64 10 one one NUM cana-2410 64 11 and and CCONJ cana-2410 64 12 only only ADV cana-2410 64 13 one one NUM cana-2410 64 14 solution solution NOUN cana-2410 64 15 p p NOUN cana-2410 64 16 mapping mapping NOUN cana-2410 64 17 from from ADP cana-2410 64 18 𝕋 𝕋 PRON cana-2410 64 19 to to ADP cana-2410 64 20 ℝ ℝ PROPN cana-2410 64 21 is be AUX cana-2410 64 22 developed develop VERB cana-2410 64 23 as as ADP cana-2410 64 24 𝑝(𝑑 𝑝(𝑑 PROPN cana-2410 64 25 ) ) PUNCT cana-2410 64 26 = = SYM cana-2410 64 27 𝑓𝑀(𝑑 𝑓𝑀(𝑑 NOUN cana-2410 64 28 , , PUNCT cana-2410 64 29 𝑑0)𝑝0 𝑑0)𝑝0 X cana-2410 64 30 + + CCONJ cana-2410 64 31 βˆ«π‘“π‘€(𝑑 βˆ«π‘“π‘€(𝑑 PROPN cana-2410 64 32 , , PUNCT cana-2410 64 33 𝜎(𝜏))𝑙(𝜏)βˆ†πœ. 𝜎(𝜏))𝑙(𝜏)βˆ†πœ. NUM cana-2410 64 34 𝑑 𝑑 PROPN cana-2410 64 35 𝑑0 𝑑0 NOUN cana-2410 64 36 theorem theorem VERB cana-2410 64 37 2.1 2.1 NUM cana-2410 64 38 . . PUNCT cana-2410 65 1 let let VERB cana-2410 65 2 z(t)=vec z(t)=vec PRON cana-2410 65 3 x(t x(t PROPN cana-2410 65 4 ) ) PUNCT cana-2410 65 5 , , PUNCT cana-2410 65 6 οΏ½ οΏ½ PROPN cana-2410 65 7 Μ‚ Μ‚ SYM cana-2410 65 8 οΏ½ οΏ½ NOUN cana-2410 65 9 (𝑑)=vec (𝑑)=vec X cana-2410 65 10 u(t u(t NOUN cana-2410 65 11 ) ) PUNCT cana-2410 65 12 , , PUNCT cana-2410 65 13 and and CCONJ cana-2410 65 14 𝑓(𝑑 𝑓(𝑑 NOUN cana-2410 65 15 , , PUNCT cana-2410 65 16 𝑧(𝑑 𝑧(𝑑 PROPN cana-2410 65 17 ) ) PUNCT cana-2410 65 18 ) ) PUNCT cana-2410 66 1 = = PUNCT cana-2410 66 2 𝑉𝑒𝑐𝐹(𝑑 𝑉𝑒𝑐𝐹(𝑑 NOUN cana-2410 66 3 , , PUNCT cana-2410 66 4 𝑋(𝑑 𝑋(𝑑 NOUN cana-2410 66 5 ) ) PUNCT cana-2410 66 6 ) ) PUNCT cana-2410 66 7 . . PUNCT cana-2410 67 1 then then ADV cana-2410 67 2 the the DET cana-2410 67 3 volterra volterra NOUN cana-2410 67 4 integro integro PROPN cana-2410 67 5 - - PUNCT cana-2410 67 6 dynamic dynamic ADJ cana-2410 67 7 sylvester sylvester NOUN cana-2410 67 8 matrix matrix NOUN cana-2410 67 9 with with ADP cana-2410 67 10 an an DET cana-2410 67 11 impulse impulse ADJ cana-2410 67 12 control control NOUN cana-2410 67 13 system system NOUN cana-2410 67 14 ( ( PUNCT cana-2410 67 15 1.1 1.1 NUM cana-2410 67 16 ) ) PUNCT cana-2410 67 17 , , PUNCT cana-2410 67 18 ( ( PUNCT cana-2410 67 19 1.2 1.2 NUM cana-2410 67 20 ) ) PUNCT cana-2410 67 21 is be AUX cana-2410 67 22 equivalent equivalent ADJ cana-2410 67 23 the the DET cana-2410 67 24 system system NOUN cana-2410 67 25 zβˆ†(t zβˆ†(t NOUN cana-2410 67 26 ) ) PUNCT cana-2410 67 27 = = SYM cana-2410 67 28 p(t)z(t p(t)z(t NOUN cana-2410 67 29 ) ) PUNCT cana-2410 68 1 + + CCONJ cana-2410 68 2 ∫k(t ∫k(t NOUN cana-2410 68 3 , , PUNCT cana-2410 68 4 s)z(s)βˆ†s s)z(s)βˆ†s PROPN cana-2410 68 5 t t NOUN cana-2410 68 6 0 0 PUNCT cana-2410 69 1 + + NOUN cana-2410 69 2 q(t)uΜ‚(t q(t)uΜ‚(t PRON cana-2410 69 3 ) ) PUNCT cana-2410 69 4 + + CCONJ cana-2410 69 5 f(t f(t NOUN cana-2410 69 6 , , PUNCT cana-2410 69 7 z(t z(t NOUN cana-2410 69 8 ) ) PUNCT cana-2410 69 9 ) ) PUNCT cana-2410 69 10 ( ( PUNCT cana-2410 69 11 2.1 2.1 NUM cana-2410 69 12 ) ) PUNCT cana-2410 69 13 z(t z(t NOUN cana-2410 69 14 ) ) PUNCT cana-2410 70 1 = = PUNCT cana-2410 71 1 [ [ X cana-2410 71 2 in in ADP cana-2410 71 3 βŠ— βŠ— PROPN cana-2410 71 4 r𝑗]z(𝑑𝑗 r𝑗]z(𝑑𝑗 PROPN cana-2410 71 5 βˆ’ βˆ’ NOUN cana-2410 71 6 ) ) PUNCT cana-2410 71 7 ( ( PUNCT cana-2410 71 8 2.2 2.2 NUM cana-2410 71 9 ) ) PUNCT cana-2410 71 10 where where SCONJ cana-2410 71 11 𝑃(𝑑 𝑃(𝑑 VERB cana-2410 71 12 ) ) PUNCT cana-2410 71 13 = = PUNCT cana-2410 72 1 [ [ X cana-2410 72 2 π΅βˆ—βŠ— π΅βˆ—βŠ— NOUN cana-2410 72 3 in in ADP cana-2410 72 4 + + CCONJ cana-2410 72 5 in in ADP cana-2410 72 6 βŠ—π΄ βŠ—π΄ NOUN cana-2410 72 7 ] ] X cana-2410 72 8 , , PUNCT cana-2410 72 9 𝑄(𝑑 𝑄(𝑑 X cana-2410 72 10 ) ) PUNCT cana-2410 72 11 = = SYM cana-2410 73 1 [ [ X cana-2410 73 2 in in ADP cana-2410 73 3 βŠ—πΆ βŠ—πΆ NOUN cana-2410 73 4 ] ] PUNCT cana-2410 73 5 , , PUNCT cana-2410 73 6 𝐾(𝑑 𝐾(𝑑 PROPN cana-2410 73 7 , , PUNCT cana-2410 73 8 𝑠 𝑠 X cana-2410 73 9 ) ) PUNCT cana-2410 73 10 = = NOUN cana-2410 74 1 [ [ X cana-2410 74 2 𝐾2 𝐾2 NOUN cana-2410 74 3 βˆ—βŠ— βˆ—βŠ— PROPN cana-2410 74 4 𝐼𝑛 𝐼𝑛 PROPN cana-2410 74 5 ) ) PUNCT cana-2410 74 6 + + CCONJ cana-2410 74 7 ( ( PUNCT cana-2410 74 8 πΌπ‘›βŠ—πΎ1 πΌπ‘›βŠ—πΎ1 NUM cana-2410 74 9 ) ) PUNCT cana-2410 74 10 and and CCONJ cana-2410 74 11 𝑅𝑗 𝑅𝑗 PROPN cana-2410 74 12 = = PUNCT cana-2410 74 13 ( ( PUNCT cana-2410 74 14 1 1 NUM cana-2410 74 15 + + X cana-2410 74 16 𝐷𝑗 𝐷𝑗 NOUN cana-2410 74 17 ) ) PUNCT cana-2410 74 18 and and CCONJ cana-2410 74 19 in in ADP cana-2410 74 20 is be AUX cana-2410 74 21 the the DET cana-2410 74 22 identity identity NOUN cana-2410 74 23 matrix matrix NOUN cana-2410 74 24 . . PUNCT cana-2410 75 1 proof proof NOUN cana-2410 75 2 : : PUNCT cana-2410 75 3 we we PRON cana-2410 75 4 apply apply VERB cana-2410 75 5 the the DET cana-2410 75 6 vec vec NOUN cana-2410 75 7 operator operator NOUN cana-2410 75 8 to to ADP cana-2410 75 9 the the DET cana-2410 75 10 equation equation NOUN cana-2410 75 11 ( ( PUNCT cana-2410 75 12 1.1 1.1 NUM cana-2410 75 13 ) ) PUNCT cana-2410 75 14 , , PUNCT cana-2410 75 15 ( ( PUNCT cana-2410 75 16 1.2 1.2 NUM cana-2410 75 17 ) ) PUNCT cana-2410 75 18 and and CCONJ cana-2410 75 19 using use VERB cana-2410 75 20 the the DET cana-2410 75 21 above above ADJ cana-2410 75 22 properties property NOUN cana-2410 75 23 of of ADP cana-2410 75 24 kronecker kronecker NOUN cana-2410 75 25 product product NOUN cana-2410 75 26 [ [ X cana-2410 75 27 3 3 NUM cana-2410 75 28 ] ] PUNCT cana-2410 75 29 , , PUNCT cana-2410 75 30 we we PRON cana-2410 75 31 have have VERB cana-2410 75 32 zβˆ†(t zβˆ†(t NOUN cana-2410 75 33 ) ) PUNCT cana-2410 75 34 = = SYM cana-2410 75 35 p(t)z(t p(t)z(t NOUN cana-2410 75 36 ) ) PUNCT cana-2410 76 1 + + CCONJ cana-2410 76 2 ∫k(t ∫k(t NOUN cana-2410 76 3 , , PUNCT cana-2410 76 4 s)z(s)βˆ†s s)z(s)βˆ†s PROPN cana-2410 76 5 t t NOUN cana-2410 76 6 0 0 PUNCT cana-2410 77 1 + + NOUN cana-2410 77 2 q(t)uΜ‚(t q(t)uΜ‚(t PRON cana-2410 77 3 ) ) PUNCT cana-2410 77 4 + + CCONJ cana-2410 77 5 f(t f(t NOUN cana-2410 77 6 , , PUNCT cana-2410 77 7 z(t z(t NOUN cana-2410 77 8 ) ) PUNCT cana-2410 77 9 ) ) PUNCT cana-2410 77 10 communications communication NOUN cana-2410 77 11 on on ADP cana-2410 77 12 applied apply VERB cana-2410 77 13 nonlinear nonlinear ADJ cana-2410 77 14 analysis analysis NOUN cana-2410 77 15 issn issn NOUN cana-2410 77 16 : : PUNCT cana-2410 77 17 1074 1074 NUM cana-2410 77 18 - - PUNCT cana-2410 77 19 133x 133x NUM cana-2410 77 20 vol vol NOUN cana-2410 77 21 32 32 NUM cana-2410 77 22 no no NOUN cana-2410 77 23 . . PUNCT cana-2410 78 1 2s 2s NUM cana-2410 78 2 ( ( PUNCT cana-2410 78 3 2025 2025 NUM cana-2410 78 4 ) ) PUNCT cana-2410 78 5 353 353 NUM cana-2410 78 6 https://internationalpubls.com https://internationalpubls.com X cana-2410 78 7 z(t z(t NOUN cana-2410 78 8 ) ) PUNCT cana-2410 78 9 = = PUNCT cana-2410 79 1 [ [ X cana-2410 79 2 in in ADP cana-2410 79 3 βŠ— βŠ— PROPN cana-2410 79 4 r𝑗]z(𝑑𝑗 r𝑗]z(𝑑𝑗 PROPN cana-2410 79 5 βˆ’ βˆ’ PROPN cana-2410 79 6 ) ) PUNCT cana-2410 79 7 lemma lemma PROPN cana-2410 79 8 2.3 2.3 NUM cana-2410 80 1 [ [ X cana-2410 80 2 7 7 NUM cana-2410 80 3 ] ] PUNCT cana-2410 80 4 . . PUNCT cana-2410 81 1 for for ADP cana-2410 81 2 the the DET cana-2410 81 3 system system NOUN cana-2410 81 4 ( ( PUNCT cana-2410 81 5 2.2 2.2 NUM cana-2410 81 6 ) ) PUNCT cana-2410 81 7 with with ADP cana-2410 81 8 𝑃 𝑃 PROPN cana-2410 81 9 ∈ ∈ PROPN cana-2410 81 10 𝑀𝑛2(ℝ 𝑀𝑛2(ℝ NOUN cana-2410 81 11 ) ) PUNCT cana-2410 81 12 is be AUX cana-2410 81 13 a a DET cana-2410 81 14 constant constant ADJ cana-2410 81 15 , , PUNCT cana-2410 81 16 there there PRON cana-2410 81 17 exists exist VERB cana-2410 81 18 a a DET cana-2410 81 19 scalar scalar ADJ cana-2410 81 20 function function NOUN cana-2410 81 21 𝛾0(𝑑 𝛾0(𝑑 PROPN cana-2410 81 22 , , PUNCT cana-2410 81 23 𝑠 𝑠 PROPN cana-2410 81 24 ) ) PUNCT cana-2410 81 25 , , PUNCT cana-2410 81 26 … … PUNCT cana-2410 81 27 , , PUNCT cana-2410 81 28 𝛾𝑛2βˆ’1(𝑑 𝛾𝑛2βˆ’1(𝑑 PROPN cana-2410 81 29 , , PUNCT cana-2410 81 30 𝑠 𝑠 X cana-2410 81 31 ) ) PUNCT cana-2410 81 32 ∈ ∈ PROPN cana-2410 81 33 ( ( PUNCT cana-2410 81 34 𝕋 𝕋 NOUN cana-2410 81 35 + + PROPN cana-2410 81 36 , , PUNCT cana-2410 81 37 ℝ ℝ PROPN cana-2410 81 38 ) ) PUNCT cana-2410 81 39 such such ADJ cana-2410 81 40 that that SCONJ cana-2410 81 41 the the DET cana-2410 81 42 only only ADJ cana-2410 81 43 one one NUM cana-2410 81 44 solution solution NOUN cana-2410 81 45 has have VERB cana-2410 81 46 representation representation NOUN cana-2410 81 47 . . PUNCT cana-2410 82 1 𝑒𝑃(𝑑 𝑒𝑃(𝑑 ADJ cana-2410 82 2 , , PUNCT cana-2410 82 3 𝑠 𝑠 PROPN cana-2410 82 4 ) ) PUNCT cana-2410 82 5 = = PUNCT cana-2410 82 6 βˆ‘ βˆ‘ PUNCT cana-2410 82 7 π›Ύπ‘˜(𝑑 π›Ύπ‘˜(𝑑 NOUN cana-2410 82 8 , , PUNCT cana-2410 82 9 𝑠)𝑃 𝑠)𝑃 X cana-2410 82 10 π‘˜.𝑛2βˆ’1 π‘˜.𝑛2βˆ’1 ADP cana-2410 82 11 π‘˜=0 π‘˜=0 PROPN cana-2410 82 12 theorem theorem VERB cana-2410 82 13 2.2 2.2 NUM cana-2410 82 14 . . PUNCT cana-2410 83 1 each each DET cana-2410 83 2 βˆ€ βˆ€ NOUN cana-2410 83 3 𝑑 𝑑 X cana-2410 83 4 ∈ ∈ PROPN cana-2410 83 5 ( ( PUNCT cana-2410 83 6 𝑠𝑗 𝑠𝑗 INTJ cana-2410 83 7 , , PUNCT cana-2410 83 8 𝑑𝑗+1]𝕋 𝑑𝑗+1]𝕋 ADV cana-2410 83 9 , , PUNCT cana-2410 83 10 𝑗 𝑗 NOUN cana-2410 83 11 = = SYM cana-2410 83 12 1,2 1,2 NUM cana-2410 83 13 , , PUNCT cana-2410 83 14 . . PUNCT cana-2410 83 15 . . PUNCT cana-2410 83 16 , , PUNCT cana-2410 83 17 implies imply VERB cana-2410 83 18 the the DET cana-2410 83 19 satisfying satisfy VERB cana-2410 83 20 function function NOUN cana-2410 83 21 is be AUX cana-2410 83 22 known know VERB cana-2410 83 23 as as ADP cana-2410 83 24 the the DET cana-2410 83 25 solution solution NOUN cana-2410 83 26 of of ADP cana-2410 83 27 a a DET cana-2410 83 28 system system NOUN cana-2410 83 29 ( ( PUNCT cana-2410 83 30 2.1 2.1 NUM cana-2410 83 31 ) ) PUNCT cana-2410 83 32 represented represent VERB cana-2410 83 33 by by ADP cana-2410 83 34 𝑧(𝑑 𝑧(𝑑 PROPN cana-2410 83 35 ) ) PUNCT cana-2410 83 36 = = SYM cana-2410 84 1 ψ(𝑑 ψ(𝑑 PROPN cana-2410 84 2 , , PUNCT cana-2410 84 3 𝑠𝑗)[𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 𝑠𝑗)[𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 X cana-2410 84 4 βˆ’ βˆ’ NOUN cana-2410 84 5 ) ) PUNCT cana-2410 85 1 + + CCONJ cana-2410 85 2 ∫ψ(𝑑 ∫ψ(𝑑 ADJ cana-2410 85 3 , , PUNCT cana-2410 85 4 𝜎(𝑠))k(t 𝜎(𝑠))k(t NOUN cana-2410 85 5 , , PUNCT cana-2410 85 6 s)z(s)βˆ†s s)z(s)βˆ†s PROPN cana-2410 85 7 𝑑 𝑑 VERB cana-2410 85 8 𝑠𝑗 𝑠𝑗 ADP cana-2410 85 9 + + SYM cana-2410 85 10 ∫ψ(𝑑 ∫ψ(𝑑 ADJ cana-2410 85 11 , , PUNCT cana-2410 85 12 𝜎(𝜏 𝜎(𝜏 NOUN cana-2410 85 13 ) ) PUNCT cana-2410 85 14 ) ) PUNCT cana-2410 86 1 𝑑 𝑑 NOUN cana-2410 86 2 𝑠𝑗 𝑠𝑗 ADP cana-2410 86 3 [ [ X cana-2410 86 4 𝑄(𝜏) 𝑄(𝜏) NUM cana-2410 86 5 οΏ½ οΏ½ NOUN cana-2410 86 6 Μ‚ Μ‚ NOUN cana-2410 86 7 οΏ½ οΏ½ NOUN cana-2410 86 8 (𝜏 (𝜏 VERB cana-2410 86 9 ) ) PUNCT cana-2410 86 10 + + CCONJ cana-2410 86 11 𝑓(𝜏 𝑓(𝜏 PROPN cana-2410 86 12 , , PUNCT cana-2410 86 13 𝑧(𝜏))]βˆ†πœ 𝑧(𝜏))]βˆ†πœ X cana-2410 86 14 , , PUNCT cana-2410 86 15 ( ( PUNCT cana-2410 86 16 2.3 2.3 NUM cana-2410 86 17 ) ) PUNCT cana-2410 86 18 proof proof NOUN cana-2410 86 19 : : PUNCT cana-2410 86 20 if if SCONJ cana-2410 86 21 𝑑 𝑑 PROPN cana-2410 86 22 ∈ ∈ PROPN cana-2410 86 23 [ [ X cana-2410 86 24 𝑑0 𝑑0 NOUN cana-2410 86 25 , , PUNCT cana-2410 86 26 𝑑1]𝕋 𝑑1]𝕋 PROPN cana-2410 86 27 , , PUNCT cana-2410 86 28 then then ADV cana-2410 86 29 there there PRON cana-2410 86 30 exists exist VERB cana-2410 86 31 an an DET cana-2410 86 32 only only ADJ cana-2410 86 33 one one NUM cana-2410 86 34 solution solution NOUN cana-2410 86 35 of of ADP cana-2410 86 36 ( ( PUNCT cana-2410 86 37 2.1 2.1 NUM cana-2410 86 38 ) ) PUNCT cana-2410 86 39 , , PUNCT cana-2410 86 40 we we PRON cana-2410 86 41 have have VERB cana-2410 86 42 𝑧(𝑑 𝑧(𝑑 NOUN cana-2410 86 43 ) ) PUNCT cana-2410 86 44 = = SYM cana-2410 87 1 ψ(𝑑 ψ(𝑑 PROPN cana-2410 87 2 , , PUNCT cana-2410 87 3 𝑑0)𝑧0 𝑑0)𝑧0 NOUN cana-2410 87 4 + + X cana-2410 87 5 ∫ψ(𝑑 ∫ψ(𝑑 ADJ cana-2410 87 6 , , PUNCT cana-2410 87 7 𝜎(𝑠))k(t 𝜎(𝑠))k(t NOUN cana-2410 87 8 , , PUNCT cana-2410 87 9 s)z(s)βˆ†s s)z(s)βˆ†s PROPN cana-2410 87 10 𝑑 𝑑 PROPN cana-2410 87 11 𝑑0 𝑑0 NOUN cana-2410 87 12 + + CCONJ cana-2410 87 13 ∫ψ(𝑑 ∫ψ(𝑑 ADJ cana-2410 87 14 , , PUNCT cana-2410 87 15 𝜎(𝜏 𝜎(𝜏 PROPN cana-2410 87 16 ) ) PUNCT cana-2410 87 17 𝑑 𝑑 PROPN cana-2410 87 18 𝑑0 𝑑0 PROPN cana-2410 87 19 𝑄(𝜏) 𝑄(𝜏) NUM cana-2410 87 20 οΏ½ οΏ½ PROPN cana-2410 87 21 Μ‚ Μ‚ NOUN cana-2410 87 22 οΏ½ οΏ½ NOUN cana-2410 87 23 (𝜏)βˆ†πœ (𝜏)βˆ†πœ X cana-2410 87 24 + + X cana-2410 87 25 ∫ψ(𝑑 ∫ψ(𝑑 ADJ cana-2410 87 26 , , PUNCT cana-2410 87 27 𝜎(𝜏 𝜎(𝜏 NUM cana-2410 87 28 ) ) PUNCT cana-2410 87 29 𝑑 𝑑 PROPN cana-2410 87 30 𝑑0 𝑑0 PROPN cana-2410 87 31 𝑓(𝜏 𝑓(𝜏 PROPN cana-2410 87 32 , , PUNCT cana-2410 87 33 𝑧(𝜏))βˆ†πœ 𝑧(𝜏))βˆ†πœ NOUN cana-2410 87 34 next next ADV cana-2410 87 35 , , PUNCT cana-2410 87 36 j=1 j=1 PROPN cana-2410 87 37 then then ADV cana-2410 87 38 𝑑 𝑑 PROPN cana-2410 87 39 ∈ ∈ PROPN cana-2410 87 40 ( ( PUNCT cana-2410 87 41 𝑠1 𝑠1 PROPN cana-2410 87 42 , , PUNCT cana-2410 87 43 𝑑2]𝕋 𝑑2]𝕋 NUM cana-2410 87 44 we we PRON cana-2410 87 45 have have VERB cana-2410 87 46 𝑧(𝑑 𝑧(𝑑 NOUN cana-2410 87 47 ) ) PUNCT cana-2410 87 48 = = SYM cana-2410 88 1 ψ(𝑑 ψ(𝑑 NOUN cana-2410 88 2 , , PUNCT cana-2410 88 3 𝑑0)𝑧(𝑠1 𝑑0)𝑧(𝑠1 PROPN cana-2410 88 4 ) ) PUNCT cana-2410 89 1 + + CCONJ cana-2410 89 2 ∫ψ(𝑑 ∫ψ(𝑑 ADJ cana-2410 89 3 , , PUNCT cana-2410 89 4 𝜎(𝑠))k(t 𝜎(𝑠))k(t NOUN cana-2410 89 5 , , PUNCT cana-2410 89 6 s)z(s)βˆ†s s)z(s)βˆ†s PROPN cana-2410 89 7 𝑑 𝑑 PROPN cana-2410 89 8 𝑠1 𝑠1 PROPN cana-2410 89 9 + + PROPN cana-2410 89 10 ∫ ∫ PROPN cana-2410 89 11 ψ(𝑑 ψ(𝑑 PROPN cana-2410 89 12 , , PUNCT cana-2410 89 13 𝜎(𝜏 𝜎(𝜏 PROPN cana-2410 89 14 ) ) PUNCT cana-2410 89 15 𝑑 𝑑 PROPN cana-2410 89 16 𝑠1 𝑠1 PROPN cana-2410 89 17 𝑄(𝜏) 𝑄(𝜏) NOUN cana-2410 89 18 οΏ½ οΏ½ PROPN cana-2410 89 19 Μ‚ Μ‚ NOUN cana-2410 89 20 οΏ½ οΏ½ NOUN cana-2410 89 21 (𝜏)βˆ†πœ (𝜏)βˆ†πœ SYM cana-2410 89 22 + + CCONJ cana-2410 89 23 ∫ ∫ PROPN cana-2410 89 24 ψ(𝑑 ψ(𝑑 PROPN cana-2410 89 25 , , PUNCT cana-2410 89 26 𝜎(𝜏 𝜎(𝜏 PROPN cana-2410 89 27 ) ) PUNCT cana-2410 89 28 𝑑 𝑑 PROPN cana-2410 89 29 𝑠1 𝑠1 PROPN cana-2410 89 30 𝑓(𝜏 𝑓(𝜏 PROPN cana-2410 89 31 , , PUNCT cana-2410 89 32 𝑧(𝜏))βˆ†πœ. 𝑧(𝜏))βˆ†πœ. NOUN cana-2410 89 33 also also ADV cana-2410 89 34 , , PUNCT cana-2410 89 35 for for ADP cana-2410 89 36 𝑧(𝑠1 𝑧(𝑠1 ADV cana-2410 89 37 ) ) PUNCT cana-2410 89 38 = = PUNCT cana-2410 90 1 [ [ X cana-2410 90 2 𝐼𝑛⨂𝑅1]𝑧(𝑑1 𝐼𝑛⨂𝑅1]𝑧(𝑑1 PROPN cana-2410 90 3 ) ) PUNCT cana-2410 90 4 substitute substitute NOUN cana-2410 90 5 above above ADP cana-2410 90 6 equation equation NOUN cana-2410 90 7 , , PUNCT cana-2410 90 8 we we PRON cana-2410 90 9 get get VERB cana-2410 90 10 𝑧(𝑑 𝑧(𝑑 NOUN cana-2410 90 11 ) ) PUNCT cana-2410 91 1 = = SYM cana-2410 91 2 ψ(𝑑 ψ(𝑑 NOUN cana-2410 91 3 , , PUNCT cana-2410 91 4 𝑑0)[𝐼𝑛⨂𝑅1]𝑧(𝑑1 𝑑0)[𝐼𝑛⨂𝑅1]𝑧(𝑑1 PROPN cana-2410 91 5 ) ) PUNCT cana-2410 92 1 + + CCONJ cana-2410 92 2 ∫ψ(𝑑 ∫ψ(𝑑 ADJ cana-2410 92 3 , , PUNCT cana-2410 92 4 𝜎(𝑠))k(t 𝜎(𝑠))k(t NOUN cana-2410 92 5 , , PUNCT cana-2410 92 6 s)z(s)βˆ†s s)z(s)βˆ†s PROPN cana-2410 92 7 𝑑 𝑑 PROPN cana-2410 92 8 𝑠1 𝑠1 PROPN cana-2410 92 9 + + PROPN cana-2410 92 10 ∫ ∫ PROPN cana-2410 92 11 ψ(𝑑 ψ(𝑑 PROPN cana-2410 92 12 , , PUNCT cana-2410 92 13 𝜎(𝜏 𝜎(𝜏 PROPN cana-2410 92 14 ) ) PUNCT cana-2410 92 15 𝑑 𝑑 PROPN cana-2410 92 16 𝑠1 𝑠1 PROPN cana-2410 92 17 𝑄(𝜏) 𝑄(𝜏) NOUN cana-2410 92 18 οΏ½ οΏ½ PROPN cana-2410 92 19 Μ‚ Μ‚ NOUN cana-2410 92 20 οΏ½ οΏ½ NOUN cana-2410 92 21 (𝜏)βˆ†πœ (𝜏)βˆ†πœ SYM cana-2410 92 22 + + CCONJ cana-2410 92 23 ∫ ∫ PROPN cana-2410 92 24 ψ(𝑑 ψ(𝑑 PROPN cana-2410 92 25 , , PUNCT cana-2410 92 26 𝜎(𝜏 𝜎(𝜏 PROPN cana-2410 92 27 ) ) PUNCT cana-2410 92 28 𝑑 𝑑 PROPN cana-2410 92 29 𝑠1 𝑠1 PROPN cana-2410 92 30 𝑓(𝜏 𝑓(𝜏 PROPN cana-2410 92 31 , , PUNCT cana-2410 92 32 𝑧(𝜏))βˆ†πœ. 𝑧(𝜏))βˆ†πœ. NOUN cana-2410 92 33 similarly similarly ADV cana-2410 92 34 , , PUNCT cana-2410 92 35 we we PRON cana-2410 92 36 are be AUX cana-2410 92 37 repeating repeat VERB cana-2410 92 38 the the DET cana-2410 92 39 above above ADJ cana-2410 92 40 same same ADJ cana-2410 92 41 process process NOUN cana-2410 92 42 for for ADP cana-2410 92 43 𝑑 𝑑 PROPN cana-2410 92 44 ∈ ∈ PROPN cana-2410 92 45 ( ( PUNCT cana-2410 92 46 𝑠𝑗 𝑠𝑗 NOUN cana-2410 92 47 , , PUNCT cana-2410 92 48 𝑑𝑗+1]𝕋 𝑑𝑗+1]𝕋 NOUN cana-2410 92 49 , , PUNCT cana-2410 92 50 𝑗 𝑗 NOUN cana-2410 92 51 = = SYM cana-2410 92 52 1,2 1,2 NUM cana-2410 92 53 , , PUNCT cana-2410 92 54 … … PUNCT cana-2410 92 55 , , PUNCT cana-2410 92 56 π‘š π‘š X cana-2410 92 57 , , PUNCT cana-2410 92 58 we we PRON cana-2410 92 59 get get VERB cana-2410 92 60 𝑧(𝑑 𝑧(𝑑 NOUN cana-2410 92 61 ) ) PUNCT cana-2410 92 62 = = SYM cana-2410 93 1 ψ(𝑑 ψ(𝑑 PROPN cana-2410 93 2 , , PUNCT cana-2410 93 3 𝑠𝑗)[𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 𝑠𝑗)[𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 X cana-2410 93 4 βˆ’ βˆ’ NOUN cana-2410 93 5 ) ) PUNCT cana-2410 94 1 + + CCONJ cana-2410 94 2 ∫ψ(𝑑 ∫ψ(𝑑 ADJ cana-2410 94 3 , , PUNCT cana-2410 94 4 𝜎(𝑠))k(t 𝜎(𝑠))k(t NOUN cana-2410 94 5 , , PUNCT cana-2410 94 6 s)z(s)βˆ†s s)z(s)βˆ†s PROPN cana-2410 94 7 𝑑 𝑑 VERB cana-2410 94 8 𝑠𝑗 𝑠𝑗 INTJ cana-2410 94 9 communications communication NOUN cana-2410 94 10 on on ADP cana-2410 94 11 applied apply VERB cana-2410 94 12 nonlinear nonlinear ADJ cana-2410 94 13 analysis analysis NOUN cana-2410 94 14 issn issn NOUN cana-2410 94 15 : : PUNCT cana-2410 94 16 1074 1074 NUM cana-2410 94 17 - - PUNCT cana-2410 94 18 133x 133x NUM cana-2410 94 19 vol vol NOUN cana-2410 94 20 32 32 NUM cana-2410 94 21 no no NOUN cana-2410 94 22 . . PUNCT cana-2410 95 1 2s 2s NUM cana-2410 95 2 ( ( PUNCT cana-2410 95 3 2025 2025 NUM cana-2410 95 4 ) ) PUNCT cana-2410 95 5 354 354 NUM cana-2410 95 6 https://internationalpubls.com https://internationalpubls.com X cana-2410 95 7 + + PUNCT cana-2410 95 8 ∫ψ(𝑑 ∫ψ(𝑑 ADJ cana-2410 95 9 , , PUNCT cana-2410 95 10 𝜎(𝜏 𝜎(𝜏 NOUN cana-2410 95 11 ) ) PUNCT cana-2410 95 12 ) ) PUNCT cana-2410 96 1 𝑑 𝑑 NOUN cana-2410 96 2 𝑠𝑗 𝑠𝑗 ADP cana-2410 96 3 [ [ X cana-2410 96 4 𝑄(𝜏) 𝑄(𝜏) NUM cana-2410 96 5 οΏ½ οΏ½ NOUN cana-2410 96 6 Μ‚ Μ‚ NOUN cana-2410 96 7 οΏ½ οΏ½ NOUN cana-2410 96 8 (𝑑 (𝑑 NOUN cana-2410 96 9 ) ) PUNCT cana-2410 96 10 + + CCONJ cana-2410 96 11 𝑓(𝜏 𝑓(𝜏 PROPN cana-2410 96 12 , , PUNCT cana-2410 96 13 𝑧(𝜏 𝑧(𝜏 PROPN cana-2410 96 14 ) ) PUNCT cana-2410 96 15 ) ) PUNCT cana-2410 96 16 ] ] PUNCT cana-2410 97 1 βˆ†πœ βˆ†πœ NOUN cana-2410 97 2 therefore therefore ADV cana-2410 97 3 , , PUNCT cana-2410 97 4 the the DET cana-2410 97 5 equation equation NOUN cana-2410 97 6 ( ( PUNCT cana-2410 97 7 2.3 2.3 NUM cana-2410 97 8 ) ) PUNCT cana-2410 97 9 was be AUX cana-2410 97 10 derived derive VERB cana-2410 97 11 . . PUNCT cana-2410 98 1 3 3 X cana-2410 98 2 . . X cana-2410 98 3 controllability controllability NOUN cana-2410 98 4 in in ADP cana-2410 98 5 this this DET cana-2410 98 6 section section NOUN cana-2410 98 7 , , PUNCT cana-2410 98 8 we we PRON cana-2410 98 9 provide provide VERB cana-2410 98 10 necessary necessary ADJ cana-2410 98 11 and and CCONJ cana-2410 98 12 sufficient sufficient ADJ cana-2410 98 13 conditions condition NOUN cana-2410 98 14 for for ADP cana-2410 98 15 complete complete ADJ cana-2410 98 16 controllability controllability NOUN cana-2410 98 17 in in ADP cana-2410 98 18 the the DET cana-2410 98 19 following follow VERB cana-2410 98 20 system system NOUN cana-2410 98 21 . . PUNCT cana-2410 99 1 { { PUNCT cana-2410 99 2 π‘§βˆ†(𝑑 π‘§βˆ†(𝑑 X cana-2410 99 3 ) ) PUNCT cana-2410 99 4 = = SYM cana-2410 99 5 𝑃(𝑑)𝑧(𝑑 𝑃(𝑑)𝑧(𝑑 NOUN cana-2410 99 6 ) ) PUNCT cana-2410 100 1 + + CCONJ cana-2410 100 2 ∫ ∫ PROPN cana-2410 100 3 𝐾(𝑑 𝐾(𝑑 PROPN cana-2410 100 4 , , PUNCT cana-2410 100 5 𝑠)𝑧(𝑠)βˆ†π‘  𝑠)𝑧(𝑠)βˆ†π‘  PROPN cana-2410 100 6 𝑑 𝑑 PROPN cana-2410 100 7 0 0 NUM cana-2410 100 8 + + CCONJ cana-2410 100 9 𝑄(𝑑) 𝑄(𝑑) PRON cana-2410 100 10 οΏ½ οΏ½ NOUN cana-2410 100 11 Μ‚ Μ‚ NOUN cana-2410 100 12 οΏ½ οΏ½ NOUN cana-2410 100 13 (𝑑 (𝑑 NOUN cana-2410 100 14 ) ) PUNCT cana-2410 101 1 + + CCONJ cana-2410 101 2 𝑓(𝑑 𝑓(𝑑 PROPN cana-2410 101 3 , , PUNCT cana-2410 101 4 𝑧(𝑑 𝑧(𝑑 PROPN cana-2410 101 5 ) ) PUNCT cana-2410 101 6 ) ) PUNCT cana-2410 101 7 , , PUNCT cana-2410 101 8 𝑑 𝑑 PROPN cana-2410 101 9 ∈ ∈ PROPN cana-2410 101 10 ( ( PUNCT cana-2410 101 11 𝑠𝑗 𝑠𝑗 NOUN cana-2410 101 12 , , PUNCT cana-2410 101 13 𝑑𝑗+1]𝕋 𝑑𝑗+1]𝕋 NOUN cana-2410 101 14 , , PUNCT cana-2410 101 15 𝑗 𝑗 NOUN cana-2410 101 16 = = SYM cana-2410 101 17 1 1 NUM cana-2410 101 18 , , PUNCT cana-2410 101 19 2 2 NUM cana-2410 101 20 , , PUNCT cana-2410 101 21 … … PUNCT cana-2410 101 22 𝑧(𝑑 𝑧(𝑑 NOUN cana-2410 101 23 ) ) PUNCT cana-2410 102 1 = = PUNCT cana-2410 103 1 [ [ X cana-2410 103 2 πΌπ‘›βŠ—π‘…π‘—]𝑧(𝑑𝑗 πΌπ‘›βŠ—π‘…π‘—]𝑧(𝑑𝑗 ADP cana-2410 103 3 βˆ’ βˆ’ NOUN cana-2410 103 4 ) ) PUNCT cana-2410 103 5 , , PUNCT cana-2410 103 6 𝑑 𝑑 PROPN cana-2410 103 7 ∈ ∈ PROPN cana-2410 103 8 ( ( PUNCT cana-2410 103 9 𝑠𝑗 𝑠𝑗 INTJ cana-2410 103 10 , , PUNCT cana-2410 103 11 𝑑𝑗+1]𝕋 𝑑𝑗+1]𝕋 ADV cana-2410 103 12 , , PUNCT cana-2410 103 13 𝑗 𝑗 NOUN cana-2410 103 14 = = SYM cana-2410 103 15 0,1 0,1 NUM cana-2410 103 16 , , PUNCT cana-2410 103 17 … … PUNCT cana-2410 103 18 𝑧(𝑑0 𝑧(𝑑0 NUM cana-2410 103 19 ) ) PUNCT cana-2410 103 20 = = SYM cana-2410 103 21 𝑧0 𝑧0 PROPN cana-2410 103 22 , , PUNCT cana-2410 103 23 𝑑0 𝑑0 PROPN cana-2410 103 24 ∈ ∈ PROPN cana-2410 103 25 𝕋 𝕋 PROPN cana-2410 103 26 ( ( PUNCT cana-2410 103 27 3.1 3.1 NUM cana-2410 103 28 ) ) PUNCT cana-2410 103 29 definition definition NOUN cana-2410 103 30 3.1 3.1 NUM cana-2410 103 31 : : PUNCT cana-2410 103 32 for for ADP cana-2410 103 33 any any DET cana-2410 103 34 𝑧0 𝑧0 NOUN cana-2410 103 35 and and CCONJ cana-2410 103 36 𝑧𝑇 𝑧𝑇 SYM cana-2410 103 37 ∈ ∈ PROPN cana-2410 104 1 ℝ𝑛 ℝ𝑛 ADJ cana-2410 104 2 2 2 NUM cana-2410 104 3 , , PUNCT cana-2410 104 4 there there PRON cana-2410 104 5 must must AUX cana-2410 104 6 be be AUX cana-2410 104 7 a a DET cana-2410 104 8 piece piece NOUN cana-2410 104 9 - - PUNCT cana-2410 104 10 wise wise ADJ cana-2410 104 11 rd rd NOUN cana-2410 104 12 - - ADJ cana-2410 104 13 continuous continuous ADJ cana-2410 104 14 control control NOUN cana-2410 104 15 function function NOUN cana-2410 104 16 uΜ‚(𝑑 uΜ‚(𝑑 NOUN cana-2410 104 17 ): ): PUNCT cana-2410 104 18 [ [ X cana-2410 104 19 𝑑0 𝑑0 NOUN cana-2410 104 20 , , PUNCT cana-2410 104 21 𝑇]𝕋 𝑇]𝕋 NUM cana-2410 104 22 β†’ β†’ SYM cana-2410 104 23 ℝ ℝ PROPN cana-2410 104 24 𝑛2 𝑛2 NOUN cana-2410 104 25 , , PUNCT cana-2410 104 26 so so SCONJ cana-2410 104 27 that that SCONJ cana-2410 104 28 the the DET cana-2410 104 29 solution solution NOUN cana-2410 104 30 of of ADP cana-2410 104 31 the the DET cana-2410 104 32 system system NOUN cana-2410 104 33 ( ( PUNCT cana-2410 104 34 3.1 3.1 NUM cana-2410 104 35 ) ) PUNCT cana-2410 104 36 satisfies satisfie NOUN cana-2410 104 37 𝑧(𝑑0 𝑧(𝑑0 NUM cana-2410 104 38 ) ) PUNCT cana-2410 104 39 = = NOUN cana-2410 104 40 𝑧0 𝑧0 NOUN cana-2410 104 41 and and CCONJ cana-2410 104 42 𝑧(𝑇 𝑧(𝑇 NUM cana-2410 104 43 ) ) PUNCT cana-2410 104 44 = = PUNCT cana-2410 104 45 𝑧𝑇 𝑧𝑇 X cana-2410 104 46 . . PUNCT cana-2410 105 1 this this DET cana-2410 105 2 system system NOUN cana-2410 105 3 is be AUX cana-2410 105 4 called call VERB cana-2410 105 5 controllability controllability NOUN cana-2410 105 6 on on ADP cana-2410 105 7 [ [ X cana-2410 105 8 𝑑0 𝑑0 NOUN cana-2410 105 9 , , PUNCT cana-2410 105 10 𝑇]𝕋 𝑇]𝕋 NUM cana-2410 105 11 with with ADP cana-2410 105 12 𝑑0 𝑑0 PROPN cana-2410 105 13 < < X cana-2410 105 14 𝑇. 𝑇. PROPN cana-2410 105 15 definition definition NOUN cana-2410 105 16 3.2 3.2 NUM cana-2410 105 17 : : PUNCT cana-2410 105 18 for for ADP cana-2410 105 19 𝑗 𝑗 NOUN cana-2410 105 20 = = SYM cana-2410 105 21 1,2 1,2 NUM cana-2410 105 22 , , PUNCT cana-2410 105 23 … … PUNCT cana-2410 105 24 , , PUNCT cana-2410 105 25 π‘š π‘š X cana-2410 105 26 , , PUNCT cana-2410 105 27 it it PRON cana-2410 105 28 is be AUX cana-2410 105 29 controllable controllable ADJ cana-2410 105 30 on on ADP cana-2410 105 31 both both DET cana-2410 105 32 [ [ X cana-2410 105 33 𝑑0 𝑑0 NOUN cana-2410 105 34 , , PUNCT cana-2410 105 35 𝑑1]𝕋 𝑑1]𝕋 PRON cana-2410 106 1 and and CCONJ cana-2410 106 2 [ [ X cana-2410 106 3 𝑠𝑗 𝑠𝑗 INTJ cana-2410 106 4 , , PUNCT cana-2410 106 5 𝑑𝑗+1]𝕋 𝑑𝑗+1]𝕋 AUX cana-2410 106 6 , , PUNCT cana-2410 106 7 then then ADV cana-2410 106 8 system system NOUN cana-2410 106 9 ( ( PUNCT cana-2410 106 10 3.1 3.1 NUM cana-2410 106 11 ) ) PUNCT cana-2410 106 12 is be AUX cana-2410 106 13 known know VERB cana-2410 106 14 as as ADP cana-2410 106 15 complete complete ADJ cana-2410 106 16 controllable controllable NOUN cana-2410 106 17 in in ADP cana-2410 106 18 [ [ X cana-2410 106 19 𝑑0 𝑑0 NOUN cana-2410 106 20 , , PUNCT cana-2410 106 21 𝑇]𝕋 𝑇]𝕋 NUM cana-2410 106 22 with with ADP cana-2410 106 23 𝑑0 𝑑0 PROPN cana-2410 106 24 < < X cana-2410 106 25 𝑇. 𝑇. PROPN cana-2410 106 26 the the DET cana-2410 106 27 corresponding corresponding ADJ cana-2410 106 28 gramian gramian ADJ cana-2410 106 29 matrices matrix NOUN cana-2410 106 30 are be AUX cana-2410 106 31 defined define VERB cana-2410 106 32 by by ADP cana-2410 106 33 . . PUNCT cana-2410 107 1 𝒩0(𝑑0 𝒩0(𝑑0 ADJ cana-2410 107 2 , , PUNCT cana-2410 107 3 𝑑1 𝑑1 NOUN cana-2410 107 4 ) ) PUNCT cana-2410 107 5 = = SYM cana-2410 107 6 ∫ψ(𝑑0,Οƒ(Ο„))𝑄(Ο„)𝑄 ∫ψ(𝑑0,Οƒ(Ο„))𝑄(Ο„)𝑄 NOUN cana-2410 108 1 βˆ—(Ο„)Οˆβˆ—(𝑑0 βˆ—(Ο„)Οˆβˆ—(𝑑0 PROPN cana-2410 108 2 , , PUNCT cana-2410 108 3 Οƒ(Ο„))δτ Οƒ(Ο„))δτ PROPN cana-2410 108 4 t t PROPN cana-2410 108 5 t0 t0 PROPN cana-2410 108 6 ( ( PUNCT cana-2410 108 7 3.2 3.2 NUM cana-2410 108 8 ) ) PUNCT cana-2410 108 9 𝒩𝑗(𝑠𝑗 𝒩𝑗(𝑠𝑗 NOUN cana-2410 108 10 , , PUNCT cana-2410 108 11 𝑑𝑗+1 𝑑𝑗+1 X cana-2410 108 12 ) ) PUNCT cana-2410 108 13 = = SYM cana-2410 109 1 ∫ ∫ PROPN cana-2410 109 2 ψ ψ X cana-2410 109 3 ( ( PUNCT cana-2410 109 4 𝑠𝑗 𝑠𝑗 INTJ cana-2410 109 5 , , PUNCT cana-2410 109 6 Οƒ(Ο„))𝑄(Ο„)𝑄 Οƒ(Ο„))𝑄(Ο„)𝑄 ADJ cana-2410 109 7 βˆ—(Ο„)Οˆβˆ— βˆ—(Ο„)Οˆβˆ— PROPN cana-2410 109 8 ( ( PUNCT cana-2410 109 9 𝑠𝑗 𝑠𝑗 INTJ cana-2410 109 10 , , PUNCT cana-2410 109 11 Οƒ(Ο„))δτ Οƒ(Ο„))δτ PROPN cana-2410 109 12 , , PUNCT cana-2410 109 13 𝑗 𝑗 NOUN cana-2410 109 14 = = SYM cana-2410 109 15 1,2 1,2 NUM cana-2410 109 16 , , PUNCT cana-2410 109 17 … … PUNCT cana-2410 109 18 , , PUNCT cana-2410 109 19 π‘š π‘š X cana-2410 109 20 , , PUNCT cana-2410 109 21 𝑑𝑗+1 𝑑𝑗+1 PROPN cana-2410 109 22 s𝑗 s𝑗 PROPN cana-2410 109 23 ( ( PUNCT cana-2410 109 24 3.3 3.3 NUM cana-2410 109 25 ) ) PUNCT cana-2410 109 26 if if SCONJ cana-2410 109 27 𝑃(𝑑 𝑃(𝑑 NUM cana-2410 109 28 ) ) PUNCT cana-2410 109 29 = = SYM cana-2410 109 30 𝑃 𝑃 NOUN cana-2410 109 31 and and CCONJ cana-2410 109 32 𝑄(𝑑 𝑄(𝑑 PRON cana-2410 109 33 ) ) PUNCT cana-2410 110 1 = = SYM cana-2410 110 2 𝑄 𝑄 NOUN cana-2410 110 3 are be AUX cana-2410 110 4 constant constant ADJ cana-2410 110 5 matrices matrix NOUN cana-2410 110 6 , , PUNCT cana-2410 110 7 then then ADV cana-2410 110 8 𝒩0(𝑑0 𝒩0(𝑑0 PROPN cana-2410 110 9 , , PUNCT cana-2410 110 10 𝑑1 𝑑1 NOUN cana-2410 110 11 ) ) PUNCT cana-2410 110 12 = = SYM cana-2410 110 13 ∫ ∫ PROPN cana-2410 110 14 𝑒𝑃(𝑑0,Οƒ(Ο„))𝑄𝑄 𝑒𝑃(𝑑0,Οƒ(Ο„))𝑄𝑄 ADV cana-2410 110 15 βˆ—π‘’π‘ƒ βˆ—π‘’π‘ƒ PROPN cana-2410 110 16 βˆ—(𝑑0,Οƒ(Ο„))δτ βˆ—(𝑑0,Οƒ(Ο„))δτ PROPN cana-2410 110 17 t t NOUN cana-2410 110 18 t0 t0 PROPN cana-2410 110 19 ( ( PUNCT cana-2410 110 20 3.4 3.4 NUM cana-2410 110 21 ) ) PUNCT cana-2410 110 22 𝒩𝑗(𝑠𝑗 𝒩𝑗(𝑠𝑗 NOUN cana-2410 110 23 , , PUNCT cana-2410 110 24 𝑑𝑗+1 𝑑𝑗+1 X cana-2410 110 25 ) ) PUNCT cana-2410 110 26 = = SYM cana-2410 111 1 ∫ ∫ PROPN cana-2410 111 2 𝑒𝑃 𝑒𝑃 NOUN cana-2410 111 3 ( ( PUNCT cana-2410 111 4 𝑠𝑗 𝑠𝑗 INTJ cana-2410 111 5 , , PUNCT cana-2410 111 6 Οƒ(Ο„))𝑄𝑄 Οƒ(Ο„))𝑄𝑄 ADJ cana-2410 111 7 βˆ—π‘’π‘ƒ βˆ—π‘’π‘ƒ PROPN cana-2410 111 8 βˆ— βˆ— NOUN cana-2410 111 9 ( ( PUNCT cana-2410 111 10 𝑠𝑗 𝑠𝑗 INTJ cana-2410 111 11 , , PUNCT cana-2410 111 12 Οƒ(Ο„))δτ Οƒ(Ο„))δτ PROPN cana-2410 111 13 , , PUNCT cana-2410 111 14 𝑗 𝑗 NOUN cana-2410 111 15 = = SYM cana-2410 111 16 1,2 1,2 NUM cana-2410 111 17 , , PUNCT cana-2410 111 18 … … PUNCT cana-2410 111 19 , , PUNCT cana-2410 111 20 π‘š π‘š ADP cana-2410 111 21 𝑑𝑗+1 𝑑𝑗+1 PROPN cana-2410 111 22 s𝑗 s𝑗 X cana-2410 111 23 ( ( PUNCT cana-2410 111 24 3.5 3.5 NUM cana-2410 111 25 ) ) PUNCT cana-2410 111 26 here here ADV cana-2410 111 27 ( ( PUNCT cana-2410 111 28 . . PUNCT cana-2410 111 29 ) ) PUNCT cana-2410 111 30 βˆ—is βˆ—is NUM cana-2410 111 31 represented represent VERB cana-2410 111 32 as as ADP cana-2410 111 33 transpose transpose NOUN cana-2410 111 34 of of ADP cana-2410 111 35 a a DET cana-2410 111 36 matrix matrix NOUN cana-2410 111 37 ( ( PUNCT cana-2410 111 38 . . PUNCT cana-2410 111 39 ) ) PUNCT cana-2410 111 40 . . PUNCT cana-2410 112 1 we we PRON cana-2410 112 2 define define VERB cana-2410 112 3 uΜ‚(𝑑 uΜ‚(𝑑 SYM cana-2410 112 4 ) ) PUNCT cana-2410 112 5 as as ADP cana-2410 112 6 uΜ‚(𝑑 uΜ‚(𝑑 NOUN cana-2410 112 7 ) ) PUNCT cana-2410 112 8 = = PRON cana-2410 113 1 { { PUNCT cana-2410 113 2 βˆ’π‘„βˆ—(𝑑)π›Ήβˆ—(𝑑0 βˆ’π‘„βˆ—(𝑑)π›Ήβˆ—(𝑑0 PROPN cana-2410 113 3 , , PUNCT cana-2410 113 4 𝜎(𝜏))Ο†0 𝜎(𝜏))Ο†0 NOUN cana-2410 113 5 , , PUNCT cana-2410 113 6 𝑑 𝑑 PROPN cana-2410 113 7 ∈ ∈ PROPN cana-2410 114 1 [ [ X cana-2410 114 2 𝑑0 𝑑0 NOUN cana-2410 114 3 , , PUNCT cana-2410 114 4 𝑑1]𝕋 𝑑1]𝕋 PROPN cana-2410 114 5 βˆ’π‘„βˆ—(t)Οˆβˆ—(π‘ π‘˜,Οƒ(Ο„))Ο†π‘˜ βˆ’π‘„βˆ—(t)Οˆβˆ—(π‘ π‘˜,Οƒ(Ο„))Ο†π‘˜ NUM cana-2410 114 6 , , PUNCT cana-2410 114 7 𝑑 𝑑 PROPN cana-2410 114 8 ∈ ∈ PROPN cana-2410 114 9 ( ( PUNCT cana-2410 114 10 𝑠𝑗 𝑠𝑗 INTJ cana-2410 114 11 , , PUNCT cana-2410 114 12 𝑑𝑗+1]𝕋 𝑑𝑗+1]𝕋 ADV cana-2410 114 13 , , PUNCT cana-2410 114 14 𝑗 𝑗 NOUN cana-2410 114 15 = = SYM cana-2410 114 16 1,2 1,2 NUM cana-2410 114 17 , , PUNCT cana-2410 114 18 … … PUNCT cana-2410 114 19 , , PUNCT cana-2410 114 20 π‘š. π‘š. ADJ cana-2410 114 21 ( ( PUNCT cana-2410 114 22 3.6 3.6 NUM cana-2410 114 23 ) ) PUNCT cana-2410 114 24 where where SCONJ cana-2410 114 25 communications communication NOUN cana-2410 114 26 on on ADP cana-2410 114 27 applied apply VERB cana-2410 114 28 nonlinear nonlinear ADJ cana-2410 114 29 analysis analysis NOUN cana-2410 114 30 issn issn NOUN cana-2410 114 31 : : PUNCT cana-2410 114 32 1074 1074 NUM cana-2410 114 33 - - PUNCT cana-2410 114 34 133x 133x NUM cana-2410 114 35 vol vol NOUN cana-2410 114 36 32 32 NUM cana-2410 114 37 no no NOUN cana-2410 114 38 . . PUNCT cana-2410 115 1 2s 2s NUM cana-2410 115 2 ( ( PUNCT cana-2410 115 3 2025 2025 NUM cana-2410 115 4 ) ) PUNCT cana-2410 115 5 355 355 NUM cana-2410 115 6 https://internationalpubls.com https://internationalpubls.com X cana-2410 115 7 Ο†0 Ο†0 PROPN cana-2410 115 8 = = PROPN cana-2410 115 9 𝒩0 𝒩0 PROPN cana-2410 115 10 βˆ’1(𝑑0 βˆ’1(𝑑0 X cana-2410 115 11 , , PUNCT cana-2410 115 12 𝑑1 𝑑1 NOUN cana-2410 115 13 ) ) PUNCT cana-2410 116 1 [ [ X cana-2410 116 2 𝑧0 𝑧0 PROPN cana-2410 116 3 βˆ’ βˆ’ PROPN cana-2410 116 4 ψ(𝑑0 ψ(𝑑0 NOUN cana-2410 116 5 , , PUNCT cana-2410 116 6 𝑑1)𝑧𝑑1 𝑑1)𝑧𝑑1 PUNCT cana-2410 116 7 + + CCONJ cana-2410 116 8 ∫ ∫ PROPN cana-2410 116 9 ψ(𝑑0 ψ(𝑑0 NOUN cana-2410 116 10 , , PUNCT cana-2410 116 11 𝜎(𝑠))k(t 𝜎(𝑠))k(t NOUN cana-2410 116 12 , , PUNCT cana-2410 116 13 s)z(s)βˆ†s s)z(s)βˆ†s PROPN cana-2410 116 14 𝑑1 𝑑1 NOUN cana-2410 116 15 t0 t0 PROPN cana-2410 117 1 + + NOUN cana-2410 117 2 ∫ ∫ PROPN cana-2410 117 3 ψ(𝑑0,Οƒ(Ο„))f(Ο„ ψ(𝑑0,Οƒ(Ο„))f(Ο„ NOUN cana-2410 117 4 , , PUNCT cana-2410 117 5 z(Ο„))δτ z(Ο„))δτ PROPN cana-2410 117 6 𝑑1 𝑑1 PROPN cana-2410 117 7 t0 t0 PROPN cana-2410 117 8 ] ] PUNCT cana-2410 117 9 , , PUNCT cana-2410 117 10 and and CCONJ cana-2410 117 11 φ𝑗 φ𝑗 NOUN cana-2410 117 12 = = VERB cana-2410 117 13 𝒩𝑗 𝒩𝑗 NOUN cana-2410 117 14 βˆ’1(𝑠𝑗 βˆ’1(𝑠𝑗 NOUN cana-2410 117 15 , , PUNCT cana-2410 117 16 𝑑𝑗+1 𝑑𝑗+1 NOUN cana-2410 117 17 ) ) PUNCT cana-2410 118 1 [ [ X cana-2410 118 2 [ [ X cana-2410 118 3 𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 X cana-2410 118 4 βˆ’ βˆ’ NOUN cana-2410 118 5 ) ) PUNCT cana-2410 118 6 βˆ’Οˆ(𝑠𝑗 βˆ’Οˆ(𝑠𝑗 VERB cana-2410 118 7 , , PUNCT cana-2410 118 8 𝑑𝑗+1)𝑧𝑑𝑗+1 𝑑𝑗+1)𝑧𝑑𝑗+1 ADJ cana-2410 119 1 + + NOUN cana-2410 119 2 ∫ ∫ PROPN cana-2410 119 3 ψ ψ X cana-2410 119 4 ( ( PUNCT cana-2410 119 5 𝑑𝑗 𝑑𝑗 PROPN cana-2410 119 6 , , PUNCT cana-2410 119 7 Οƒ(𝑠))k(t Οƒ(𝑠))k(t NOUN cana-2410 119 8 , , PUNCT cana-2410 119 9 s)z(s)βˆ†s s)z(s)βˆ†s VERB cana-2410 119 10 𝑑𝑗+1 𝑑𝑗+1 NUM cana-2410 119 11 s𝑗 s𝑗 PROPN cana-2410 119 12 + + PROPN cana-2410 119 13 ∫ ∫ PROPN cana-2410 119 14 ψ ψ X cana-2410 119 15 ( ( PUNCT cana-2410 119 16 𝑠𝑗 𝑠𝑗 INTJ cana-2410 119 17 , , PUNCT cana-2410 119 18 Οƒ(Ο„ Οƒ(Ο„ PROPN cana-2410 119 19 ) ) PUNCT cana-2410 119 20 ) ) PUNCT cana-2410 119 21 f(Ο„ f(Ο„ PROPN cana-2410 119 22 , , PUNCT cana-2410 119 23 z(Ο„))δτ z(Ο„))δτ VERB cana-2410 119 24 𝑑𝑗+1 𝑑𝑗+1 PRON cana-2410 119 25 s𝑗 s𝑗 PART cana-2410 119 26 ] ] PUNCT cana-2410 119 27 . . PUNCT cana-2410 120 1 we we PRON cana-2410 120 2 need need VERB cana-2410 120 3 following follow VERB cana-2410 120 4 the the DET cana-2410 120 5 conditions condition NOUN cana-2410 120 6 : : PUNCT cana-2410 120 7 ( ( PUNCT cana-2410 120 8 h1 h1 PROPN cana-2410 120 9 ): ): PUNCT cana-2410 120 10 the the DET cana-2410 120 11 nonlinear nonlinear ADJ cana-2410 120 12 function function NOUN cana-2410 120 13 𝑓 𝑓 NOUN cana-2410 120 14 : : PUNCT cana-2410 120 15 𝐽1 𝐽1 ADP cana-2410 120 16 Γ— Γ— PROPN cana-2410 120 17 ℝ𝑛 ℝ𝑛 ADJ cana-2410 120 18 2 2 NUM cana-2410 120 19 β†’ β†’ SYM cana-2410 120 20 ℝ𝑛 ℝ𝑛 ADP cana-2410 120 21 2 2 NUM cana-2410 120 22 , , PUNCT cana-2410 120 23 𝐽1 𝐽1 NOUN cana-2410 120 24 = = SYM cana-2410 120 25 ⋃ ⋃ PROPN cana-2410 121 1 [ [ X cana-2410 121 2 𝑠𝑗 𝑠𝑗 INTJ cana-2410 121 3 , , PUNCT cana-2410 121 4 𝑑𝑗+1]𝕋 𝑑𝑗+1]𝕋 AUX cana-2410 121 5 π‘š π‘š PROPN cana-2410 121 6 𝑗=0 𝑗=0 PROPN cana-2410 121 7 is be AUX cana-2410 121 8 rd rd NOUN cana-2410 121 9 - - ADJ cana-2410 121 10 continuous continuous ADJ cana-2410 121 11 and and CCONJ cana-2410 121 12 there there PRON cana-2410 121 13 is be VERB cana-2410 121 14 exists exist NOUN cana-2410 122 1 𝑀𝑓 𝑀𝑓 PROPN cana-2410 122 2 > > X cana-2410 122 3 0 0 NUM cana-2410 122 4 such such ADJ cana-2410 122 5 that that SCONJ cana-2410 122 6 ‖𝑓(𝑑 ‖𝑓(𝑑 SYM cana-2410 122 7 , , PUNCT cana-2410 122 8 𝑧 𝑧 NOUN cana-2410 122 9 ) ) PUNCT cana-2410 122 10 βˆ’ βˆ’ PROPN cana-2410 123 1 𝑓(𝑑 𝑓(𝑑 NOUN cana-2410 123 2 , , PUNCT cana-2410 123 3 π‘₯)β€– π‘₯)β€– ADJ cana-2410 123 4 ≀ ≀ NUM cana-2410 123 5 𝑀𝑓‖𝑧 𝑀𝑓‖𝑧 NOUN cana-2410 123 6 βˆ’ βˆ’ PROPN cana-2410 123 7 π‘₯β€– π‘₯β€– NOUN cana-2410 123 8 , , PUNCT cana-2410 123 9 βˆ€ βˆ€ X cana-2410 123 10 𝑧 𝑧 VERB cana-2410 123 11 , , PUNCT cana-2410 123 12 π‘₯ π‘₯ PRON cana-2410 123 13 ∈ ∈ PROPN cana-2410 124 1 ℝ𝑛 ℝ𝑛 ADJ cana-2410 124 2 2 2 NUM cana-2410 124 3 , , PUNCT cana-2410 124 4 𝑑 𝑑 PROPN cana-2410 124 5 ∈ ∈ PROPN cana-2410 124 6 𝐽1 𝐽1 NOUN cana-2410 124 7 . . PUNCT cana-2410 125 1 also also ADV cana-2410 125 2 , , PUNCT cana-2410 125 3 there there PRON cana-2410 125 4 is be VERB cana-2410 125 5 exists exist VERB cana-2410 126 1 𝐿𝑓 𝐿𝑓 PROPN cana-2410 126 2 > > X cana-2410 126 3 0 0 NUM cana-2410 127 1 such such ADJ cana-2410 127 2 that that SCONJ cana-2410 127 3 ‖𝑓(𝑑 ‖𝑓(𝑑 SYM cana-2410 127 4 , , PUNCT cana-2410 127 5 𝑧)β€– 𝑧)β€– ADJ cana-2410 127 6 ≀ ≀ ADJ cana-2410 127 7 𝐿𝑓 𝐿𝑓 PROPN cana-2410 127 8 , , PUNCT cana-2410 127 9 βˆ€π‘‘ βˆ€π‘‘ PROPN cana-2410 127 10 ∈ ∈ PROPN cana-2410 127 11 𝐽1 𝐽1 VERB cana-2410 127 12 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ cana-2410 128 1 𝑧 𝑧 PRON cana-2410 128 2 ∈ ∈ PROPN cana-2410 129 1 ℝ𝑛 ℝ𝑛 ADJ cana-2410 129 2 2 2 NUM cana-2410 129 3 . . PUNCT cana-2410 130 1 ( ( PUNCT cana-2410 130 2 h2 h2 NOUN cana-2410 130 3 ): ): PUNCT cana-2410 130 4 the the DET cana-2410 130 5 function function NOUN cana-2410 130 6 [ [ X cana-2410 130 7 𝐼𝑛⨂𝑅𝑗]:= 𝐼𝑛⨂𝑅𝑗]:= NOUN cana-2410 130 8 [ [ X cana-2410 130 9 𝑑𝑗 𝑑𝑗 ADP cana-2410 130 10 , , PUNCT cana-2410 130 11 𝑠𝑗]𝕋 𝑠𝑗]𝕋 PROPN cana-2410 130 12 Γ— Γ— NOUN cana-2410 130 13 ℝ𝑛 ℝ𝑛 PROPN cana-2410 130 14 2 2 NUM cana-2410 130 15 β†’ β†’ SYM cana-2410 130 16 ℝ𝑛 ℝ𝑛 ADJ cana-2410 130 17 2 2 NUM cana-2410 130 18 are be AUX cana-2410 130 19 rd rd NOUN cana-2410 130 20 - - NOUN cana-2410 130 21 continuous continuous ADJ cana-2410 130 22 there there PRON cana-2410 130 23 is be VERB cana-2410 130 24 exists exist NOUN cana-2410 130 25 𝑀[𝐼𝑛⨂𝑅𝑗 𝑀[𝐼𝑛⨂𝑅𝑗 PROPN cana-2410 130 26 ] ] X cana-2410 130 27 > > X cana-2410 130 28 0 0 NUM cana-2410 130 29 such such ADJ cana-2410 130 30 that that SCONJ cana-2410 130 31 β€–[𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 β€–[𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 PROPN cana-2410 130 32 βˆ’ βˆ’ NOUN cana-2410 130 33 ) ) PUNCT cana-2410 130 34 βˆ’ βˆ’ PROPN cana-2410 131 1 [ [ X cana-2410 131 2 𝐼𝑛⨂𝑅𝑗]π‘₯(𝑑𝑗 𝐼𝑛⨂𝑅𝑗]π‘₯(𝑑𝑗 X cana-2410 131 3 βˆ’)β€– βˆ’)β€– ADJ cana-2410 131 4 ≀ ≀ NUM cana-2410 131 5 𝑀[𝐼𝑛⨂𝑅𝑗 𝑀[𝐼𝑛⨂𝑅𝑗 PROPN cana-2410 131 6 ] ] X cana-2410 131 7 ‖𝑧 ‖𝑧 PROPN cana-2410 131 8 βˆ’ βˆ’ PROPN cana-2410 131 9 π‘₯β€– π‘₯β€– PROPN cana-2410 131 10 , , PUNCT cana-2410 131 11 βˆ€ βˆ€ X cana-2410 131 12 𝑧 𝑧 VERB cana-2410 131 13 , , PUNCT cana-2410 131 14 π‘₯ π‘₯ PRON cana-2410 131 15 ∈ ∈ PROPN cana-2410 132 1 ℝ𝑛 ℝ𝑛 ADJ cana-2410 132 2 2 2 NUM cana-2410 132 3 , , PUNCT cana-2410 132 4 𝑑 𝑑 PROPN cana-2410 132 5 ∈ ∈ PROPN cana-2410 132 6 𝐼𝑗 𝐼𝑗 NOUN cana-2410 132 7 . . PUNCT cana-2410 133 1 also also ADV cana-2410 133 2 , , PUNCT cana-2410 133 3 there there PRON cana-2410 133 4 is be VERB cana-2410 133 5 exists exist NOUN cana-2410 133 6 𝐿[𝐼𝑛⨂𝑅𝑗 𝐿[𝐼𝑛⨂𝑅𝑗 PROPN cana-2410 133 7 ] ] PUNCT cana-2410 133 8 > > X cana-2410 133 9 0 0 NUM cana-2410 134 1 such such ADJ cana-2410 134 2 that that SCONJ cana-2410 134 3 β€–[𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗)β€– β€–[𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗)β€– PROPN cana-2410 134 4 ≀ ≀ PROPN cana-2410 134 5 𝐿[𝐼𝑛⨂𝑅𝑗 𝐿[𝐼𝑛⨂𝑅𝑗 PROPN cana-2410 134 6 ] ] PUNCT cana-2410 134 7 , , PUNCT cana-2410 134 8 βˆ€π‘‘ βˆ€π‘‘ PROPN cana-2410 134 9 ∈ ∈ PROPN cana-2410 134 10 𝐼𝑗 𝐼𝑗 NOUN cana-2410 134 11 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ cana-2410 134 12 𝑧 𝑧 PRON cana-2410 134 13 ∈ ∈ NOUN cana-2410 134 14 ℝ𝑛 ℝ𝑛 ADJ cana-2410 134 15 2 2 NUM cana-2410 134 16 . . PUNCT cana-2410 135 1 ( ( PUNCT cana-2410 135 2 h3 h3 NOUN cana-2410 135 3 ): ): PUNCT cana-2410 135 4 𝑀𝛼 𝑀𝛼 PROPN cana-2410 135 5 = = PUNCT cana-2410 135 6 max max PROPN cana-2410 135 7 1β‰€π‘˜β‰€π‘š 1β‰€π‘˜β‰€π‘š NUM cana-2410 135 8 { { PUNCT cana-2410 135 9 𝑀𝛼1 𝑀𝛼1 NOUN cana-2410 135 10 0 0 NUM cana-2410 135 11 , , PUNCT cana-2410 135 12 𝑀𝛼1 𝑀𝛼1 NOUN cana-2410 135 13 𝑗 𝑗 INTJ cana-2410 135 14 , , PUNCT cana-2410 135 15 𝑀[𝐼𝑛⨂𝑅𝑗 𝑀[𝐼𝑛⨂𝑅𝑗 PROPN cana-2410 135 16 ] ] X cana-2410 135 17 } } PUNCT cana-2410 135 18 < < X cana-2410 135 19 1 1 NUM cana-2410 135 20 , , PUNCT cana-2410 135 21 were be AUX cana-2410 135 22 𝑀𝛼1 𝑀𝛼1 ADV cana-2410 135 23 0 0 X cana-2410 136 1 = = SYM cana-2410 136 2 πΏπ‘€π‘˜π‘€π‘“π‘‘1(1 πΏπ‘€π‘˜π‘€π‘“π‘‘1(1 ADJ cana-2410 136 3 + + CCONJ cana-2410 136 4 𝐿 𝐿 PROPN cana-2410 136 5 2𝐿𝑄 2𝐿𝑄 NUM cana-2410 136 6 2 2 NUM cana-2410 136 7 𝑑1𝛿0 𝑑1𝛿0 NOUN cana-2410 136 8 ) ) PUNCT cana-2410 136 9 , , PUNCT cana-2410 136 10 𝑀𝛼1 𝑀𝛼1 NOUN cana-2410 136 11 𝑗 𝑗 X cana-2410 136 12 = = SYM cana-2410 136 13 𝐿𝑀[𝐼𝑛⨂𝑅𝑗 𝐿𝑀[𝐼𝑛⨂𝑅𝑗 NUM cana-2410 136 14 ] ] X cana-2410 136 15 + + CCONJ cana-2410 136 16 𝐿2𝐿𝑄 𝐿2𝐿𝑄 NUM cana-2410 136 17 2 2 NUM cana-2410 136 18 𝑀[𝐼𝑛⨂𝑅𝑗 𝑀[𝐼𝑛⨂𝑅𝑗 PROPN cana-2410 136 19 ] ] X cana-2410 136 20 𝑇𝛿𝑗 𝑇𝛿𝑗 PROPN cana-2410 136 21 + + CCONJ cana-2410 136 22 πΏπ‘€π‘˜π‘€π‘“π‘‡(1 πΏπ‘€π‘˜π‘€π‘“π‘‡(1 PROPN cana-2410 136 23 + + CCONJ cana-2410 136 24 𝐿 𝐿 PROPN cana-2410 136 25 2𝐿𝑄 2𝐿𝑄 NUM cana-2410 136 26 2 2 NUM cana-2410 136 27 𝑇𝛿𝑗 𝑇𝛿𝑗 PROPN cana-2410 136 28 ) ) PUNCT cana-2410 136 29 , , PUNCT cana-2410 136 30 𝑗 𝑗 NOUN cana-2410 136 31 = = SYM cana-2410 136 32 1,2 1,2 NUM cana-2410 136 33 , , PUNCT cana-2410 136 34 … … PUNCT cana-2410 136 35 , , PUNCT cana-2410 136 36 π‘š π‘š X cana-2410 136 37 for for ADP cana-2410 136 38 notational notational ADJ cana-2410 136 39 accommodation accommodation NOUN cana-2410 136 40 , , PUNCT cana-2410 136 41 we we PRON cana-2410 136 42 get get VERB cana-2410 136 43 𝛿0 𝛿0 NOUN cana-2410 136 44 = = SYM cana-2410 136 45 ‖𝒩0 ‖𝒩0 PUNCT cana-2410 136 46 βˆ’1(𝑑0 βˆ’1(𝑑0 X cana-2410 136 47 , , PUNCT cana-2410 136 48 𝑑1)β€– 𝑑1)β€– NOUN cana-2410 136 49 , , PUNCT cana-2410 136 50 𝛿𝑗 𝛿𝑗 PRON cana-2410 136 51 = = PUNCT cana-2410 136 52 ‖𝒩𝑗 ‖𝒩𝑗 ADJ cana-2410 136 53 βˆ’1(𝑠𝑗 βˆ’1(𝑠𝑗 NOUN cana-2410 136 54 , , PUNCT cana-2410 136 55 𝑑𝑗+1)β€–. 𝑑𝑗+1)β€–. VERB cana-2410 136 56 𝐿 𝐿 PROPN cana-2410 136 57 = = PUNCT cana-2410 136 58 ψ(𝑑 ψ(𝑑 PROPN cana-2410 136 59 , , PUNCT cana-2410 136 60 𝑠)(𝑑,𝑠)∈𝐼 𝑠)(𝑑,𝑠)∈𝐼 VERB cana-2410 136 61 π‘šπ‘Žπ‘₯ π‘šπ‘Žπ‘₯ ADJ cana-2410 136 62 , , PUNCT cana-2410 136 63 𝐿𝑄 𝐿𝑄 NOUN cana-2410 136 64 = = SYM cana-2410 136 65 ‖𝑄(t)β€–π‘‘βˆˆπΌ ‖𝑄(t)β€–π‘‘βˆˆπΌ NOUN cana-2410 136 66 π‘šπ‘Žπ‘₯ π‘šπ‘Žπ‘₯ NOUN cana-2410 136 67 , , PUNCT cana-2410 136 68 𝐿𝐾 𝐿𝐾 PROPN cana-2410 136 69 = = SYM cana-2410 136 70 ‖𝐾(t ‖𝐾(t PROPN cana-2410 136 71 , , PUNCT cana-2410 136 72 s)β€–π‘‘βˆˆπΌ s)β€–π‘‘βˆˆπΌ VERB cana-2410 136 73 π‘šπ‘Žπ‘₯ π‘šπ‘Žπ‘₯ PRON cana-2410 136 74 β„‹0 β„‹0 NOUN cana-2410 137 1 = = PUNCT cana-2410 138 1 𝐿‖𝑧0β€– 𝐿‖𝑧0β€– PROPN cana-2410 139 1 + + CCONJ cana-2410 140 1 𝐿𝐿𝐾𝑑1 𝐿𝐿𝐾𝑑1 VERB cana-2410 140 2 + + PUNCT cana-2410 140 3 𝐿𝐿𝑓𝑑1 𝐿𝐿𝑓𝑑1 X cana-2410 140 4 + + CCONJ cana-2410 140 5 πΏπΏπ‘„πΏπ‘ˆ πΏπΏπ‘„πΏπ‘ˆ PROPN cana-2410 140 6 0 0 NUM cana-2410 140 7 𝑑1 𝑑1 NOUN cana-2410 140 8 . . PUNCT cana-2410 141 1 πΏπ‘ˆ πΏπ‘ˆ PROPN cana-2410 141 2 0 0 NUM cana-2410 141 3 = = SYM cana-2410 141 4 𝐿𝐿𝑄𝛿0(‖𝑧0β€– 𝐿𝐿𝑄𝛿0(‖𝑧0β€– PROPN cana-2410 141 5 + + NOUN cana-2410 141 6 𝐿‖𝑧𝑑1β€– 𝐿‖𝑧𝑑1β€– PROPN cana-2410 142 1 + + CCONJ cana-2410 142 2 𝐿𝐿𝐾𝑑1 𝐿𝐿𝐾𝑑1 VERB cana-2410 142 3 + + ADJ cana-2410 142 4 𝐿𝐿𝑓𝑑1 𝐿𝐿𝑓𝑑1 NUM cana-2410 142 5 ) ) PUNCT cana-2410 142 6 . . PUNCT cana-2410 143 1 β„‹1 β„‹1 PROPN cana-2410 143 2 𝑗 𝑗 PROPN cana-2410 143 3 = = SYM cana-2410 143 4 𝐿𝐿[𝐼𝑛⨂𝑅𝑗 𝐿𝐿[𝐼𝑛⨂𝑅𝑗 PROPN cana-2410 143 5 ] ] X cana-2410 144 1 + + CCONJ cana-2410 144 2 𝐿𝐿𝐾𝑇 𝐿𝐿𝐾𝑇 PROPN cana-2410 144 3 + + CCONJ cana-2410 144 4 𝐿𝐿𝑓𝑇 𝐿𝐿𝑓𝑇 PROPN cana-2410 144 5 + + CCONJ cana-2410 144 6 πΏπΏπ‘„πΏπ‘ˆ πΏπΏπ‘„πΏπ‘ˆ PROPN cana-2410 144 7 𝑇 𝑇 PROPN cana-2410 144 8 𝑇. 𝑇. PROPN cana-2410 144 9 𝐿 𝐿 PROPN cana-2410 144 10 π‘ˆ π‘ˆ PROPN cana-2410 144 11 𝑗 𝑗 NOUN cana-2410 144 12 = = NOUN cana-2410 144 13 𝐿𝐿𝑄𝛿0 𝐿𝐿𝑄𝛿0 NOUN cana-2410 144 14 ( ( PUNCT cana-2410 144 15 𝐿[𝐼𝑛⨂𝑅𝑗 𝐿[𝐼𝑛⨂𝑅𝑗 PROPN cana-2410 144 16 ] ] PUNCT cana-2410 144 17 + + CCONJ cana-2410 144 18 𝐿‖𝑧𝑑𝑗+1β€– 𝐿‖𝑧𝑑𝑗+1β€– NOUN cana-2410 144 19 + + CCONJ cana-2410 144 20 𝐿𝐿𝐾𝑑𝑗+1 𝐿𝐿𝐾𝑑𝑗+1 ADP cana-2410 144 21 + + NUM cana-2410 144 22 𝐿𝐿𝑓𝑑𝑗+1 𝐿𝐿𝑓𝑑𝑗+1 NOUN cana-2410 144 23 ) ) PUNCT cana-2410 144 24 . . PUNCT cana-2410 145 1 𝛾 𝛾 PROPN cana-2410 145 2 β‰₯ β‰₯ NOUN cana-2410 145 3 max max PROPN cana-2410 145 4 1β‰€π‘˜β‰€π‘š 1β‰€π‘˜β‰€π‘š NUM cana-2410 145 5 { { PUNCT cana-2410 145 6 β„‹0,β„‹1 β„‹0,β„‹1 NOUN cana-2410 145 7 𝑗 𝑗 NOUN cana-2410 145 8 , , PUNCT cana-2410 145 9 𝐿[𝐼𝑛⨂𝑅𝑗 𝐿[𝐼𝑛⨂𝑅𝑗 PROPN cana-2410 145 10 ] ] PUNCT cana-2410 145 11 } } PUNCT cana-2410 145 12 . . PUNCT cana-2410 146 1 communications communication NOUN cana-2410 146 2 on on ADP cana-2410 146 3 applied apply VERB cana-2410 146 4 nonlinear nonlinear ADJ cana-2410 146 5 analysis analysis NOUN cana-2410 146 6 issn issn NOUN cana-2410 146 7 : : PUNCT cana-2410 146 8 1074 1074 NUM cana-2410 146 9 - - PUNCT cana-2410 146 10 133x 133x NUM cana-2410 146 11 vol vol NOUN cana-2410 146 12 32 32 NUM cana-2410 146 13 no no NOUN cana-2410 146 14 . . PUNCT cana-2410 147 1 2s 2s NUM cana-2410 147 2 ( ( PUNCT cana-2410 147 3 2025 2025 NUM cana-2410 147 4 ) ) PUNCT cana-2410 147 5 356 356 NUM cana-2410 147 6 https://internationalpubls.com https://internationalpubls.com X cana-2410 147 7 theorem theorem VERB cana-2410 147 8 3.1 3.1 NUM cana-2410 147 9 : : PUNCT cana-2410 147 10 assuming assume VERB cana-2410 147 11 that that SCONJ cana-2410 147 12 requirements requirement NOUN cana-2410 147 13 ( ( PUNCT cana-2410 147 14 h1 h1 PROPN cana-2410 147 15 ) ) PUNCT cana-2410 147 16 ( ( PUNCT cana-2410 147 17 h3 h3 NOUN cana-2410 147 18 ) ) PUNCT cana-2410 147 19 are be AUX cana-2410 147 20 satisfied satisfied ADJ cana-2410 147 21 , , PUNCT cana-2410 147 22 then then ADV cana-2410 147 23 there there PRON cana-2410 147 24 is be VERB cana-2410 147 25 only only ADV cana-2410 147 26 one one NUM cana-2410 147 27 solution solution NOUN cana-2410 147 28 to to ADP cana-2410 147 29 system system NOUN cana-2410 147 30 ( ( PUNCT cana-2410 147 31 3.1 3.1 NUM cana-2410 147 32 ) ) PUNCT cana-2410 147 33 . . PUNCT cana-2410 148 1 proof proof NOUN cana-2410 148 2 : : PUNCT cana-2410 148 3 the the DET cana-2410 148 4 subset subset NOUN cana-2410 148 5 π’Ÿ π’Ÿ NOUN cana-2410 148 6 βŠ† βŠ† NUM cana-2410 148 7 𝑃𝐢 𝑃𝐢 PROPN cana-2410 148 8 is be AUX cana-2410 148 9 defined define VERB cana-2410 148 10 as as ADP cana-2410 148 11 the the DET cana-2410 148 12 set set NOUN cana-2410 148 13 π’Ÿ π’Ÿ NOUN cana-2410 148 14 = = PUNCT cana-2410 148 15 { { PUNCT cana-2410 148 16 𝑧 𝑧 PRON cana-2410 148 17 ∈ ∈ PROPN cana-2410 148 18 𝑃𝐢 𝑃𝐢 PROPN cana-2410 148 19 : : PUNCT cana-2410 148 20 ‖𝑧‖𝑃𝐢 ‖𝑧‖𝑃𝐢 PROPN cana-2410 148 21 ≀ ≀ PROPN cana-2410 148 22 𝛾 𝛾 ADP cana-2410 148 23 } } PUNCT cana-2410 148 24 . . PUNCT cana-2410 149 1 currently currently ADV cana-2410 149 2 , , PUNCT cana-2410 149 3 we we PRON cana-2410 149 4 are be AUX cana-2410 149 5 defining define VERB cana-2410 149 6 the the DET cana-2410 149 7 function function NOUN cana-2410 149 8 𝒒:π’Ÿ 𝒒:π’Ÿ PROPN cana-2410 149 9 ⟢ ⟢ PROPN cana-2410 149 10 π’Ÿ π’Ÿ PROPN cana-2410 149 11 , , PUNCT cana-2410 149 12 which which PRON cana-2410 149 13 means mean VERB cana-2410 149 14 that that SCONJ cana-2410 149 15 for for ADP cana-2410 149 16 , , PUNCT cana-2410 149 17 𝑑 𝑑 PROPN cana-2410 149 18 ∈ ∈ PROPN cana-2410 150 1 [ [ X cana-2410 150 2 0 0 NUM cana-2410 150 3 , , PUNCT cana-2410 150 4 𝑑1]𝕋 𝑑1]𝕋 PROPN cana-2410 150 5 ( ( PUNCT cana-2410 150 6 𝒒𝑧)(𝑑 𝒒𝑧)(𝑑 PROPN cana-2410 150 7 ) ) PUNCT cana-2410 150 8 = = SYM cana-2410 150 9 ψ(𝑑 ψ(𝑑 PROPN cana-2410 150 10 , , PUNCT cana-2410 150 11 𝑑0)𝑧0 𝑑0)𝑧0 VERB cana-2410 150 12 + + NOUN cana-2410 150 13 ∫ ∫ PROPN cana-2410 150 14 ψ(𝑑 ψ(𝑑 NOUN cana-2410 150 15 , , PUNCT cana-2410 150 16 𝜎(𝑠))𝐾(𝑑 𝜎(𝑠))𝐾(𝑑 NOUN cana-2410 150 17 , , PUNCT cana-2410 150 18 𝑠)𝑧(𝑠)βˆ†π‘  𝑠)𝑧(𝑠)βˆ†π‘  PROPN cana-2410 150 19 𝑑 𝑑 PROPN cana-2410 150 20 𝑑0 𝑑0 PROPN cana-2410 150 21 + + PROPN cana-2410 150 22 ∫ ∫ PROPN cana-2410 150 23 ψ(𝑑 ψ(𝑑 PROPN cana-2410 150 24 , , PUNCT cana-2410 150 25 𝜎(𝜏))(𝑓(𝜏 𝜎(𝜏))(𝑓(𝜏 NUM cana-2410 150 26 , , PUNCT cana-2410 150 27 𝑧(𝜏 𝑧(𝜏 PROPN cana-2410 150 28 ) ) PUNCT cana-2410 150 29 ) ) PUNCT cana-2410 151 1 + + CCONJ cana-2410 151 2 𝑄(𝜏))βˆ†πœ. 𝑄(𝜏))βˆ†πœ. PUNCT cana-2410 151 3 𝑑 𝑑 PROPN cana-2410 151 4 𝑑0 𝑑0 PROPN cana-2410 151 5 ( ( PUNCT cana-2410 151 6 3.8 3.8 NUM cana-2410 151 7 ) ) PUNCT cana-2410 151 8 for for ADP cana-2410 151 9 𝑑 𝑑 PROPN cana-2410 151 10 ∈ ∈ PROPN cana-2410 151 11 ( ( PUNCT cana-2410 151 12 𝑑𝑗 𝑑𝑗 PROPN cana-2410 151 13 , , PUNCT cana-2410 151 14 𝑠𝑗]𝕋 𝑠𝑗]𝕋 PROPN cana-2410 151 15 , , PUNCT cana-2410 151 16 𝑗 𝑗 NOUN cana-2410 151 17 = = SYM cana-2410 151 18 1,2 1,2 NUM cana-2410 151 19 , , PUNCT cana-2410 151 20 … … PUNCT cana-2410 151 21 , , PUNCT cana-2410 151 22 π‘š π‘š X cana-2410 151 23 ( ( PUNCT cana-2410 151 24 𝒒𝑧)(𝑑 𝒒𝑧)(𝑑 PROPN cana-2410 151 25 ) ) PUNCT cana-2410 151 26 = = PUNCT cana-2410 152 1 [ [ X cana-2410 152 2 𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 X cana-2410 152 3 βˆ’ βˆ’ NOUN cana-2410 152 4 ) ) PUNCT cana-2410 152 5 , , PUNCT cana-2410 152 6 ( ( PUNCT cana-2410 152 7 3.8 3.8 NUM cana-2410 152 8 ) ) PUNCT cana-2410 152 9 for for ADP cana-2410 152 10 βˆ€π‘‘ βˆ€π‘‘ PROPN cana-2410 152 11 ∈ ∈ PROPN cana-2410 152 12 ( ( PUNCT cana-2410 152 13 𝑠𝑗 𝑠𝑗 NOUN cana-2410 152 14 , , PUNCT cana-2410 152 15 𝑑𝑗+1]𝕋 𝑑𝑗+1]𝕋 NOUN cana-2410 152 16 , , PUNCT cana-2410 152 17 𝑗 𝑗 NOUN cana-2410 152 18 = = SYM cana-2410 152 19 1,2 1,2 NUM cana-2410 152 20 , , PUNCT cana-2410 152 21 … … PUNCT cana-2410 152 22 , , PUNCT cana-2410 152 23 π‘š π‘š X cana-2410 152 24 ( ( PUNCT cana-2410 152 25 𝒒𝑧)(𝑑 𝒒𝑧)(𝑑 PROPN cana-2410 152 26 ) ) PUNCT cana-2410 152 27 = = SYM cana-2410 152 28 ψ(𝑑 ψ(𝑑 PROPN cana-2410 152 29 , , PUNCT cana-2410 152 30 𝑠𝑗)[𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 𝑠𝑗)[𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 X cana-2410 152 31 βˆ’ βˆ’ NOUN cana-2410 152 32 ) ) PUNCT cana-2410 153 1 + + CCONJ cana-2410 153 2 ∫ ∫ PROPN cana-2410 153 3 ψ(𝑑 ψ(𝑑 NOUN cana-2410 153 4 , , PUNCT cana-2410 153 5 𝜎(𝑠))𝐾(𝑑 𝜎(𝑠))𝐾(𝑑 NOUN cana-2410 153 6 , , PUNCT cana-2410 153 7 𝑠)𝑧(𝑠)βˆ†π‘  𝑠)𝑧(𝑠)βˆ†π‘  PROPN cana-2410 153 8 𝑑 𝑑 PROPN cana-2410 153 9 𝑠𝑗 𝑠𝑗 PROPN cana-2410 153 10 + + NOUN cana-2410 153 11 ∫ ∫ PROPN cana-2410 153 12 ψ(𝑑 ψ(𝑑 PROPN cana-2410 153 13 , , PUNCT cana-2410 153 14 𝜎(𝜏 𝜎(𝜏 PROPN cana-2410 153 15 ) ) PUNCT cana-2410 153 16 ) ) PUNCT cana-2410 153 17 ( ( PUNCT cana-2410 153 18 𝑓(𝜏 𝑓(𝜏 PROPN cana-2410 153 19 , , PUNCT cana-2410 153 20 𝑧(𝜏 𝑧(𝜏 PROPN cana-2410 153 21 ) ) PUNCT cana-2410 153 22 ) ) PUNCT cana-2410 154 1 + + CCONJ cana-2410 154 2 𝑄(𝜏) 𝑄(𝜏) NUM cana-2410 154 3 οΏ½ οΏ½ NOUN cana-2410 154 4 Μ‚ Μ‚ NOUN cana-2410 154 5 οΏ½ οΏ½ NOUN cana-2410 154 6 (𝜏 (𝜏 VERB cana-2410 154 7 ) ) PUNCT cana-2410 154 8 ) ) PUNCT cana-2410 154 9 βˆ†πœ. βˆ†πœ. PROPN cana-2410 154 10 𝑑 𝑑 SYM cana-2410 154 11 𝑠𝑗 𝑠𝑗 PROPN cana-2410 154 12 ( ( PUNCT cana-2410 154 13 3.9 3.9 NUM cana-2410 154 14 ) ) PUNCT cana-2410 154 15 it it PRON cana-2410 154 16 is be AUX cana-2410 154 17 evident evident ADJ cana-2410 154 18 that that SCONJ cana-2410 154 19 the the DET cana-2410 154 20 solution solution NOUN cana-2410 154 21 is be AUX cana-2410 154 22 the the DET cana-2410 154 23 banach banach ADV cana-2410 154 24 fixed fix VERB cana-2410 154 25 point point NOUN cana-2410 154 26 for for ADP cana-2410 154 27 𝒒. 𝒒. PROPN cana-2410 154 28 let let VERB cana-2410 154 29 us we PRON cana-2410 154 30 now now ADV cana-2410 154 31 consider consider VERB cana-2410 154 32 𝑑 𝑑 NOUN cana-2410 154 33 ∈ ∈ PROPN cana-2410 154 34 ( ( PUNCT cana-2410 154 35 𝑠𝑗 𝑠𝑗 INTJ cana-2410 154 36 , , PUNCT cana-2410 154 37 𝑑𝑗+1]𝕋 𝑑𝑗+1]𝕋 ADV cana-2410 154 38 , , PUNCT cana-2410 154 39 𝑗 𝑗 NOUN cana-2410 154 40 = = SYM cana-2410 154 41 1,2 1,2 NUM cana-2410 154 42 , , PUNCT cana-2410 154 43 … … PUNCT cana-2410 154 44 , , PUNCT cana-2410 154 45 π‘š π‘š NOUN cana-2410 154 46 , , PUNCT cana-2410 154 47 and and CCONJ cana-2410 154 48 𝑧 𝑧 DET cana-2410 154 49 ∈ ∈ PROPN cana-2410 154 50 π’Ÿ π’Ÿ PROPN cana-2410 154 51 , , PUNCT cana-2410 154 52 we we PRON cana-2410 154 53 obtain obtain VERB cana-2410 154 54 β€–(𝒒𝑧)(𝑑)β€– β€–(𝒒𝑧)(𝑑)β€– ADJ cana-2410 154 55 ≀ ≀ NUM cana-2410 154 56 β€–Οˆ(𝑑 β€–Οˆ(𝑑 NOUN cana-2410 154 57 , , PUNCT cana-2410 154 58 𝑠𝑗)β€–β€–[𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 𝑠𝑗)β€–β€–[𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 PRON cana-2410 154 59 βˆ’)β€– βˆ’)β€– ADJ cana-2410 155 1 + + ADJ cana-2410 155 2 ∫ ∫ NOUN cana-2410 155 3 β€–Οˆ(𝑑 β€–Οˆ(𝑑 NUM cana-2410 155 4 , , PUNCT cana-2410 155 5 𝜎(𝑠))‖‖𝐾(𝑑 𝜎(𝑠))‖‖𝐾(𝑑 NUM cana-2410 155 6 , , PUNCT cana-2410 155 7 𝑠)‖‖𝑧(𝑠)β€–βˆ†π‘  𝑠)‖‖𝑧(𝑠)β€–βˆ†π‘  NOUN cana-2410 155 8 𝑑 𝑑 PROPN cana-2410 155 9 𝑠𝑗 𝑠𝑗 PROPN cana-2410 155 10 + + NOUN cana-2410 155 11 ∫ ∫ PROPN cana-2410 155 12 β€–Οˆ(𝑑,𝑑(𝑑))‖‖𝑑(𝑑,𝑑(𝑑))β€–βˆ†π‘‘ β€–Οˆ(𝑑,𝑑(𝑑))‖‖𝑑(𝑑,𝑑(𝑑))β€–βˆ†π‘‘ CCONJ cana-2410 155 13 𝑑 𝑑 PROPN cana-2410 155 14 𝑑𝑑 𝑑𝑑 PROPN cana-2410 155 15 + + NOUN cana-2410 155 16 ∫ ∫ NOUN cana-2410 155 17 β€–Οˆ(𝑑 β€–Οˆ(𝑑 NOUN cana-2410 155 18 , , PUNCT cana-2410 155 19 𝜎(𝜏))β€–β€–uΜ‚(𝜏)‖‖𝑄(𝜏)β€– 𝜎(𝜏))β€–β€–uΜ‚(𝜏)‖‖𝑄(𝜏)β€– ADJ cana-2410 155 20 𝑑 𝑑 SYM cana-2410 155 21 𝑠𝑗 𝑠𝑗 INTJ cana-2410 155 22 βˆ†πœ βˆ†πœ NOUN cana-2410 155 23 ( ( PUNCT cana-2410 155 24 3.10 3.10 NUM cana-2410 155 25 ) ) PUNCT cana-2410 155 26 ≀ ≀ NOUN cana-2410 155 27 𝐿𝐿[𝐼𝑛⨂𝑅𝑗 𝐿𝐿[𝐼𝑛⨂𝑅𝑗 PROPN cana-2410 155 28 ] ] X cana-2410 156 1 + + CCONJ cana-2410 156 2 𝐿𝐿𝐾𝑑𝑗+1 𝐿𝐿𝐾𝑑𝑗+1 ADP cana-2410 156 3 + + CCONJ cana-2410 156 4 𝐿𝐿𝑓𝑑𝑗+1 𝐿𝐿𝑓𝑑𝑗+1 NOUN cana-2410 156 5 + + CCONJ cana-2410 156 6 πΏπΏπ‘„πΏπ‘ˆ πΏπΏπ‘„πΏπ‘ˆ PROPN cana-2410 156 7 𝑗 𝑗 VERB cana-2410 156 8 𝑑𝑗+1 𝑑𝑗+1 PROPN cana-2410 156 9 ≀ ≀ PUNCT cana-2410 156 10 β„‹1 β„‹1 NOUN cana-2410 156 11 𝑗 𝑗 PRON cana-2410 156 12 ≀ ≀ X cana-2410 156 13 𝛾. 𝛾. ADV cana-2410 156 14 similarly similarly ADV cana-2410 156 15 , , PUNCT cana-2410 156 16 for for ADP cana-2410 156 17 𝑑 𝑑 PROPN cana-2410 156 18 ∈ ∈ PROPN cana-2410 157 1 [ [ X cana-2410 157 2 0 0 NUM cana-2410 157 3 , , PUNCT cana-2410 157 4 𝑑1]𝕋 𝑑1]𝕋 PRON cana-2410 157 5 and and CCONJ cana-2410 157 6 𝑧 𝑧 DET cana-2410 157 7 ∈ ∈ PROPN cana-2410 157 8 π’Ÿ π’Ÿ PROPN cana-2410 157 9 , , PUNCT cana-2410 157 10 then then ADV cana-2410 157 11 communications communication NOUN cana-2410 157 12 on on ADP cana-2410 157 13 applied apply VERB cana-2410 157 14 nonlinear nonlinear ADJ cana-2410 157 15 analysis analysis NOUN cana-2410 157 16 issn issn NOUN cana-2410 157 17 : : PUNCT cana-2410 157 18 1074 1074 NUM cana-2410 157 19 - - PUNCT cana-2410 157 20 133x 133x NUM cana-2410 157 21 vol vol NOUN cana-2410 157 22 32 32 NUM cana-2410 157 23 no no NOUN cana-2410 157 24 . . PUNCT cana-2410 158 1 2s 2s NUM cana-2410 158 2 ( ( PUNCT cana-2410 158 3 2025 2025 NUM cana-2410 158 4 ) ) PUNCT cana-2410 158 5 357 357 NUM cana-2410 158 6 https://internationalpubls.com https://internationalpubls.com X cana-2410 158 7 β€–(𝒒𝑧)(𝑑)β€– β€–(𝒒𝑧)(𝑑)β€– ADJ cana-2410 158 8 ≀ ≀ NUM cana-2410 158 9 β€–Οˆ(𝑑 β€–Οˆ(𝑑 NOUN cana-2410 158 10 , , PUNCT cana-2410 158 11 𝑑0)‖‖𝑧0β€– 𝑑0)‖‖𝑧0β€– X cana-2410 158 12 + + X cana-2410 158 13 ∫ ∫ PROPN cana-2410 158 14 β€–Οˆ(𝑑 β€–Οˆ(𝑑 NUM cana-2410 158 15 , , PUNCT cana-2410 158 16 𝜎(𝑠))‖‖𝐾(𝑑 𝜎(𝑠))‖‖𝐾(𝑑 NUM cana-2410 158 17 , , PUNCT cana-2410 158 18 𝑠)‖‖𝑧(𝑠)β€–βˆ†π‘  𝑠)‖‖𝑧(𝑠)β€–βˆ†π‘  PROPN cana-2410 158 19 𝑑 𝑑 PROPN cana-2410 158 20 𝑑0 𝑑0 PROPN cana-2410 158 21 + + CCONJ cana-2410 158 22 ∫ ∫ PROPN cana-2410 158 23 β€–Οˆ(𝑑 β€–Οˆ(𝑑 NOUN cana-2410 158 24 , , PUNCT cana-2410 158 25 𝜎(𝜏))‖‖𝑓(𝜏 𝜎(𝜏))‖‖𝑓(𝜏 VERB cana-2410 158 26 , , PUNCT cana-2410 158 27 𝑧(𝜏))β€–βˆ†πœ 𝑧(𝜏))β€–βˆ†πœ PROPN cana-2410 158 28 𝑑 𝑑 PROPN cana-2410 158 29 𝑑0 𝑑0 PROPN cana-2410 158 30 + + CCONJ cana-2410 158 31 ∫ ∫ PROPN cana-2410 158 32 β€–Οˆ(𝑑 β€–Οˆ(𝑑 NOUN cana-2410 158 33 , , PUNCT cana-2410 158 34 𝜎(𝜏))β€–β€–uΜ‚(𝜏)‖‖𝑄(𝜏)β€– 𝜎(𝜏))β€–β€–uΜ‚(𝜏)‖‖𝑄(𝜏)β€– ADJ cana-2410 158 35 𝑑 𝑑 PROPN cana-2410 158 36 𝑑0 𝑑0 PROPN cana-2410 158 37 βˆ†πœ βˆ†πœ NOUN cana-2410 158 38 ( ( PUNCT cana-2410 158 39 3.11 3.11 NUM cana-2410 158 40 ) ) PUNCT cana-2410 158 41 ≀ ≀ NOUN cana-2410 158 42 𝐿‖𝑧0β€– 𝐿‖𝑧0β€– PROPN cana-2410 158 43 + + CCONJ cana-2410 158 44 𝐿𝐿𝐾𝑑 𝐿𝐿𝐾𝑑 NOUN cana-2410 158 45 + + CCONJ cana-2410 158 46 𝐿𝐿𝑓𝑑 𝐿𝐿𝑓𝑑 PROPN cana-2410 159 1 + + CCONJ cana-2410 159 2 πΏπΏπ‘„πΏπ‘ˆ πΏπΏπ‘„πΏπ‘ˆ PROPN cana-2410 159 3 0 0 PUNCT cana-2410 159 4 𝑑 𝑑 PROPN cana-2410 159 5 ≀ ≀ PROPN cana-2410 159 6 β„‹0 β„‹0 PROPN cana-2410 159 7 ≀ ≀ PROPN cana-2410 159 8 𝛾. 𝛾. ADV cana-2410 159 9 similarly similarly ADV cana-2410 159 10 , , PUNCT cana-2410 159 11 for for ADP cana-2410 159 12 𝑑 𝑑 PROPN cana-2410 159 13 ∈ ∈ PROPN cana-2410 159 14 ( ( PUNCT cana-2410 159 15 𝑠𝑗 𝑠𝑗 NOUN cana-2410 159 16 , , PUNCT cana-2410 159 17 𝑑𝑗]𝕋 𝑑𝑗]𝕋 NOUN cana-2410 159 18 , , PUNCT cana-2410 159 19 and and CCONJ cana-2410 159 20 𝑧 𝑧 DET cana-2410 159 21 ∈ ∈ PROPN cana-2410 159 22 π’Ÿ π’Ÿ PROPN cana-2410 159 23 , , PUNCT cana-2410 159 24 we we PRON cana-2410 159 25 get get VERB cana-2410 159 26 β€–(𝒒𝑧)‖𝑃𝐢 β€–(𝒒𝑧)‖𝑃𝐢 PRON cana-2410 159 27 ≀ ≀ NUM cana-2410 159 28 𝐿[𝐼𝑛⨂𝑅𝑗 𝐿[𝐼𝑛⨂𝑅𝑗 PROPN cana-2410 159 29 ] ] PUNCT cana-2410 159 30 ≀ ≀ NUM cana-2410 159 31 𝛾. 𝛾. NOUN cana-2410 159 32 ( ( PUNCT cana-2410 159 33 3.12 3.12 NUM cana-2410 159 34 ) ) PUNCT cana-2410 159 35 after after ADP cana-2410 159 36 succinct succinct ADJ cana-2410 159 37 the the DET cana-2410 159 38 above above ADJ cana-2410 159 39 inequalities inequality NOUN cana-2410 159 40 ( ( PUNCT cana-2410 159 41 3.10 3.10 NUM cana-2410 159 42 ) ) PUNCT cana-2410 159 43 ( ( PUNCT cana-2410 159 44 3.12 3.12 NUM cana-2410 159 45 ) ) PUNCT cana-2410 159 46 , , PUNCT cana-2410 159 47 we we PRON cana-2410 159 48 have have VERB cana-2410 159 49 β€–(𝒒𝑧)‖𝑃𝐢 β€–(𝒒𝑧)‖𝑃𝐢 NOUN cana-2410 159 50 ≀ ≀ NOUN cana-2410 159 51 𝛾. 𝛾. ADV cana-2410 159 52 since since ADV cana-2410 159 53 , , PUNCT cana-2410 159 54 𝒒 𝒒 PROPN cana-2410 159 55 : : PUNCT cana-2410 159 56 π’Ÿ π’Ÿ PROPN cana-2410 159 57 ⟢ ⟢ PROPN cana-2410 159 58 π’Ÿ π’Ÿ PROPN cana-2410 159 59 , , PUNCT cana-2410 159 60 for for ADP cana-2410 159 61 any any DET cana-2410 159 62 𝑧 𝑧 NOUN cana-2410 159 63 , , PUNCT cana-2410 159 64 π‘₯ π‘₯ PROPN cana-2410 159 65 ∈ ∈ PROPN cana-2410 159 66 π’Ÿ π’Ÿ PROPN cana-2410 159 67 , , PUNCT cana-2410 159 68 𝑑 𝑑 PROPN cana-2410 159 69 ∈ ∈ PROPN cana-2410 159 70 ( ( PUNCT cana-2410 159 71 𝑠𝑗 𝑠𝑗 INTJ cana-2410 159 72 , , PUNCT cana-2410 159 73 𝑑𝑗+1]𝕋 𝑑𝑗+1]𝕋 ADV cana-2410 159 74 , , PUNCT cana-2410 159 75 𝑗 𝑗 NOUN cana-2410 159 76 = = SYM cana-2410 159 77 1,2 1,2 NUM cana-2410 159 78 , , PUNCT cana-2410 159 79 … … PUNCT cana-2410 159 80 , , PUNCT cana-2410 159 81 π‘š π‘š X cana-2410 159 82 , , PUNCT cana-2410 159 83 we we PRON cana-2410 159 84 get get VERB cana-2410 159 85 β€–(𝒒𝑧)(𝑑 β€–(𝒒𝑧)(𝑑 ADJ cana-2410 159 86 ) ) PUNCT cana-2410 159 87 βˆ’ βˆ’ PROPN cana-2410 160 1 ( ( PUNCT cana-2410 160 2 𝒒π‘₯)(𝑑)β€– 𝒒π‘₯)(𝑑)β€– PROPN cana-2410 160 3 ≀ ≀ NOUN cana-2410 160 4 β€–Οˆ(𝑑 β€–Οˆ(𝑑 NOUN cana-2410 160 5 , , PUNCT cana-2410 160 6 𝑠𝑗)β€–β€–[𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 𝑠𝑗)β€–β€–[𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 ADJ cana-2410 160 7 βˆ’ βˆ’ NOUN cana-2410 160 8 ) ) PUNCT cana-2410 160 9 βˆ’ βˆ’ PROPN cana-2410 161 1 [ [ X cana-2410 161 2 𝐼𝑛⨂𝑅𝑗]π‘₯(𝑑𝑗 𝐼𝑛⨂𝑅𝑗]π‘₯(𝑑𝑗 X cana-2410 161 3 βˆ’)β€– βˆ’)β€– NUM cana-2410 161 4 + + NOUN cana-2410 161 5 ∫ ∫ NOUN cana-2410 161 6 β€–Οˆ(𝑑 β€–Οˆ(𝑑 NUM cana-2410 161 7 , , PUNCT cana-2410 161 8 𝜎(𝑠))‖‖𝐾(𝑑 𝜎(𝑠))‖‖𝐾(𝑑 NUM cana-2410 161 9 , , PUNCT cana-2410 161 10 𝑠)‖‖𝑧(𝑠 𝑠)‖‖𝑧(𝑠 NOUN cana-2410 161 11 ) ) PUNCT cana-2410 161 12 βˆ’ βˆ’ PROPN cana-2410 161 13 π‘₯(𝑠)β€–βˆ†π‘  π‘₯(𝑠)β€–βˆ†π‘  NUM cana-2410 161 14 𝑑 𝑑 NOUN cana-2410 161 15 𝑠𝑗 𝑠𝑗 INTJ cana-2410 161 16 + + NOUN cana-2410 161 17 ∫ ∫ NOUN cana-2410 161 18 β€–Οˆ(𝑑 β€–Οˆ(𝑑 NOUN cana-2410 161 19 , , PUNCT cana-2410 161 20 𝜎(𝜏))‖‖𝑓(𝜏 𝜎(𝜏))‖‖𝑓(𝜏 NUM cana-2410 161 21 , , PUNCT cana-2410 161 22 𝑧(𝜏 𝑧(𝜏 PROPN cana-2410 161 23 ) ) PUNCT cana-2410 161 24 ) ) PUNCT cana-2410 162 1 βˆ’ βˆ’ NOUN cana-2410 163 1 𝑓(𝜏 𝑓(𝜏 PROPN cana-2410 163 2 , , PUNCT cana-2410 163 3 π‘₯(𝜏))β€–βˆ†πœ π‘₯(𝜏))β€–βˆ†πœ PROPN cana-2410 163 4 𝑑 𝑑 PROPN cana-2410 163 5 𝑠𝑗 𝑠𝑗 PROPN cana-2410 163 6 + + NOUN cana-2410 163 7 ∫ ∫ PROPN cana-2410 163 8 [ [ X cana-2410 163 9 β€–Οˆ(𝑑 β€–Οˆ(𝑑 NOUN cana-2410 163 10 , , PUNCT cana-2410 163 11 𝜎(𝜏))‖‖𝑄(Ο„)β€–β€–π‘„βˆ—(𝜏)β€–β€–Οˆβˆ—(𝑑 𝜎(𝜏))‖‖𝑄(Ο„)β€–β€–π‘„βˆ—(𝜏)β€–β€–Οˆβˆ—(𝑑 PROPN cana-2410 163 12 , , PUNCT cana-2410 163 13 𝜎(𝜏))β€– 𝜎(𝜏))β€– PROPN cana-2410 163 14 𝑑 𝑑 NOUN cana-2410 163 15 𝑠𝑗 𝑠𝑗 INTJ cana-2410 163 16 Γ— Γ— NOUN cana-2410 163 17 ‖𝒩𝑗 ‖𝒩𝑗 ADJ cana-2410 163 18 βˆ’1(𝑠𝑗 βˆ’1(𝑠𝑗 NOUN cana-2410 163 19 , , PUNCT cana-2410 163 20 𝑑𝑗+1)β€–β€–[𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 𝑑𝑗+1)β€–β€–[𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 PROPN cana-2410 163 21 βˆ’ βˆ’ NOUN cana-2410 163 22 ) ) PUNCT cana-2410 163 23 βˆ’ βˆ’ PROPN cana-2410 164 1 [ [ X cana-2410 164 2 𝐼𝑛⨂𝑅𝑗]π‘₯(𝑑𝑗 𝐼𝑛⨂𝑅𝑗]π‘₯(𝑑𝑗 X cana-2410 164 3 βˆ’)β€– βˆ’)β€– PROPN cana-2410 164 4 + + NUM cana-2410 164 5 ∫ ∫ NOUN cana-2410 164 6 β€–Οˆ(𝑑,𝑑(𝑑))‖‖𝑑(𝑑,𝑑)‖‖𝑑(𝑑 β€–Οˆ(𝑑,𝑑(𝑑))‖‖𝑑(𝑑,𝑑)‖‖𝑑(𝑑 PRON cana-2410 164 7 ) ) PUNCT cana-2410 164 8 βˆ’π‘‘(𝑑)β€–βˆ†π‘‘ βˆ’π‘‘(𝑑)β€–βˆ†π‘‘ PUNCT cana-2410 164 9 𝑑𝑑+1 𝑑𝑑+1 VERB cana-2410 164 10 𝑑𝑑 𝑑𝑑 NOUN cana-2410 164 11 + + NOUN cana-2410 164 12 ∫ ∫ NOUN cana-2410 164 13 β€–Οˆ β€–Οˆ NOUN cana-2410 164 14 ( ( PUNCT cana-2410 164 15 𝑠𝑗 𝑠𝑗 INTJ cana-2410 164 16 , , PUNCT cana-2410 164 17 𝜎(𝑠))β€– 𝜎(𝑠))β€– PROPN cana-2410 164 18 ‖𝑓(𝑠 ‖𝑓(𝑠 NOUN cana-2410 164 19 , , PUNCT cana-2410 164 20 𝑧(𝑠 𝑧(𝑠 NOUN cana-2410 164 21 ) ) PUNCT cana-2410 164 22 ) ) PUNCT cana-2410 164 23 βˆ’ βˆ’ PROPN cana-2410 164 24 𝑓(𝑠 𝑓(𝑠 NOUN cana-2410 164 25 , , PUNCT cana-2410 164 26 π‘₯(𝑠))β€–βˆ†π‘  π‘₯(𝑠))β€–βˆ†π‘  PROPN cana-2410 164 27 𝑑𝑗+1 𝑑𝑗+1 NUM cana-2410 164 28 𝑠𝑗 𝑠𝑗 NOUN cana-2410 164 29 ] ] PUNCT cana-2410 164 30 βˆ†πœ βˆ†πœ PROPN cana-2410 164 31 ≀ ≀ NUM cana-2410 164 32 𝐿𝑀[𝐼𝑛⨂𝑅𝑗 𝐿𝑀[𝐼𝑛⨂𝑅𝑗 NUM cana-2410 164 33 ] ] X cana-2410 164 34 ‖𝑧(𝑑𝑗 ‖𝑧(𝑑𝑗 PROPN cana-2410 164 35 βˆ’ βˆ’ PROPN cana-2410 164 36 ) ) PUNCT cana-2410 164 37 βˆ’ βˆ’ PROPN cana-2410 164 38 π‘₯(𝑑𝑗 π‘₯(𝑑𝑗 NOUN cana-2410 164 39 βˆ’)β€– βˆ’)β€– NOUN cana-2410 165 1 + + PUNCT cana-2410 166 1 πΏπ‘€πΎβˆ« πΏπ‘€πΎβˆ« PROPN cana-2410 166 2 ‖𝑧(𝑠 ‖𝑧(𝑠 PUNCT cana-2410 166 3 ) ) PUNCT cana-2410 166 4 βˆ’ βˆ’ PROPN cana-2410 166 5 π‘₯(𝑠)β€–βˆ†π‘  π‘₯(𝑠)β€–βˆ†π‘  NUM cana-2410 166 6 𝑑 𝑑 NOUN cana-2410 166 7 𝑠𝑗 𝑠𝑗 ADP cana-2410 166 8 + + ADV cana-2410 166 9 πΏπ‘€π‘“βˆ« πΏπ‘€π‘“βˆ« NOUN cana-2410 166 10 ‖𝑧(𝜏 ‖𝑧(𝜏 PRON cana-2410 166 11 ) ) PUNCT cana-2410 166 12 βˆ’ βˆ’ PROPN cana-2410 167 1 π‘₯(𝜏)β€–βˆ†πœ π‘₯(𝜏)β€–βˆ†πœ PUNCT cana-2410 167 2 + + CCONJ cana-2410 167 3 𝐿2𝐿𝑄 𝐿2𝐿𝑄 NUM cana-2410 167 4 2 2 NUM cana-2410 167 5 𝛿𝑗 𝛿𝑗 PROPN cana-2410 167 6 𝑑 𝑑 X cana-2410 167 7 𝑠𝑗 𝑠𝑗 PRON cana-2410 167 8 Γ—βˆ« Γ—βˆ« VERB cana-2410 167 9 [ [ X cana-2410 167 10 𝑀[𝐼𝑛⨂𝑅𝑗 𝑀[𝐼𝑛⨂𝑅𝑗 X cana-2410 167 11 ] ] X cana-2410 167 12 ‖𝑧(𝑑𝑗 ‖𝑧(𝑑𝑗 PROPN cana-2410 167 13 βˆ’ βˆ’ PROPN cana-2410 167 14 ) ) PUNCT cana-2410 167 15 βˆ’ βˆ’ PROPN cana-2410 167 16 π‘₯(𝑑𝑗 π‘₯(𝑑𝑗 PROPN cana-2410 167 17 βˆ’)β€– βˆ’)β€– NUM cana-2410 167 18 𝑑 𝑑 NOUN cana-2410 167 19 𝑠𝑗 𝑠𝑗 INTJ cana-2410 167 20 + + NOUN cana-2410 167 21 π‘€πΎβˆ« π‘€πΎβˆ« NOUN cana-2410 167 22 ‖𝑧(𝑠 ‖𝑧(𝑠 X cana-2410 167 23 ) ) PUNCT cana-2410 167 24 βˆ’ βˆ’ PROPN cana-2410 167 25 π‘₯(𝑠)β€–βˆ†π‘  π‘₯(𝑠)β€–βˆ†π‘  NUM cana-2410 167 26 𝑑𝑗+1 𝑑𝑗+1 NUM cana-2410 167 27 𝑠𝑗 𝑠𝑗 NOUN cana-2410 167 28 + + ADJ cana-2410 167 29 π‘€π‘“βˆ« π‘€π‘“βˆ« ADJ cana-2410 167 30 ‖𝑧(𝑠 ‖𝑧(𝑠 X cana-2410 167 31 ) ) PUNCT cana-2410 167 32 βˆ’ βˆ’ PROPN cana-2410 167 33 π‘₯(𝑠)β€–βˆ†π‘  π‘₯(𝑠)β€–βˆ†π‘  NUM cana-2410 167 34 𝑑𝑗+1 𝑑𝑗+1 NUM cana-2410 167 35 𝑠𝑗 𝑠𝑗 INTJ cana-2410 167 36 ] ] PUNCT cana-2410 167 37 βˆ†πœ βˆ†πœ PROPN cana-2410 167 38 ≀ ≀ NUM cana-2410 167 39 𝐿𝑀[𝐼𝑛⨂𝑅𝑗 𝐿𝑀[𝐼𝑛⨂𝑅𝑗 NUM cana-2410 167 40 ] ] X cana-2410 167 41 ‖𝑧 ‖𝑧 NOUN cana-2410 168 1 βˆ’ βˆ’ PROPN cana-2410 169 1 π‘₯‖𝑃𝐢 π‘₯‖𝑃𝐢 NOUN cana-2410 169 2 + + CCONJ cana-2410 169 3 𝐿𝑀𝐾‖𝑧 𝐿𝑀𝐾‖𝑧 NOUN cana-2410 169 4 βˆ’ βˆ’ PROPN cana-2410 169 5 π‘₯‖𝑃𝐢(𝑑 π‘₯‖𝑃𝐢(𝑑 ADJ cana-2410 169 6 βˆ’ βˆ’ NOUN cana-2410 169 7 𝑠𝑗 𝑠𝑗 NOUN cana-2410 169 8 ) ) PUNCT cana-2410 170 1 + + NUM cana-2410 170 2 𝐿𝑀𝑓‖𝑧 𝐿𝑀𝑓‖𝑧 NUM cana-2410 170 3 βˆ’ βˆ’ NOUN cana-2410 170 4 π‘₯‖𝑃𝐢(𝑑 π‘₯‖𝑃𝐢(𝑑 ADJ cana-2410 170 5 βˆ’ βˆ’ NOUN cana-2410 170 6 𝑠𝑗 𝑠𝑗 NOUN cana-2410 170 7 ) ) PUNCT cana-2410 171 1 + + NOUN cana-2410 171 2 𝐿2𝐿𝑄 𝐿2𝐿𝑄 NUM cana-2410 171 3 2 2 NUM cana-2410 171 4 ( ( PUNCT cana-2410 171 5 𝑑 𝑑 NOUN cana-2410 171 6 βˆ’ βˆ’ PROPN cana-2410 171 7 𝑠𝑗)𝛿𝑗[𝑀[𝐼𝑛⨂𝑅𝑗 𝑠𝑗)𝛿𝑗[𝑀[𝐼𝑛⨂𝑅𝑗 NOUN cana-2410 171 8 ] ] X cana-2410 171 9 + + NUM cana-2410 171 10 𝐿𝑀𝐾(𝑑𝑗+1 𝐿𝑀𝐾(𝑑𝑗+1 PROPN cana-2410 171 11 βˆ’ βˆ’ NOUN cana-2410 171 12 𝑠𝑗 𝑠𝑗 NOUN cana-2410 171 13 ) ) PUNCT cana-2410 171 14 ] ] PUNCT cana-2410 171 15 ‖𝑧 ‖𝑧 PROPN cana-2410 172 1 βˆ’ βˆ’ PROPN cana-2410 172 2 π‘₯‖𝑃𝐢 π‘₯‖𝑃𝐢 NOUN cana-2410 172 3 + + CCONJ cana-2410 172 4 𝐿𝑀𝑓(𝑑𝑗+1 𝐿𝑀𝑓(𝑑𝑗+1 ADP cana-2410 172 5 βˆ’ βˆ’ NOUN cana-2410 172 6 𝑠𝑗 𝑠𝑗 NOUN cana-2410 172 7 ) ) PUNCT cana-2410 172 8 ] ] PUNCT cana-2410 172 9 ‖𝑧 ‖𝑧 PROPN cana-2410 173 1 βˆ’ βˆ’ PROPN cana-2410 173 2 π‘₯‖𝑃𝐢 π‘₯‖𝑃𝐢 NOUN cana-2410 173 3 communications communication NOUN cana-2410 173 4 on on ADP cana-2410 173 5 applied apply VERB cana-2410 173 6 nonlinear nonlinear ADJ cana-2410 173 7 analysis analysis NOUN cana-2410 173 8 issn issn NOUN cana-2410 173 9 : : PUNCT cana-2410 173 10 1074 1074 NUM cana-2410 173 11 - - PUNCT cana-2410 173 12 133x 133x NUM cana-2410 173 13 vol vol NOUN cana-2410 173 14 32 32 NUM cana-2410 173 15 no no NOUN cana-2410 173 16 . . PUNCT cana-2410 174 1 2s 2s NUM cana-2410 174 2 ( ( PUNCT cana-2410 174 3 2025 2025 NUM cana-2410 174 4 ) ) PUNCT cana-2410 174 5 358 358 NUM cana-2410 174 6 https://internationalpubls.com https://internationalpubls.com X cana-2410 174 7 ≀ ≀ NUM cana-2410 174 8 𝑀𝛼1 𝑀𝛼1 NOUN cana-2410 174 9 𝑗 𝑗 PROPN cana-2410 174 10 ‖𝑧 ‖𝑧 PROPN cana-2410 174 11 βˆ’ βˆ’ PROPN cana-2410 175 1 π‘₯‖𝑃𝐢 π‘₯‖𝑃𝐢 NOUN cana-2410 175 2 ≀ ≀ NUM cana-2410 175 3 𝑀𝛼‖𝑧 𝑀𝛼‖𝑧 PROPN cana-2410 175 4 βˆ’ βˆ’ PROPN cana-2410 175 5 π‘₯‖𝑃𝐢 π‘₯‖𝑃𝐢 NOUN cana-2410 175 6 ( ( PUNCT cana-2410 175 7 3.13 3.13 NUM cana-2410 175 8 ) ) PUNCT cana-2410 175 9 for for ADP cana-2410 175 10 any any DET cana-2410 175 11 𝑧 𝑧 NOUN cana-2410 175 12 , , PUNCT cana-2410 175 13 π‘₯ π‘₯ PROPN cana-2410 175 14 ∈ ∈ PROPN cana-2410 175 15 π’Ÿ π’Ÿ PROPN cana-2410 175 16 , , PUNCT cana-2410 175 17 𝑑 𝑑 PROPN cana-2410 175 18 ∈ ∈ PROPN cana-2410 176 1 [ [ X cana-2410 176 2 0 0 NUM cana-2410 176 3 , , PUNCT cana-2410 176 4 𝑑1]𝕋 𝑑1]𝕋 NUM cana-2410 176 5 , , PUNCT cana-2410 176 6 we we PRON cana-2410 176 7 get get VERB cana-2410 176 8 β€–(𝑑𝑑)(𝑑 β€–(𝑑𝑑)(𝑑 ADJ cana-2410 176 9 ) ) PUNCT cana-2410 177 1 βˆ’ βˆ’ PROPN cana-2410 177 2 ( ( PUNCT cana-2410 177 3 𝑑𝑑)(𝑑)β€– 𝑑𝑑)(𝑑)β€– PROPN cana-2410 177 4 ≀ ≀ PROPN cana-2410 177 5 ∫ ∫ PROPN cana-2410 177 6 β€–Οˆ(𝑑,𝑑(𝑑))‖‖𝑑(𝑑,𝑑)‖‖𝑑(𝑑 β€–Οˆ(𝑑,𝑑(𝑑))‖‖𝑑(𝑑,𝑑)‖‖𝑑(𝑑 X cana-2410 177 7 ) ) PUNCT cana-2410 177 8 βˆ’π‘‘(𝑑)β€–βˆ†π‘‘ βˆ’π‘‘(𝑑)β€–βˆ†π‘‘ PUNCT cana-2410 178 1 𝑑 𝑑 PROPN cana-2410 178 2 0 0 NUM cana-2410 179 1 + + NOUN cana-2410 179 2 ∫ ∫ NOUN cana-2410 179 3 β€–Οˆ(𝑑 β€–Οˆ(𝑑 NOUN cana-2410 179 4 , , PUNCT cana-2410 179 5 𝜎(𝜏))‖‖𝑓(𝜏 𝜎(𝜏))‖‖𝑓(𝜏 NUM cana-2410 179 6 , , PUNCT cana-2410 179 7 𝑧(𝜏 𝑧(𝜏 PROPN cana-2410 179 8 ) ) PUNCT cana-2410 179 9 ) ) PUNCT cana-2410 180 1 βˆ’ βˆ’ NOUN cana-2410 181 1 𝑓(𝜏 𝑓(𝜏 PROPN cana-2410 181 2 , , PUNCT cana-2410 181 3 π‘₯(𝜏))β€–βˆ†πœ π‘₯(𝜏))β€–βˆ†πœ PROPN cana-2410 181 4 𝑑 𝑑 PROPN cana-2410 181 5 0 0 PUNCT cana-2410 182 1 + + NUM cana-2410 182 2 ∫ ∫ PROPN cana-2410 182 3 [ [ X cana-2410 182 4 β€–Οˆ(𝑑 β€–Οˆ(𝑑 NOUN cana-2410 182 5 , , PUNCT cana-2410 182 6 𝜎(𝜏))‖‖𝑄(Ο„)β€–β€–π‘„βˆ—(𝜏)β€–β€–Οˆβˆ—(𝑑 𝜎(𝜏))‖‖𝑄(Ο„)β€–β€–π‘„βˆ—(𝜏)β€–β€–Οˆβˆ—(𝑑 PROPN cana-2410 182 7 , , PUNCT cana-2410 182 8 𝜎(𝜏))β€– 𝜎(𝜏))β€– PROPN cana-2410 182 9 𝑑 𝑑 PROPN cana-2410 182 10 0 0 NUM cana-2410 182 11 Γ— Γ— NOUN cana-2410 182 12 ‖𝒩0 ‖𝒩0 ADJ cana-2410 182 13 βˆ’1(𝑑0 βˆ’1(𝑑0 X cana-2410 182 14 , , PUNCT cana-2410 182 15 𝑑1)β€–[∫ 𝑑1)β€–[∫ ADP cana-2410 182 16 β€–Οˆ(𝑑 β€–Οˆ(𝑑 NOUN cana-2410 182 17 , , PUNCT cana-2410 182 18 𝜎(𝑠))‖‖𝐾(𝑑 𝜎(𝑠))‖‖𝐾(𝑑 NUM cana-2410 182 19 , , PUNCT cana-2410 182 20 𝑠)‖‖𝑧(𝑠 𝑠)‖‖𝑧(𝑠 NOUN cana-2410 182 21 ) ) PUNCT cana-2410 182 22 βˆ’ βˆ’ PROPN cana-2410 182 23 π‘₯(𝑠)β€–βˆ†π‘  π‘₯(𝑠)β€–βˆ†π‘  NUM cana-2410 182 24 𝑑1 𝑑1 NOUN cana-2410 182 25 0 0 PUNCT cana-2410 183 1 + + NUM cana-2410 183 2 ∫ ∫ PROPN cana-2410 183 3 β€–Οˆ(π‘ π‘˜ β€–Οˆ(π‘ π‘˜ PROPN cana-2410 183 4 , , PUNCT cana-2410 183 5 𝜎(𝑠))‖‖𝑓(𝑠 𝜎(𝑠))‖‖𝑓(𝑠 NOUN cana-2410 183 6 , , PUNCT cana-2410 183 7 𝑧(𝑠 𝑧(𝑠 NOUN cana-2410 183 8 ) ) PUNCT cana-2410 183 9 ) ) PUNCT cana-2410 183 10 βˆ’ βˆ’ PROPN cana-2410 183 11 𝑓(𝑠 𝑓(𝑠 NOUN cana-2410 183 12 , , PUNCT cana-2410 183 13 π‘₯(𝑠))β€–βˆ†π‘ ]βˆ†πœ π‘₯(𝑠))β€–βˆ†π‘ ]βˆ†πœ PROPN cana-2410 183 14 𝑑1 𝑑1 PROPN cana-2410 183 15 𝑑0 𝑑0 PROPN cana-2410 183 16 ≀ ≀ PROPN cana-2410 183 17 𝑀𝛼1 𝑀𝛼1 NOUN cana-2410 183 18 0 0 NUM cana-2410 183 19 ‖𝑧 ‖𝑧 NUM cana-2410 184 1 βˆ’ βˆ’ PROPN cana-2410 185 1 π‘₯‖𝑃𝐢 π‘₯‖𝑃𝐢 NOUN cana-2410 185 2 ≀ ≀ NUM cana-2410 185 3 𝑀𝛼‖𝑧 𝑀𝛼‖𝑧 PROPN cana-2410 185 4 βˆ’ βˆ’ PROPN cana-2410 185 5 π‘₯‖𝑃𝐢 π‘₯‖𝑃𝐢 NOUN cana-2410 185 6 . . PUNCT cana-2410 186 1 ( ( PUNCT cana-2410 186 2 3.14 3.14 NUM cana-2410 186 3 ) ) PUNCT cana-2410 186 4 similarly similarly ADV cana-2410 186 5 , , PUNCT cana-2410 186 6 for for ADP cana-2410 186 7 𝑑 𝑑 PROPN cana-2410 186 8 ∈ ∈ PROPN cana-2410 186 9 ( ( PUNCT cana-2410 186 10 𝑠𝑗 𝑠𝑗 NOUN cana-2410 186 11 , , PUNCT cana-2410 186 12 𝑑𝑗]𝕋 𝑑𝑗]𝕋 NOUN cana-2410 186 13 , , PUNCT cana-2410 186 14 we we PRON cana-2410 186 15 have have AUX cana-2410 186 16 β€–(𝒒𝑧)(𝑑 β€–(𝒒𝑧)(𝑑 VERB cana-2410 186 17 ) ) PUNCT cana-2410 187 1 βˆ’ βˆ’ PROPN cana-2410 187 2 ( ( PUNCT cana-2410 187 3 𝒒π‘₯)(𝑑)β€– 𝒒π‘₯)(𝑑)β€– PROPN cana-2410 187 4 ≀ ≀ NUM cana-2410 187 5 𝑀[𝐼𝑛⨂𝑅𝑗 𝑀[𝐼𝑛⨂𝑅𝑗 PROPN cana-2410 187 6 ] ] X cana-2410 187 7 ‖𝑧 ‖𝑧 PROPN cana-2410 187 8 βˆ’ βˆ’ PROPN cana-2410 187 9 π‘₯‖𝑃𝐢 π‘₯‖𝑃𝐢 NOUN cana-2410 187 10 ≀ ≀ NUM cana-2410 187 11 𝑀𝛼‖𝑧 𝑀𝛼‖𝑧 PROPN cana-2410 187 12 βˆ’ βˆ’ PROPN cana-2410 187 13 π‘₯‖𝑃𝐢 π‘₯‖𝑃𝐢 NOUN cana-2410 187 14 ( ( PUNCT cana-2410 187 15 3.15 3.15 NUM cana-2410 187 16 ) ) PUNCT cana-2410 187 17 after after ADP cana-2410 187 18 succinct succinct ADJ cana-2410 187 19 the the DET cana-2410 187 20 inequalities inequality NOUN cana-2410 187 21 ( ( PUNCT cana-2410 187 22 3.13 3.13 NUM cana-2410 187 23 ) ) PUNCT cana-2410 187 24 ( ( PUNCT cana-2410 187 25 3.15 3.15 NUM cana-2410 187 26 ) ) PUNCT cana-2410 187 27 , , PUNCT cana-2410 187 28 for for ADP cana-2410 187 29 𝑑 𝑑 PROPN cana-2410 187 30 ∈ ∈ PROPN cana-2410 187 31 𝐼 𝐼 PROPN cana-2410 187 32 , , PUNCT cana-2410 187 33 we we PRON cana-2410 187 34 have have AUX cana-2410 187 35 β€–(𝒒𝑧 β€–(𝒒𝑧 NOUN cana-2410 187 36 ) ) PUNCT cana-2410 187 37 βˆ’ βˆ’ PROPN cana-2410 187 38 ( ( PUNCT cana-2410 187 39 𝒒π‘₯)‖𝑃𝐢 𝒒π‘₯)‖𝑃𝐢 PROPN cana-2410 187 40 ≀ ≀ X cana-2410 187 41 𝑀𝛼‖𝑧 𝑀𝛼‖𝑧 PROPN cana-2410 187 42 βˆ’ βˆ’ PROPN cana-2410 187 43 π‘₯‖𝑃𝐢 π‘₯‖𝑃𝐢 NOUN cana-2410 187 44 . . PUNCT cana-2410 188 1 thus thus ADV cana-2410 188 2 , , PUNCT cana-2410 188 3 according accord VERB cana-2410 188 4 to to ADP cana-2410 188 5 banach banach NOUN cana-2410 188 6 's 's PART cana-2410 188 7 fixed fix VERB cana-2410 188 8 point point NOUN cana-2410 188 9 theorem theorem VERB cana-2410 188 10 , , PUNCT cana-2410 188 11 there there PRON cana-2410 188 12 is be VERB cana-2410 188 13 only only ADV cana-2410 188 14 one one NUM cana-2410 188 15 solution solution NOUN cana-2410 188 16 to to ADP cana-2410 188 17 system system NOUN cana-2410 188 18 ( ( PUNCT cana-2410 188 19 3.1 3.1 NUM cana-2410 188 20 ) ) PUNCT cana-2410 188 21 . . PUNCT cana-2410 189 1 because because SCONJ cana-2410 189 2 of of ADP cana-2410 189 3 this this PRON cana-2410 189 4 , , PUNCT cana-2410 189 5 𝒒 𝒒 PROPN cana-2410 189 6 is be AUX cana-2410 189 7 a a DET cana-2410 189 8 mapping mapping NOUN cana-2410 189 9 that that PRON cana-2410 189 10 strictly strictly ADV cana-2410 189 11 contracts contract NOUN cana-2410 189 12 . . PUNCT cana-2410 190 1 theorem theorem ADJ cana-2410 190 2 3.2 3.2 NUM cana-2410 190 3 : : PUNCT cana-2410 190 4 assuming assume VERB cana-2410 190 5 that that SCONJ cana-2410 190 6 requirements requirement NOUN cana-2410 190 7 ( ( PUNCT cana-2410 190 8 h1 h1 PROPN cana-2410 190 9 ) ) PUNCT cana-2410 190 10 ( ( PUNCT cana-2410 190 11 h3 h3 NOUN cana-2410 190 12 ) ) PUNCT cana-2410 190 13 are be AUX cana-2410 190 14 satisfied satisfied ADJ cana-2410 190 15 ; ; PUNCT cana-2410 190 16 the the DET cana-2410 190 17 system system NOUN cana-2410 190 18 ( ( PUNCT cana-2410 190 19 3.1 3.1 NUM cana-2410 190 20 ) ) PUNCT cana-2410 190 21 is be AUX cana-2410 190 22 complete complete ADJ cana-2410 190 23 controllable controllable ADJ cana-2410 190 24 in in ADP cana-2410 190 25 [ [ X cana-2410 190 26 𝑑0 𝑑0 NOUN cana-2410 190 27 , , PUNCT cana-2410 190 28 𝑇]𝕋 𝑇]𝕋 ADJ cana-2410 190 29 if if SCONJ cana-2410 190 30 and and CCONJ cana-2410 190 31 only only ADV cana-2410 190 32 if if SCONJ cana-2410 190 33 the the DET cana-2410 190 34 matrices matrix NOUN cana-2410 190 35 𝒩0(𝑑0 𝒩0(𝑑0 NOUN cana-2410 190 36 , , PUNCT cana-2410 190 37 𝑑1 𝑑1 NOUN cana-2410 190 38 ) ) PUNCT cana-2410 190 39 and and CCONJ cana-2410 190 40 𝒩𝑗(𝑠𝑗 𝒩𝑗(𝑠𝑗 PROPN cana-2410 190 41 , , PUNCT cana-2410 190 42 𝑑𝑗+1 𝑑𝑗+1 NOUN cana-2410 190 43 ) ) PUNCT cana-2410 190 44 are be AUX cana-2410 190 45 invertible invertible ADJ cana-2410 190 46 . . PUNCT cana-2410 191 1 proof proof NOUN cana-2410 191 2 : : PUNCT cana-2410 191 3 let let VERB cana-2410 191 4 𝒩0(𝑑0 𝒩0(𝑑0 NOUN cana-2410 191 5 , , PUNCT cana-2410 191 6 𝑑1 𝑑1 NOUN cana-2410 191 7 ) ) PUNCT cana-2410 191 8 and and CCONJ cana-2410 191 9 𝒩𝑗(𝑠𝑗 𝒩𝑗(𝑠𝑗 NOUN cana-2410 191 10 , , PUNCT cana-2410 191 11 𝑑𝑗+1 𝑑𝑗+1 X cana-2410 191 12 ) ) PUNCT cana-2410 191 13 are be AUX cana-2410 191 14 invertible invertible ADJ cana-2410 191 15 . . PUNCT cana-2410 192 1 then then ADV cana-2410 192 2 , , PUNCT cana-2410 192 3 for for ADP cana-2410 192 4 the the DET cana-2410 192 5 given give VERB cana-2410 192 6 𝑧𝑑1and 𝑧𝑑1and PROPN cana-2410 192 7 𝑧𝑑𝑗+1 𝑧𝑑𝑗+1 NOUN cana-2410 192 8 , , PUNCT cana-2410 192 9 and and CCONJ cana-2410 192 10 the the DET cana-2410 192 11 input input NOUN cana-2410 192 12 control control NOUN cana-2410 192 13 uΜ‚ uΜ‚ X cana-2410 192 14 ( ( PUNCT cana-2410 192 15 t t NOUN cana-2410 192 16 ) ) PUNCT cana-2410 192 17 given give VERB cana-2410 192 18 by by ADP cana-2410 192 19 ( ( PUNCT cana-2410 192 20 3.6 3.6 NUM cana-2410 192 21 ) ) PUNCT cana-2410 192 22 . . PUNCT cana-2410 193 1 now now ADV cana-2410 193 2 , , PUNCT cana-2410 193 3 put put VERB cana-2410 193 4 𝑑 𝑑 NOUN cana-2410 193 5 = = NOUN cana-2410 193 6 𝑑1 𝑑1 NOUN cana-2410 193 7 , , PUNCT cana-2410 193 8 in in ADP cana-2410 193 9 the the DET cana-2410 193 10 system system NOUN cana-2410 193 11 ( ( PUNCT cana-2410 193 12 3.1 3.1 NUM cana-2410 193 13 ) ) PUNCT cana-2410 193 14 , , PUNCT cana-2410 193 15 we we PRON cana-2410 193 16 have have AUX cana-2410 193 17 𝑧(𝑑1 𝑧(𝑑1 VERB cana-2410 193 18 ) ) PUNCT cana-2410 194 1 = = SYM cana-2410 194 2 ψ(𝑑1 ψ(𝑑1 NOUN cana-2410 194 3 , , PUNCT cana-2410 194 4 𝑑0)𝑧0 𝑑0)𝑧0 VERB cana-2410 194 5 + + NOUN cana-2410 194 6 ∫ ∫ PROPN cana-2410 194 7 ψ(𝑑1 ψ(𝑑1 NOUN cana-2410 194 8 , , PUNCT cana-2410 194 9 𝜎(𝑠))𝐾(𝑑 𝜎(𝑠))𝐾(𝑑 NOUN cana-2410 194 10 , , PUNCT cana-2410 194 11 𝑠)𝑧(𝑠)βˆ†π‘  𝑠)𝑧(𝑠)βˆ†π‘  PROPN cana-2410 194 12 𝑑1 𝑑1 PROPN cana-2410 194 13 𝑑0 𝑑0 PROPN cana-2410 194 14 + + CCONJ cana-2410 194 15 ∫ ∫ PROPN cana-2410 194 16 ψ(𝑑1 ψ(𝑑1 PROPN cana-2410 194 17 , , PUNCT cana-2410 194 18 𝜎(𝜏))𝑓(𝜏 𝜎(𝜏))𝑓(𝜏 PROPN cana-2410 194 19 , , PUNCT cana-2410 194 20 𝑧(𝜏))βˆ†πœ 𝑧(𝜏))βˆ†πœ NOUN cana-2410 194 21 𝑑1 𝑑1 PROPN cana-2410 194 22 𝑑0 𝑑0 PROPN cana-2410 194 23 βˆ’βˆ« βˆ’βˆ« NOUN cana-2410 194 24 β€–Οˆ(𝑑1 β€–Οˆ(𝑑1 PROPN cana-2410 194 25 , , PUNCT cana-2410 194 26 𝜎(𝜏))‖‖𝑄(Ο„)‖‖𝑄 𝜎(𝜏))‖‖𝑄(Ο„)‖‖𝑄 NUM cana-2410 194 27 βˆ—(𝜏)β€–β€–Οˆβˆ—(𝑑0 βˆ—(𝜏)β€–β€–Οˆβˆ—(𝑑0 PROPN cana-2410 194 28 , , PUNCT cana-2410 194 29 𝜎(𝜏))β€– 𝜎(𝜏))β€– PROPN cana-2410 194 30 𝑑1 𝑑1 NOUN cana-2410 194 31 𝑑0 𝑑0 NOUN cana-2410 194 32 𝑧0βˆ†πœ 𝑧0βˆ†πœ NOUN cana-2410 194 33 communications communication NOUN cana-2410 194 34 on on ADP cana-2410 194 35 applied apply VERB cana-2410 194 36 nonlinear nonlinear ADJ cana-2410 194 37 analysis analysis NOUN cana-2410 194 38 issn issn NOUN cana-2410 194 39 : : PUNCT cana-2410 194 40 1074 1074 NUM cana-2410 194 41 - - PUNCT cana-2410 194 42 133x 133x NUM cana-2410 194 43 vol vol NOUN cana-2410 194 44 32 32 NUM cana-2410 194 45 no no NOUN cana-2410 194 46 . . PUNCT cana-2410 195 1 2s 2s NUM cana-2410 195 2 ( ( PUNCT cana-2410 195 3 2025 2025 NUM cana-2410 195 4 ) ) PUNCT cana-2410 195 5 359 359 NUM cana-2410 195 6 https://internationalpubls.com https://internationalpubls.com X cana-2410 195 7 = = SYM cana-2410 195 8 ψ(𝑑1 ψ(𝑑1 PROPN cana-2410 195 9 , , PUNCT cana-2410 195 10 𝑑0)𝑧0 𝑑0)𝑧0 VERB cana-2410 195 11 + + NOUN cana-2410 195 12 ∫ ∫ PROPN cana-2410 195 13 ψ(𝑑1 ψ(𝑑1 NOUN cana-2410 195 14 , , PUNCT cana-2410 195 15 𝜎(𝑠))𝐾(𝑑 𝜎(𝑠))𝐾(𝑑 NOUN cana-2410 195 16 , , PUNCT cana-2410 195 17 𝑠)𝑧(𝑠)βˆ†π‘  𝑠)𝑧(𝑠)βˆ†π‘  PROPN cana-2410 195 18 𝑑1 𝑑1 PROPN cana-2410 195 19 𝑑0 𝑑0 PROPN cana-2410 196 1 + + CCONJ cana-2410 196 2 ∫ ∫ PROPN cana-2410 196 3 ψ(𝑑1 ψ(𝑑1 PROPN cana-2410 196 4 , , PUNCT cana-2410 196 5 𝜎(𝜏))𝑓(𝜏 𝜎(𝜏))𝑓(𝜏 PROPN cana-2410 196 6 , , PUNCT cana-2410 196 7 𝑧(𝜏))βˆ†πœ 𝑧(𝜏))βˆ†πœ NOUN cana-2410 196 8 𝑑1 𝑑1 PROPN cana-2410 196 9 𝑑0 𝑑0 PROPN cana-2410 196 10 βˆ’ βˆ’ PROPN cana-2410 196 11 ψ(𝑑1 ψ(𝑑1 PROPN cana-2410 196 12 , , PUNCT cana-2410 196 13 𝑑0)𝒩0(𝑑0 𝑑0)𝒩0(𝑑0 NOUN cana-2410 196 14 , , PUNCT cana-2410 196 15 𝑑1)𝒩0 𝑑1)𝒩0 VERB cana-2410 196 16 βˆ’1(𝑑0 βˆ’1(𝑑0 X cana-2410 196 17 , , PUNCT cana-2410 196 18 𝑑1 𝑑1 NOUN cana-2410 196 19 ) ) PUNCT cana-2410 197 1 [ [ X cana-2410 197 2 𝑧0 𝑧0 PROPN cana-2410 197 3 βˆ’ βˆ’ PROPN cana-2410 197 4 ψ(𝑑0 ψ(𝑑0 NOUN cana-2410 197 5 , , PUNCT cana-2410 197 6 𝑑1)𝑧𝑑1 𝑑1)𝑧𝑑1 PUNCT cana-2410 197 7 + + NOUN cana-2410 197 8 ∫ ∫ PROPN cana-2410 197 9 ψ(𝑑1 ψ(𝑑1 NOUN cana-2410 197 10 , , PUNCT cana-2410 197 11 𝜎(𝑠))𝐾(𝑑 𝜎(𝑠))𝐾(𝑑 NOUN cana-2410 197 12 , , PUNCT cana-2410 197 13 𝑠)𝑧(𝑠)βˆ†π‘  𝑠)𝑧(𝑠)βˆ†π‘  PROPN cana-2410 197 14 𝑑1 𝑑1 PROPN cana-2410 197 15 𝑑0 𝑑0 PROPN cana-2410 197 16 + + CCONJ cana-2410 197 17 ∫ ∫ PROPN cana-2410 197 18 ψ(𝑑0 ψ(𝑑0 NOUN cana-2410 197 19 , , PUNCT cana-2410 197 20 𝜎(𝜏))𝑓(𝜏 𝜎(𝜏))𝑓(𝜏 PROPN cana-2410 197 21 , , PUNCT cana-2410 197 22 𝑧(𝜏))βˆ†πœ 𝑧(𝜏))βˆ†πœ NOUN cana-2410 197 23 𝑑1 𝑑1 NOUN cana-2410 197 24 𝑑0 𝑑0 NOUN cana-2410 197 25 ] ] PUNCT cana-2410 198 1 = = PUNCT cana-2410 198 2 𝑧𝑑1 𝑧𝑑1 NOUN cana-2410 198 3 similarly similarly ADV cana-2410 198 4 , , PUNCT cana-2410 198 5 for for ADP cana-2410 198 6 𝑑 𝑑 PROPN cana-2410 198 7 ∈ ∈ PROPN cana-2410 198 8 ( ( PUNCT cana-2410 198 9 𝑠𝑗 𝑠𝑗 NOUN cana-2410 198 10 , , PUNCT cana-2410 198 11 𝑑𝑗+1]𝕋 𝑑𝑗+1]𝕋 NOUN cana-2410 198 12 , , PUNCT cana-2410 198 13 , , PUNCT cana-2410 198 14 𝑗 𝑗 NOUN cana-2410 198 15 = = SYM cana-2410 198 16 1,2 1,2 NUM cana-2410 198 17 , , PUNCT cana-2410 198 18 … … PUNCT cana-2410 198 19 , , PUNCT cana-2410 198 20 π‘š π‘š NOUN cana-2410 198 21 , , PUNCT cana-2410 198 22 . . PUNCT cana-2410 199 1 we we PRON cana-2410 199 2 replace replace VERB cana-2410 199 3 𝑑 𝑑 PROPN cana-2410 199 4 = = SYM cana-2410 199 5 𝑑𝑗+1 𝑑𝑗+1 PROPN cana-2410 199 6 , , PUNCT cana-2410 199 7 in in ADP cana-2410 199 8 the the DET cana-2410 199 9 solution solution NOUN cana-2410 199 10 of of ADP cana-2410 199 11 ( ( PUNCT cana-2410 199 12 3.1 3.1 NUM cana-2410 199 13 ) ) PUNCT cana-2410 199 14 , , PUNCT cana-2410 199 15 we we PRON cana-2410 199 16 have have VERB cana-2410 199 17 𝑧(𝑑𝑗+1 𝑧(𝑑𝑗+1 PRON cana-2410 199 18 ) ) PUNCT cana-2410 199 19 = = SYM cana-2410 199 20 ψ(𝑑𝑗+1 ψ(𝑑𝑗+1 NOUN cana-2410 199 21 , , PUNCT cana-2410 199 22 𝑠𝑗)[𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 𝑠𝑗)[𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 X cana-2410 199 23 βˆ’ βˆ’ NOUN cana-2410 199 24 ) ) PUNCT cana-2410 200 1 + + NUM cana-2410 200 2 ∫ ∫ X cana-2410 200 3 ψ ψ X cana-2410 200 4 ( ( PUNCT cana-2410 200 5 𝑑𝑗+1 𝑑𝑗+1 PROPN cana-2410 200 6 , , PUNCT cana-2410 200 7 𝜎(𝑠))𝐾(𝑑 𝜎(𝑠))𝐾(𝑑 NOUN cana-2410 200 8 , , PUNCT cana-2410 200 9 𝑠)𝑧(𝑠)βˆ†π‘  𝑠)𝑧(𝑠)βˆ†π‘  PROPN cana-2410 200 10 𝑑𝑗+1 𝑑𝑗+1 NUM cana-2410 200 11 𝑠𝑗 𝑠𝑗 PROPN cana-2410 200 12 + + NOUN cana-2410 200 13 ∫ ∫ PROPN cana-2410 200 14 ψ ψ X cana-2410 200 15 ( ( PUNCT cana-2410 200 16 𝑑𝑗+1 𝑑𝑗+1 PROPN cana-2410 200 17 , , PUNCT cana-2410 200 18 𝜎(𝜏 𝜎(𝜏 PROPN cana-2410 200 19 ) ) PUNCT cana-2410 200 20 ) ) PUNCT cana-2410 201 1 [ [ X cana-2410 201 2 𝑓(𝜏 𝑓(𝜏 PROPN cana-2410 201 3 , , PUNCT cana-2410 201 4 𝑧(𝜏 𝑧(𝜏 PROPN cana-2410 201 5 ) ) PUNCT cana-2410 201 6 ) ) PUNCT cana-2410 201 7 βˆ’ βˆ’ PROPN cana-2410 201 8 𝑄(𝜏)𝑄 𝑄(𝜏)𝑄 ADJ cana-2410 201 9 βˆ—(𝜏)Οˆβˆ— βˆ—(𝜏)Οˆβˆ— PROPN cana-2410 201 10 ( ( PUNCT cana-2410 201 11 𝑠𝑗 𝑠𝑗 INTJ cana-2410 201 12 , , PUNCT cana-2410 201 13 𝜎(𝜏 𝜎(𝜏 PROPN cana-2410 201 14 ) ) PUNCT cana-2410 201 15 ) ) PUNCT cana-2410 201 16 𝑧𝑗 𝑧𝑗 ADP cana-2410 201 17 ] ] X cana-2410 201 18 𝑑𝑗+1 𝑑𝑗+1 NUM cana-2410 201 19 𝑠𝑗 𝑠𝑗 PROPN cana-2410 201 20 βˆ†πœ βˆ†πœ PROPN cana-2410 201 21 = = SYM cana-2410 201 22 ψ(𝑑𝑗+1 ψ(𝑑𝑗+1 PROPN cana-2410 201 23 , , PUNCT cana-2410 201 24 𝑠𝑗)[𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 𝑠𝑗)[𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 X cana-2410 201 25 βˆ’ βˆ’ NOUN cana-2410 201 26 ) ) PUNCT cana-2410 202 1 + + NUM cana-2410 202 2 ∫ ∫ X cana-2410 202 3 ψ ψ X cana-2410 202 4 ( ( PUNCT cana-2410 202 5 𝑑𝑗+1 𝑑𝑗+1 PROPN cana-2410 202 6 , , PUNCT cana-2410 202 7 𝜎(𝑠))𝐾(𝑑 𝜎(𝑠))𝐾(𝑑 NOUN cana-2410 202 8 , , PUNCT cana-2410 202 9 𝑠)𝑧(𝑠)βˆ†π‘  𝑠)𝑧(𝑠)βˆ†π‘  PROPN cana-2410 202 10 𝑑𝑗+1 𝑑𝑗+1 NUM cana-2410 202 11 𝑠𝑗 𝑠𝑗 PROPN cana-2410 202 12 + + NOUN cana-2410 202 13 ∫ ∫ NOUN cana-2410 202 14 ψ(𝑑𝑑+1,𝑑(𝑑))𝑑(𝑑,𝑑(𝑑))βˆ†π‘‘ ψ(𝑑𝑑+1,𝑑(𝑑))𝑑(𝑑,𝑑(𝑑))βˆ†π‘‘ PROPN cana-2410 202 15 𝑑𝑑+1 𝑑𝑑+1 PROPN cana-2410 202 16 𝑑𝑑 𝑑𝑑 NOUN cana-2410 202 17 βˆ’ βˆ’ NOUN cana-2410 202 18 ψ(𝑑𝑗+1 ψ(𝑑𝑗+1 NOUN cana-2410 202 19 , , PUNCT cana-2410 202 20 𝑠𝑗)𝒩𝑗(𝑠𝑗 𝑠𝑗)𝒩𝑗(𝑠𝑗 NOUN cana-2410 202 21 , , PUNCT cana-2410 202 22 𝑑𝑗+1)𝒩𝑗 𝑑𝑗+1)𝒩𝑗 PROPN cana-2410 202 23 βˆ’1(𝑠𝑗 βˆ’1(𝑠𝑗 NOUN cana-2410 202 24 , , PUNCT cana-2410 202 25 𝑑𝑗+1 𝑑𝑗+1 NOUN cana-2410 202 26 ) ) PUNCT cana-2410 202 27 Γ— Γ— NOUN cana-2410 203 1 [ [ X cana-2410 203 2 [ [ X cana-2410 203 3 𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 X cana-2410 203 4 βˆ’ βˆ’ NOUN cana-2410 203 5 ) ) PUNCT cana-2410 204 1 βˆ’ βˆ’ PROPN cana-2410 204 2 ψ(𝑠𝑗 ψ(𝑠𝑗 PROPN cana-2410 204 3 , , PUNCT cana-2410 204 4 𝑑𝑗+1)𝑧𝑑𝑗+1 𝑑𝑗+1)𝑧𝑑𝑗+1 ADJ cana-2410 204 5 + + NOUN cana-2410 204 6 ∫ ∫ PROPN cana-2410 204 7 ψ ψ X cana-2410 204 8 ( ( PUNCT cana-2410 204 9 𝑑𝑗+1 𝑑𝑗+1 PROPN cana-2410 204 10 , , PUNCT cana-2410 204 11 𝜎(𝑠))𝐾(𝑑 𝜎(𝑠))𝐾(𝑑 NOUN cana-2410 204 12 , , PUNCT cana-2410 204 13 𝑠)𝑧(𝑠)βˆ†π‘  𝑠)𝑧(𝑠)βˆ†π‘  PROPN cana-2410 204 14 𝑑𝑗+1 𝑑𝑗+1 NUM cana-2410 204 15 𝑠𝑗 𝑠𝑗 PROPN cana-2410 204 16 + + NOUN cana-2410 204 17 ∫ ∫ PROPN cana-2410 204 18 ψ ψ X cana-2410 204 19 ( ( PUNCT cana-2410 204 20 𝑑𝑗+1 𝑑𝑗+1 PROPN cana-2410 204 21 , , PUNCT cana-2410 204 22 𝜎(𝜏 𝜎(𝜏 PROPN cana-2410 204 23 ) ) PUNCT cana-2410 204 24 ) ) PUNCT cana-2410 205 1 𝑓(𝜏 𝑓(𝜏 PROPN cana-2410 205 2 , , PUNCT cana-2410 205 3 𝑧(𝜏))βˆ†πœ 𝑧(𝜏))βˆ†πœ NOUN cana-2410 205 4 𝑑𝑗+1 𝑑𝑗+1 X cana-2410 205 5 𝑠𝑗 𝑠𝑗 NOUN cana-2410 205 6 ] ] PUNCT cana-2410 205 7 = = SYM cana-2410 205 8 𝑧𝑑𝑗+1 𝑧𝑑𝑗+1 NOUN cana-2410 205 9 . . PUNCT cana-2410 206 1 hence hence ADV cana-2410 206 2 , , PUNCT cana-2410 206 3 for for ADP cana-2410 206 4 in in ADP cana-2410 206 5 [ [ X cana-2410 206 6 𝑑0 𝑑0 NOUN cana-2410 206 7 , , PUNCT cana-2410 206 8 𝑇]𝕋 𝑇]𝕋 NUM cana-2410 206 9 the the DET cana-2410 206 10 system system NOUN cana-2410 206 11 ( ( PUNCT cana-2410 206 12 3.1 3.1 NUM cana-2410 206 13 ) ) PUNCT cana-2410 206 14 is be AUX cana-2410 206 15 complete complete ADJ cana-2410 206 16 controllable controllable ADJ cana-2410 206 17 conversely conversely ADV cana-2410 206 18 , , PUNCT cana-2410 206 19 on on ADP cana-2410 206 20 the the DET cana-2410 206 21 interval interval NOUN cana-2410 206 22 [ [ X cana-2410 206 23 𝑑0 𝑑0 NOUN cana-2410 206 24 , , PUNCT cana-2410 206 25 𝑇]𝕋 𝑇]𝕋 AUX cana-2410 206 26 , , PUNCT cana-2410 206 27 we we PRON cana-2410 206 28 presume presume VERB cana-2410 206 29 that that DET cana-2410 206 30 system system NOUN cana-2410 206 31 ( ( PUNCT cana-2410 206 32 3.1 3.1 NUM cana-2410 206 33 ) ) PUNCT cana-2410 206 34 is be AUX cana-2410 206 35 complete complete ADJ cana-2410 206 36 controllable controllable ADJ cana-2410 206 37 . . PUNCT cana-2410 207 1 therefore therefore ADV cana-2410 207 2 , , PUNCT cana-2410 207 3 the the DET cana-2410 207 4 matrices matrix NOUN cana-2410 207 5 𝒩0(𝑑0 𝒩0(𝑑0 NOUN cana-2410 207 6 , , PUNCT cana-2410 207 7 𝑑1 𝑑1 NOUN cana-2410 207 8 ) ) PUNCT cana-2410 207 9 and and CCONJ cana-2410 207 10 𝒩𝑗(𝑠𝑗 𝒩𝑗(𝑠𝑗 NOUN cana-2410 207 11 , , PUNCT cana-2410 207 12 𝑑𝑗+1 𝑑𝑗+1 X cana-2410 207 13 ) ) PUNCT cana-2410 207 14 are be AUX cana-2410 207 15 not not PART cana-2410 207 16 invertible invertible ADJ cana-2410 207 17 . . PUNCT cana-2410 208 1 then then ADV cana-2410 208 2 , , PUNCT cana-2410 208 3 there there PRON cana-2410 208 4 exists exist VERB cana-2410 208 5 a a DET cana-2410 208 6 non non ADJ cana-2410 208 7 - - ADJ cana-2410 208 8 zero zero NUM cana-2410 208 9 vector vector NOUN cana-2410 208 10 𝑧𝛼 𝑧𝛼 NOUN cana-2410 208 11 , , PUNCT cana-2410 208 12 𝑧𝛼𝑗 𝑧𝛼𝑗 PROPN cana-2410 208 13 ∈ ∈ PROPN cana-2410 209 1 ℝ𝑛 ℝ𝑛 ADP cana-2410 209 2 2 2 NUM cana-2410 209 3 such such ADJ cana-2410 209 4 that that PRON cana-2410 209 5 𝑧𝛼 𝑧𝛼 ADP cana-2410 209 6 βˆ—π’©0(𝑑0 βˆ—π’©0(𝑑0 NUM cana-2410 209 7 , , PUNCT cana-2410 209 8 𝑑1)𝑧𝛼 𝑑1)𝑧𝛼 PROPN cana-2410 209 9 = = SYM cana-2410 209 10 0 0 NUM cana-2410 209 11 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ PROPN cana-2410 209 12 𝑧𝛼𝑗 𝑧𝛼𝑗 PROPN cana-2410 209 13 βˆ— βˆ— NOUN cana-2410 209 14 𝒩𝑗(𝑠𝑗 𝒩𝑗(𝑠𝑗 PROPN cana-2410 209 15 , , PUNCT cana-2410 209 16 𝑑𝑗+1)𝑧𝛼𝑗 𝑑𝑗+1)𝑧𝛼𝑗 PROPN cana-2410 209 17 = = PUNCT cana-2410 209 18 0 0 X cana-2410 209 19 . . PUNCT cana-2410 210 1 ( ( PUNCT cana-2410 210 2 3.16 3.16 NUM cana-2410 210 3 ) ) PUNCT cana-2410 210 4 from from ADP cana-2410 210 5 the the DET cana-2410 210 6 equations equation NOUN cana-2410 210 7 ( ( PUNCT cana-2410 210 8 3.2 3.2 NUM cana-2410 210 9 ) ) PUNCT cana-2410 210 10 , , PUNCT cana-2410 210 11 ( ( PUNCT cana-2410 210 12 3.3 3.3 NUM cana-2410 210 13 ) ) PUNCT cana-2410 210 14 and and CCONJ cana-2410 210 15 ( ( PUNCT cana-2410 210 16 3.16 3.16 NUM cana-2410 210 17 ) ) PUNCT cana-2410 210 18 , , PUNCT cana-2410 210 19 we we PRON cana-2410 210 20 get get VERB cana-2410 210 21 βˆ«π‘§π›Ό βˆ«π‘§π›Ό PROPN cana-2410 210 22 βˆ—Οˆ(𝑑0 βˆ—Οˆ(𝑑0 NOUN cana-2410 210 23 , , PUNCT cana-2410 210 24 Οƒ(Ο„))𝑄(𝜏)𝑄 Οƒ(Ο„))𝑄(𝜏)𝑄 ADJ cana-2410 210 25 βˆ—(𝜏)Οˆβˆ—(𝑑0,Οƒ(Ο„))𝑧𝛼δτ βˆ—(𝜏)Οˆβˆ—(𝑑0,Οƒ(Ο„))𝑧𝛼δτ NOUN cana-2410 210 26 = = SYM cana-2410 210 27 0 0 PROPN cana-2410 210 28 t t PROPN cana-2410 210 29 t0 t0 PROPN cana-2410 210 30 . . PUNCT cana-2410 211 1 ( ( PUNCT cana-2410 211 2 3.17 3.17 NUM cana-2410 211 3 ) ) PUNCT cana-2410 211 4 communications communication NOUN cana-2410 211 5 on on ADP cana-2410 211 6 applied apply VERB cana-2410 211 7 nonlinear nonlinear ADJ cana-2410 211 8 analysis analysis NOUN cana-2410 211 9 issn issn NOUN cana-2410 211 10 : : PUNCT cana-2410 211 11 1074 1074 NUM cana-2410 211 12 - - PUNCT cana-2410 211 13 133x 133x NUM cana-2410 211 14 vol vol NOUN cana-2410 211 15 32 32 NUM cana-2410 211 16 no no NOUN cana-2410 211 17 . . PUNCT cana-2410 212 1 2s 2s NUM cana-2410 212 2 ( ( PUNCT cana-2410 212 3 2025 2025 NUM cana-2410 212 4 ) ) PUNCT cana-2410 212 5 360 360 NUM cana-2410 212 6 https://internationalpubls.com https://internationalpubls.com X cana-2410 212 7 ∫ ∫ PROPN cana-2410 213 1 𝑧𝛼𝑗 𝑧𝛼𝑗 PROPN cana-2410 213 2 βˆ— βˆ— VERB cana-2410 213 3 ψ(𝑑0,Οƒ(Ο„))𝑄(Ο„)𝑄 ψ(𝑑0,Οƒ(Ο„))𝑄(Ο„)𝑄 PROPN cana-2410 213 4 βˆ—(𝜏)Οˆβˆ—(𝑑0,Οƒ(Ο„))𝑧𝛼𝑗δτ βˆ—(𝜏)Οˆβˆ—(𝑑0,Οƒ(Ο„))𝑧𝛼𝑗δτ PROPN cana-2410 213 5 = = PUNCT cana-2410 213 6 0 0 NUM cana-2410 213 7 𝑑𝑗+1 𝑑𝑗+1 NUM cana-2410 213 8 s𝑗 s𝑗 X cana-2410 213 9 . . PUNCT cana-2410 214 1 ( ( PUNCT cana-2410 214 2 3.18 3.18 NUM cana-2410 214 3 ) ) PUNCT cana-2410 214 4 on on ADP cana-2410 214 5 solving solve VERB cana-2410 214 6 the the DET cana-2410 214 7 above above ADJ cana-2410 214 8 equations equation NOUN cana-2410 214 9 ( ( PUNCT cana-2410 214 10 3.17 3.17 NUM cana-2410 214 11 ) ) PUNCT cana-2410 214 12 and and CCONJ cana-2410 214 13 ( ( PUNCT cana-2410 214 14 3.18 3.18 NUM cana-2410 214 15 ) ) PUNCT cana-2410 214 16 , , PUNCT cana-2410 214 17 we we PRON cana-2410 214 18 have have VERB cana-2410 214 19 𝑧𝛼 𝑧𝛼 ADP cana-2410 214 20 βˆ—Οˆ(𝑑0 βˆ—Οˆ(𝑑0 NOUN cana-2410 214 21 , , PUNCT cana-2410 214 22 Οƒ(Ο„))𝑄(𝜏 Οƒ(Ο„))𝑄(𝜏 NOUN cana-2410 214 23 ) ) PUNCT cana-2410 215 1 = = SYM cana-2410 215 2 0 0 NUM cana-2410 215 3 , , PUNCT cana-2410 215 4 𝜏 𝜏 PROPN cana-2410 215 5 ∈ ∈ PROPN cana-2410 216 1 [ [ X cana-2410 216 2 𝑑0 𝑑0 NOUN cana-2410 216 3 , , PUNCT cana-2410 216 4 𝑑1]𝕋 𝑑1]𝕋 NUM cana-2410 216 5 𝑧𝛼𝑗 𝑧𝛼𝑗 PROPN cana-2410 216 6 βˆ— βˆ— NOUN cana-2410 216 7 ψ ψ X cana-2410 216 8 ( ( PUNCT cana-2410 216 9 𝑠𝑗 𝑠𝑗 INTJ cana-2410 216 10 , , PUNCT cana-2410 216 11 Οƒ(Ο„))𝑄(𝜏 Οƒ(Ο„))𝑄(𝜏 NOUN cana-2410 216 12 ) ) PUNCT cana-2410 216 13 = = SYM cana-2410 216 14 0 0 NUM cana-2410 216 15 , , PUNCT cana-2410 216 16 𝜏 𝜏 PROPN cana-2410 216 17 ∈ ∈ NOUN cana-2410 216 18 ( ( PUNCT cana-2410 216 19 𝑠𝑗 𝑠𝑗 INTJ cana-2410 216 20 , , PUNCT cana-2410 216 21 𝑑𝑗+1]𝕋 𝑑𝑗+1]𝕋 ADV cana-2410 216 22 , , PUNCT cana-2410 216 23 𝑗 𝑗 NOUN cana-2410 216 24 = = SYM cana-2410 216 25 1,2 1,2 NUM cana-2410 216 26 , , PUNCT cana-2410 216 27 … … PUNCT cana-2410 216 28 , , PUNCT cana-2410 216 29 π‘š. π‘š. ADV cana-2410 216 30 therefore therefore ADV cana-2410 216 31 , , PUNCT cana-2410 216 32 the the DET cana-2410 216 33 system system NOUN cana-2410 216 34 ( ( PUNCT cana-2410 216 35 3.1 3.1 NUM cana-2410 216 36 ) ) PUNCT cana-2410 216 37 is be AUX cana-2410 216 38 complete complete ADJ cana-2410 216 39 controllable controllable ADJ cana-2410 216 40 on[𝑑0 on[𝑑0 PROPN cana-2410 216 41 , , PUNCT cana-2410 216 42 𝑑1]𝕋 𝑑1]𝕋 ADV cana-2410 216 43 , , PUNCT cana-2410 216 44 so so CCONJ cana-2410 216 45 , , PUNCT cana-2410 216 46 if if SCONJ cana-2410 216 47 we we PRON cana-2410 216 48 choose choose VERB cana-2410 216 49 𝑧0 𝑧0 PROPN cana-2410 216 50 = = SYM cana-2410 216 51 𝑧𝛼 𝑧𝛼 PROPN cana-2410 216 52 + + NUM cana-2410 216 53 ψ(𝑑0 ψ(𝑑0 NOUN cana-2410 216 54 , , PUNCT cana-2410 216 55 𝑑1)𝑧𝛼𝑗 𝑑1)𝑧𝛼𝑗 PROPN cana-2410 216 56 βˆ’βˆ« βˆ’βˆ« NOUN cana-2410 216 57 ψ(𝑑0 ψ(𝑑0 NOUN cana-2410 216 58 , , PUNCT cana-2410 216 59 𝜎(𝑠))𝐾(𝑑 𝜎(𝑠))𝐾(𝑑 NOUN cana-2410 216 60 , , PUNCT cana-2410 216 61 𝑠)𝑧(𝑠)βˆ†π‘  𝑠)𝑧(𝑠)βˆ†π‘  PROPN cana-2410 216 62 𝑑1 𝑑1 PROPN cana-2410 216 63 𝑑0 𝑑0 PROPN cana-2410 216 64 βˆ’βˆ« βˆ’βˆ« NOUN cana-2410 216 65 ψ(𝑑0 ψ(𝑑0 NOUN cana-2410 216 66 , , PUNCT cana-2410 216 67 Οƒ(Ο„))𝑓(𝜏 Οƒ(Ο„))𝑓(𝜏 NOUN cana-2410 216 68 , , PUNCT cana-2410 216 69 𝑧(𝜏))βˆ†πœ 𝑧(𝜏))βˆ†πœ NOUN cana-2410 216 70 , , PUNCT cana-2410 216 71 𝑑1 𝑑1 NOUN cana-2410 216 72 𝑑0 𝑑0 NOUN cana-2410 216 73 in in ADP cana-2410 216 74 [ [ X cana-2410 216 75 𝑑0 𝑑0 NOUN cana-2410 216 76 , , PUNCT cana-2410 216 77 𝑑1]𝕋. 𝑑1]𝕋. PROPN cana-2410 216 78 in in ADP cana-2410 216 79 that that DET cana-2410 216 80 case case NOUN cana-2410 216 81 , , PUNCT cana-2410 216 82 there there PRON cana-2410 216 83 exist exist VERB cana-2410 216 84 a a DET cana-2410 216 85 piece piece NOUN cana-2410 216 86 - - PUNCT cana-2410 216 87 wise wise ADJ cana-2410 216 88 rd rd NOUN cana-2410 216 89 - - ADJ cana-2410 216 90 continuous continuous ADJ cana-2410 216 91 control control NOUN cana-2410 216 92 uΜ‚(t uΜ‚(t NOUN cana-2410 216 93 ) ) PUNCT cana-2410 216 94 that that PRON cana-2410 216 95 𝑧𝛼1 𝑧𝛼1 NOUN cana-2410 216 96 = = SYM cana-2410 216 97 ψ(𝑑1 ψ(𝑑1 NOUN cana-2410 216 98 , , PUNCT cana-2410 216 99 𝑑0 𝑑0 PROPN cana-2410 216 100 ) ) PUNCT cana-2410 216 101 ( ( PUNCT cana-2410 216 102 𝑧𝛼 𝑧𝛼 X cana-2410 216 103 + + NUM cana-2410 216 104 ψ(𝑑0 ψ(𝑑0 NUM cana-2410 216 105 , , PUNCT cana-2410 216 106 𝑑1)𝑧𝛼1 𝑑1)𝑧𝛼1 VERB cana-2410 216 107 βˆ’βˆ« βˆ’βˆ« NOUN cana-2410 216 108 ψ(𝑑0 ψ(𝑑0 NOUN cana-2410 216 109 , , PUNCT cana-2410 216 110 𝜎(𝑠))𝐾(𝑑 𝜎(𝑠))𝐾(𝑑 NOUN cana-2410 216 111 , , PUNCT cana-2410 216 112 𝑠)𝑧(𝑠)βˆ†π‘  𝑠)𝑧(𝑠)βˆ†π‘  PROPN cana-2410 216 113 𝑑1 𝑑1 PROPN cana-2410 216 114 𝑑0 𝑑0 PROPN cana-2410 216 115 βˆ’βˆ« βˆ’βˆ« NOUN cana-2410 216 116 ψ(𝑑0,Οƒ(Ο„))𝑓(𝜏 ψ(𝑑0,Οƒ(Ο„))𝑓(𝜏 NOUN cana-2410 216 117 , , PUNCT cana-2410 216 118 𝑧(𝜏))βˆ†πœ 𝑧(𝜏))βˆ†πœ NOUN cana-2410 216 119 𝑑1 𝑑1 NOUN cana-2410 216 120 𝑑0 𝑑0 NOUN cana-2410 216 121 ) ) PUNCT cana-2410 217 1 + + CCONJ cana-2410 217 2 ∫ ∫ PROPN cana-2410 217 3 ψ(𝑑0 ψ(𝑑0 NOUN cana-2410 217 4 , , PUNCT cana-2410 217 5 𝜎(𝑠))𝐾(𝑑 𝜎(𝑠))𝐾(𝑑 NOUN cana-2410 217 6 , , PUNCT cana-2410 217 7 𝑠)𝑧(𝑠)βˆ†π‘  𝑠)𝑧(𝑠)βˆ†π‘  PROPN cana-2410 217 8 𝑑1 𝑑1 PROPN cana-2410 217 9 𝑑0 𝑑0 PROPN cana-2410 217 10 + + CCONJ cana-2410 217 11 ∫ ∫ PROPN cana-2410 217 12 ψ(𝑑0,Οƒ(Ο„ ψ(𝑑0,Οƒ(Ο„ NOUN cana-2410 217 13 ) ) PUNCT cana-2410 217 14 ) ) PUNCT cana-2410 217 15 ( ( PUNCT cana-2410 217 16 𝑓(𝜏 𝑓(𝜏 PROPN cana-2410 217 17 , , PUNCT cana-2410 217 18 𝑧(𝜏 𝑧(𝜏 PROPN cana-2410 217 19 ) ) PUNCT cana-2410 217 20 ) ) PUNCT cana-2410 218 1 + + CCONJ cana-2410 218 2 𝑄(Ο„)uΜ‚(𝜏))βˆ†πœ 𝑄(Ο„)uΜ‚(𝜏))βˆ†πœ NUM cana-2410 218 3 𝑑1 𝑑1 NOUN cana-2410 218 4 𝑑0 𝑑0 NOUN cana-2410 218 5 , , PUNCT cana-2410 218 6 which which PRON cana-2410 218 7 gives give VERB cana-2410 218 8 𝑧𝛼 𝑧𝛼 ADP cana-2410 218 9 βˆ—π‘§π›Ό βˆ—π‘§π›Ό NOUN cana-2410 218 10 = = NOUN cana-2410 218 11 0 0 X cana-2410 218 12 . . PUNCT cana-2410 219 1 similarly similarly ADV cana-2410 219 2 , , PUNCT cana-2410 219 3 we we PRON cana-2410 219 4 have have VERB cana-2410 219 5 [ [ X cana-2410 219 6 𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 𝐼𝑛⨂𝑅𝑗]𝑧(𝑑𝑗 X cana-2410 219 7 βˆ’ βˆ’ NOUN cana-2410 219 8 ) ) PUNCT cana-2410 220 1 = = SYM cana-2410 220 2 𝑧𝛼𝑗 𝑧𝛼𝑗 NOUN cana-2410 220 3 + + CCONJ cana-2410 220 4 ψ(𝑠𝑗 ψ(𝑠𝑗 PROPN cana-2410 220 5 , , PUNCT cana-2410 220 6 𝑑𝑗+1)𝑧𝑑𝑗+1 𝑑𝑗+1)𝑧𝑑𝑗+1 ADP cana-2410 220 7 βˆ’βˆ« βˆ’βˆ« X cana-2410 220 8 ψ(𝑑0 ψ(𝑑0 NOUN cana-2410 220 9 , , PUNCT cana-2410 220 10 𝜎(𝑠))𝐾(𝑑 𝜎(𝑠))𝐾(𝑑 NOUN cana-2410 220 11 , , PUNCT cana-2410 220 12 𝑠)𝑧(𝑠)βˆ†π‘  𝑠)𝑧(𝑠)βˆ†π‘  PROPN cana-2410 220 13 𝑑𝑗+1 𝑑𝑗+1 NUM cana-2410 220 14 𝑠𝑗 𝑠𝑗 NOUN cana-2410 220 15 βˆ’βˆ« βˆ’βˆ« NOUN cana-2410 220 16 ψ(𝑑0,Οƒ(Ο„))𝑓(𝜏 ψ(𝑑0,Οƒ(Ο„))𝑓(𝜏 NOUN cana-2410 220 17 , , PUNCT cana-2410 220 18 𝑧(𝜏))βˆ†πœ 𝑧(𝜏))βˆ†πœ NOUN cana-2410 220 19 𝑑𝑗+1 𝑑𝑗+1 X cana-2410 220 20 𝑠𝑗 𝑠𝑗 NOUN cana-2410 220 21 . . PUNCT cana-2410 221 1 it it PRON cana-2410 221 2 can can AUX cana-2410 221 3 be be AUX cana-2410 221 4 shown show VERB cana-2410 221 5 that that SCONJ cana-2410 221 6 𝑧𝛼𝑗 𝑧𝛼𝑗 PROPN cana-2410 221 7 βˆ— βˆ— VERB cana-2410 221 8 𝑧𝛼𝑗 𝑧𝛼𝑗 NOUN cana-2410 221 9 = = SYM cana-2410 221 10 0 0 PROPN cana-2410 221 11 , , PUNCT cana-2410 221 12 which which PRON cana-2410 221 13 contradicts contradict VERB cana-2410 221 14 the the DET cana-2410 221 15 fact fact NOUN cana-2410 221 16 that that SCONJ cana-2410 221 17 𝑧𝛼 𝑧𝛼 ADP cana-2410 221 18 βˆ—π‘§π›Ό βˆ—π‘§π›Ό VERB cana-2410 221 19 β‰  β‰  PROPN cana-2410 221 20 0,therefore 0,therefore NOUN cana-2410 221 21 , , PUNCT cana-2410 221 22 the the DET cana-2410 221 23 matrices matrix NOUN cana-2410 221 24 𝒩0(𝑑0 𝒩0(𝑑0 NOUN cana-2410 221 25 , , PUNCT cana-2410 221 26 𝑑1 𝑑1 NOUN cana-2410 221 27 ) ) PUNCT cana-2410 221 28 and and CCONJ cana-2410 221 29 𝒩𝑗(𝑠𝑗 𝒩𝑗(𝑠𝑗 NOUN cana-2410 221 30 , , PUNCT cana-2410 221 31 𝑑𝑗+1 𝑑𝑗+1 X cana-2410 221 32 ) ) PUNCT cana-2410 221 33 are be AUX cana-2410 221 34 invertible invertible ADJ cana-2410 221 35 . . PUNCT cana-2410 222 1 theorem theorem VERB cana-2410 222 2 3.3 3.3 NUM cana-2410 222 3 : : PUNCT cana-2410 222 4 assuming assume VERB cana-2410 222 5 that that SCONJ cana-2410 222 6 requirements requirement NOUN cana-2410 222 7 ( ( PUNCT cana-2410 222 8 h1 h1 PROPN cana-2410 222 9 ) ) PUNCT cana-2410 222 10 ( ( PUNCT cana-2410 222 11 h3 h3 NOUN cana-2410 222 12 ) ) PUNCT cana-2410 222 13 are be AUX cana-2410 222 14 satisfied satisfied ADJ cana-2410 222 15 ; ; PUNCT cana-2410 222 16 the the DET cana-2410 222 17 time time NOUN cana-2410 222 18 - - PUNCT cana-2410 222 19 invariant invariant ADJ cana-2410 222 20 case case NOUN cana-2410 222 21 of of ADP cana-2410 222 22 system system NOUN cana-2410 222 23 ( ( PUNCT cana-2410 222 24 3.1 3.1 NUM cana-2410 222 25 ) ) PUNCT cana-2410 222 26 is be AUX cana-2410 222 27 said say VERB cana-2410 222 28 to to PART cana-2410 222 29 be be AUX cana-2410 222 30 complete complete ADJ cana-2410 222 31 controllable controllable ADJ cana-2410 222 32 in in ADP cana-2410 222 33 interval interval NOUN cana-2410 222 34 [ [ X cana-2410 222 35 𝑑0 𝑑0 NOUN cana-2410 222 36 , , PUNCT cana-2410 222 37 𝑇]𝕋 𝑇]𝕋 ADJ cana-2410 222 38 if if SCONJ cana-2410 222 39 and and CCONJ cana-2410 222 40 only only ADV cana-2410 222 41 if if SCONJ cana-2410 222 42 the the DET cana-2410 222 43 rank rank NOUN cana-2410 222 44 of of ADP cana-2410 222 45 the the DET cana-2410 222 46 matrix matrix NOUN cana-2410 222 47 [ [ X cana-2410 222 48 𝑄 𝑄 PROPN cana-2410 222 49 𝑃𝑄 𝑃𝑄 PROPN cana-2410 222 50 𝑃2𝑄 𝑃2𝑄 NUM cana-2410 222 51 … … PUNCT cana-2410 222 52 π‘ƒπ‘›βˆ’1𝑄 π‘ƒπ‘›βˆ’1𝑄 X cana-2410 222 53 ] ] X cana-2410 222 54 = = SYM cana-2410 222 55 𝑛2 𝑛2 NOUN cana-2410 222 56 ( ( PUNCT cana-2410 222 57 3.19 3.19 NUM cana-2410 222 58 ) ) PUNCT cana-2410 222 59 proof proof NOUN cana-2410 222 60 : : PUNCT cana-2410 222 61 assume assume VERB cana-2410 222 62 that that SCONJ cana-2410 222 63 system system NOUN cana-2410 222 64 ( ( PUNCT cana-2410 222 65 3.1 3.1 NUM cana-2410 222 66 ) ) PUNCT cana-2410 222 67 is be AUX cana-2410 222 68 to to PART cana-2410 222 69 be be AUX cana-2410 222 70 complete complete ADJ cana-2410 222 71 controllable controllable ADJ cana-2410 222 72 in in ADP cana-2410 222 73 [ [ X cana-2410 222 74 𝑑0 𝑑0 NOUN cana-2410 222 75 , , PUNCT cana-2410 222 76 𝑇]𝕋. 𝑇]𝕋. PROPN cana-2410 222 77 but but CCONJ cana-2410 222 78 the the DET cana-2410 222 79 rank rank NOUN cana-2410 222 80 of of ADP cana-2410 222 81 𝐢 𝐢 PROPN cana-2410 222 82 β‰  β‰  PROPN cana-2410 222 83 𝑛2(∡ 𝑛2(∡ PROPN cana-2410 223 1 [ [ X cana-2410 223 2 𝑄 𝑄 PROPN cana-2410 223 3 𝑃𝑄 𝑃𝑄 PROPN cana-2410 223 4 𝑃2𝑄 𝑃2𝑄 NUM cana-2410 223 5 … … PUNCT cana-2410 223 6 𝑃𝑛 𝑃𝑛 PROPN cana-2410 223 7 2βˆ’1𝑄 2βˆ’1𝑄 PROPN cana-2410 223 8 ] ] X cana-2410 223 9 = = SYM cana-2410 223 10 𝐢 𝐢 PROPN cana-2410 223 11 ) ) PUNCT cana-2410 223 12 , , PUNCT cana-2410 223 13 then then ADV cana-2410 223 14 there there PRON cana-2410 223 15 exists exist VERB cana-2410 223 16 non non ADJ cana-2410 223 17 - - ADJ cana-2410 223 18 zero zero NUM cana-2410 223 19 vector vector NOUN cana-2410 223 20 𝑧𝛼 𝑧𝛼 X cana-2410 223 21 ∈ ∈ PROPN cana-2410 224 1 ℝ𝑛 ℝ𝑛 PROPN cana-2410 224 2 2 2 NUM cana-2410 224 3 such such ADJ cana-2410 224 4 that that DET cana-2410 224 5 𝑧𝛼 𝑧𝛼 ADP cana-2410 224 6 βˆ—π‘ƒπ‘–π΅ βˆ—π‘ƒπ‘–π΅ PROPN cana-2410 224 7 = = SYM cana-2410 224 8 0 0 NUM cana-2410 224 9 , , PUNCT cana-2410 224 10 𝑖 𝑖 NOUN cana-2410 224 11 = = SYM cana-2410 224 12 0,1 0,1 NUM cana-2410 224 13 , , PUNCT cana-2410 224 14 … … PUNCT cana-2410 224 15 , , PUNCT cana-2410 224 16 𝑛2 𝑛2 NOUN cana-2410 224 17 βˆ’ βˆ’ PROPN cana-2410 224 18 1 1 NUM cana-2410 224 19 . . PUNCT cana-2410 225 1 ( ( PUNCT cana-2410 225 2 3.20 3.20 NUM cana-2410 225 3 ) ) PUNCT cana-2410 225 4 furthermore furthermore ADV cana-2410 225 5 , , PUNCT cana-2410 225 6 based base VERB cana-2410 225 7 on on ADP cana-2410 225 8 equations equation NOUN cana-2410 225 9 ( ( PUNCT cana-2410 225 10 3.4 3.4 NUM cana-2410 225 11 ) ) PUNCT cana-2410 225 12 and and CCONJ cana-2410 225 13 ( ( PUNCT cana-2410 225 14 3.5 3.5 NUM cana-2410 225 15 ) ) PUNCT cana-2410 225 16 , , PUNCT cana-2410 225 17 we we PRON cana-2410 225 18 can can AUX cana-2410 225 19 deduce deduce VERB cana-2410 225 20 𝑧𝛼 𝑧𝛼 ADP cana-2410 225 21 βˆ—π’©0(𝑑0 βˆ—π’©0(𝑑0 NUM cana-2410 225 22 , , PUNCT cana-2410 225 23 𝑑1)𝑧𝛼 𝑑1)𝑧𝛼 PROPN cana-2410 225 24 = = SYM cana-2410 226 1 ∫ ∫ PROPN cana-2410 226 2 𝑧𝛼 𝑧𝛼 ADP cana-2410 226 3 βˆ—π‘’π‘ƒ(𝑑0,Οƒ(Ο„))𝑄𝑄 βˆ—π‘’π‘ƒ(𝑑0,Οƒ(Ο„))𝑄𝑄 ADJ cana-2410 226 4 βˆ—π‘’π‘ƒ βˆ—π‘’π‘ƒ PROPN cana-2410 226 5 βˆ—(𝑑0,Οƒ(Ο„))𝑧𝛼δτ βˆ—(𝑑0,Οƒ(Ο„))𝑧𝛼δτ NOUN cana-2410 226 6 t t PROPN cana-2410 226 7 t0 t0 PROPN cana-2410 226 8 ( ( PUNCT cana-2410 226 9 3.21 3.21 NUM cana-2410 226 10 ) ) PUNCT cana-2410 226 11 communications communication NOUN cana-2410 226 12 on on ADP cana-2410 226 13 applied apply VERB cana-2410 226 14 nonlinear nonlinear ADJ cana-2410 226 15 analysis analysis NOUN cana-2410 226 16 issn issn NOUN cana-2410 226 17 : : PUNCT cana-2410 226 18 1074 1074 NUM cana-2410 226 19 - - PUNCT cana-2410 226 20 133x 133x NUM cana-2410 226 21 vol vol NOUN cana-2410 226 22 32 32 NUM cana-2410 226 23 no no NOUN cana-2410 226 24 . . PUNCT cana-2410 227 1 2s 2s NUM cana-2410 227 2 ( ( PUNCT cana-2410 227 3 2025 2025 NUM cana-2410 227 4 ) ) PUNCT cana-2410 227 5 361 361 NUM cana-2410 227 6 https://internationalpubls.com https://internationalpubls.com X cana-2410 227 7 𝑧𝛼 𝑧𝛼 X cana-2410 227 8 βˆ—π’©π‘˜(𝑠𝑗 βˆ—π’©π‘˜(𝑠𝑗 PROPN cana-2410 227 9 , , PUNCT cana-2410 227 10 𝑑𝑗+1)𝑧𝛼 𝑑𝑗+1)𝑧𝛼 PROPN cana-2410 227 11 = = SYM cana-2410 227 12 ∫ ∫ PROPN cana-2410 227 13 𝑧𝛼 𝑧𝛼 X cana-2410 227 14 βˆ—π‘’π‘ƒ βˆ—π‘’π‘ƒ PROPN cana-2410 227 15 ( ( PUNCT cana-2410 227 16 𝑠𝑗 𝑠𝑗 INTJ cana-2410 227 17 , , PUNCT cana-2410 227 18 Οƒ(Ο„))𝑄𝑄 Οƒ(Ο„))𝑄𝑄 ADJ cana-2410 227 19 βˆ—π‘’π‘ƒ βˆ—π‘’π‘ƒ PROPN cana-2410 227 20 βˆ— βˆ— NOUN cana-2410 227 21 ( ( PUNCT cana-2410 227 22 𝑠𝑗 𝑠𝑗 INTJ cana-2410 227 23 , , PUNCT cana-2410 227 24 Οƒ(Ο„ Οƒ(Ο„ PROPN cana-2410 227 25 ) ) PUNCT cana-2410 227 26 ) ) PUNCT cana-2410 227 27 𝑧𝛼δτ 𝑧𝛼δτ VERB cana-2410 227 28 𝑑𝑗+1 𝑑𝑗+1 NUM cana-2410 227 29 s𝑗 s𝑗 PROPN cana-2410 227 30 . . PUNCT cana-2410 228 1 ( ( PUNCT cana-2410 228 2 3.22 3.22 NUM cana-2410 228 3 ) ) PUNCT cana-2410 228 4 now now ADV cana-2410 228 5 , , PUNCT cana-2410 228 6 we we PRON cana-2410 228 7 are be AUX cana-2410 228 8 using use VERB cana-2410 228 9 theorem theorem ADJ cana-2410 228 10 2.4 2.4 NUM cana-2410 228 11 . . PUNCT cana-2410 229 1 and and CCONJ cana-2410 229 2 from from ADP cana-2410 229 3 equation equation NOUN cana-2410 229 4 ( ( PUNCT cana-2410 229 5 3.20 3.20 NUM cana-2410 229 6 ) ) PUNCT cana-2410 229 7 in in ADP cana-2410 229 8 the the DET cana-2410 229 9 above above ADJ cana-2410 229 10 equation equation NOUN cana-2410 229 11 ( ( PUNCT cana-2410 229 12 3.21 3.21 NUM cana-2410 229 13 ) ) PUNCT cana-2410 229 14 and and CCONJ cana-2410 229 15 ( ( PUNCT cana-2410 229 16 3.22 3.22 NUM cana-2410 229 17 ) ) PUNCT cana-2410 229 18 , , PUNCT cana-2410 229 19 we we PRON cana-2410 229 20 have have VERB cana-2410 229 21 𝑧𝛼 𝑧𝛼 NUM cana-2410 229 22 βˆ—π’©0(𝑑0 βˆ—π’©0(𝑑0 NOUN cana-2410 229 23 , , PUNCT cana-2410 229 24 𝑑1)𝑧𝛼 𝑑1)𝑧𝛼 PROPN cana-2410 229 25 = = SYM cana-2410 230 1 ∫ ∫ PROPN cana-2410 231 1 [ [ X cana-2410 231 2 βˆ‘ βˆ‘ INTJ cana-2410 231 3 𝛾𝑗 𝛾𝑗 INTJ cana-2410 231 4 𝑛2βˆ’1 𝑛2βˆ’1 INTJ cana-2410 231 5 𝑗=0 𝑗=0 PROPN cana-2410 231 6 ( ( PUNCT cana-2410 231 7 𝑑0 𝑑0 PROPN cana-2410 231 8 , , PUNCT cana-2410 231 9 Οƒ(Ο„))𝑧𝛼 Οƒ(Ο„))𝑧𝛼 PROPN cana-2410 231 10 βˆ—π‘ƒπ‘–π‘„]π‘„βˆ—π‘’π‘ƒ βˆ—π‘ƒπ‘–π‘„]π‘„βˆ—π‘’π‘ƒ PROPN cana-2410 231 11 βˆ—(𝑑0 βˆ—(𝑑0 PROPN cana-2410 231 12 , , PUNCT cana-2410 231 13 Οƒ(Ο„))𝑧𝛼δτ Οƒ(Ο„))𝑧𝛼δτ PROPN cana-2410 231 14 = = SYM cana-2410 231 15 0 0 PROPN cana-2410 231 16 t t PROPN cana-2410 231 17 t0 t0 PROPN cana-2410 231 18 𝑧𝛼 𝑧𝛼 ADP cana-2410 231 19 βˆ—π’©π‘—(𝑠𝑗 βˆ—π’©π‘—(𝑠𝑗 PROPN cana-2410 231 20 , , PUNCT cana-2410 231 21 𝑑𝑗+1)𝑧𝛼 𝑑𝑗+1)𝑧𝛼 PROPN cana-2410 231 22 = = SYM cana-2410 231 23 ∫ ∫ PROPN cana-2410 232 1 [ [ X cana-2410 232 2 βˆ‘ βˆ‘ INTJ cana-2410 232 3 𝛾𝑗 𝛾𝑗 INTJ cana-2410 232 4 𝑛2βˆ’1 𝑛2βˆ’1 INTJ cana-2410 232 5 𝑗=0 𝑗=0 PROPN cana-2410 232 6 ( ( PUNCT cana-2410 232 7 𝑠𝑗 𝑠𝑗 INTJ cana-2410 232 8 , , PUNCT cana-2410 232 9 Οƒ(Ο„ Οƒ(Ο„ PROPN cana-2410 232 10 ) ) PUNCT cana-2410 232 11 ) ) PUNCT cana-2410 233 1 𝑧𝛼 𝑧𝛼 CCONJ cana-2410 233 2 βˆ—π‘ƒπ‘–π‘„]π‘„βˆ—π‘’π‘ƒ βˆ—π‘ƒπ‘–π‘„]π‘„βˆ—π‘’π‘ƒ PROPN cana-2410 233 3 βˆ— βˆ— NOUN cana-2410 233 4 ( ( PUNCT cana-2410 233 5 𝑠𝑗 𝑠𝑗 INTJ cana-2410 233 6 , , PUNCT cana-2410 233 7 Οƒ(Ο„ Οƒ(Ο„ PROPN cana-2410 233 8 ) ) PUNCT cana-2410 233 9 ) ) PUNCT cana-2410 233 10 𝑧𝛼δτ 𝑧𝛼δτ NOUN cana-2410 233 11 = = NOUN cana-2410 233 12 0 0 X cana-2410 233 13 . . X cana-2410 233 14 𝑑𝑗+1 𝑑𝑗+1 X cana-2410 233 15 s𝑗 s𝑗 PROPN cana-2410 233 16 thus thus ADV cana-2410 233 17 , , PUNCT cana-2410 233 18 𝒩0(𝑑0 𝒩0(𝑑0 NOUN cana-2410 233 19 , , PUNCT cana-2410 233 20 𝑑1 𝑑1 NOUN cana-2410 233 21 ) ) PUNCT cana-2410 233 22 and and CCONJ cana-2410 233 23 𝒩𝑗(𝑠𝑗 𝒩𝑗(𝑠𝑗 NOUN cana-2410 233 24 , , PUNCT cana-2410 233 25 𝑑𝑗+1 𝑑𝑗+1 PROPN cana-2410 233 26 ) ) PUNCT cana-2410 233 27 , , PUNCT cana-2410 233 28 are be AUX cana-2410 233 29 not not PART cana-2410 233 30 invertible invertible ADJ cana-2410 233 31 . . PUNCT cana-2410 234 1 theorem theorem VERB cana-2410 234 2 3.1states 3.1states NUM cana-2410 234 3 that that DET cana-2410 234 4 system system NOUN cana-2410 234 5 ( ( PUNCT cana-2410 234 6 3.1 3.1 NUM cana-2410 234 7 ) ) PUNCT cana-2410 234 8 is be AUX cana-2410 234 9 not not PART cana-2410 234 10 completely completely ADV cana-2410 234 11 controllable controllable ADJ cana-2410 234 12 . . PUNCT cana-2410 235 1 therefore therefore ADV cana-2410 235 2 , , PUNCT cana-2410 235 3 it it PRON cana-2410 235 4 contradicts contradict VERB cana-2410 235 5 . . PUNCT cana-2410 236 1 the the DET cana-2410 236 2 rank rank NOUN cana-2410 236 3 of of ADP cana-2410 236 4 𝐢 𝐢 PROPN cana-2410 236 5 = = PUNCT cana-2410 236 6 𝑛2 𝑛2 NOUN cana-2410 236 7 . . PUNCT cana-2410 237 1 conversely conversely ADV cana-2410 237 2 , , PUNCT cana-2410 237 3 the the DET cana-2410 237 4 matrices matrix NOUN cana-2410 237 5 𝒩0(𝑑0 𝒩0(𝑑0 NOUN cana-2410 237 6 , , PUNCT cana-2410 237 7 𝑑1 𝑑1 NOUN cana-2410 237 8 ) ) PUNCT cana-2410 237 9 and and CCONJ cana-2410 237 10 𝒩𝑗(𝑠𝑗 𝒩𝑗(𝑠𝑗 NOUN cana-2410 237 11 , , PUNCT cana-2410 237 12 𝑑𝑗+1 𝑑𝑗+1 PROPN cana-2410 237 13 ) ) PUNCT cana-2410 237 14 , , PUNCT cana-2410 237 15 are be AUX cana-2410 237 16 not not PART cana-2410 237 17 invertible invertible ADJ cana-2410 237 18 and and CCONJ cana-2410 237 19 we we PRON cana-2410 237 20 assume assume VERB cana-2410 237 21 that that SCONJ cana-2410 237 22 the the DET cana-2410 237 23 rank rank NOUN cana-2410 237 24 of of ADP cana-2410 237 25 𝐢 𝐢 PROPN cana-2410 237 26 = = PUNCT cana-2410 237 27 𝑛2 𝑛2 NOUN cana-2410 237 28 . . PUNCT cana-2410 238 1 the the DET cana-2410 238 2 system system NOUN cana-2410 238 3 ( ( PUNCT cana-2410 238 4 3.1 3.1 NUM cana-2410 238 5 ) ) PUNCT cana-2410 238 6 is be AUX cana-2410 238 7 not not PART cana-2410 238 8 to to PART cana-2410 238 9 be be AUX cana-2410 238 10 complete complete ADJ cana-2410 238 11 controllable controllable ADJ cana-2410 238 12 . . PUNCT cana-2410 239 1 this this PRON cana-2410 239 2 means mean VERB cana-2410 239 3 that that SCONJ cana-2410 239 4 there there PRON cana-2410 239 5 exists exist VERB cana-2410 239 6 non non ADJ cana-2410 239 7 - - ADJ cana-2410 239 8 zero zero NUM cana-2410 239 9 vectors vector NOUN cana-2410 239 10 𝑧𝛼 𝑧𝛼 ADP cana-2410 239 11 , , PUNCT cana-2410 239 12 𝑧𝛼𝑗 𝑧𝛼𝑗 PROPN cana-2410 239 13 ∈ ∈ PROPN cana-2410 240 1 ℝ𝑛 ℝ𝑛 ADJ cana-2410 240 2 2 2 NUM cana-2410 240 3 , , PUNCT cana-2410 240 4 such such ADJ cana-2410 240 5 that that PRON cana-2410 240 6 𝑧𝛼 𝑧𝛼 ADP cana-2410 240 7 βˆ—π’©0(𝑑0 βˆ—π’©0(𝑑0 NUM cana-2410 240 8 , , PUNCT cana-2410 240 9 𝑑1)𝑧𝛼 𝑑1)𝑧𝛼 PROPN cana-2410 240 10 = = SYM cana-2410 240 11 0 0 NUM cana-2410 240 12 . . PUNCT cana-2410 241 1 ( ( PUNCT cana-2410 241 2 3.23 3.23 NUM cana-2410 241 3 ) ) PUNCT cana-2410 241 4 and and CCONJ cana-2410 241 5 𝑧𝛼𝑗 𝑧𝛼𝑗 PROPN cana-2410 241 6 βˆ— βˆ— NOUN cana-2410 241 7 𝒩𝑗(𝑠𝑗 𝒩𝑗(𝑠𝑗 PROPN cana-2410 241 8 , , PUNCT cana-2410 241 9 𝑑𝑗+1)𝑧𝛼𝑗 𝑑𝑗+1)𝑧𝛼𝑗 PROPN cana-2410 241 10 = = SYM cana-2410 241 11 0 0 NUM cana-2410 241 12 , , PUNCT cana-2410 241 13 𝑗 𝑗 NOUN cana-2410 241 14 = = SYM cana-2410 241 15 1,2 1,2 NUM cana-2410 241 16 … … PUNCT cana-2410 241 17 , , PUNCT cana-2410 241 18 π‘š π‘š NOUN cana-2410 241 19 , , PUNCT cana-2410 241 20 ( ( PUNCT cana-2410 241 21 3.24 3.24 NUM cana-2410 241 22 ) ) PUNCT cana-2410 241 23 now now ADV cana-2410 241 24 , , PUNCT cana-2410 241 25 from from ADP cana-2410 241 26 the the DET cana-2410 241 27 equations equation NOUN cana-2410 241 28 ( ( PUNCT cana-2410 241 29 3.4 3.4 NUM cana-2410 241 30 ) ) PUNCT cana-2410 241 31 , , PUNCT cana-2410 241 32 ( ( PUNCT cana-2410 241 33 3.5 3.5 NUM cana-2410 241 34 ) ) PUNCT cana-2410 241 35 , , PUNCT cana-2410 241 36 ( ( PUNCT cana-2410 241 37 3.23 3.23 NUM cana-2410 241 38 ) ) PUNCT cana-2410 241 39 and and CCONJ cana-2410 241 40 ( ( PUNCT cana-2410 241 41 3.24 3.24 NUM cana-2410 241 42 ) ) PUNCT cana-2410 241 43 , , PUNCT cana-2410 241 44 we we PRON cana-2410 241 45 have have VERB cana-2410 241 46 𝑧𝛼 𝑧𝛼 ADP cana-2410 241 47 βˆ—π‘’π‘ƒ(𝑑0 βˆ—π‘’π‘ƒ(𝑑0 NOUN cana-2410 241 48 , , PUNCT cana-2410 241 49 𝑑1)𝑄 𝑑1)𝑄 X cana-2410 241 50 = = SYM cana-2410 241 51 0 0 NUM cana-2410 241 52 , , PUNCT cana-2410 241 53 βˆ€ βˆ€ PUNCT cana-2410 241 54 𝑑 𝑑 X cana-2410 241 55 ∈ ∈ PROPN cana-2410 241 56 [ [ X cana-2410 241 57 𝑑0 𝑑0 NOUN cana-2410 241 58 , , PUNCT cana-2410 241 59 𝑑1]𝕋 𝑑1]𝕋 PROPN cana-2410 241 60 ( ( PUNCT cana-2410 241 61 3.25 3.25 NUM cana-2410 241 62 ) ) PUNCT cana-2410 241 63 and and CCONJ cana-2410 241 64 𝑧𝛼 𝑧𝛼 ADP cana-2410 241 65 βˆ—π‘’π‘ƒ(𝑠𝑗 βˆ—π‘’π‘ƒ(𝑠𝑗 NOUN cana-2410 241 66 , , PUNCT cana-2410 241 67 𝑑𝑗+1)𝑄 𝑑𝑗+1)𝑄 NOUN cana-2410 241 68 = = SYM cana-2410 241 69 0 0 NUM cana-2410 241 70 , , PUNCT cana-2410 241 71 βˆ€ βˆ€ PUNCT cana-2410 241 72 𝑑 𝑑 NOUN cana-2410 241 73 ∈ ∈ PROPN cana-2410 241 74 ( ( PUNCT cana-2410 241 75 𝑠𝑗 𝑠𝑗 INTJ cana-2410 241 76 , , PUNCT cana-2410 241 77 𝑑𝑗+1]𝕋 𝑑𝑗+1]𝕋 ADV cana-2410 241 78 , , PUNCT cana-2410 241 79 , , PUNCT cana-2410 241 80 𝑗 𝑗 NOUN cana-2410 241 81 = = SYM cana-2410 241 82 1,2 1,2 NUM cana-2410 241 83 , , PUNCT cana-2410 241 84 … … PUNCT cana-2410 241 85 , , PUNCT cana-2410 241 86 π‘š π‘š NOUN cana-2410 241 87 , , PUNCT cana-2410 241 88 ( ( PUNCT cana-2410 241 89 3.26 3.26 NUM cana-2410 241 90 ) ) PUNCT cana-2410 241 91 now now ADV cana-2410 241 92 , , PUNCT cana-2410 241 93 for for ADP cana-2410 241 94 𝑗 𝑗 NOUN cana-2410 241 95 = = SYM cana-2410 241 96 1,2 1,2 NUM cana-2410 241 97 , , PUNCT cana-2410 241 98 … … PUNCT cana-2410 241 99 , , PUNCT cana-2410 241 100 π‘š π‘š NOUN cana-2410 241 101 , , PUNCT cana-2410 241 102 the the DET cana-2410 241 103 𝑒𝑃(𝑑0 𝑒𝑃(𝑑0 NOUN cana-2410 241 104 , , PUNCT cana-2410 241 105 . . PUNCT cana-2410 241 106 ) ) PUNCT cana-2410 241 107 , , PUNCT cana-2410 241 108 𝑒𝑃(𝑠𝑗 𝑒𝑃(𝑠𝑗 PROPN cana-2410 241 109 , , PUNCT cana-2410 241 110 . . PUNCT cana-2410 241 111 ) ) PUNCT cana-2410 242 1 are be AUX cana-2410 242 2 rd rd NOUN cana-2410 242 3 - - ADJ cana-2410 242 4 continuous continuous ADJ cana-2410 242 5 and and CCONJ cana-2410 242 6 𝜎([𝑑0 𝜎([𝑑0 NUM cana-2410 242 7 , , PUNCT cana-2410 242 8 𝑑1]𝕋 𝑑1]𝕋 NUM cana-2410 242 9 ) ) PUNCT cana-2410 242 10 , , PUNCT cana-2410 242 11 𝜎((𝑠𝑗 𝜎((𝑠𝑗 PROPN cana-2410 242 12 , , PUNCT cana-2410 242 13 𝑑𝑗+1]𝕋 𝑑𝑗+1]𝕋 NOUN cana-2410 242 14 ) ) PUNCT cana-2410 242 15 are be AUX cana-2410 242 16 density density NOUN cana-2410 242 17 argument argument NOUN cana-2410 243 1 [ [ X cana-2410 243 2 𝜎(𝑑0 𝜎(𝑑0 NOUN cana-2410 243 3 ) ) PUNCT cana-2410 243 4 , , PUNCT cana-2410 244 1 𝜎(𝑑1)]𝕋 𝜎(𝑑1)]𝕋 NOUN cana-2410 244 2 = = PUNCT cana-2410 245 1 [ [ X cana-2410 245 2 𝑑0 𝑑0 NOUN cana-2410 245 3 , , PUNCT cana-2410 245 4 𝑑1]𝕋 𝑑1]𝕋 PROPN cana-2410 245 5 , , PUNCT cana-2410 245 6 ( ( PUNCT cana-2410 245 7 𝜎(𝑠𝑗 𝜎(𝑠𝑗 NUM cana-2410 245 8 ) ) PUNCT cana-2410 245 9 , , PUNCT cana-2410 245 10 𝜎(𝑑𝑗+1)]𝕋 𝜎(𝑑𝑗+1)]𝕋 X cana-2410 245 11 = = SYM cana-2410 245 12 ( ( PUNCT cana-2410 245 13 𝑠𝑗 𝑠𝑗 ADP cana-2410 245 14 , , PUNCT cana-2410 245 15 𝑑𝑗+1]𝕋 𝑑𝑗+1]𝕋 NOUN cana-2410 245 16 , , PUNCT cana-2410 245 17 hence hence ADV cana-2410 245 18 , , PUNCT cana-2410 245 19 from from ADP cana-2410 245 20 the the DET cana-2410 245 21 above above ADJ cana-2410 245 22 equations equation NOUN cana-2410 245 23 ( ( PUNCT cana-2410 245 24 3.25 3.25 NUM cana-2410 245 25 ) ) PUNCT cana-2410 245 26 and and CCONJ cana-2410 245 27 ( ( PUNCT cana-2410 245 28 3.26 3.26 NUM cana-2410 245 29 ) ) PUNCT cana-2410 245 30 , , PUNCT cana-2410 245 31 we we PRON cana-2410 245 32 have have VERB cana-2410 245 33 𝑧𝛼 𝑧𝛼 ADP cana-2410 245 34 βˆ—π‘’π‘ƒ(𝑑0 βˆ—π‘’π‘ƒ(𝑑0 NOUN cana-2410 245 35 , , PUNCT cana-2410 245 36 𝑑)𝑄 𝑑)𝑄 ADJ cana-2410 245 37 = = SYM cana-2410 245 38 0 0 NUM cana-2410 245 39 , , PUNCT cana-2410 245 40 βˆ€ βˆ€ PUNCT cana-2410 246 1 𝑑 𝑑 X cana-2410 246 2 ∈ ∈ PROPN cana-2410 247 1 [ [ X cana-2410 247 2 𝑑0 𝑑0 NOUN cana-2410 247 3 , , PUNCT cana-2410 247 4 𝑑1]𝕋. 𝑑1]𝕋. PROPN cana-2410 247 5 ( ( PUNCT cana-2410 247 6 3.27 3.27 NUM cana-2410 247 7 ) ) PUNCT cana-2410 247 8 𝑧𝛼 𝑧𝛼 ADP cana-2410 247 9 βˆ—π‘’π‘ƒ(𝑠𝑗 βˆ—π‘’π‘ƒ(𝑠𝑗 NOUN cana-2410 247 10 , , PUNCT cana-2410 247 11 𝑑)𝑄 𝑑)𝑄 ADJ cana-2410 247 12 = = SYM cana-2410 247 13 0 0 NUM cana-2410 247 14 , , PUNCT cana-2410 247 15 βˆ€ βˆ€ PUNCT cana-2410 247 16 𝑑 𝑑 NOUN cana-2410 247 17 ∈ ∈ PROPN cana-2410 247 18 ( ( PUNCT cana-2410 247 19 𝑠𝑗 𝑠𝑗 INTJ cana-2410 247 20 , , PUNCT cana-2410 247 21 𝑑𝑗+1]𝕋 𝑑𝑗+1]𝕋 ADV cana-2410 247 22 , , PUNCT cana-2410 247 23 𝑗 𝑗 NOUN cana-2410 247 24 = = SYM cana-2410 247 25 1,2 1,2 NUM cana-2410 247 26 , , PUNCT cana-2410 247 27 … … PUNCT cana-2410 247 28 , , PUNCT cana-2410 247 29 π‘š π‘š NOUN cana-2410 247 30 , , PUNCT cana-2410 247 31 ( ( PUNCT cana-2410 247 32 3.28 3.28 NUM cana-2410 247 33 ) ) PUNCT cana-2410 247 34 at at ADP cana-2410 247 35 𝑑 𝑑 PROPN cana-2410 247 36 = = SYM cana-2410 247 37 𝑑0 𝑑0 PROPN cana-2410 247 38 , , PUNCT cana-2410 247 39 an an DET cana-2410 247 40 equation equation NOUN cana-2410 247 41 ( ( PUNCT cana-2410 247 42 3.27 3.27 NUM cana-2410 247 43 ) ) PUNCT cana-2410 247 44 becomes become VERB cana-2410 247 45 𝑧𝛼 𝑧𝛼 ADV cana-2410 247 46 βˆ—π‘„ βˆ—π‘„ VERB cana-2410 247 47 = = NOUN cana-2410 247 48 0 0 X cana-2410 247 49 . . PUNCT cana-2410 248 1 also also ADV cana-2410 248 2 , , PUNCT cana-2410 248 3 𝑒𝑃(𝑑0 𝑒𝑃(𝑑0 PROPN cana-2410 248 4 , , PUNCT cana-2410 248 5 . . PUNCT cana-2410 248 6 ) ) PUNCT cana-2410 249 1 is be AUX cana-2410 249 2 delta delta NOUN cana-2410 249 3 differentiable differentiable ADJ cana-2410 249 4 , , PUNCT cana-2410 249 5 we we PRON cana-2410 249 6 get get VERB cana-2410 249 7 𝑒𝑃 𝑒𝑃 ADJ cana-2410 249 8 βˆ†π‘‘(𝑑0 βˆ†π‘‘(𝑑0 PROPN cana-2410 249 9 , , PUNCT cana-2410 249 10 𝑑 𝑑 X cana-2410 249 11 ) ) PUNCT cana-2410 249 12 = = SYM cana-2410 250 1 βˆ’π‘’π‘ƒ(𝑑0 βˆ’π‘’π‘ƒ(𝑑0 PROPN cana-2410 250 2 , , PUNCT cana-2410 250 3 𝜎(𝑑))𝑃. 𝜎(𝑑))𝑃. VERB cana-2410 250 4 then then ADV cana-2410 250 5 subsequent subsequent ADJ cana-2410 250 6 derivatives derivative NOUN cana-2410 250 7 and and CCONJ cana-2410 250 8 the the DET cana-2410 250 9 density density NOUN cana-2410 250 10 equations equation NOUN cana-2410 250 11 of of ADP cana-2410 250 12 ( ( PUNCT cana-2410 250 13 3.27 3.27 NUM cana-2410 250 14 ) ) PUNCT cana-2410 250 15 give give NOUN cana-2410 250 16 ( ( PUNCT cana-2410 250 17 βˆ’1)𝑖𝑧𝛼 βˆ’1)𝑖𝑧𝛼 PROPN cana-2410 250 18 βˆ—π‘’π‘ƒ(𝑑0 βˆ—π‘’π‘ƒ(𝑑0 NOUN cana-2410 250 19 , , PUNCT cana-2410 250 20 𝑑)𝑃 𝑑)𝑃 NUM cana-2410 250 21 π‘–βˆ’1𝑄 π‘–βˆ’1𝑄 X cana-2410 250 22 = = SYM cana-2410 250 23 0 0 NUM cana-2410 250 24 , , PUNCT cana-2410 250 25 𝑖 𝑖 NOUN cana-2410 250 26 = = NOUN cana-2410 250 27 0,1,2 0,1,2 NUM cana-2410 250 28 , , PUNCT cana-2410 250 29 … … PUNCT cana-2410 250 30 , , PUNCT cana-2410 250 31 𝑛2 𝑛2 NOUN cana-2410 250 32 βˆ’ βˆ’ PROPN cana-2410 250 33 1 1 NUM cana-2410 250 34 , , PUNCT cana-2410 250 35 𝑑 𝑑 PROPN cana-2410 250 36 ∈ ∈ PROPN cana-2410 250 37 [ [ X cana-2410 250 38 𝑑0 𝑑0 NOUN cana-2410 250 39 , , PUNCT cana-2410 250 40 𝑑1]𝕋. 𝑑1]𝕋. PROPN cana-2410 250 41 ( ( PUNCT cana-2410 250 42 3.29 3.29 NUM cana-2410 250 43 ) ) PUNCT cana-2410 250 44 put put VERB cana-2410 250 45 𝑑 𝑑 NOUN cana-2410 250 46 = = NOUN cana-2410 250 47 𝑑0 𝑑0 PROPN cana-2410 250 48 in in ADP cana-2410 250 49 the the DET cana-2410 250 50 above above ADJ cana-2410 250 51 equation equation NOUN cana-2410 250 52 ( ( PUNCT cana-2410 250 53 3.29 3.29 NUM cana-2410 250 54 ) ) PUNCT cana-2410 250 55 , , PUNCT cana-2410 250 56 we we PRON cana-2410 250 57 have have VERB cana-2410 250 58 communications communication NOUN cana-2410 250 59 on on ADP cana-2410 250 60 applied apply VERB cana-2410 250 61 nonlinear nonlinear ADJ cana-2410 250 62 analysis analysis NOUN cana-2410 250 63 issn issn NOUN cana-2410 250 64 : : PUNCT cana-2410 250 65 1074 1074 NUM cana-2410 250 66 - - PUNCT cana-2410 250 67 133x 133x NUM cana-2410 250 68 vol vol NOUN cana-2410 250 69 32 32 NUM cana-2410 250 70 no no NOUN cana-2410 250 71 . . PUNCT cana-2410 251 1 2s 2s NUM cana-2410 251 2 ( ( PUNCT cana-2410 251 3 2025 2025 NUM cana-2410 251 4 ) ) PUNCT cana-2410 251 5 362 362 NUM cana-2410 251 6 https://internationalpubls.com https://internationalpubls.com X cana-2410 251 7 𝑧𝛼 𝑧𝛼 ADP cana-2410 251 8 βˆ—π‘ƒπ‘–βˆ’1𝑄 βˆ—π‘ƒπ‘–βˆ’1𝑄 NUM cana-2410 251 9 = = SYM cana-2410 251 10 0 0 NUM cana-2410 251 11 , , PUNCT cana-2410 251 12 𝑖 𝑖 NOUN cana-2410 251 13 = = NOUN cana-2410 251 14 0,1,2 0,1,2 NUM cana-2410 251 15 , , PUNCT cana-2410 251 16 … … PUNCT cana-2410 251 17 , , PUNCT cana-2410 251 18 𝑛2 𝑛2 NOUN cana-2410 251 19 βˆ’ βˆ’ PROPN cana-2410 251 20 1 1 NUM cana-2410 251 21 . . PUNCT cana-2410 252 1 therefore therefore ADV cana-2410 252 2 , , PUNCT cana-2410 252 3 𝑧𝛼 𝑧𝛼 ADV cana-2410 252 4 βˆ— βˆ— NOUN cana-2410 253 1 [ [ X cana-2410 253 2 𝑄 𝑄 PROPN cana-2410 253 3 𝑃𝑄 𝑃𝑄 PROPN cana-2410 253 4 𝑃2𝑄 𝑃2𝑄 NUM cana-2410 253 5 … … PUNCT cana-2410 253 6 𝑃𝑛 𝑃𝑛 PROPN cana-2410 253 7 2βˆ’1𝑄 2βˆ’1𝑄 NUM cana-2410 253 8 ] ] X cana-2410 253 9 = = SYM cana-2410 253 10 0 0 NUM cana-2410 253 11 , , PUNCT cana-2410 253 12 hence hence ADV cana-2410 253 13 , , PUNCT cana-2410 253 14 our our PRON cana-2410 253 15 assumption assumption NOUN cana-2410 253 16 is be AUX cana-2410 253 17 wrong wrong ADJ cana-2410 253 18 : : PUNCT cana-2410 253 19 therefore therefore ADV cana-2410 253 20 , , PUNCT cana-2410 253 21 it it PRON cana-2410 253 22 is be AUX cana-2410 253 23 a a DET cana-2410 253 24 contradictory contradictory ADJ cana-2410 253 25 that that SCONJ cana-2410 253 26 the the DET cana-2410 253 27 rank rank NOUN cana-2410 253 28 of of ADP cana-2410 253 29 𝐢 𝐢 PROPN cana-2410 253 30 = = PROPN cana-2410 253 31 𝑛2.similarly 𝑛2.similarly ADV cana-2410 253 32 , , PUNCT cana-2410 253 33 we we PRON cana-2410 253 34 iterate iterate VERB cana-2410 253 35 the the DET cana-2410 253 36 procedure procedure NOUN cana-2410 253 37 on on ADP cana-2410 253 38 equation equation NOUN cana-2410 253 39 ( ( PUNCT cana-2410 253 40 3.28 3.28 NUM cana-2410 253 41 ) ) PUNCT cana-2410 253 42 , , PUNCT cana-2410 253 43 yielding yield VERB cana-2410 253 44 π‘§π›Όπ‘˜ π‘§π›Όπ‘˜ NOUN cana-2410 253 45 βˆ— βˆ— NOUN cana-2410 254 1 [ [ X cana-2410 254 2 𝑄 𝑄 PROPN cana-2410 254 3 𝑃𝑄 𝑃𝑄 PROPN cana-2410 254 4 𝑃2𝑄 𝑃2𝑄 NUM cana-2410 254 5 … … PUNCT cana-2410 254 6 𝑃𝑛 𝑃𝑛 PROPN cana-2410 254 7 2βˆ’1𝑄 2βˆ’1𝑄 NUM cana-2410 254 8 ] ] X cana-2410 255 1 = = SYM cana-2410 255 2 0 0 PUNCT cana-2410 255 3 once once ADV cana-2410 255 4 again again ADV cana-2410 255 5 , , PUNCT cana-2410 255 6 the the DET cana-2410 255 7 contradiction contradiction NOUN cana-2410 255 8 demonstrate demonstrate VERB cana-2410 255 9 that that SCONJ cana-2410 255 10 system system NOUN cana-2410 255 11 ( ( PUNCT cana-2410 255 12 3.1 3.1 NUM cana-2410 255 13 ) ) PUNCT cana-2410 255 14 is be AUX cana-2410 255 15 completely completely ADV cana-2410 255 16 controllable controllable ADJ cana-2410 255 17 throughout throughout ADP cana-2410 255 18 the the DET cana-2410 255 19 time time NOUN cana-2410 255 20 interval interval NOUN cana-2410 255 21 [ [ X cana-2410 255 22 𝑑0 𝑑0 NOUN cana-2410 255 23 , , PUNCT cana-2410 255 24 𝑇]𝕋 𝑇]𝕋 NUM cana-2410 255 25 . . PUNCT cana-2410 256 1 example example NOUN cana-2410 256 2 3.1 3.1 NUM cana-2410 256 3 : : PUNCT cana-2410 256 4 the the DET cana-2410 256 5 following follow VERB cana-2410 256 6 non non ADJ cana-2410 256 7 - - ADJ cana-2410 256 8 linear linear ADJ cana-2410 256 9 kreneker kreneker NOUN cana-2410 256 10 product product NOUN cana-2410 256 11 of of ADP cana-2410 256 12 volterra volterra PROPN cana-2410 256 13 integro integro PROPN cana-2410 256 14 - - PUNCT cana-2410 256 15 dynamic dynamic NOUN cana-2410 256 16 with with ADP cana-2410 256 17 an an DET cana-2410 256 18 impulse impulse ADJ cana-2410 256 19 control control NOUN cana-2410 256 20 system system NOUN cana-2410 256 21 { { PUNCT cana-2410 256 22 π‘§βˆ†(𝑑 π‘§βˆ†(𝑑 NOUN cana-2410 256 23 ) ) PUNCT cana-2410 256 24 = = SYM cana-2410 256 25 𝑃(𝑑)𝑧(𝑑 𝑃(𝑑)𝑧(𝑑 NOUN cana-2410 256 26 ) ) PUNCT cana-2410 257 1 + + CCONJ cana-2410 257 2 ∫ ∫ PROPN cana-2410 257 3 𝐾(𝑑 𝐾(𝑑 PROPN cana-2410 257 4 , , PUNCT cana-2410 257 5 𝑠)𝑧(𝑠)βˆ†π‘  𝑠)𝑧(𝑠)βˆ†π‘  PROPN cana-2410 257 6 𝑑 𝑑 PROPN cana-2410 257 7 0 0 NUM cana-2410 257 8 + + CCONJ cana-2410 257 9 𝑄(𝑑) 𝑄(𝑑) PRON cana-2410 257 10 οΏ½ οΏ½ NOUN cana-2410 257 11 Μ‚ Μ‚ NOUN cana-2410 257 12 οΏ½ οΏ½ NOUN cana-2410 257 13 (𝑑 (𝑑 NOUN cana-2410 257 14 ) ) PUNCT cana-2410 258 1 + + CCONJ cana-2410 258 2 𝑓(𝑑 𝑓(𝑑 PROPN cana-2410 258 3 , , PUNCT cana-2410 258 4 𝑧(𝑑 𝑧(𝑑 PROPN cana-2410 258 5 ) ) PUNCT cana-2410 258 6 ) ) PUNCT cana-2410 258 7 , , PUNCT cana-2410 258 8 𝑑 𝑑 PROPN cana-2410 258 9 ∈ ∈ PROPN cana-2410 258 10 ( ( PUNCT cana-2410 258 11 𝑠𝑗 𝑠𝑗 INTJ cana-2410 258 12 , , PUNCT cana-2410 258 13 𝑑𝑗+1]𝕋 𝑑𝑗+1]𝕋 ADV cana-2410 258 14 , , PUNCT cana-2410 258 15 𝑗 𝑗 NOUN cana-2410 258 16 = = SYM cana-2410 258 17 0,1 0,1 NUM cana-2410 258 18 , , PUNCT cana-2410 258 19 2 2 NUM cana-2410 258 20 , , PUNCT cana-2410 258 21 𝑧(𝑑 𝑧(𝑑 NOUN cana-2410 258 22 ) ) PUNCT cana-2410 258 23 = = PUNCT cana-2410 259 1 [ [ X cana-2410 259 2 πΌπ‘›βŠ—π‘…π‘˜]𝑧(𝑑𝑗 πΌπ‘›βŠ—π‘…π‘˜]𝑧(𝑑𝑗 NUM cana-2410 259 3 βˆ’ βˆ’ NOUN cana-2410 259 4 ) ) PUNCT cana-2410 259 5 , , PUNCT cana-2410 259 6 𝑑 𝑑 PROPN cana-2410 259 7 ∈ ∈ PROPN cana-2410 259 8 ( ( PUNCT cana-2410 259 9 𝑠𝑗 𝑠𝑗 NOUN cana-2410 259 10 , , PUNCT cana-2410 259 11 𝑑𝑗+1]𝕋 𝑑𝑗+1]𝕋 NOUN cana-2410 259 12 , , PUNCT cana-2410 259 13 𝑗 𝑗 NOUN cana-2410 259 14 = = SYM cana-2410 259 15 1,2 1,2 NUM cana-2410 259 16 , , PUNCT cana-2410 259 17 … … PUNCT cana-2410 259 18 𝑧(𝑑0 𝑧(𝑑0 NUM cana-2410 259 19 ) ) PUNCT cana-2410 259 20 = = SYM cana-2410 259 21 𝑧0 𝑧0 PROPN cana-2410 259 22 , , PUNCT cana-2410 259 23 𝑧0 𝑧0 PROPN cana-2410 259 24 ∈ ∈ PROPN cana-2410 259 25 ℝ2 ℝ2 X cana-2410 259 26 ( ( PUNCT cana-2410 259 27 3.30 3.30 NUM cana-2410 259 28 ) ) PUNCT cana-2410 259 29 where where SCONJ cana-2410 259 30 𝑧(𝑑 𝑧(𝑑 NOUN cana-2410 259 31 ) ) PUNCT cana-2410 259 32 = = SYM cana-2410 259 33 [ [ PUNCT cana-2410 259 34 𝑧11(𝑑 𝑧11(𝑑 NOUN cana-2410 259 35 ) ) PUNCT cana-2410 259 36 𝑧12(𝑑 𝑧12(𝑑 NOUN cana-2410 259 37 ) ) PUNCT cana-2410 259 38 𝑧21(𝑑 𝑧21(𝑑 NOUN cana-2410 259 39 ) ) PUNCT cana-2410 259 40 𝑧22(𝑑 𝑧22(𝑑 NOUN cana-2410 259 41 ) ) PUNCT cana-2410 259 42 ] ] PUNCT cana-2410 259 43 , , PUNCT cana-2410 259 44 𝑑0 𝑑0 NOUN cana-2410 259 45 = = SYM cana-2410 259 46 𝑠0 𝑠0 PROPN cana-2410 259 47 = = SYM cana-2410 259 48 0 0 NUM cana-2410 259 49 , , PUNCT cana-2410 259 50 𝑑1 𝑑1 NOUN cana-2410 259 51 = = SYM cana-2410 259 52 0.8 0.8 NUM cana-2410 259 53 , , PUNCT cana-2410 259 54 𝑠1 𝑠1 PROPN cana-2410 259 55 = = SYM cana-2410 259 56 0.9 0.9 NUM cana-2410 259 57 , , PUNCT cana-2410 259 58 𝑑2 𝑑2 NOUN cana-2410 259 59 = = NOUN cana-2410 259 60 2.1 2.1 NUM cana-2410 259 61 , , PUNCT cana-2410 259 62 𝑠2 𝑠2 NOUN cana-2410 259 63 = = SYM cana-2410 259 64 2.2 2.2 NUM cana-2410 259 65 , , PUNCT cana-2410 259 66 𝑑3 𝑑3 PROPN cana-2410 259 67 = = PUNCT cana-2410 259 68 𝑇 𝑇 PROPN cana-2410 259 69 = = SYM cana-2410 259 70 3 3 NUM cana-2410 259 71 , , PUNCT cana-2410 259 72 𝑃(𝑑 𝑃(𝑑 NUM cana-2410 259 73 ) ) PUNCT cana-2410 259 74 = = PUNCT cana-2410 260 1 [ [ X cana-2410 260 2 π΅βˆ—βŠ— π΅βˆ—βŠ— NOUN cana-2410 260 3 in in ADP cana-2410 260 4 + + CCONJ cana-2410 260 5 in in ADP cana-2410 260 6 βŠ—π΄ βŠ—π΄ NOUN cana-2410 260 7 ] ] X cana-2410 261 1 = = PUNCT cana-2410 261 2 [ [ PUNCT cana-2410 261 3 βˆ’2 βˆ’2 NOUN cana-2410 261 4 0 0 NUM cana-2410 261 5 0 0 NUM cana-2410 261 6 0 0 NUM cana-2410 261 7 0 0 NUM cana-2410 261 8 βˆ’2 βˆ’2 NOUN cana-2410 261 9 0 0 NUM cana-2410 261 10 0 0 NUM cana-2410 261 11 0 0 NUM cana-2410 261 12 0 0 NUM cana-2410 261 13 0 0 NUM cana-2410 261 14 0 0 NUM cana-2410 262 1 βˆ’3 βˆ’3 NOUN cana-2410 262 2 0 0 NUM cana-2410 262 3 0 0 NUM cana-2410 263 1 βˆ’3 βˆ’3 NOUN cana-2410 263 2 ] ] X cana-2410 263 3 , , PUNCT cana-2410 263 4 𝐾(𝑑 𝐾(𝑑 PROPN cana-2410 263 5 , , PUNCT cana-2410 263 6 𝑠 𝑠 X cana-2410 263 7 ) ) PUNCT cana-2410 263 8 = = NOUN cana-2410 264 1 [ [ X cana-2410 264 2 𝐾2 𝐾2 NOUN cana-2410 264 3 βˆ—βŠ— βˆ—βŠ— PROPN cana-2410 264 4 𝐼𝑛 𝐼𝑛 PROPN cana-2410 264 5 ) ) PUNCT cana-2410 264 6 + + CCONJ cana-2410 264 7 ( ( PUNCT cana-2410 264 8 πΌπ‘›βŠ—πΎ1 πΌπ‘›βŠ—πΎ1 NUM cana-2410 264 9 ) ) PUNCT cana-2410 264 10 = = SYM cana-2410 265 1 [ [ PUNCT cana-2410 265 2 𝑠𝑖𝑛𝑑 𝑠𝑖𝑛𝑑 NOUN cana-2410 265 3 0 0 NUM cana-2410 265 4 0 0 NUM cana-2410 265 5 0 0 NUM cana-2410 265 6 0 0 NUM cana-2410 265 7 π‘π‘œπ‘ π‘‘ π‘π‘œπ‘ π‘‘ NOUN cana-2410 265 8 0 0 NUM cana-2410 265 9 0 0 NUM cana-2410 265 10 0 0 NUM cana-2410 265 11 0 0 NUM cana-2410 265 12 0 0 NUM cana-2410 265 13 0 0 NUM cana-2410 265 14 𝑠𝑖𝑛𝑑 𝑠𝑖𝑛𝑑 NOUN cana-2410 265 15 0 0 NUM cana-2410 265 16 0 0 NUM cana-2410 265 17 π‘π‘œπ‘ π‘‘ π‘π‘œπ‘ π‘‘ NOUN cana-2410 265 18 ] ] PUNCT cana-2410 265 19 , , PUNCT cana-2410 265 20 𝑄(𝑑 𝑄(𝑑 X cana-2410 265 21 ) ) PUNCT cana-2410 265 22 = = SYM cana-2410 266 1 [ [ X cana-2410 266 2 in in ADP cana-2410 266 3 βŠ—πΆ βŠ—πΆ NOUN cana-2410 266 4 ] ] PUNCT cana-2410 267 1 = = PUNCT cana-2410 267 2 [ [ PUNCT cana-2410 267 3 1 1 NUM cana-2410 267 4 0 0 NUM cana-2410 267 5 2 2 NUM cana-2410 267 6 25 25 NUM cana-2410 267 7 𝑒1(𝜎(𝑑 𝑒1(𝜎(𝑑 NUM cana-2410 267 8 ) ) PUNCT cana-2410 267 9 , , PUNCT cana-2410 267 10 0 0 NUM cana-2410 267 11 ) ) PUNCT cana-2410 267 12 0 0 NUM cana-2410 267 13 0 0 NUM cana-2410 267 14 0 0 NUM cana-2410 267 15 1 1 NUM cana-2410 267 16 2 2 NUM cana-2410 267 17 25 25 NUM cana-2410 267 18 𝑒1(𝜎(𝑑 𝑒1(𝜎(𝑑 NUM cana-2410 267 19 ) ) PUNCT cana-2410 267 20 , , PUNCT cana-2410 267 21 0 0 NUM cana-2410 267 22 ) ) PUNCT cana-2410 267 23 ] ] PUNCT cana-2410 267 24 , , PUNCT cana-2410 267 25 𝑓(𝑑 𝑓(𝑑 PROPN cana-2410 267 26 , , PUNCT cana-2410 267 27 𝑧(𝑑 𝑧(𝑑 PROPN cana-2410 267 28 ) ) PUNCT cana-2410 267 29 = = SYM cana-2410 268 1 1 1 NUM cana-2410 268 2 35 35 NUM cana-2410 268 3 [ [ PUNCT cana-2410 268 4 sin(𝑧22(𝑑 sin(𝑧22(𝑑 NOUN cana-2410 268 5 ) ) PUNCT cana-2410 268 6 ) ) PUNCT cana-2410 268 7 𝑒𝑑 𝑒𝑑 ADP cana-2410 268 8 2 2 NUM cana-2410 268 9 + + NOUN cana-2410 268 10 2 2 NUM cana-2410 268 11 0 0 NUM cana-2410 268 12 0 0 NUM cana-2410 268 13 cos(𝑧11(𝑑 cos(𝑧11(𝑑 NOUN cana-2410 268 14 ) ) PUNCT cana-2410 268 15 ) ) PUNCT cana-2410 268 16 𝑒𝑑 𝑒𝑑 ADP cana-2410 268 17 2 2 NUM cana-2410 268 18 + + NOUN cana-2410 268 19 2 2 NUM cana-2410 268 20 ] ] PUNCT cana-2410 268 21 , , PUNCT cana-2410 269 1 [ [ X cana-2410 269 2 πΌπ‘›βŠ—π‘…π‘—]𝑧(𝑑𝑗 πΌπ‘›βŠ—π‘…π‘—]𝑧(𝑑𝑗 X cana-2410 269 3 βˆ’ βˆ’ NOUN cana-2410 269 4 ) ) PUNCT cana-2410 270 1 = = SYM cana-2410 270 2 1 1 NUM cana-2410 270 3 20 20 NUM cana-2410 270 4 [ [ PUNCT cana-2410 270 5 𝑧2(π‘‘π‘˜ 𝑧2(π‘‘π‘˜ PROPN cana-2410 270 6 βˆ’ βˆ’ NOUN cana-2410 270 7 ) ) PUNCT cana-2410 270 8 𝑒𝑑 𝑒𝑑 NOUN cana-2410 270 9 2 2 NUM cana-2410 270 10 + + NOUN cana-2410 270 11 2(1 2(1 NUM cana-2410 270 12 + + NOUN cana-2410 270 13 𝑖𝑑 𝑖𝑑 NOUN cana-2410 270 14 ) ) PUNCT cana-2410 270 15 0 0 NUM cana-2410 271 1 𝑧1(π‘‘π‘˜ 𝑧1(π‘‘π‘˜ NUM cana-2410 271 2 βˆ’ βˆ’ NOUN cana-2410 271 3 ) ) PUNCT cana-2410 271 4 𝑒𝑑 𝑒𝑑 NOUN cana-2410 271 5 2 2 NUM cana-2410 271 6 + + NUM cana-2410 271 7 3(1 3(1 NUM cana-2410 271 8 + + NUM cana-2410 271 9 𝑖𝑑2 𝑖𝑑2 NOUN cana-2410 271 10 ) ) PUNCT cana-2410 271 11 0 0 NUM cana-2410 272 1 0 0 NUM cana-2410 272 2 0 0 NUM cana-2410 273 1 𝑧2(π‘‘π‘˜ 𝑧2(π‘‘π‘˜ PROPN cana-2410 273 2 βˆ’ βˆ’ NOUN cana-2410 273 3 ) ) PUNCT cana-2410 273 4 𝑒𝑑 𝑒𝑑 NOUN cana-2410 273 5 2 2 NUM cana-2410 273 6 + + NOUN cana-2410 273 7 2(1 2(1 NUM cana-2410 273 8 + + NOUN cana-2410 273 9 𝑖𝑑 𝑖𝑑 NOUN cana-2410 273 10 ) ) PUNCT cana-2410 273 11 𝑧1(π‘‘π‘˜ 𝑧1(π‘‘π‘˜ NUM cana-2410 273 12 βˆ’ βˆ’ NOUN cana-2410 273 13 ) ) PUNCT cana-2410 273 14 𝑒𝑑 𝑒𝑑 NOUN cana-2410 273 15 2 2 NUM cana-2410 273 16 + + NUM cana-2410 273 17 3(1 3(1 NUM cana-2410 273 18 + + SYM cana-2410 273 19 𝑖𝑑2 𝑖𝑑2 NOUN cana-2410 273 20 ) ) PUNCT cana-2410 273 21 ] ] PUNCT cana-2410 273 22 , , PUNCT cana-2410 273 23 𝑗 𝑗 NOUN cana-2410 273 24 = = SYM cana-2410 273 25 1,2 1,2 NUM cana-2410 273 26 . . PUNCT cana-2410 273 27 communications communication NOUN cana-2410 273 28 on on ADP cana-2410 273 29 applied apply VERB cana-2410 273 30 nonlinear nonlinear ADJ cana-2410 273 31 analysis analysis NOUN cana-2410 273 32 issn issn NOUN cana-2410 273 33 : : PUNCT cana-2410 273 34 1074 1074 NUM cana-2410 273 35 - - PUNCT cana-2410 273 36 133x 133x NUM cana-2410 273 37 vol vol NOUN cana-2410 273 38 32 32 NUM cana-2410 273 39 no no NOUN cana-2410 273 40 . . PUNCT cana-2410 274 1 2s 2s NUM cana-2410 274 2 ( ( PUNCT cana-2410 274 3 2025 2025 NUM cana-2410 274 4 ) ) PUNCT cana-2410 274 5 363 363 NUM cana-2410 274 6 https://internationalpubls.com https://internationalpubls.com X cana-2410 274 7 the the DET cana-2410 274 8 matrix matrix NOUN cana-2410 274 9 that that PRON cana-2410 274 10 provides provide VERB cana-2410 274 11 the the DET cana-2410 274 12 fundamental fundamental ADJ cana-2410 274 13 solution solution NOUN cana-2410 274 14 to to ADP cana-2410 274 15 the the DET cana-2410 274 16 system system NOUN cana-2410 274 17 ( ( PUNCT cana-2410 274 18 3.30 3.30 NUM cana-2410 274 19 ) ) PUNCT cana-2410 274 20 is be AUX cana-2410 274 21 𝑒𝑃(𝑑 𝑒𝑃(𝑑 ADJ cana-2410 274 22 , , PUNCT cana-2410 274 23 0 0 NUM cana-2410 274 24 ) ) PUNCT cana-2410 274 25 = = NOUN cana-2410 275 1 [ [ PUNCT cana-2410 275 2 π‘’βˆ’2(𝑑 π‘’βˆ’2(𝑑 PROPN cana-2410 275 3 , , PUNCT cana-2410 275 4 0 0 NUM cana-2410 275 5 ) ) PUNCT cana-2410 275 6 0 0 NUM cana-2410 275 7 0 0 NUM cana-2410 275 8 0 0 NUM cana-2410 275 9 0 0 NUM cana-2410 276 1 π‘’βˆ’2(𝑑 π‘’βˆ’2(𝑑 PROPN cana-2410 276 2 , , PUNCT cana-2410 276 3 0 0 NUM cana-2410 276 4 ) ) PUNCT cana-2410 276 5 0 0 NUM cana-2410 276 6 0 0 NUM cana-2410 276 7 0 0 NUM cana-2410 276 8 0 0 NUM cana-2410 276 9 0 0 NUM cana-2410 276 10 0 0 NUM cana-2410 276 11 π‘’βˆ’3(𝑑 π‘’βˆ’3(𝑑 PROPN cana-2410 276 12 , , PUNCT cana-2410 276 13 0 0 NUM cana-2410 276 14 ) ) PUNCT cana-2410 276 15 0 0 NUM cana-2410 276 16 0 0 NUM cana-2410 276 17 π‘’βˆ’3(𝑑 π‘’βˆ’3(𝑑 PROPN cana-2410 276 18 , , PUNCT cana-2410 276 19 0 0 NUM cana-2410 276 20 ) ) PUNCT cana-2410 276 21 ] ] PUNCT cana-2410 276 22 . . PUNCT cana-2410 277 1 therefore therefore ADV cana-2410 277 2 𝑒𝑃(0 𝑒𝑃(0 NOUN cana-2410 277 3 , , PUNCT cana-2410 277 4 𝑑 𝑑 NOUN cana-2410 277 5 ) ) PUNCT cana-2410 277 6 = = SYM cana-2410 277 7 [ [ PUNCT cana-2410 277 8 𝑒2(0 𝑒2(0 PROPN cana-2410 277 9 , , PUNCT cana-2410 277 10 𝑑 𝑑 PROPN cana-2410 277 11 ) ) PUNCT cana-2410 277 12 0 0 NUM cana-2410 277 13 0 0 NUM cana-2410 277 14 0 0 NUM cana-2410 277 15 0 0 NUM cana-2410 277 16 𝑒2(0 𝑒2(0 PROPN cana-2410 277 17 , , PUNCT cana-2410 277 18 𝑑 𝑑 PROPN cana-2410 277 19 ) ) PUNCT cana-2410 277 20 0 0 NUM cana-2410 277 21 0 0 NUM cana-2410 277 22 0 0 NUM cana-2410 277 23 0 0 NUM cana-2410 277 24 0 0 NUM cana-2410 277 25 0 0 NUM cana-2410 278 1 𝑒3(0 𝑒3(0 PROPN cana-2410 278 2 , , PUNCT cana-2410 278 3 𝑑 𝑑 PROPN cana-2410 278 4 ) ) PUNCT cana-2410 278 5 0 0 NUM cana-2410 278 6 0 0 NUM cana-2410 278 7 𝑒3(0 𝑒3(0 PROPN cana-2410 278 8 , , PUNCT cana-2410 278 9 𝑑 𝑑 PROPN cana-2410 278 10 ) ) PUNCT cana-2410 278 11 ] ] PUNCT cana-2410 278 12 . . PUNCT cana-2410 279 1 also also ADV cana-2410 279 2 we we PRON cana-2410 279 3 can can AUX cana-2410 279 4 easily easily ADV cana-2410 279 5 compute compute VERB cana-2410 279 6 𝒩(0 𝒩(0 NUM cana-2410 279 7 , , PUNCT cana-2410 279 8 Οƒ(Ο„ Οƒ(Ο„ PROPN cana-2410 279 9 ) ) PUNCT cana-2410 279 10 ) ) PUNCT cana-2410 280 1 = = PUNCT cana-2410 280 2 ψ(0 ψ(0 PROPN cana-2410 280 3 , , PUNCT cana-2410 280 4 Οƒ(Ο„))𝑄(Ο„)π‘„βˆ—(Ο„)Οˆβˆ—(0,Οƒ(Ο„ Οƒ(Ο„))𝑄(Ο„)π‘„βˆ—(Ο„)Οˆβˆ—(0,Οƒ(Ο„ PROPN cana-2410 280 5 ) ) PUNCT cana-2410 280 6 ) ) PUNCT cana-2410 281 1 = = PUNCT cana-2410 281 2 [ [ PUNCT cana-2410 281 3 ( ( PUNCT cana-2410 281 4 π‘’βˆ’2(0,Οƒ(Ο„ π‘’βˆ’2(0,Οƒ(Ο„ PROPN cana-2410 281 5 ) ) PUNCT cana-2410 281 6 ) ) PUNCT cana-2410 282 1 + + CCONJ cana-2410 282 2 8 8 NUM cana-2410 282 3 625 625 NUM cana-2410 282 4 ) ) PUNCT cana-2410 282 5 2 2 NUM cana-2410 282 6 0 0 NUM cana-2410 282 7 0 0 NUM cana-2410 282 8 0 0 NUM cana-2410 282 9 0 0 NUM cana-2410 282 10 ( ( PUNCT cana-2410 282 11 π‘’βˆ’2(0,Οƒ(Ο„ π‘’βˆ’2(0,Οƒ(Ο„ PROPN cana-2410 282 12 ) ) PUNCT cana-2410 282 13 ) ) PUNCT cana-2410 283 1 + + CCONJ cana-2410 283 2 8 8 NUM cana-2410 283 3 625 625 NUM cana-2410 283 4 ) ) PUNCT cana-2410 283 5 2 2 NUM cana-2410 283 6 0 0 NUM cana-2410 283 7 0 0 NUM cana-2410 283 8 0 0 NUM cana-2410 283 9 0 0 NUM cana-2410 283 10 0 0 NUM cana-2410 283 11 0 0 NUM cana-2410 283 12 ( ( PUNCT cana-2410 283 13 π‘’βˆ’3(0 π‘’βˆ’3(0 PROPN cana-2410 283 14 , , PUNCT cana-2410 283 15 Οƒ(Ο„ Οƒ(Ο„ PROPN cana-2410 283 16 ) ) PUNCT cana-2410 283 17 ) ) PUNCT cana-2410 284 1 + + CCONJ cana-2410 284 2 8 8 NUM cana-2410 284 3 625 625 NUM cana-2410 284 4 ) ) PUNCT cana-2410 284 5 2 2 NUM cana-2410 284 6 0 0 NUM cana-2410 284 7 0 0 NUM cana-2410 284 8 ( ( PUNCT cana-2410 284 9 π‘’βˆ’3(0,Οƒ(Ο„ π‘’βˆ’3(0,Οƒ(Ο„ PROPN cana-2410 284 10 ) ) PUNCT cana-2410 284 11 ) ) PUNCT cana-2410 285 1 + + CCONJ cana-2410 285 2 8 8 NUM cana-2410 285 3 625 625 NUM cana-2410 285 4 ) ) PUNCT cana-2410 285 5 2 2 NUM cana-2410 285 6 ] ] PUNCT cana-2410 285 7 . . PUNCT cana-2410 286 1 𝒩(𝑠1,Οƒ(Ο„ 𝒩(𝑠1,Οƒ(Ο„ NOUN cana-2410 286 2 ) ) PUNCT cana-2410 286 3 ) ) PUNCT cana-2410 287 1 = = SYM cana-2410 287 2 ψ(𝑠1 ψ(𝑠1 NOUN cana-2410 287 3 , , PUNCT cana-2410 287 4 Οƒ(Ο„))𝑄(Ο„)𝑄 Οƒ(Ο„))𝑄(Ο„)𝑄 ADJ cana-2410 287 5 βˆ—(Ο„)Οˆβˆ—(𝑠1,Οƒ(Ο„ βˆ—(Ο„)Οˆβˆ—(𝑠1,Οƒ(Ο„ NOUN cana-2410 287 6 ) ) PUNCT cana-2410 287 7 ) ) PUNCT cana-2410 288 1 = = PUNCT cana-2410 288 2 [ [ PUNCT cana-2410 288 3 ( ( PUNCT cana-2410 288 4 π‘’βˆ’2(𝑠1,Οƒ(Ο„ π‘’βˆ’2(𝑠1,Οƒ(Ο„ PROPN cana-2410 288 5 ) ) PUNCT cana-2410 288 6 ) ) PUNCT cana-2410 289 1 + + CCONJ cana-2410 289 2 8 8 NUM cana-2410 289 3 625 625 NUM cana-2410 289 4 ) ) PUNCT cana-2410 289 5 2 2 NUM cana-2410 289 6 0 0 NUM cana-2410 289 7 0 0 NUM cana-2410 289 8 0 0 NUM cana-2410 289 9 0 0 NUM cana-2410 289 10 ( ( PUNCT cana-2410 289 11 π‘’βˆ’2(𝑠1,Οƒ(Ο„ π‘’βˆ’2(𝑠1,Οƒ(Ο„ PROPN cana-2410 289 12 ) ) PUNCT cana-2410 289 13 ) ) PUNCT cana-2410 290 1 + + CCONJ cana-2410 290 2 8 8 NUM cana-2410 290 3 625 625 NUM cana-2410 290 4 ) ) PUNCT cana-2410 290 5 2 2 NUM cana-2410 290 6 0 0 NUM cana-2410 290 7 0 0 NUM cana-2410 290 8 0 0 NUM cana-2410 290 9 0 0 NUM cana-2410 290 10 0 0 NUM cana-2410 290 11 0 0 NUM cana-2410 290 12 ( ( PUNCT cana-2410 290 13 π‘’βˆ’3(𝑠1 π‘’βˆ’3(𝑠1 PROPN cana-2410 290 14 , , PUNCT cana-2410 290 15 Οƒ(Ο„ Οƒ(Ο„ PROPN cana-2410 290 16 ) ) PUNCT cana-2410 290 17 ) ) PUNCT cana-2410 291 1 + + CCONJ cana-2410 291 2 8 8 NUM cana-2410 291 3 625 625 NUM cana-2410 291 4 ) ) PUNCT cana-2410 292 1 2 2 NUM cana-2410 292 2 0 0 NUM cana-2410 292 3 0 0 NUM cana-2410 292 4 ( ( PUNCT cana-2410 292 5 π‘’βˆ’3(𝑠1 π‘’βˆ’3(𝑠1 PROPN cana-2410 292 6 , , PUNCT cana-2410 292 7 Οƒ(Ο„ Οƒ(Ο„ PROPN cana-2410 292 8 ) ) PUNCT cana-2410 292 9 ) ) PUNCT cana-2410 293 1 + + CCONJ cana-2410 293 2 8 8 NUM cana-2410 293 3 625 625 NUM cana-2410 293 4 ) ) PUNCT cana-2410 293 5 2 2 NUM cana-2410 293 6 ] ] PUNCT cana-2410 293 7 . . PUNCT cana-2410 294 1 𝒩(𝑠2,Οƒ(Ο„ 𝒩(𝑠2,Οƒ(Ο„ NOUN cana-2410 294 2 ) ) PUNCT cana-2410 294 3 ) ) PUNCT cana-2410 295 1 = = SYM cana-2410 295 2 ψ(𝑠2 ψ(𝑠2 NOUN cana-2410 295 3 , , PUNCT cana-2410 295 4 Οƒ(Ο„))𝑄(Ο„)𝑄 Οƒ(Ο„))𝑄(Ο„)𝑄 ADJ cana-2410 295 5 βˆ—(Ο„)Οˆβˆ—(𝑠2,Οƒ(Ο„ βˆ—(Ο„)Οˆβˆ—(𝑠2,Οƒ(Ο„ NOUN cana-2410 295 6 ) ) PUNCT cana-2410 295 7 ) ) PUNCT cana-2410 296 1 = = PUNCT cana-2410 296 2 [ [ PUNCT cana-2410 296 3 ( ( PUNCT cana-2410 296 4 π‘’βˆ’2(𝑠2,Οƒ(Ο„ π‘’βˆ’2(𝑠2,Οƒ(Ο„ PROPN cana-2410 296 5 ) ) PUNCT cana-2410 296 6 ) ) PUNCT cana-2410 297 1 + + CCONJ cana-2410 297 2 8 8 NUM cana-2410 297 3 625 625 NUM cana-2410 297 4 ) ) PUNCT cana-2410 297 5 2 2 NUM cana-2410 297 6 0 0 NUM cana-2410 297 7 0 0 NUM cana-2410 297 8 0 0 NUM cana-2410 297 9 0 0 NUM cana-2410 297 10 ( ( PUNCT cana-2410 297 11 π‘’βˆ’2(𝑠2,Οƒ(Ο„ π‘’βˆ’2(𝑠2,Οƒ(Ο„ PROPN cana-2410 297 12 ) ) PUNCT cana-2410 297 13 ) ) PUNCT cana-2410 298 1 + + CCONJ cana-2410 298 2 8 8 NUM cana-2410 298 3 625 625 NUM cana-2410 298 4 ) ) PUNCT cana-2410 298 5 2 2 NUM cana-2410 298 6 0 0 NUM cana-2410 298 7 0 0 NUM cana-2410 298 8 0 0 NUM cana-2410 298 9 0 0 NUM cana-2410 298 10 0 0 NUM cana-2410 298 11 0 0 NUM cana-2410 298 12 ( ( PUNCT cana-2410 298 13 π‘’βˆ’3(𝑠2 π‘’βˆ’3(𝑠2 PROPN cana-2410 298 14 , , PUNCT cana-2410 298 15 Οƒ(Ο„ Οƒ(Ο„ PROPN cana-2410 298 16 ) ) PUNCT cana-2410 298 17 ) ) PUNCT cana-2410 299 1 + + CCONJ cana-2410 299 2 8 8 NUM cana-2410 299 3 625 625 NUM cana-2410 299 4 ) ) PUNCT cana-2410 300 1 2 2 NUM cana-2410 300 2 0 0 NUM cana-2410 300 3 0 0 NUM cana-2410 300 4 ( ( PUNCT cana-2410 300 5 π‘’βˆ’3(𝑠2 π‘’βˆ’3(𝑠2 PROPN cana-2410 300 6 , , PUNCT cana-2410 300 7 Οƒ(Ο„ Οƒ(Ο„ PROPN cana-2410 300 8 ) ) PUNCT cana-2410 300 9 ) ) PUNCT cana-2410 301 1 + + CCONJ cana-2410 301 2 8 8 NUM cana-2410 301 3 625 625 NUM cana-2410 301 4 ) ) PUNCT cana-2410 301 5 2 2 NUM cana-2410 301 6 ] ] PUNCT cana-2410 301 7 . . PUNCT cana-2410 302 1 now now ADV cana-2410 302 2 consider consider VERB cana-2410 302 3 the the DET cana-2410 302 4 following follow VERB cana-2410 302 5 two two NUM cana-2410 302 6 cases case NOUN cana-2410 302 7 : : PUNCT cana-2410 302 8 case case NOUN cana-2410 302 9 ( ( PUNCT cana-2410 302 10 1 1 NUM cana-2410 302 11 ): ): PUNCT cana-2410 302 12 if if SCONJ cana-2410 302 13 𝕋 𝕋 PROPN cana-2410 302 14 = = SYM cana-2410 302 15 ℝ ℝ PROPN cana-2410 302 16 , , PUNCT cana-2410 302 17 then then ADV cana-2410 302 18 π‘’π‘Ž(𝑑 π‘’π‘Ž(𝑑 NOUN cana-2410 302 19 , , PUNCT cana-2410 302 20 0 0 NUM cana-2410 302 21 ) ) PUNCT cana-2410 303 1 = = NOUN cana-2410 303 2 π‘’π‘Žπ‘‘. π‘’π‘Žπ‘‘. X cana-2410 303 3 therefore therefore ADV cana-2410 303 4 , , PUNCT cana-2410 303 5 𝒩0(0 𝒩0(0 ADV cana-2410 303 6 , , PUNCT cana-2410 303 7 𝑑1 𝑑1 NOUN cana-2410 303 8 ) ) PUNCT cana-2410 303 9 = = PUNCT cana-2410 303 10 βˆ«π’©(0 βˆ«π’©(0 PROPN cana-2410 303 11 , , PUNCT cana-2410 303 12 Οƒ(Ο„))dΟ„ Οƒ(Ο„))dΟ„ VERB cana-2410 303 13 t t NOUN cana-2410 303 14 0 0 NUM cana-2410 304 1 = = PUNCT cana-2410 304 2 [ [ PUNCT cana-2410 304 3 7.2665 7.2665 NUM cana-2410 304 4 0 0 NUM cana-2410 304 5 0 0 NUM cana-2410 304 6 0 0 NUM cana-2410 304 7 0 0 NUM cana-2410 305 1 7.2665 7.2665 NUM cana-2410 305 2 0 0 NUM cana-2410 305 3 0 0 NUM cana-2410 305 4 0 0 NUM cana-2410 305 5 0 0 NUM cana-2410 305 6 0 0 NUM cana-2410 305 7 0 0 NUM cana-2410 305 8 24.156 24.156 NUM cana-2410 305 9 0 0 NUM cana-2410 305 10 0 0 NUM cana-2410 305 11 24.156 24.156 NUM cana-2410 305 12 ] ] PUNCT cana-2410 305 13 . . PUNCT cana-2410 306 1 communications communication NOUN cana-2410 306 2 on on ADP cana-2410 306 3 applied apply VERB cana-2410 306 4 nonlinear nonlinear ADJ cana-2410 306 5 analysis analysis NOUN cana-2410 306 6 issn issn NOUN cana-2410 306 7 : : PUNCT cana-2410 306 8 1074 1074 NUM cana-2410 306 9 - - PUNCT cana-2410 306 10 133x 133x NUM cana-2410 306 11 vol vol NOUN cana-2410 306 12 32 32 NUM cana-2410 306 13 no no NOUN cana-2410 306 14 . . PUNCT cana-2410 307 1 2s 2s NUM cana-2410 307 2 ( ( PUNCT cana-2410 307 3 2025 2025 NUM cana-2410 307 4 ) ) PUNCT cana-2410 307 5 364 364 NUM cana-2410 307 6 https://internationalpubls.com https://internationalpubls.com X cana-2410 307 7 𝒩1(𝑠1 𝒩1(𝑠1 PROPN cana-2410 307 8 , , PUNCT cana-2410 307 9 𝑑2 𝑑2 NOUN cana-2410 307 10 ) ) PUNCT cana-2410 307 11 = = SYM cana-2410 308 1 ∫ ∫ PROPN cana-2410 308 2 𝒩(𝑠1,Οƒ(Ο„))dΟ„ 𝒩(𝑠1,Οƒ(Ο„))dΟ„ PROPN cana-2410 308 3 𝑑2 𝑑2 PROPN cana-2410 308 4 𝑠1 𝑠1 PROPN cana-2410 309 1 = = PUNCT cana-2410 310 1 [ [ PUNCT cana-2410 310 2 16.513 16.513 NUM cana-2410 310 3 0 0 NUM cana-2410 310 4 0 0 NUM cana-2410 310 5 0 0 NUM cana-2410 310 6 0 0 NUM cana-2410 311 1 16.513 16.513 NUM cana-2410 311 2 0 0 NUM cana-2410 311 3 0 0 NUM cana-2410 311 4 0 0 NUM cana-2410 311 5 0 0 NUM cana-2410 311 6 0 0 NUM cana-2410 311 7 0 0 NUM cana-2410 311 8 74.52 74.52 NUM cana-2410 311 9 0 0 NUM cana-2410 311 10 0 0 NUM cana-2410 311 11 74.52 74.52 NUM cana-2410 311 12 ] ] PUNCT cana-2410 311 13 . . PUNCT cana-2410 312 1 𝒩2(𝑠2 𝒩2(𝑠2 NUM cana-2410 312 2 , , PUNCT cana-2410 312 3 𝑇 𝑇 PROPN cana-2410 312 4 ) ) PUNCT cana-2410 312 5 = = SYM cana-2410 313 1 ∫ ∫ PROPN cana-2410 313 2 𝒩(𝑠2,Οƒ(Ο„))dΟ„ 𝒩(𝑠2,Οƒ(Ο„))dΟ„ PROPN cana-2410 313 3 𝑇 𝑇 PROPN cana-2410 313 4 𝑠2 𝑠2 NOUN cana-2410 313 5 = = PUNCT cana-2410 313 6 [ [ PUNCT cana-2410 313 7 53.762 53.762 NUM cana-2410 313 8 0 0 NUM cana-2410 313 9 0 0 NUM cana-2410 313 10 0 0 NUM cana-2410 313 11 0 0 NUM cana-2410 313 12 53.762 53.762 NUM cana-2410 313 13 0 0 NUM cana-2410 313 14 0 0 NUM cana-2410 313 15 0 0 NUM cana-2410 313 16 0 0 NUM cana-2410 313 17 0 0 NUM cana-2410 313 18 0 0 NUM cana-2410 313 19 137.24 137.24 NUM cana-2410 313 20 0 0 NUM cana-2410 313 21 0 0 NUM cana-2410 313 22 137.24 137.24 NUM cana-2410 313 23 ] ] PUNCT cana-2410 313 24 . . PUNCT cana-2410 314 1 it it PRON cana-2410 314 2 follows follow VERB cana-2410 314 3 that that SCONJ cana-2410 314 4 the the DET cana-2410 314 5 matrices matrix NOUN cana-2410 314 6 𝒩0(0 𝒩0(0 NOUN cana-2410 314 7 , , PUNCT cana-2410 314 8 𝑑1 𝑑1 NOUN cana-2410 314 9 ) ) PUNCT cana-2410 314 10 , , PUNCT cana-2410 314 11 𝒩1(𝑠1 𝒩1(𝑠1 PROPN cana-2410 314 12 , , PUNCT cana-2410 314 13 𝑑2 𝑑2 PROPN cana-2410 314 14 ) ) PUNCT cana-2410 314 15 , , PUNCT cana-2410 314 16 and and CCONJ cana-2410 314 17 𝒩2(𝑠2 𝒩2(𝑠2 NUM cana-2410 314 18 , , PUNCT cana-2410 314 19 𝑇 𝑇 PROPN cana-2410 314 20 ) ) PUNCT cana-2410 314 21 are be AUX cana-2410 314 22 all all ADV cana-2410 314 23 invertible invertible ADJ cana-2410 314 24 . . PUNCT cana-2410 315 1 in in ADP cana-2410 315 2 addition addition NOUN cana-2410 315 3 , , PUNCT cana-2410 315 4 all all DET cana-2410 315 5 three three NUM cana-2410 315 6 assumptions assumption NOUN cana-2410 315 7 ( ( PUNCT cana-2410 315 8 h1 h1 PROPN cana-2410 315 9 ) ) PUNCT cana-2410 315 10 βˆ’ βˆ’ PROPN cana-2410 315 11 ( ( PUNCT cana-2410 315 12 h3 h3 NOUN cana-2410 315 13 ) ) PUNCT cana-2410 315 14 hold hold VERB cana-2410 315 15 . . PUNCT cana-2410 316 1 with with ADP cana-2410 316 2 m𝛼 m𝛼 NOUN cana-2410 316 3 = = SYM cana-2410 316 4 max max PROPN cana-2410 316 5 { { PUNCT cana-2410 316 6 0.9104 0.9104 NUM cana-2410 316 7 , , PUNCT cana-2410 316 8 0.2328 0.2328 NUM cana-2410 316 9 , , PUNCT cana-2410 316 10 0.7531 0.7531 NUM cana-2410 316 11 , , PUNCT cana-2410 316 12 0.01718 0.01718 NUM cana-2410 316 13 } } PUNCT cana-2410 316 14 < < X cana-2410 316 15 1 1 NUM cana-2410 316 16 . . PUNCT cana-2410 316 17 the the DET cana-2410 316 18 system system NOUN cana-2410 316 19 ( ( PUNCT cana-2410 316 20 3.30 3.30 NUM cana-2410 316 21 ) ) PUNCT cana-2410 316 22 is be AUX cana-2410 316 23 complete complete ADJ cana-2410 316 24 controllable controllable ADJ cana-2410 316 25 , , PUNCT cana-2410 316 26 since since SCONJ cana-2410 316 27 all all DET cana-2410 316 28 the the DET cana-2410 316 29 criteria criterion NOUN cana-2410 316 30 of of ADP cana-2410 316 31 theorem theorem NOUN cana-2410 316 32 3.2 3.2 NUM cana-2410 316 33 , , PUNCT cana-2410 316 34 are be AUX cana-2410 316 35 satisfied satisfied ADJ cana-2410 316 36 . . PUNCT cana-2410 317 1 case case NOUN cana-2410 317 2 ( ( PUNCT cana-2410 317 3 2 2 NUM cana-2410 317 4 ): ): PUNCT cana-2410 317 5 if if SCONJ cana-2410 317 6 𝕋 𝕋 NOUN cana-2410 317 7 = = SYM cana-2410 317 8 β„™1,1 β„™1,1 NOUN cana-2410 318 1 = = NOUN cana-2410 318 2 βˆͺ𝑗=0 βˆͺ𝑗=0 NOUN cana-2410 318 3 ∞ ∞ PROPN cana-2410 319 1 [ [ X cana-2410 319 2 2𝑗. 2𝑗. NOUN cana-2410 319 3 2𝑗 2𝑗 NOUN cana-2410 319 4 + + NOUN cana-2410 319 5 1 1 NUM cana-2410 319 6 ] ] PUNCT cana-2410 319 7 , , PUNCT cana-2410 319 8 then then ADV cana-2410 319 9 π‘’π‘Ž(𝑑 π‘’π‘Ž(𝑑 NOUN cana-2410 319 10 , , PUNCT cana-2410 319 11 0 0 NUM cana-2410 319 12 ) ) PUNCT cana-2410 319 13 = = SYM cana-2410 319 14 ( ( PUNCT cana-2410 319 15 1 1 NUM cana-2410 319 16 + + NUM cana-2410 319 17 π‘Ž)π‘—π‘’π‘Ž(π‘‘βˆ’π‘— π‘Ž)π‘—π‘’π‘Ž(π‘‘βˆ’π‘— NOUN cana-2410 319 18 ) ) PUNCT cana-2410 319 19 . . PUNCT cana-2410 320 1 therefore therefore ADV cana-2410 320 2 , , PUNCT cana-2410 320 3 𝒩0(0 𝒩0(0 ADV cana-2410 320 4 , , PUNCT cana-2410 320 5 𝑑1 𝑑1 NOUN cana-2410 320 6 ) ) PUNCT cana-2410 320 7 = = PUNCT cana-2410 321 1 βˆ«π’©(0 βˆ«π’©(0 PROPN cana-2410 321 2 , , PUNCT cana-2410 321 3 Οƒ(Ο„))dΟ„ Οƒ(Ο„))dΟ„ VERB cana-2410 321 4 t t NOUN cana-2410 321 5 0 0 NUM cana-2410 322 1 = = PUNCT cana-2410 322 2 [ [ PUNCT cana-2410 322 3 7.2665 7.2665 NUM cana-2410 322 4 0 0 NUM cana-2410 322 5 0 0 NUM cana-2410 322 6 0 0 NUM cana-2410 322 7 0 0 NUM cana-2410 323 1 7.2665 7.2665 NUM cana-2410 323 2 0 0 NUM cana-2410 323 3 0 0 NUM cana-2410 323 4 0 0 NUM cana-2410 323 5 0 0 NUM cana-2410 323 6 0 0 NUM cana-2410 323 7 0 0 NUM cana-2410 323 8 24.156 24.156 NUM cana-2410 323 9 0 0 NUM cana-2410 323 10 0 0 NUM cana-2410 323 11 24.156 24.156 NUM cana-2410 323 12 ] ] PUNCT cana-2410 323 13 . . PUNCT cana-2410 324 1 𝒩1(𝑠1 𝒩1(𝑠1 NOUN cana-2410 324 2 , , PUNCT cana-2410 324 3 𝑑2 𝑑2 NOUN cana-2410 324 4 ) ) PUNCT cana-2410 324 5 = = SYM cana-2410 325 1 ∫ ∫ PROPN cana-2410 325 2 𝒩(𝑠1,Οƒ(Ο„))dΟ„ 𝒩(𝑠1,Οƒ(Ο„))dΟ„ PROPN cana-2410 325 3 𝑑2 𝑑2 PROPN cana-2410 325 4 𝑠1 𝑠1 PROPN cana-2410 326 1 = = PUNCT cana-2410 326 2 [ [ PUNCT cana-2410 326 3 6.781 6.781 NUM cana-2410 326 4 0 0 NUM cana-2410 326 5 0 0 NUM cana-2410 326 6 0 0 NUM cana-2410 326 7 0 0 NUM cana-2410 327 1 6.781 6.781 NUM cana-2410 327 2 0 0 NUM cana-2410 327 3 0 0 NUM cana-2410 327 4 0 0 NUM cana-2410 327 5 0 0 NUM cana-2410 327 6 0 0 NUM cana-2410 327 7 0 0 NUM cana-2410 328 1 15.167 15.167 NUM cana-2410 328 2 0 0 NUM cana-2410 328 3 0 0 NUM cana-2410 329 1 15.167 15.167 NUM cana-2410 329 2 ] ] PUNCT cana-2410 329 3 . . PUNCT cana-2410 330 1 𝒩2(𝑠2 𝒩2(𝑠2 NUM cana-2410 330 2 , , PUNCT cana-2410 330 3 𝑇 𝑇 PROPN cana-2410 330 4 ) ) PUNCT cana-2410 330 5 = = SYM cana-2410 331 1 ∫ ∫ PROPN cana-2410 331 2 𝒩(𝑠2,Οƒ(Ο„))dΟ„ 𝒩(𝑠2,Οƒ(Ο„))dΟ„ PROPN cana-2410 331 3 𝑇 𝑇 PROPN cana-2410 331 4 𝑠2 𝑠2 NOUN cana-2410 331 5 = = PUNCT cana-2410 331 6 [ [ PUNCT cana-2410 331 7 73.2665 73.2665 NUM cana-2410 331 8 0 0 NUM cana-2410 331 9 0 0 NUM cana-2410 331 10 0 0 NUM cana-2410 331 11 0 0 NUM cana-2410 331 12 73.2665 73.2665 NUM cana-2410 331 13 0 0 NUM cana-2410 331 14 0 0 NUM cana-2410 331 15 0 0 NUM cana-2410 331 16 0 0 NUM cana-2410 331 17 0 0 NUM cana-2410 331 18 0 0 NUM cana-2410 331 19 827.24 827.24 NUM cana-2410 331 20 0 0 NUM cana-2410 331 21 0 0 NUM cana-2410 331 22 827.24 827.24 NUM cana-2410 331 23 ] ] PUNCT cana-2410 331 24 . . PUNCT cana-2410 332 1 it it PRON cana-2410 332 2 follows follow VERB cana-2410 332 3 that that SCONJ cana-2410 332 4 the the DET cana-2410 332 5 matrices matrix NOUN cana-2410 332 6 𝒩0(0 𝒩0(0 NOUN cana-2410 332 7 , , PUNCT cana-2410 332 8 𝑑1 𝑑1 NOUN cana-2410 332 9 ) ) PUNCT cana-2410 332 10 , , PUNCT cana-2410 332 11 𝒩1(𝑠1 𝒩1(𝑠1 PROPN cana-2410 332 12 , , PUNCT cana-2410 332 13 𝑑2 𝑑2 PROPN cana-2410 332 14 ) ) PUNCT cana-2410 332 15 , , PUNCT cana-2410 332 16 and and CCONJ cana-2410 332 17 𝒩2(𝑠2 𝒩2(𝑠2 NUM cana-2410 332 18 , , PUNCT cana-2410 332 19 𝑇 𝑇 PROPN cana-2410 332 20 ) ) PUNCT cana-2410 332 21 are be AUX cana-2410 332 22 all all ADV cana-2410 332 23 invertible invertible ADJ cana-2410 332 24 . . PUNCT cana-2410 333 1 in in ADP cana-2410 333 2 addition addition NOUN cana-2410 333 3 , , PUNCT cana-2410 333 4 all all DET cana-2410 333 5 three three NUM cana-2410 333 6 assumptions assumption NOUN cana-2410 333 7 ( ( PUNCT cana-2410 333 8 h1 h1 PROPN cana-2410 333 9 ) ) PUNCT cana-2410 333 10 βˆ’ βˆ’ PROPN cana-2410 333 11 ( ( PUNCT cana-2410 333 12 h3 h3 NOUN cana-2410 333 13 ) ) PUNCT cana-2410 333 14 hold hold VERB cana-2410 333 15 . . PUNCT cana-2410 334 1 with with ADP cana-2410 334 2 m𝛼 m𝛼 NOUN cana-2410 334 3 = = SYM cana-2410 334 4 max max PROPN cana-2410 334 5 { { PUNCT cana-2410 334 6 0.9480 0.9480 NOUN cana-2410 334 7 , , PUNCT cana-2410 334 8 0.2107 0.2107 NUM cana-2410 334 9 , , PUNCT cana-2410 334 10 0.9513 0.9513 NUM cana-2410 334 11 , , PUNCT cana-2410 334 12 0.00367 0.00367 NUM cana-2410 334 13 } } PUNCT cana-2410 334 14 < < X cana-2410 334 15 1 1 NUM cana-2410 334 16 . . PUNCT cana-2410 334 17 the the DET cana-2410 334 18 system system NOUN cana-2410 334 19 ( ( PUNCT cana-2410 334 20 3.30 3.30 NUM cana-2410 334 21 ) ) PUNCT cana-2410 334 22 is be AUX cana-2410 334 23 complete complete ADJ cana-2410 334 24 controllable controllable ADJ cana-2410 334 25 , , PUNCT cana-2410 334 26 since since SCONJ cana-2410 334 27 all all DET cana-2410 334 28 the the DET cana-2410 334 29 criteria criterion NOUN cana-2410 334 30 of of ADP cana-2410 334 31 theorem theorem NOUN cana-2410 334 32 3.1 3.1 NUM cana-2410 334 33 , , PUNCT cana-2410 334 34 are be AUX cana-2410 334 35 satisfied satisfied ADJ cana-2410 334 36 . . PUNCT cana-2410 335 1 declarations declaration NOUN cana-2410 335 2 conflict conflict NOUN cana-2410 335 3 of of ADP cana-2410 335 4 interest interest NOUN cana-2410 335 5 : : PUNCT cana-2410 335 6 the the DET cana-2410 335 7 authors author NOUN cana-2410 335 8 declare declare VERB cana-2410 335 9 that that SCONJ cana-2410 335 10 they they PRON cana-2410 335 11 have have VERB cana-2410 335 12 no no DET cana-2410 335 13 competing compete VERB cana-2410 335 14 interests interest NOUN cana-2410 335 15 . . PUNCT cana-2410 336 1 author author NOUN cana-2410 336 2 contributions contribution NOUN cana-2410 336 3 : : PUNCT cana-2410 336 4 all all DET cana-2410 336 5 authors author NOUN cana-2410 336 6 contributed contribute VERB cana-2410 336 7 equally equally ADV cana-2410 336 8 to to ADP cana-2410 336 9 this this DET cana-2410 336 10 article article NOUN cana-2410 336 11 . . PUNCT cana-2410 337 1 all all DET cana-2410 337 2 authors author NOUN cana-2410 337 3 read read VERB cana-2410 337 4 and and CCONJ cana-2410 337 5 approved approve VERB cana-2410 337 6 the the DET cana-2410 337 7 final final ADJ cana-2410 337 8 manuscript manuscript NOUN cana-2410 337 9 . . PUNCT cana-2410 338 1 data datum NOUN cana-2410 338 2 availability availability NOUN cana-2410 338 3 : : PUNCT cana-2410 338 4 not not PART cana-2410 338 5 applicable applicable ADJ cana-2410 338 6 . . PUNCT cana-2410 339 1 funding funding NOUN cana-2410 339 2 : : PUNCT cana-2410 339 3 not not PART cana-2410 339 4 applicable applicable ADJ cana-2410 339 5 . . PUNCT cana-2410 340 1 references reference NOUN cana-2410 340 2 : : PUNCT cana-2410 341 1 [ [ X cana-2410 341 2 1 1 NUM cana-2410 341 3 ] ] X cana-2410 341 4 agarwal agarwal PROPN cana-2410 341 5 , , PUNCT cana-2410 341 6 r. r. PROPN cana-2410 341 7 p. p. PROPN cana-2410 341 8 , , PUNCT cana-2410 341 9 bohner bohner NOUN cana-2410 341 10 , , PUNCT cana-2410 341 11 m. m. NOUN cana-2410 341 12 , , PUNCT cana-2410 341 13 regan regan PROPN cana-2410 341 14 , , PUNCT cana-2410 341 15 d.o d.o PROPN cana-2410 341 16 . . PROPN cana-2410 341 17 , , PUNCT cana-2410 341 18 and and CCONJ cana-2410 341 19 peterson peterson PROPN cana-2410 341 20 , , PUNCT cana-2410 341 21 a. a. NOUN cana-2410 341 22 : : PUNCT cana-2410 341 23 dynamic dynamic ADJ cana-2410 341 24 equations equation NOUN cana-2410 341 25 on on ADP cana-2410 341 26 time time NOUN cana-2410 341 27 [ [ X cana-2410 341 28 2 2 NUM cana-2410 341 29 ] ] PUNCT cana-2410 341 30 scales scale NOUN cana-2410 341 31 . . PUNCT cana-2410 342 1 a a DET cana-2410 342 2 survey survey NOUN cana-2410 342 3 , , PUNCT cana-2410 342 4 j j PROPN cana-2410 342 5 comput comput NOUN cana-2410 342 6 . . PUNCT cana-2410 343 1 appl appl PROPN cana-2410 343 2 . . PROPN cana-2410 343 3 math math PROPN cana-2410 343 4 , , PUNCT cana-2410 343 5 no.4 no.4 PROPN cana-2410 343 6 , , PUNCT cana-2410 343 7 1 1 NUM cana-2410 343 8 - - SYM cana-2410 343 9 26 26 NUM cana-2410 343 10 , , PUNCT cana-2410 343 11 ( ( PUNCT cana-2410 343 12 2002 2002 NUM cana-2410 343 13 ) ) PUNCT cana-2410 343 14 . . PUNCT cana-2410 344 1 [ [ X cana-2410 344 2 3 3 NUM cana-2410 344 3 ] ] X cana-2410 344 4 alexander alexander NOUN cana-2410 344 5 , , PUNCT cana-2410 344 6 g. g. NOUN cana-2410 344 7 : : PUNCT cana-2410 344 8 kronecker kronecker NOUN cana-2410 344 9 products product NOUN cana-2410 344 10 and and CCONJ cana-2410 344 11 matrix matrix NOUN cana-2410 344 12 calculus calculus NOUN cana-2410 344 13 ; ; PUNCT cana-2410 344 14 with with ADP cana-2410 344 15 applications application NOUN cana-2410 344 16 , , PUNCT cana-2410 344 17 ellis ellis PROPN cana-2410 344 18 hordwood hordwood PROPN cana-2410 344 19 ltd ltd PROPN cana-2410 344 20 . . PROPN cana-2410 344 21 , , PUNCT cana-2410 344 22 england england PROPN cana-2410 344 23 , , PUNCT cana-2410 344 24 ( ( PUNCT cana-2410 344 25 1981 1981 NUM cana-2410 344 26 ) ) PUNCT cana-2410 344 27 . . PUNCT cana-2410 345 1 [ [ X cana-2410 345 2 4 4 X cana-2410 345 3 ] ] X cana-2410 345 4 appa appa PROPN cana-2410 345 5 rao rao PROPN cana-2410 345 6 , , PUNCT cana-2410 345 7 b. b. PROPN cana-2410 345 8 v. v. ADV cana-2410 345 9 , , PUNCT cana-2410 345 10 and and CCONJ cana-2410 345 11 prasad prasad PROPN cana-2410 345 12 , , PUNCT cana-2410 345 13 kasnv kasnv PROPN cana-2410 345 14 . . PUNCT cana-2410 345 15 : : PUNCT cana-2410 346 1 study study NOUN cana-2410 346 2 of of ADP cana-2410 346 3 controllability controllability NOUN cana-2410 346 4 of of ADP cana-2410 346 5 matrix matrix NOUN cana-2410 346 6 integrodifferential integrodifferential ADJ cana-2410 346 7 equations equation NOUN cana-2410 346 8 on on ADP cana-2410 346 9 time time NOUN cana-2410 346 10 scales scale NOUN cana-2410 346 11 . . PUNCT cana-2410 347 1 international international ADJ cana-2410 347 2 journal journal PROPN cana-2410 347 3 of of ADP cana-2410 347 4 chemical chemical PROPN cana-2410 347 5 sciences sciences PROPN cana-2410 347 6 , , PUNCT cana-2410 347 7 13 13 NUM cana-2410 347 8 ( ( PUNCT cana-2410 347 9 3 3 NUM cana-2410 347 10 ) ) PUNCT cana-2410 347 11 , , PUNCT cana-2410 347 12 pp pp ADP cana-2410 347 13 . . PUNCT cana-2410 347 14 1324 1324 NUM cana-2410 347 15 – – PUNCT cana-2410 347 16 1332 1332 NUM cana-2410 347 17 , , PUNCT cana-2410 347 18 ( ( PUNCT cana-2410 347 19 2015 2015 NUM cana-2410 347 20 ) ) PUNCT cana-2410 347 21 . . PUNCT cana-2410 348 1 communications communication NOUN cana-2410 348 2 on on ADP cana-2410 348 3 applied apply VERB cana-2410 348 4 nonlinear nonlinear ADJ cana-2410 348 5 analysis analysis NOUN cana-2410 348 6 issn issn NOUN cana-2410 348 7 : : PUNCT cana-2410 348 8 1074 1074 NUM cana-2410 348 9 - - PUNCT cana-2410 348 10 133x 133x NUM cana-2410 348 11 vol vol NOUN cana-2410 348 12 32 32 NUM cana-2410 348 13 no no NOUN cana-2410 348 14 . . PUNCT cana-2410 349 1 2s 2s NUM cana-2410 349 2 ( ( PUNCT cana-2410 349 3 2025 2025 NUM cana-2410 349 4 ) ) PUNCT cana-2410 349 5 365 365 NUM cana-2410 349 6 https://internationalpubls.com https://internationalpubls.com X cana-2410 350 1 [ [ X cana-2410 350 2 5 5 NUM cana-2410 350 3 ] ] PUNCT cana-2410 350 4 appa appa PROPN cana-2410 350 5 rao rao PROPN cana-2410 350 6 , , PUNCT cana-2410 350 7 b. b. PROPN cana-2410 351 1 v. v. ADV cana-2410 351 2 , , PUNCT cana-2410 351 3 and and CCONJ cana-2410 351 4 prasad prasad PROPN cana-2410 351 5 , , PUNCT cana-2410 351 6 kasnv kasnv PROPN cana-2410 351 7 . . PUNCT cana-2410 351 8 : : PUNCT cana-2410 352 1 controllability controllability NOUN cana-2410 352 2 and and CCONJ cana-2410 352 3 observability observability NOUN cana-2410 352 4 of of ADP cana-2410 352 5 sylvester sylvester ADJ cana-2410 352 6 matrix matrix NOUN cana-2410 352 7 dynamical dynamical ADJ cana-2410 352 8 systems system NOUN cana-2410 352 9 on on ADP cana-2410 352 10 time time NOUN cana-2410 352 11 scales scale NOUN cana-2410 352 12 kyungpook kyungpook PROPN cana-2410 352 13 math.j math.j PROPN cana-2410 352 14 . . PROPN cana-2410 352 15 56 56 NUM cana-2410 352 16 , , PUNCT cana-2410 352 17 529 529 NUM cana-2410 352 18 - - SYM cana-2410 352 19 539 539 NUM cana-2410 352 20 , , PUNCT cana-2410 352 21 ( ( PUNCT cana-2410 352 22 2016 2016 NUM cana-2410 352 23 ) ) PUNCT cana-2410 352 24 . . PUNCT cana-2410 353 1 [ [ X cana-2410 353 2 6 6 NUM cana-2410 353 3 ] ] SYM cana-2410 353 4 bhoner bhoner NOUN cana-2410 353 5 , , PUNCT cana-2410 353 6 m. m. NOUN cana-2410 353 7 , , PUNCT cana-2410 353 8 and and CCONJ cana-2410 353 9 peterson peterson PROPN cana-2410 353 10 , , PUNCT cana-2410 353 11 a. a. NOUN cana-2410 353 12 : : PUNCT cana-2410 353 13 dynamic dynamic ADJ cana-2410 353 14 equations equation NOUN cana-2410 353 15 on on ADP cana-2410 353 16 time time NOUN cana-2410 353 17 scales scale NOUN cana-2410 353 18 , , PUNCT cana-2410 353 19 birkhauser birkhauser PROPN cana-2410 353 20 , , PUNCT cana-2410 353 21 boston boston PROPN cana-2410 353 22 , , PUNCT cana-2410 353 23 ( ( PUNCT cana-2410 353 24 2001 2001 NUM cana-2410 353 25 ) ) PUNCT cana-2410 353 26 . . PUNCT cana-2410 354 1 [ [ X cana-2410 354 2 7 7 NUM cana-2410 354 3 ] ] X cana-2410 354 4 bhoner bhoner NOUN cana-2410 354 5 , , PUNCT cana-2410 354 6 m. m. NOUN cana-2410 354 7 , , PUNCT cana-2410 354 8 and and CCONJ cana-2410 354 9 peterson peterson PROPN cana-2410 354 10 , , PUNCT cana-2410 354 11 a. a. NOUN cana-2410 354 12 : : PUNCT cana-2410 354 13 advances advance NOUN cana-2410 354 14 in in ADP cana-2410 354 15 dynamic dynamic ADJ cana-2410 354 16 equations equation NOUN cana-2410 354 17 on on ADP cana-2410 354 18 time time NOUN cana-2410 354 19 scales scale NOUN cana-2410 354 20 , , PUNCT cana-2410 354 21 birkhauser birkhauser PROPN cana-2410 354 22 , , PUNCT cana-2410 354 23 boston boston PROPN cana-2410 354 24 , , PUNCT cana-2410 354 25 ( ( PUNCT cana-2410 354 26 2003 2003 NUM cana-2410 354 27 ) ) PUNCT cana-2410 354 28 . . PUNCT cana-2410 355 1 [ [ X cana-2410 355 2 8 8 NUM cana-2410 355 3 ] ] PUNCT cana-2410 355 4 dacunha dacunha VERB cana-2410 355 5 j.j j.j PROPN cana-2410 355 6 . . PROPN cana-2410 355 7 : : PUNCT cana-2410 355 8 transition transition NOUN cana-2410 355 9 matrix matrix NOUN cana-2410 355 10 and and CCONJ cana-2410 355 11 generalized generalized ADJ cana-2410 355 12 matrix matrix NOUN cana-2410 355 13 exponential exponential NOUN cana-2410 355 14 via via ADP cana-2410 355 15 the the DET cana-2410 355 16 peano peano PROPN cana-2410 355 17 - - PUNCT cana-2410 355 18 baker baker PROPN cana-2410 355 19 series series PROPN cana-2410 355 20 , , PUNCT cana-2410 355 21 j. j. PROPN cana-2410 355 22 differ differ VERB cana-2410 355 23 . . PUNCT cana-2410 356 1 equ equ PROPN cana-2410 356 2 . . PUNCT cana-2410 356 3 appl.11 appl.11 PROPN cana-2410 356 4 , , PUNCT cana-2410 356 5 ( ( PUNCT cana-2410 356 6 15 15 NUM cana-2410 356 7 ) ) PUNCT cana-2410 356 8 , , PUNCT cana-2410 356 9 1245 1245 NUM cana-2410 356 10 - - SYM cana-2410 356 11 1264 1264 NUM cana-2410 356 12 , , PUNCT cana-2410 356 13 ( ( PUNCT cana-2410 356 14 2005 2005 NUM cana-2410 356 15 ) ) PUNCT cana-2410 356 16 . . PUNCT cana-2410 357 1 [ [ X cana-2410 357 2 9 9 X cana-2410 357 3 ] ] X cana-2410 357 4 davis davis PROPN cana-2410 357 5 jhon jhon PROPN cana-2410 357 6 , , PUNCT cana-2410 357 7 m. m. NOUN cana-2410 357 8 , , PUNCT cana-2410 357 9 gravagre gravagre NOUN cana-2410 357 10 ian ian PROPN cana-2410 357 11 , , PUNCT cana-2410 357 12 a. a. PROPN cana-2410 357 13 , , PUNCT cana-2410 357 14 jackson jackson PROPN cana-2410 357 15 billy billy PROPN cana-2410 357 16 , , PUNCT cana-2410 357 17 j. j. PROPN cana-2410 357 18 , , PUNCT cana-2410 357 19 and and CCONJ cana-2410 357 20 marks mark VERB cana-2410 357 21 robert robert PROPN cana-2410 357 22 . . PROPN cana-2410 357 23 j. j. PROPN cana-2410 357 24 : : PUNCT cana-2410 357 25 controllability controllability NOUN cana-2410 357 26 , , PUNCT cana-2410 357 27 observability observability NOUN cana-2410 357 28 , , PUNCT cana-2410 357 29 realizability realizability NOUN cana-2410 357 30 and and CCONJ cana-2410 357 31 stability stability NOUN cana-2410 357 32 of of ADP cana-2410 357 33 dynamic dynamic ADJ cana-2410 357 34 linear linear NOUN cana-2410 357 35 systems system NOUN cana-2410 357 36 , , PUNCT cana-2410 357 37 electronic electronic ADJ cana-2410 357 38 journal journal NOUN cana-2410 357 39 of of ADP cana-2410 357 40 differential differential ADJ cana-2410 357 41 equations equation NOUN cana-2410 357 42 vol vol VERB cana-2410 357 43 2009 2009 NUM cana-2410 357 44 no.37 no.37 NOUN cana-2410 357 45 , , PUNCT cana-2410 357 46 1 1 NUM cana-2410 357 47 - - SYM cana-2410 357 48 32 32 NUM cana-2410 357 49 , , PUNCT cana-2410 357 50 ( ( PUNCT cana-2410 357 51 2009 2009 NUM cana-2410 357 52 ) ) PUNCT cana-2410 357 53 . . PUNCT cana-2410 358 1 [ [ X cana-2410 358 2 10 10 NUM cana-2410 358 3 ] ] X cana-2410 358 4 fausett fausett NOUN cana-2410 358 5 , , PUNCT cana-2410 358 6 l. l. PROPN cana-2410 358 7 v. v. PROPN cana-2410 358 8 , , PUNCT cana-2410 358 9 and and CCONJ cana-2410 358 10 murty murty NOUN cana-2410 358 11 , , PUNCT cana-2410 358 12 k.n k.n PROPN cana-2410 358 13 . . PROPN cana-2410 358 14 : : PUNCT cana-2410 359 1 controllability controllability NOUN cana-2410 359 2 , , PUNCT cana-2410 359 3 observability observability NOUN cana-2410 359 4 , , PUNCT cana-2410 359 5 and and CCONJ cana-2410 359 6 realizability realizability NOUN cana-2410 359 7 criteria criterion NOUN cana-2410 359 8 on on ADP cana-2410 359 9 time time NOUN cana-2410 359 10 scale scale NOUN cana-2410 359 11 dynamical dynamical ADJ cana-2410 359 12 systems system NOUN cana-2410 359 13 , , PUNCT cana-2410 359 14 nonlinear nonlinear ADJ cana-2410 359 15 stud.11 stud.11 PROPN cana-2410 359 16 , , PUNCT cana-2410 359 17 627–638 627–638 NUM cana-2410 359 18 , , PUNCT cana-2410 359 19 ( ( PUNCT cana-2410 359 20 2004 2004 NUM cana-2410 359 21 ) ) PUNCT cana-2410 359 22 . . PUNCT cana-2410 360 1 [ [ X cana-2410 360 2 11 11 NUM cana-2410 360 3 ] ] PUNCT cana-2410 360 4 hilger hilger NOUN cana-2410 360 5 , , PUNCT cana-2410 360 6 s. s. PROPN cana-2410 360 7 : : PUNCT cana-2410 360 8 analysis analysis NOUN cana-2410 360 9 on on ADP cana-2410 360 10 measure measure NOUN cana-2410 360 11 chains chain NOUN cana-2410 360 12 a a DET cana-2410 360 13 unified unified ADJ cana-2410 360 14 approach approach NOUN cana-2410 360 15 to to ADP cana-2410 360 16 continuous continuous ADJ cana-2410 360 17 and and CCONJ cana-2410 360 18 discrete discrete ADJ cana-2410 360 19 calculus calculus NOUN cana-2410 360 20 , , PUNCT cana-2410 360 21 results result VERB cana-2410 360 22 math math NOUN cana-2410 360 23 . . PUNCT cana-2410 361 1 18 18 NUM cana-2410 361 2 , , PUNCT cana-2410 361 3 18 18 NUM cana-2410 361 4 – – SYM cana-2410 361 5 56 56 NUM cana-2410 361 6 , , PUNCT cana-2410 361 7 ( ( PUNCT cana-2410 361 8 1990 1990 NUM cana-2410 361 9 ) ) PUNCT cana-2410 361 10 . . PUNCT cana-2410 362 1 [ [ X cana-2410 362 2 12 12 NUM cana-2410 362 3 ] ] X cana-2410 362 4 kostic kostic NOUN cana-2410 362 5 , , PUNCT cana-2410 362 6 m. m. NOUN cana-2410 362 7 , , PUNCT cana-2410 362 8 kumar kumar PROPN cana-2410 362 9 , , PUNCT cana-2410 362 10 v. v. PROPN cana-2410 362 11 , , PUNCT cana-2410 362 12 and and CCONJ cana-2410 362 13 pinto pinto NOUN cana-2410 362 14 , , PUNCT cana-2410 362 15 m. m. NOUN cana-2410 362 16 : : PUNCT cana-2410 362 17 stepanov stepanov VERB cana-2410 362 18 multi multi ADJ cana-2410 362 19 - - ADJ cana-2410 362 20 dimensional dimensional ADJ cana-2410 362 21 almost almost ADV cana-2410 362 22 automorphic automorphic ADJ cana-2410 362 23 type type NOUN cana-2410 362 24 functions function NOUN cana-2410 362 25 and and CCONJ cana-2410 362 26 applications application NOUN cana-2410 362 27 , , PUNCT cana-2410 362 28 journal journal NOUN cana-2410 362 29 of of ADP cana-2410 362 30 nonlinear nonlinear ADJ cana-2410 362 31 evolution evolution NOUN cana-2410 362 32 equations equation NOUN cana-2410 362 33 and and CCONJ cana-2410 362 34 applications application NOUN cana-2410 362 35 volume volume NOUN cana-2410 362 36 2022 2022 NUM cana-2410 362 37 , , PUNCT cana-2410 362 38 number number NOUN cana-2410 362 39 1 1 NUM cana-2410 362 40 , , PUNCT cana-2410 362 41 pp pp PROPN cana-2410 362 42 . . PUNCT cana-2410 363 1 1–24 1–24 PROPN cana-2410 363 2 ( ( PUNCT cana-2410 363 3 june june PROPN cana-2410 363 4 2022 2022 NUM cana-2410 363 5 ) ) PUNCT cana-2410 363 6 . . PUNCT cana-2410 364 1 [ [ X cana-2410 364 2 13 13 NUM cana-2410 364 3 ] ] X cana-2410 364 4 kumar kumar PROPN cana-2410 364 5 , , PUNCT cana-2410 364 6 v. v. PROPN cana-2410 364 7 , , PUNCT cana-2410 364 8 malik malik PROPN cana-2410 364 9 , , PUNCT cana-2410 364 10 m. m. NOUN cana-2410 364 11 , , PUNCT cana-2410 364 12 mohamed mohamed PROPN cana-2410 364 13 , , PUNCT cana-2410 364 14 d. d. PROPN cana-2410 364 15 : : PUNCT cana-2410 364 16 results result VERB cana-2410 364 17 on on ADP cana-2410 364 18 abstract abstract ADJ cana-2410 364 19 integro integro ADJ cana-2410 364 20 hybrid hybrid ADJ cana-2410 364 21 evolution evolution NOUN cana-2410 364 22 system system NOUN cana-2410 364 23 with with ADP cana-2410 364 24 impulses impulse NOUN cana-2410 364 25 on on ADP cana-2410 364 26 time time NOUN cana-2410 364 27 scales scale NOUN cana-2410 364 28 , , PUNCT cana-2410 364 29 nonlinear nonlinear ADJ cana-2410 364 30 analysis analysis NOUN cana-2410 364 31 : : PUNCT cana-2410 364 32 hybrid hybrid ADJ cana-2410 364 33 systems system NOUN cana-2410 364 34 , , PUNCT cana-2410 364 35 [ [ X cana-2410 364 36 14 14 NUM cana-2410 364 37 ] ] PUNCT cana-2410 364 38 volume volume NOUN cana-2410 364 39 39 39 NUM cana-2410 364 40 , , PUNCT cana-2410 364 41 ( ( PUNCT cana-2410 364 42 2021 2021 NUM cana-2410 364 43 ) ) PUNCT cana-2410 364 44 . . PUNCT cana-2410 365 1 https://doi.org/10.1016/j.nahs.2020.100986 https://doi.org/10.1016/j.nahs.2020.100986 PROPN cana-2410 365 2 . . PUNCT cana-2410 366 1 [ [ X cana-2410 366 2 15 15 NUM cana-2410 366 3 ] ] X cana-2410 366 4 kumar kumar PROPN cana-2410 366 5 , , PUNCT cana-2410 366 6 v. v. PROPN cana-2410 366 7 , , PUNCT cana-2410 366 8 mohamed mohamed PROPN cana-2410 366 9 , , PUNCT cana-2410 366 10 d. d. PROPN cana-2410 366 11 : : PUNCT cana-2410 366 12 existence existence NOUN cana-2410 366 13 , , PUNCT cana-2410 366 14 stability stability NOUN cana-2410 366 15 and and CCONJ cana-2410 366 16 controllability controllability NOUN cana-2410 366 17 of of ADP cana-2410 366 18 piecewise piecewise NOUN cana-2410 366 19 impulsive impulsive ADJ cana-2410 366 20 dynamic dynamic ADJ cana-2410 366 21 systems system NOUN cana-2410 366 22 on on ADP cana-2410 366 23 arbitrary arbitrary ADJ cana-2410 366 24 time time NOUN cana-2410 366 25 domain domain NOUN cana-2410 366 26 , , PUNCT cana-2410 366 27 applied apply VERB cana-2410 366 28 mathematical mathematical ADJ cana-2410 366 29 modelling modelling NOUN cana-2410 366 30 , , PUNCT cana-2410 366 31 volume volume NOUN cana-2410 366 32 117 117 NUM cana-2410 366 33 , , PUNCT cana-2410 366 34 pages page NOUN cana-2410 366 35 529 529 NUM cana-2410 366 36 - - SYM cana-2410 366 37 548 548 NUM cana-2410 366 38 , , PUNCT cana-2410 366 39 ( ( PUNCT cana-2410 366 40 2023 2023 NUM cana-2410 366 41 ) ) PUNCT cana-2410 366 42 . . PUNCT cana-2410 367 1 https://doi.org/10.1016/j.apm.2022.12.027 https://doi.org/10.1016/j.apm.2022.12.027 PROPN cana-2410 367 2 . . PUNCT cana-2410 368 1 [ [ X cana-2410 368 2 16 16 NUM cana-2410 368 3 ] ] X cana-2410 368 4 liu liu PROPN cana-2410 368 5 , , PUNCT cana-2410 368 6 h. h. PROPN cana-2410 368 7 , , PUNCT cana-2410 368 8 xiang xiang PROPN cana-2410 368 9 , , PUNCT cana-2410 368 10 x. x. NOUN cana-2410 368 11 : : PUNCT cana-2410 368 12 a a DET cana-2410 368 13 class class NOUN cana-2410 368 14 of of ADP cana-2410 368 15 the the DET cana-2410 368 16 first first ADJ cana-2410 368 17 order order NOUN cana-2410 368 18 impulsive impulsive ADJ cana-2410 368 19 dynamic dynamic ADJ cana-2410 368 20 equations equation NOUN cana-2410 368 21 on on ADP cana-2410 368 22 time time NOUN cana-2410 368 23 scales scale NOUN cana-2410 368 24 , , PUNCT cana-2410 368 25 nonlinear nonlinear ADJ cana-2410 368 26 analysis analysis NOUN cana-2410 368 27 , , PUNCT cana-2410 368 28 69 69 NUM cana-2410 368 29 , , PUNCT cana-2410 368 30 28032811 28032811 NUM cana-2410 368 31 , , PUNCT cana-2410 368 32 ( ( PUNCT cana-2410 368 33 2008 2008 NUM cana-2410 368 34 ) ) PUNCT cana-2410 368 35 . . PUNCT cana-2410 369 1 [ [ X cana-2410 369 2 17 17 NUM cana-2410 369 3 ] ] X cana-2410 369 4 murty murty NOUN cana-2410 369 5 , , PUNCT cana-2410 369 6 m.s.n m.s.n NOUN cana-2410 369 7 . . PUNCT cana-2410 369 8 , , PUNCT cana-2410 369 9 kumar kumar PROPN cana-2410 369 10 , , PUNCT cana-2410 369 11 g.s g.s PROPN cana-2410 369 12 . . PROPN cana-2410 369 13 , , PUNCT cana-2410 369 14 appa appa PROPN cana-2410 369 15 rao rao PROPN cana-2410 369 16 , , PUNCT cana-2410 369 17 b. b. PROPN cana-2410 369 18 v. v. ADV cana-2410 369 19 , , PUNCT cana-2410 369 20 and and CCONJ cana-2410 369 21 prasad prasad PROPN cana-2410 369 22 , , PUNCT cana-2410 369 23 kasnv kasnv PROPN cana-2410 369 24 . . PUNCT cana-2410 369 25 : : PUNCT cana-2410 370 1 on on ADP cana-2410 370 2 controllability controllability NOUN cana-2410 370 3 of of ADP cana-2410 370 4 fuzzy fuzzy ADJ cana-2410 370 5 dynamical dynamical ADJ cana-2410 370 6 matrix matrix NOUN cana-2410 370 7 lyapunov lyapunov NOUN cana-2410 370 8 systems system NOUN cana-2410 370 9 , , PUNCT cana-2410 370 10 analele analele ADP cana-2410 370 11 university university PROPN cana-2410 370 12 , , PUNCT cana-2410 370 13 di di X cana-2410 370 14 vest vest PROPN cana-2410 370 15 timsora timsora PROPN cana-2410 370 16 , , PUNCT cana-2410 370 17 seria seria PROPN cana-2410 370 18 mathematica mathematica PROPN cana-2410 370 19 informatica informatica PROPN cana-2410 370 20 li li PROPN cana-2410 370 21 , , PUNCT cana-2410 370 22 2(13 2(13 NUM cana-2410 370 23 ) ) PUNCT cana-2410 370 24 , , PUNCT cana-2410 370 25 73 73 NUM cana-2410 370 26 - - SYM cana-2410 370 27 86 86 NUM cana-2410 370 28 , , PUNCT cana-2410 370 29 ( ( PUNCT cana-2410 370 30 2013 2013 NUM cana-2410 370 31 ) ) PUNCT cana-2410 370 32 . . PUNCT cana-2410 371 1 [ [ X cana-2410 371 2 18 18 NUM cana-2410 371 3 ] ] X cana-2410 371 4 murty murty NOUN cana-2410 371 5 , , PUNCT cana-2410 371 6 m.s.n m.s.n NOUN cana-2410 371 7 . . PUNCT cana-2410 371 8 , , PUNCT cana-2410 371 9 appa appa PROPN cana-2410 371 10 rao rao PROPN cana-2410 371 11 , , PUNCT cana-2410 371 12 b. b. PROPN cana-2410 372 1 v. v. CCONJ cana-2410 372 2 : : PUNCT cana-2410 372 3 controllability controllability NOUN cana-2410 372 4 and and CCONJ cana-2410 372 5 observability observability NOUN cana-2410 372 6 of of ADP cana-2410 372 7 matrix matrix NOUN cana-2410 372 8 lyapunov lyapunov NOUN cana-2410 372 9 systems system NOUN cana-2410 372 10 , , PUNCT cana-2410 372 11 ranchi ranchi PROPN cana-2410 372 12 univ.math univ.math PROPN cana-2410 372 13 . . PUNCT cana-2410 372 14 journal journal PROPN cana-2410 372 15 vol.no.32 vol.no.32 PROPN cana-2410 372 16 , , PUNCT cana-2410 372 17 55 55 NUM cana-2410 372 18 - - SYM cana-2410 372 19 65 65 NUM cana-2410 372 20 , , PUNCT cana-2410 372 21 ( ( PUNCT cana-2410 372 22 2005 2005 NUM cana-2410 372 23 ) ) PUNCT cana-2410 372 24 . . PUNCT cana-2410 373 1 [ [ X cana-2410 373 2 19 19 NUM cana-2410 373 3 ] ] X cana-2410 373 4 murty murty NOUN cana-2410 373 5 , , PUNCT cana-2410 373 6 m.s.n m.s.n NOUN cana-2410 373 7 . . PUNCT cana-2410 373 8 , , PUNCT cana-2410 373 9 appa appa PROPN cana-2410 373 10 rao rao PROPN cana-2410 373 11 , , PUNCT cana-2410 373 12 b. b. PROPN cana-2410 373 13 v. v. ADV cana-2410 373 14 , , PUNCT cana-2410 373 15 and and CCONJ cana-2410 373 16 suresh suresh PROPN cana-2410 373 17 kumar kumar PROPN cana-2410 373 18 g. g. PROPN cana-2410 373 19 : : PUNCT cana-2410 373 20 controllability controllability NOUN cana-2410 373 21 , , PUNCT cana-2410 373 22 observability observability NOUN cana-2410 373 23 and and CCONJ cana-2410 373 24 realizability realizability NOUN cana-2410 373 25 of of ADP cana-2410 373 26 matrix matrix NOUN cana-2410 373 27 lyapunov lyapunov NOUN cana-2410 373 28 systems system NOUN cana-2410 373 29 , , PUNCT cana-2410 373 30 bull.kor bull.kor X cana-2410 373 31 , , PUNCT cana-2410 373 32 ean ean NOUN cana-2410 373 33 math.soc math.soc NOUN cana-2410 373 34 . . NOUN cana-2410 373 35 ,43 ,43 NOUN cana-2410 373 36 , , PUNCT cana-2410 373 37 no.1 no.1 NUM cana-2410 373 38 , , PUNCT cana-2410 373 39 149 149 NUM cana-2410 373 40 - - SYM cana-2410 373 41 159 159 NUM cana-2410 373 42 , , PUNCT cana-2410 373 43 , , PUNCT cana-2410 373 44 ( ( PUNCT cana-2410 373 45 2006 2006 NUM cana-2410 373 46 ) ) PUNCT cana-2410 373 47 . . PUNCT cana-2410 374 1 [ [ X cana-2410 374 2 20 20 NUM cana-2410 374 3 ] ] X cana-2410 374 4 sreenivasulu sreenivasulu PROPN cana-2410 374 5 , , PUNCT cana-2410 374 6 a. a. PROPN cana-2410 374 7 , , PUNCT cana-2410 374 8 appa appa PROPN cana-2410 374 9 rao rao PROPN cana-2410 374 10 , , PUNCT cana-2410 374 11 b. b. PROPN cana-2410 375 1 v. v. NOUN cana-2410 375 2 : : PUNCT cana-2410 375 3 stability stability NOUN cana-2410 375 4 criteria criterion NOUN cana-2410 375 5 for for ADP cana-2410 375 6 non non ADJ cana-2410 375 7 - - ADJ cana-2410 375 8 linear linear ADJ cana-2410 375 9 volterra volterra NOUN cana-2410 375 10 integro integro PROPN cana-2410 375 11 dynamical dynamical ADJ cana-2410 375 12 matrix matrix NOUN cana-2410 375 13 sylvester sylvester NOUN cana-2410 375 14 systems system NOUN cana-2410 375 15 on on ADP cana-2410 375 16 measure measure NOUN cana-2410 375 17 chains chain NOUN cana-2410 375 18 , , PUNCT cana-2410 375 19 adv adv PROPN cana-2410 375 20 differ differ VERB cana-2410 375 21 equ2021 equ2021 NOUN cana-2410 375 22 , , PUNCT cana-2410 375 23 514 514 NUM cana-2410 375 24 , , PUNCT cana-2410 375 25 ( ( PUNCT cana-2410 375 26 2021 2021 NUM cana-2410 375 27 ) ) PUNCT cana-2410 375 28 . . PUNCT cana-2410 376 1 https://doi.org/10.1186/s13662-021-03665-6 https://doi.org/10.1186/s13662-021-03665-6 PROPN cana-2410 376 2 . . PUNCT cana-2410 377 1 [ [ X cana-2410 377 2 21 21 NUM cana-2410 377 3 ] ] X cana-2410 377 4 sreenivasulu sreenivasulu PROPN cana-2410 377 5 , , PUNCT cana-2410 377 6 a. a. PROPN cana-2410 377 7 , , PUNCT cana-2410 377 8 appa appa PROPN cana-2410 377 9 rao rao PROPN cana-2410 377 10 , , PUNCT cana-2410 377 11 b. b. PROPN cana-2410 378 1 v. v. NOUN cana-2410 378 2 : : PUNCT cana-2410 378 3 stability stability NOUN cana-2410 378 4 and and CCONJ cana-2410 378 5 controllability controllability NOUN cana-2410 378 6 for for ADP cana-2410 378 7 volterra volterra PROPN cana-2410 378 8 integro integro PROPN cana-2410 378 9 - - PUNCT cana-2410 378 10 dynamical dynamical ADJ cana-2410 378 11 matrix matrix NOUN cana-2410 378 12 sylvester sylvester NOUN cana-2410 378 13 impulsive impulsive ADJ cana-2410 378 14 system system NOUN cana-2410 378 15 on on ADP cana-2410 378 16 time time NOUN cana-2410 378 17 scales scale NOUN cana-2410 378 18 . . PUNCT cana-2410 379 1 j. j. PROPN cana-2410 379 2 appl appl PROPN cana-2410 379 3 . . PROPN cana-2410 379 4 math math PROPN cana-2410 379 5 . . PUNCT cana-2410 380 1 comput comput NOUN cana-2410 380 2 , , PUNCT cana-2410 380 3 68 68 NUM cana-2410 380 4 , , PUNCT cana-2410 380 5 3705–3720 3705–3720 NUM cana-2410 380 6 ( ( PUNCT cana-2410 380 7 2022 2022 NUM cana-2410 380 8 ) ) PUNCT cana-2410 380 9 . . PUNCT cana-2410 381 1 https://doi.org/10.1007/s12190-02101688-6 https://doi.org/10.1007/s12190-02101688-6 PROPN cana-2410 381 2 . . PUNCT cana-2410 382 1 [ [ X cana-2410 382 2 22 22 NUM cana-2410 382 3 ] ] X cana-2410 382 4 sreenivasulu sreenivasulu PROPN cana-2410 382 5 , , PUNCT cana-2410 382 6 a. a. PROPN cana-2410 382 7 , , PUNCT cana-2410 382 8 appa appa PROPN cana-2410 382 9 rao rao PROPN cana-2410 382 10 , , PUNCT cana-2410 382 11 b. b. PROPN cana-2410 383 1 v. v. NOUN cana-2410 383 2 : : PUNCT cana-2410 383 3 qualitative qualitative ADJ cana-2410 383 4 aspects aspect NOUN cana-2410 383 5 for for ADP cana-2410 383 6 volterra volterra PROPN cana-2410 383 7 integro integro PROPN cana-2410 383 8 - - PUNCT cana-2410 383 9 dynamic dynamic ADJ cana-2410 383 10 matrix matrix NOUN cana-2410 383 11 sylvester sylvester NOUN cana-2410 383 12 impulsive impulsive ADJ cana-2410 383 13 system system NOUN cana-2410 383 14 on on ADP cana-2410 383 15 time time NOUN cana-2410 383 16 scales scale NOUN cana-2410 383 17 , , PUNCT cana-2410 383 18 journal journal NOUN cana-2410 383 19 of of ADP cana-2410 383 20 applied apply VERB cana-2410 383 21 nonlinear nonlinear ADJ cana-2410 383 22 dynamics dynamic NOUN cana-2410 383 23 13(1 13(1 NUM cana-2410 383 24 ) ) PUNCT cana-2410 383 25 , , PUNCT cana-2410 383 26 71 71 NUM cana-2410 383 27 - - SYM cana-2410 383 28 87 87 NUM cana-2410 383 29 , , PUNCT cana-2410 383 30 ( ( PUNCT cana-2410 383 31 2024 2024 NUM cana-2410 383 32 ) ) PUNCT cana-2410 383 33 . . PUNCT cana-2410 384 1 doi:10.5890 doi:10.5890 NOUN cana-2410 384 2 / / SYM cana-2410 384 3 jand.2024.03.006 jand.2024.03.006 NOUN cana-2410 384 4 . . PUNCT cana-2410 385 1 [ [ X cana-2410 385 2 23 23 NUM cana-2410 385 3 ] ] X cana-2410 385 4 sreenivasulu sreenivasulu PROPN cana-2410 385 5 , , PUNCT cana-2410 385 6 a. a. PROPN cana-2410 385 7 , , PUNCT cana-2410 385 8 appa appa PROPN cana-2410 385 9 rao rao PROPN cana-2410 385 10 , , PUNCT cana-2410 385 11 b. b. PROPN cana-2410 386 1 v. v. CCONJ cana-2410 386 2 : : PUNCT cana-2410 386 3 controllability controllability NOUN cana-2410 386 4 and and CCONJ cana-2410 386 5 observability observability NOUN cana-2410 386 6 for for ADP cana-2410 386 7 matrix matrix NOUN cana-2410 386 8 sylvester sylvester NOUN cana-2410 386 9 impulsive impulsive ADJ cana-2410 386 10 non non ADJ cana-2410 386 11 - - ADJ cana-2410 386 12 linear linear ADJ cana-2410 386 13 dynamic dynamic ADJ cana-2410 386 14 system system NOUN cana-2410 386 15 with with ADP cana-2410 386 16 delta delta NOUN cana-2410 386 17 settings setting NOUN cana-2410 386 18 , , PUNCT cana-2410 386 19 aip aip PROPN cana-2410 386 20 conference conference NOUN cana-2410 386 21 proceedings proceeding NOUN cana-2410 386 22 , , PUNCT cana-2410 386 23 2707,200020 2707,200020 NUM cana-2410 386 24 , , PUNCT cana-2410 386 25 ( ( PUNCT cana-2410 386 26 2023 2023 NUM cana-2410 386 27 ) ) PUNCT cana-2410 386 28 . . PUNCT cana-2410 387 1 [ [ X cana-2410 387 2 24 24 NUM cana-2410 387 3 ] ] X cana-2410 387 4 xie xie PROPN cana-2410 387 5 , , PUNCT cana-2410 387 6 g. g. PROPN cana-2410 387 7 m m PROPN cana-2410 387 8 , , PUNCT cana-2410 387 9 wang wang PROPN cana-2410 387 10 , , PUNCT cana-2410 387 11 l. l. PROPN cana-2410 387 12 : : PUNCT cana-2410 387 13 controllability controllability NOUN cana-2410 387 14 and and CCONJ cana-2410 387 15 observability observability NOUN cana-2410 387 16 of of ADP cana-2410 387 17 a a DET cana-2410 387 18 class class NOUN cana-2410 387 19 of of ADP cana-2410 387 20 linear linear ADJ cana-2410 387 21 impulsive impulsive ADJ cana-2410 387 22 systems system NOUN cana-2410 387 23 j.math.anal.appl j.math.anal.appl NOUN cana-2410 387 24 . . PUNCT cana-2410 388 1 ,304 ,304 PROPN cana-2410 388 2 , , PUNCT cana-2410 388 3 336 336 NUM cana-2410 388 4 - - SYM cana-2410 388 5 355 355 NUM cana-2410 388 6 , , PUNCT cana-2410 388 7 ( ( PUNCT cana-2410 388 8 2005 2005 NUM cana-2410 388 9 ) ) PUNCT cana-2410 388 10 . . PUNCT cana-2410 389 1 [ [ X cana-2410 389 2 25 25 NUM cana-2410 389 3 ] ] X cana-2410 389 4 zehra zehra PROPN cana-2410 389 5 , , PUNCT cana-2410 389 6 a. a. PROPN cana-2410 389 7 , , PUNCT cana-2410 389 8 younus younus PROPN cana-2410 389 9 , , PUNCT cana-2410 389 10 a. a. PROPN cana-2410 389 11 , , PUNCT cana-2410 389 12 & & CCONJ cana-2410 389 13 tunΓ§ tunΓ§ PROPN cana-2410 389 14 , , PUNCT cana-2410 389 15 c. c. PROPN cana-2410 389 16 : : PUNCT cana-2410 389 17 controllability controllability NOUN cana-2410 389 18 and and CCONJ cana-2410 389 19 observability observability NOUN cana-2410 389 20 of of ADP cana-2410 389 21 linear linear ADJ cana-2410 389 22 impulsive impulsive ADJ cana-2410 389 23 differential differential ADJ cana-2410 389 24 algebraic algebraic ADJ cana-2410 389 25 system system NOUN cana-2410 389 26 with with ADP cana-2410 389 27 caputo caputo PROPN cana-2410 389 28 fractional fractional PROPN cana-2410 389 29 derivative derivative PROPN cana-2410 389 30 . . PUNCT cana-2410 390 1 computational computational ADJ cana-2410 390 2 methods method NOUN cana-2410 390 3 for for ADP cana-2410 390 4 differential differential ADJ cana-2410 390 5 equations equation NOUN cana-2410 390 6 , , PUNCT cana-2410 390 7 10(1 10(1 NUM cana-2410 390 8 ) ) PUNCT cana-2410 390 9 , , PUNCT cana-2410 390 10 200 200 NUM cana-2410 390 11 - - SYM cana-2410 390 12 214 214 NUM cana-2410 390 13 , , PUNCT cana-2410 390 14 ( ( PUNCT cana-2410 390 15 2022 2022 NUM cana-2410 390 16 ) ) PUNCT cana-2410 390 17 . . PUNCT cana-2410 391 1 [ [ X cana-2410 391 2 26 26 NUM cana-2410 391 3 ] ] X cana-2410 391 4 zhao zhao PROPN cana-2410 391 5 , , PUNCT cana-2410 391 6 s. s. PROPN cana-2410 391 7 , , PUNCT cana-2410 391 8 sun sun PROPN cana-2410 391 9 , , PUNCT cana-2410 391 10 j. j. PROPN cana-2410 391 11 : : PUNCT cana-2410 391 12 controllability controllability NOUN cana-2410 391 13 and and CCONJ cana-2410 391 14 observability observability NOUN cana-2410 391 15 of of ADP cana-2410 391 16 a a DET cana-2410 391 17 class class NOUN cana-2410 391 18 of of ADP cana-2410 391 19 time time NOUN cana-2410 391 20 varying vary VERB cana-2410 391 21 impulsive impulsive ADJ cana-2410 391 22 systems system NOUN cana-2410 391 23 , , PUNCT cana-2410 391 24 nonlinear nonlinear ADJ cana-2410 391 25 analysis analysis NOUN cana-2410 391 26 : : PUNCT cana-2410 391 27 real real ADJ cana-2410 391 28 world world NOUN cana-2410 391 29 applications application NOUN cana-2410 391 30 , , PUNCT cana-2410 391 31 10 10 NUM cana-2410 391 32 , , PUNCT cana-2410 391 33 1370 1370 NUM cana-2410 391 34 - - SYM cana-2410 391 35 1380 1380 NUM cana-2410 391 36 , , PUNCT cana-2410 391 37 ( ( PUNCT cana-2410 391 38 2009 2009 NUM cana-2410 391 39 ) ) PUNCT cana-2410 391 40 . . PUNCT cana-2410 392 1 https://doi.org/10.1016/j.nahs.2020.100986 https://doi.org/10.1016/j.nahs.2020.100986 PROPN cana-2410 392 2 https://doi.org/10.1186/s13662-021-03665-6 https://doi.org/10.1186/s13662-021-03665-6 PROPN cana-2410 392 3 https://doi.org/10.1007/s12190-021-01688-6 https://doi.org/10.1007/s12190-021-01688-6 NUM cana-2410 392 4 https://doi.org/10.1007/s12190-021-01688-6 https://doi.org/10.1007/s12190-021-01688-6 X