id sid tid token lemma pos ijassa-1480 1 1 microsoft microsoft PROPN ijassa-1480 1 2 word word NOUN ijassa-1480 1 3 1480 1480 NUM ijassa-1480 1 4 - - PUNCT ijassa-1480 1 5 article article NOUN ijassa-1480 1 6 text-7014 text-7014 NOUN ijassa-1480 1 7 - - PUNCT ijassa-1480 1 8 1 1 NUM ijassa-1480 1 9 - - PUNCT ijassa-1480 1 10 18 18 NUM ijassa-1480 1 11 - - SYM ijassa-1480 1 12 20240409 20240409 NUM ijassa-1480 1 13 adv adv PROPN ijassa-1480 1 14 syst syst PROPN ijassa-1480 1 15 sci sci PROPN ijassa-1480 1 16 appl appl PROPN ijassa-1480 1 17 2024 2024 NUM ijassa-1480 1 18 ; ; PUNCT ijassa-1480 1 19 02 02 NUM ijassa-1480 1 20 ; ; PUNCT ijassa-1480 1 21 187 187 NUM ijassa-1480 1 22 - - SYM ijassa-1480 1 23 191 191 NUM ijassa-1480 1 24 published publish VERB ijassa-1480 1 25 online online ADV ijassa-1480 1 26 at at ADP ijassa-1480 1 27 https://ijassa.ipu.ru https://ijassa.ipu.ru ADV ijassa-1480 1 28 . . PUNCT ijassa-1480 2 1 lyapunov lyapunov ADJ ijassa-1480 2 2 functions function NOUN ijassa-1480 2 3 for for ADP ijassa-1480 2 4 periodic periodic ADJ ijassa-1480 2 5 selector selector NOUN ijassa-1480 2 6 - - PUNCT ijassa-1480 2 7 linear linear NOUN ijassa-1480 2 8 difference difference NOUN ijassa-1480 2 9 inclusions inclusion NOUN ijassa-1480 2 10 mikhail mikhail PROPN ijassa-1480 2 11 morozov morozov PROPN ijassa-1480 2 12 v.a v.a PROPN ijassa-1480 2 13 . . PROPN ijassa-1480 2 14 trapeznikov trapeznikov PROPN ijassa-1480 2 15 institute institute PROPN ijassa-1480 2 16 of of ADP ijassa-1480 2 17 control control PROPN ijassa-1480 2 18 sciences sciences PROPN ijassa-1480 2 19 of of ADP ijassa-1480 2 20 russian russian ADJ ijassa-1480 2 21 academy academy PROPN ijassa-1480 2 22 of of ADP ijassa-1480 2 23 sciences sciences PROPN ijassa-1480 2 24 , , PUNCT ijassa-1480 2 25 moscow moscow PROPN ijassa-1480 2 26 , , PUNCT ijassa-1480 2 27 russia russia PROPN ijassa-1480 2 28 abstract abstract NOUN ijassa-1480 2 29 : : PUNCT ijassa-1480 2 30 the the DET ijassa-1480 2 31 paper paper NOUN ijassa-1480 2 32 considers consider VERB ijassa-1480 2 33 periodic periodic ADJ ijassa-1480 2 34 selector selector NOUN ijassa-1480 2 35 - - PUNCT ijassa-1480 2 36 linear linear NOUN ijassa-1480 2 37 difference difference NOUN ijassa-1480 2 38 inclusions inclusion NOUN ijassa-1480 2 39 . . PUNCT ijassa-1480 3 1 a a DET ijassa-1480 3 2 class class NOUN ijassa-1480 3 3 of of ADP ijassa-1480 3 4 timeperiodic timeperiodic ADJ ijassa-1480 3 5 quasi quasi ADJ ijassa-1480 3 6 - - ADJ ijassa-1480 3 7 quadratic quadratic ADJ ijassa-1480 3 8 lyapunov lyapunov NOUN ijassa-1480 3 9 functions function NOUN ijassa-1480 3 10 is be AUX ijassa-1480 3 11 distinguished distinguish VERB ijassa-1480 3 12 , , PUNCT ijassa-1480 3 13 as as ADV ijassa-1480 3 14 well well ADV ijassa-1480 3 15 as as ADP ijassa-1480 3 16 parametric parametric ADJ ijassa-1480 3 17 classes class NOUN ijassa-1480 3 18 of of ADP ijassa-1480 3 19 piecewise piecewise NOUN ijassa-1480 3 20 - - PUNCT ijassa-1480 3 21 quadratic quadratic ADJ ijassa-1480 3 22 and and CCONJ ijassa-1480 3 23 piecewise piecewise NOUN ijassa-1480 3 24 - - PUNCT ijassa-1480 3 25 linear linear NOUN ijassa-1480 3 26 lyapunov lyapunov NOUN ijassa-1480 3 27 functions function NOUN ijassa-1480 3 28 . . PUNCT ijassa-1480 4 1 these these DET ijassa-1480 4 2 functions function NOUN ijassa-1480 4 3 establish establish VERB ijassa-1480 4 4 necessary necessary ADJ ijassa-1480 4 5 and and CCONJ ijassa-1480 4 6 sufficient sufficient ADJ ijassa-1480 4 7 conditions condition NOUN ijassa-1480 4 8 for for ADP ijassa-1480 4 9 asymptotic asymptotic ADJ ijassa-1480 4 10 stability stability NOUN ijassa-1480 4 11 . . PUNCT ijassa-1480 5 1 an an DET ijassa-1480 5 2 example example NOUN ijassa-1480 5 3 leading lead VERB ijassa-1480 5 4 to to ADP ijassa-1480 5 5 periodic periodic ADJ ijassa-1480 5 6 selector selector NOUN ijassa-1480 5 7 - - PUNCT ijassa-1480 5 8 linear linear NOUN ijassa-1480 5 9 differential differential NOUN ijassa-1480 5 10 and and CCONJ ijassa-1480 5 11 difference difference NOUN ijassa-1480 5 12 inclusions inclusion NOUN ijassa-1480 5 13 is be AUX ijassa-1480 5 14 given give VERB ijassa-1480 5 15 . . PUNCT ijassa-1480 6 1 the the DET ijassa-1480 6 2 results result NOUN ijassa-1480 6 3 can can AUX ijassa-1480 6 4 find find VERB ijassa-1480 6 5 applications application NOUN ijassa-1480 6 6 in in ADP ijassa-1480 6 7 the the DET ijassa-1480 6 8 stability stability NOUN ijassa-1480 6 9 analysis analysis NOUN ijassa-1480 6 10 of of ADP ijassa-1480 6 11 control control NOUN ijassa-1480 6 12 systems system NOUN ijassa-1480 6 13 with with ADP ijassa-1480 6 14 periodic periodic ADJ ijassa-1480 6 15 parameters parameter NOUN ijassa-1480 6 16 , , PUNCT ijassa-1480 6 17 in in ADP ijassa-1480 6 18 particular particular ADJ ijassa-1480 6 19 , , PUNCT ijassa-1480 6 20 servomechanisms servomechanism VERB ijassa-1480 6 21 whose whose DET ijassa-1480 6 22 elements element NOUN ijassa-1480 6 23 operate operate VERB ijassa-1480 6 24 on on ADP ijassa-1480 6 25 alternating alternate VERB ijassa-1480 6 26 current current ADJ ijassa-1480 6 27 , , PUNCT ijassa-1480 6 28 control control NOUN ijassa-1480 6 29 systems system NOUN ijassa-1480 6 30 with with ADP ijassa-1480 6 31 amplitude amplitude NOUN ijassa-1480 6 32 - - PUNCT ijassa-1480 6 33 frequency frequency NOUN ijassa-1480 6 34 modulation modulation NOUN ijassa-1480 6 35 . . PUNCT ijassa-1480 7 1 keywords keyword NOUN ijassa-1480 7 2 : : PUNCT ijassa-1480 7 3 periodic periodic ADJ ijassa-1480 7 4 selector selector NOUN ijassa-1480 7 5 - - PUNCT ijassa-1480 7 6 linear linear NOUN ijassa-1480 7 7 difference difference NOUN ijassa-1480 7 8 inclusion inclusion NOUN ijassa-1480 7 9 , , PUNCT ijassa-1480 7 10 asymptotic asymptotic ADJ ijassa-1480 7 11 stability stability NOUN ijassa-1480 7 12 , , PUNCT ijassa-1480 7 13 lyapunov lyapunov NOUN ijassa-1480 7 14 functions function NOUN ijassa-1480 7 15 . . PUNCT ijassa-1480 8 1 1 1 X ijassa-1480 8 2 . . X ijassa-1480 8 3 introduction introduction NOUN ijassa-1480 8 4 the the DET ijassa-1480 8 5 problem problem NOUN ijassa-1480 8 6 of of ADP ijassa-1480 8 7 the the DET ijassa-1480 8 8 stability stability NOUN ijassa-1480 8 9 of of ADP ijassa-1480 8 10 periodic periodic ADJ ijassa-1480 8 11 difference difference NOUN ijassa-1480 8 12 inclusions inclusion NOUN ijassa-1480 8 13 arises arise VERB ijassa-1480 8 14 in in ADP ijassa-1480 8 15 the the DET ijassa-1480 8 16 study study NOUN ijassa-1480 8 17 of of ADP ijassa-1480 8 18 discrete discrete ADJ ijassa-1480 8 19 control control NOUN ijassa-1480 8 20 systems system NOUN ijassa-1480 8 21 with with ADP ijassa-1480 8 22 periodic periodic ADJ ijassa-1480 8 23 parameters parameter NOUN ijassa-1480 8 24 and and CCONJ ijassa-1480 8 25 uncertainty uncertainty NOUN ijassa-1480 8 26 , , PUNCT ijassa-1480 8 27 in in ADP ijassa-1480 8 28 particular particular ADJ ijassa-1480 8 29 , , PUNCT ijassa-1480 8 30 tracking tracking NOUN ijassa-1480 8 31 systems system NOUN ijassa-1480 8 32 that that PRON ijassa-1480 8 33 operate operate VERB ijassa-1480 8 34 on on ADP ijassa-1480 8 35 alternating alternate VERB ijassa-1480 8 36 current current ADJ ijassa-1480 8 37 and and CCONJ ijassa-1480 8 38 systems system NOUN ijassa-1480 8 39 with with ADP ijassa-1480 8 40 amplitude amplitude NOUN ijassa-1480 8 41 - - PUNCT ijassa-1480 8 42 frequency frequency NOUN ijassa-1480 8 43 modulation modulation NOUN ijassa-1480 8 44 . . PUNCT ijassa-1480 9 1 in in ADP ijassa-1480 9 2 some some DET ijassa-1480 9 3 cases case NOUN ijassa-1480 9 4 , , PUNCT ijassa-1480 9 5 such such ADJ ijassa-1480 9 6 as as ADP ijassa-1480 9 7 the the DET ijassa-1480 9 8 absolute absolute ADJ ijassa-1480 9 9 stability stability NOUN ijassa-1480 9 10 problem problem NOUN ijassa-1480 9 11 , , PUNCT ijassa-1480 9 12 the the DET ijassa-1480 9 13 study study NOUN ijassa-1480 9 14 of of ADP ijassa-1480 9 15 linear linear PROPN ijassa-1480 9 16 nonstationary nonstationary ADJ ijassa-1480 9 17 systems system NOUN ijassa-1480 9 18 whose whose DET ijassa-1480 9 19 right right ADJ ijassa-1480 9 20 part part NOUN ijassa-1480 9 21 matrix matrix NOUN ijassa-1480 9 22 satisfies satisfy VERB ijassa-1480 9 23 interval interval NOUN ijassa-1480 9 24 constraints constraint NOUN ijassa-1480 9 25 can can AUX ijassa-1480 9 26 be be AUX ijassa-1480 9 27 used use VERB ijassa-1480 9 28 selector selector NOUN ijassa-1480 9 29 - - PUNCT ijassa-1480 9 30 linear linear NOUN ijassa-1480 9 31 difference difference NOUN ijassa-1480 9 32 inclusions inclusion NOUN ijassa-1480 9 33 . . PUNCT ijassa-1480 10 1 in in ADP ijassa-1480 10 2 [ [ X ijassa-1480 10 3 5 5 NUM ijassa-1480 10 4 ] ] SYM ijassa-1480 10 5 periodic periodic ADJ ijassa-1480 10 6 difference difference NOUN ijassa-1480 10 7 inclusions inclusion NOUN ijassa-1480 10 8 are be AUX ijassa-1480 10 9 considered consider VERB ijassa-1480 10 10 . . PUNCT ijassa-1480 11 1 the the DET ijassa-1480 11 2 definitions definition NOUN ijassa-1480 11 3 of of ADP ijassa-1480 11 4 asymptotic asymptotic ADJ ijassa-1480 11 5 , , PUNCT ijassa-1480 11 6 uniform uniform ADJ ijassa-1480 11 7 asymptotic asymptotic ADJ ijassa-1480 11 8 and and CCONJ ijassa-1480 11 9 uniform uniform ADJ ijassa-1480 11 10 exponential exponential ADJ ijassa-1480 11 11 stability stability NOUN ijassa-1480 11 12 are be AUX ijassa-1480 11 13 given give VERB ijassa-1480 11 14 and and CCONJ ijassa-1480 11 15 the the DET ijassa-1480 11 16 equivalence equivalence NOUN ijassa-1480 11 17 of of ADP ijassa-1480 11 18 these these DET ijassa-1480 11 19 properties property NOUN ijassa-1480 11 20 for for ADP ijassa-1480 11 21 selector selector NOUN ijassa-1480 11 22 - - PUNCT ijassa-1480 11 23 linear linear NOUN ijassa-1480 11 24 difference difference NOUN ijassa-1480 11 25 inclusions inclusion NOUN ijassa-1480 11 26 is be AUX ijassa-1480 11 27 proved prove VERB ijassa-1480 11 28 . . PUNCT ijassa-1480 12 1 on on ADP ijassa-1480 12 2 the the DET ijassa-1480 12 3 basis basis NOUN ijassa-1480 12 4 of of ADP ijassa-1480 12 5 the the DET ijassa-1480 12 6 variational variational ADJ ijassa-1480 12 7 approach approach NOUN ijassa-1480 12 8 , , PUNCT ijassa-1480 12 9 a a DET ijassa-1480 12 10 necessary necessary ADJ ijassa-1480 12 11 and and CCONJ ijassa-1480 12 12 sufficient sufficient ADJ ijassa-1480 12 13 condition condition NOUN ijassa-1480 12 14 for for ADP ijassa-1480 12 15 uniform uniform ADJ ijassa-1480 12 16 asymptotic asymptotic ADJ ijassa-1480 12 17 stability stability NOUN ijassa-1480 12 18 in in ADP ijassa-1480 12 19 the the DET ijassa-1480 12 20 form form NOUN ijassa-1480 12 21 of of ADP ijassa-1480 12 22 some some DET ijassa-1480 12 23 limit limit NOUN ijassa-1480 12 24 relation relation NOUN ijassa-1480 12 25 is be AUX ijassa-1480 12 26 obtained obtain VERB ijassa-1480 12 27 . . PUNCT ijassa-1480 13 1 in in ADP ijassa-1480 13 2 this this DET ijassa-1480 13 3 paper paper NOUN ijassa-1480 13 4 we we PRON ijassa-1480 13 5 obtain obtain VERB ijassa-1480 13 6 asymptotic asymptotic ADJ ijassa-1480 13 7 stability stability NOUN ijassa-1480 13 8 criteria criterion NOUN ijassa-1480 13 9 for for ADP ijassa-1480 13 10 periodic periodic ADJ ijassa-1480 13 11 selectorlinear selectorlinear NOUN ijassa-1480 13 12 difference difference NOUN ijassa-1480 13 13 inclusions inclusion NOUN ijassa-1480 13 14 based base VERB ijassa-1480 13 15 on on ADP ijassa-1480 13 16 the the DET ijassa-1480 13 17 lyapunov lyapunov ADJ ijassa-1480 13 18 function function NOUN ijassa-1480 13 19 method method NOUN ijassa-1480 13 20 . . PUNCT ijassa-1480 14 1 the the DET ijassa-1480 14 2 remainder remainder NOUN ijassa-1480 14 3 of of ADP ijassa-1480 14 4 this this DET ijassa-1480 14 5 paper paper NOUN ijassa-1480 14 6 is be AUX ijassa-1480 14 7 structured structure VERB ijassa-1480 14 8 as as SCONJ ijassa-1480 14 9 follows follow VERB ijassa-1480 14 10 . . PUNCT ijassa-1480 15 1 in in ADP ijassa-1480 15 2 section section NOUN ijassa-1480 15 3 2 2 NUM ijassa-1480 15 4 , , PUNCT ijassa-1480 15 5 we we PRON ijassa-1480 15 6 consider consider VERB ijassa-1480 15 7 periodic periodic ADJ ijassa-1480 15 8 selector selector NOUN ijassa-1480 15 9 - - PUNCT ijassa-1480 15 10 linear linear NOUN ijassa-1480 15 11 difference difference NOUN ijassa-1480 15 12 inclusions inclusion NOUN ijassa-1480 15 13 and and CCONJ ijassa-1480 15 14 set set VERB ijassa-1480 15 15 the the DET ijassa-1480 15 16 problem problem NOUN ijassa-1480 15 17 of of ADP ijassa-1480 15 18 obtaining obtain VERB ijassa-1480 15 19 asymptotic asymptotic ADJ ijassa-1480 15 20 stability stability NOUN ijassa-1480 15 21 criteria criterion NOUN ijassa-1480 15 22 for for ADP ijassa-1480 15 23 such such ADJ ijassa-1480 15 24 inclusions inclusion NOUN ijassa-1480 15 25 on on ADP ijassa-1480 15 26 the the DET ijassa-1480 15 27 basis basis NOUN ijassa-1480 15 28 of of ADP ijassa-1480 15 29 the the DET ijassa-1480 15 30 lyapunov lyapunov ADJ ijassa-1480 15 31 function function NOUN ijassa-1480 15 32 method method NOUN ijassa-1480 15 33 . . PUNCT ijassa-1480 16 1 in in ADP ijassa-1480 16 2 section section NOUN ijassa-1480 16 3 3 3 NUM ijassa-1480 16 4 we we PRON ijassa-1480 16 5 distinguish distinguish VERB ijassa-1480 16 6 a a DET ijassa-1480 16 7 class class NOUN ijassa-1480 16 8 of of ADP ijassa-1480 16 9 periodic periodic NOUN ijassa-1480 16 10 in in ADP ijassa-1480 16 11 time time NOUN ijassa-1480 16 12 lyapunov lyapunov NOUN ijassa-1480 16 13 functions function NOUN ijassa-1480 16 14 of of ADP ijassa-1480 16 15 quasi quasi ADJ ijassa-1480 16 16 - - ADJ ijassa-1480 16 17 quadratic quadratic ADJ ijassa-1480 16 18 form form NOUN ijassa-1480 16 19 and and CCONJ ijassa-1480 16 20 parametric parametric ADJ ijassa-1480 16 21 classes class NOUN ijassa-1480 16 22 of of ADP ijassa-1480 16 23 piecewise piecewise NOUN ijassa-1480 16 24 quadratic quadratic ADJ ijassa-1480 16 25 and and CCONJ ijassa-1480 16 26 piecewise piecewise PROPN ijassa-1480 16 27 linear linear PROPN ijassa-1480 16 28 lyapunov lyapunov NOUN ijassa-1480 16 29 functions function NOUN ijassa-1480 16 30 . . PUNCT ijassa-1480 17 1 theorems theorems PROPN ijassa-1480 17 2 1 1 NUM ijassa-1480 17 3 - - SYM ijassa-1480 17 4 3 3 NUM ijassa-1480 17 5 are be AUX ijassa-1480 17 6 formulated formulate VERB ijassa-1480 17 7 and and CCONJ ijassa-1480 17 8 proven prove VERB ijassa-1480 17 9 . . PUNCT ijassa-1480 18 1 they they PRON ijassa-1480 18 2 establish establish VERB ijassa-1480 18 3 necessary necessary ADJ ijassa-1480 18 4 and and CCONJ ijassa-1480 18 5 sufficient sufficient ADJ ijassa-1480 18 6 conditions condition NOUN ijassa-1480 18 7 for for ADP ijassa-1480 18 8 asymptotic asymptotic ADJ ijassa-1480 18 9 stability stability NOUN ijassa-1480 18 10 of of ADP ijassa-1480 18 11 the the DET ijassa-1480 18 12 considered consider VERB ijassa-1480 18 13 inclusions inclusion NOUN ijassa-1480 18 14 . . PUNCT ijassa-1480 19 1 in in ADP ijassa-1480 19 2 section section NOUN ijassa-1480 19 3 4 4 NUM ijassa-1480 19 4 , , PUNCT ijassa-1480 19 5 an an DET ijassa-1480 19 6 example example NOUN ijassa-1480 19 7 of of ADP ijassa-1480 19 8 a a DET ijassa-1480 19 9 system system NOUN ijassa-1480 19 10 leading lead VERB ijassa-1480 19 11 to to ADP ijassa-1480 19 12 periodic periodic ADJ ijassa-1480 19 13 differential differential NOUN ijassa-1480 19 14 and and CCONJ ijassa-1480 19 15 difference difference NOUN ijassa-1480 19 16 inclusions inclusion NOUN ijassa-1480 19 17 is be AUX ijassa-1480 19 18 given give VERB ijassa-1480 19 19 . . PUNCT ijassa-1480 20 1 in in ADP ijassa-1480 20 2 section section NOUN ijassa-1480 20 3 5 5 NUM ijassa-1480 20 4 , , PUNCT ijassa-1480 20 5 we we PRON ijassa-1480 20 6 offer offer VERB ijassa-1480 20 7 concluding concluding NOUN ijassa-1480 20 8 remarks remark NOUN ijassa-1480 20 9 . . PUNCT ijassa-1480 21 1 2 2 X ijassa-1480 21 2 . . X ijassa-1480 21 3 statement statement NOUN ijassa-1480 21 4 of of ADP ijassa-1480 21 5 the the DET ijassa-1480 21 6 problem problem NOUN ijassa-1480 21 7 consider consider VERB ijassa-1480 21 8 the the DET ijassa-1480 21 9 dynamic dynamic ADJ ijassa-1480 21 10 systems system NOUN ijassa-1480 21 11 described describe VERB ijassa-1480 21 12 by by ADP ijassa-1480 21 13 periodic periodic ADJ ijassa-1480 21 14 selector selector NOUN ijassa-1480 21 15 - - PUNCT ijassa-1480 21 16 linear linear NOUN ijassa-1480 21 17 difference difference NOUN ijassa-1480 21 18 inclusion inclusion NOUN ijassa-1480 21 19 π‘₯(𝑠 π‘₯(𝑠 NOUN ijassa-1480 21 20 + + CCONJ ijassa-1480 21 21 1 1 X ijassa-1480 21 22 ) ) PUNCT ijassa-1480 21 23 ∈ ∈ PROPN ijassa-1480 21 24 𝐹(𝑠 𝐹(𝑠 NUM ijassa-1480 21 25 , , PUNCT ijassa-1480 21 26 π‘₯ π‘₯ NOUN ijassa-1480 21 27 ) ) PUNCT ijassa-1480 21 28 , , PUNCT ijassa-1480 21 29 𝑠 𝑠 PROPN ijassa-1480 21 30 = = SYM ijassa-1480 21 31 0 0 NUM ijassa-1480 21 32 , , PUNCT ijassa-1480 21 33 1 1 NUM ijassa-1480 21 34 , , PUNCT ijassa-1480 21 35 … … PUNCT ijassa-1480 21 36 , , PUNCT ijassa-1480 21 37 π‘₯ π‘₯ PROPN ijassa-1480 21 38 ∈ ∈ PROPN ijassa-1480 21 39 𝑅 𝑅 PROPN ijassa-1480 21 40 , , PUNCT ijassa-1480 21 41 ( ( PUNCT ijassa-1480 21 42 2.1 2.1 NUM ijassa-1480 21 43 ) ) PUNCT ijassa-1480 21 44 where where SCONJ ijassa-1480 21 45 the the DET ijassa-1480 21 46 set set NOUN ijassa-1480 21 47 - - PUNCT ijassa-1480 21 48 valued value VERB ijassa-1480 21 49 map map NOUN ijassa-1480 21 50 𝐹 𝐹 PROPN ijassa-1480 21 51 : : PUNCT ijassa-1480 21 52 𝑅 𝑅 PROPN ijassa-1480 21 53 β†’ β†’ SYM ijassa-1480 21 54 𝑅 𝑅 PROPN ijassa-1480 21 55 has have VERB ijassa-1480 21 56 the the DET ijassa-1480 21 57 form form NOUN ijassa-1480 21 58 𝐹(𝑠 𝐹(𝑠 NUM ijassa-1480 21 59 , , PUNCT ijassa-1480 21 60 π‘₯ π‘₯ NOUN ijassa-1480 21 61 ) ) PUNCT ijassa-1480 21 62 = = SYM ijassa-1480 21 63 { { PUNCT ijassa-1480 21 64 𝑦 𝑦 NOUN ijassa-1480 21 65 : : PUNCT ijassa-1480 21 66 𝑦 𝑦 NOUN ijassa-1480 21 67 = = SYM ijassa-1480 21 68 𝐡(𝑠)π‘₯ 𝐡(𝑠)π‘₯ PROPN ijassa-1480 21 69 , , PUNCT ijassa-1480 21 70 𝐡(𝑠 𝐡(𝑠 NOUN ijassa-1480 21 71 ) ) PUNCT ijassa-1480 21 72 ∈ ∈ PROPN ijassa-1480 21 73 Ο‰(𝑠 Ο‰(𝑠 PROPN ijassa-1480 21 74 ) ) PUNCT ijassa-1480 21 75 } } PUNCT ijassa-1480 21 76 , , PUNCT ijassa-1480 21 77 188 188 NUM ijassa-1480 21 78 m. m. NOUN ijassa-1480 21 79 morozov morozov NOUN ijassa-1480 21 80 copyright copyright NOUN ijassa-1480 21 81 Β© Β© PROPN ijassa-1480 21 82 2024 2024 NUM ijassa-1480 21 83 assa assa PROPN ijassa-1480 21 84 adv adv PROPN ijassa-1480 21 85 . . PUNCT ijassa-1480 22 1 in in ADP ijassa-1480 22 2 systems system NOUN ijassa-1480 22 3 science science NOUN ijassa-1480 22 4 and and CCONJ ijassa-1480 22 5 appl appl NOUN ijassa-1480 22 6 . . PUNCT ijassa-1480 23 1 ( ( PUNCT ijassa-1480 23 2 2024 2024 NUM ijassa-1480 23 3 ) ) PUNCT ijassa-1480 23 4 here here ADV ijassa-1480 23 5 𝛺(𝑠 𝛺(𝑠 NOUN ijassa-1480 23 6 ) ) PUNCT ijassa-1480 23 7 , , PUNCT ijassa-1480 23 8 𝛺(𝑠 𝛺(𝑠 X ijassa-1480 24 1 + + CCONJ ijassa-1480 24 2 𝑁 𝑁 PROPN ijassa-1480 24 3 ) ) PUNCT ijassa-1480 24 4 = = SYM ijassa-1480 24 5 𝛺(𝑠 𝛺(𝑠 PROPN ijassa-1480 24 6 ) ) PUNCT ijassa-1480 24 7 ( ( PUNCT ijassa-1480 24 8 𝑠 𝑠 NOUN ijassa-1480 24 9 = = SYM ijassa-1480 24 10 0,1 0,1 NUM ijassa-1480 24 11 , , PUNCT ijassa-1480 24 12 . . PUNCT ijassa-1480 25 1 .. .. PUNCT ijassa-1480 25 2 , , PUNCT ijassa-1480 25 3 𝑁is 𝑁is VERB ijassa-1480 25 4 a a DET ijassa-1480 25 5 natural natural ADJ ijassa-1480 25 6 number number NOUN ijassa-1480 25 7 ) ) PUNCT ijassa-1480 25 8 is be AUX ijassa-1480 25 9 a a DET ijassa-1480 25 10 convex convex NOUN ijassa-1480 25 11 , , PUNCT ijassa-1480 25 12 compact compact ADJ ijassa-1480 25 13 set set NOUN ijassa-1480 25 14 of of ADP ijassa-1480 25 15 real real ADJ ijassa-1480 25 16 ( ( PUNCT ijassa-1480 25 17 𝑛 𝑛 PROPN ijassa-1480 25 18 Γ— Γ— PROPN ijassa-1480 25 19 𝑛 𝑛 PROPN ijassa-1480 25 20 ) ) PUNCT ijassa-1480 25 21 – – PUNCT ijassa-1480 25 22 matrices matrice VERB ijassa-1480 25 23 𝐡. 𝐡. ADP ijassa-1480 25 24 such such ADJ ijassa-1480 25 25 set set NOUN ijassa-1480 25 26 - - PUNCT ijassa-1480 25 27 valued value VERB ijassa-1480 25 28 maps map NOUN ijassa-1480 25 29 are be AUX ijassa-1480 25 30 called call VERB ijassa-1480 25 31 selector selector NOUN ijassa-1480 25 32 - - PUNCT ijassa-1480 25 33 linear linear NOUN ijassa-1480 25 34 , , PUNCT ijassa-1480 25 35 since since SCONJ ijassa-1480 25 36 the the DET ijassa-1480 25 37 righthand righthand NOUN ijassa-1480 25 38 side side NOUN ijassa-1480 25 39 is be AUX ijassa-1480 25 40 a a DET ijassa-1480 25 41 union union NOUN ijassa-1480 25 42 of of ADP ijassa-1480 25 43 linear linear PROPN ijassa-1480 25 44 maps map NOUN ijassa-1480 25 45 . . PUNCT ijassa-1480 26 1 the the DET ijassa-1480 26 2 sequence sequence NOUN ijassa-1480 26 3 of of ADP ijassa-1480 26 4 vectors vector NOUN ijassa-1480 26 5 { { PUNCT ijassa-1480 26 6 π‘₯(𝑠 π‘₯(𝑠 PROPN ijassa-1480 26 7 ) ) PUNCT ijassa-1480 26 8 } } PUNCT ijassa-1480 26 9 , , PUNCT ijassa-1480 26 10 satisfying satisfy VERB ijassa-1480 26 11 for for ADP ijassa-1480 26 12 all all DET ijassa-1480 26 13 𝑠 𝑠 NOUN ijassa-1480 26 14 = = SYM ijassa-1480 26 15 0,1 0,1 NUM ijassa-1480 26 16 , , PUNCT ijassa-1480 26 17 … … PUNCT ijassa-1480 26 18 inclusion inclusion NOUN ijassa-1480 26 19 ( ( PUNCT ijassa-1480 26 20 2.1 2.1 NUM ijassa-1480 26 21 ) ) PUNCT ijassa-1480 26 22 , , PUNCT ijassa-1480 26 23 is be AUX ijassa-1480 26 24 the the DET ijassa-1480 26 25 solution solution NOUN ijassa-1480 26 26 of of ADP ijassa-1480 26 27 the the DET ijassa-1480 26 28 inclusion inclusion NOUN ijassa-1480 26 29 ( ( PUNCT ijassa-1480 26 30 2.1 2.1 NUM ijassa-1480 26 31 ) ) PUNCT ijassa-1480 26 32 . . PUNCT ijassa-1480 27 1 let let VERB ijassa-1480 27 2 π‘₯(𝑠 π‘₯(𝑠 PROPN ijassa-1480 27 3 , , PUNCT ijassa-1480 27 4 𝑠 𝑠 INTJ ijassa-1480 27 5 , , PUNCT ijassa-1480 27 6 π‘₯ π‘₯ PROPN ijassa-1480 27 7 ) ) PUNCT ijassa-1480 27 8 be be AUX ijassa-1480 27 9 the the DET ijassa-1480 27 10 solution solution NOUN ijassa-1480 27 11 of of ADP ijassa-1480 27 12 inclusion inclusion NOUN ijassa-1480 27 13 ( ( PUNCT ijassa-1480 27 14 2.1 2.1 NUM ijassa-1480 27 15 ) ) PUNCT ijassa-1480 27 16 with with ADP ijassa-1480 27 17 initial initial ADJ ijassa-1480 27 18 conditions condition NOUN ijassa-1480 27 19 ( ( PUNCT ijassa-1480 27 20 𝑠 𝑠 INTJ ijassa-1480 27 21 , , PUNCT ijassa-1480 27 22 π‘₯ π‘₯ PROPN ijassa-1480 27 23 ) ) PUNCT ijassa-1480 27 24 . . PUNCT ijassa-1480 28 1 due due ADP ijassa-1480 28 2 to to ADP ijassa-1480 28 3 the the DET ijassa-1480 28 4 periodicity periodicity NOUN ijassa-1480 28 5 of of ADP ijassa-1480 28 6 the the DET ijassa-1480 28 7 multivalued multivalue VERB ijassa-1480 28 8 function function NOUN ijassa-1480 28 9 𝐹(𝑠 𝐹(𝑠 NUM ijassa-1480 28 10 , , PUNCT ijassa-1480 28 11 π‘₯ π‘₯ NOUN ijassa-1480 28 12 ) ) PUNCT ijassa-1480 28 13 in in ADP ijassa-1480 28 14 𝑠 𝑠 PROPN ijassa-1480 28 15 without without ADP ijassa-1480 28 16 generality generality NOUN ijassa-1480 28 17 restriction restriction NOUN ijassa-1480 28 18 we we PRON ijassa-1480 28 19 can can AUX ijassa-1480 28 20 assume assume VERB ijassa-1480 28 21 that that SCONJ ijassa-1480 28 22 0 0 NUM ijassa-1480 28 23 ≀ ≀ NUM ijassa-1480 29 1 𝑠 𝑠 PRON ijassa-1480 29 2 ≀ ≀ NOUN ijassa-1480 29 3 𝑁. 𝑁. PROPN ijassa-1480 29 4 the the DET ijassa-1480 29 5 equivalence equivalence NOUN ijassa-1480 29 6 of of ADP ijassa-1480 29 7 the the DET ijassa-1480 29 8 properties property NOUN ijassa-1480 29 9 of of ADP ijassa-1480 29 10 asymptotic asymptotic ADJ ijassa-1480 29 11 stability stability NOUN ijassa-1480 29 12 , , PUNCT ijassa-1480 29 13 uniform uniform ADJ ijassa-1480 29 14 asymptotic asymptotic ADJ ijassa-1480 29 15 stability stability NOUN ijassa-1480 29 16 , , PUNCT ijassa-1480 29 17 and and CCONJ ijassa-1480 29 18 uniform uniform ADJ ijassa-1480 29 19 exponential exponential ADJ ijassa-1480 29 20 stability stability NOUN ijassa-1480 29 21 for for ADP ijassa-1480 29 22 inclusion inclusion NOUN ijassa-1480 29 23 ( ( PUNCT ijassa-1480 29 24 2.1 2.1 NUM ijassa-1480 29 25 ) ) PUNCT ijassa-1480 29 26 was be AUX ijassa-1480 29 27 proved prove VERB ijassa-1480 29 28 in in ADP ijassa-1480 29 29 [ [ X ijassa-1480 29 30 5 5 NUM ijassa-1480 29 31 ] ] PUNCT ijassa-1480 29 32 . . PUNCT ijassa-1480 30 1 hereinafter hereinafter NOUN ijassa-1480 30 2 we we PRON ijassa-1480 30 3 will will AUX ijassa-1480 30 4 refer refer VERB ijassa-1480 30 5 to to ADP ijassa-1480 30 6 the the DET ijassa-1480 30 7 asymptotic asymptotic ADJ ijassa-1480 30 8 stability stability NOUN ijassa-1480 30 9 of of ADP ijassa-1480 30 10 inclusion inclusion NOUN ijassa-1480 30 11 ( ( PUNCT ijassa-1480 30 12 2.1 2.1 NUM ijassa-1480 30 13 ) ) PUNCT ijassa-1480 30 14 . . PUNCT ijassa-1480 31 1 the the DET ijassa-1480 31 2 problem problem NOUN ijassa-1480 31 3 is be AUX ijassa-1480 31 4 to to PART ijassa-1480 31 5 identify identify VERB ijassa-1480 31 6 the the DET ijassa-1480 31 7 parametric parametric ADJ ijassa-1480 31 8 classes class NOUN ijassa-1480 31 9 of of ADP ijassa-1480 31 10 lyapunov lyapunov NOUN ijassa-1480 31 11 functions function NOUN ijassa-1480 31 12 establishing establish VERB ijassa-1480 31 13 necessary necessary ADJ ijassa-1480 31 14 and and CCONJ ijassa-1480 31 15 sufficient sufficient ADJ ijassa-1480 31 16 conditions condition NOUN ijassa-1480 31 17 for for ADP ijassa-1480 31 18 asymptotic asymptotic ADJ ijassa-1480 31 19 stability stability NOUN ijassa-1480 31 20 of of ADP ijassa-1480 31 21 inclusion inclusion NOUN ijassa-1480 31 22 ( ( PUNCT ijassa-1480 31 23 2.1 2.1 NUM ijassa-1480 31 24 ) ) PUNCT ijassa-1480 31 25 and and CCONJ ijassa-1480 31 26 to to PART ijassa-1480 31 27 construct construct VERB ijassa-1480 31 28 the the DET ijassa-1480 31 29 stability stability NOUN ijassa-1480 31 30 criteria criterion NOUN ijassa-1480 31 31 for for ADP ijassa-1480 31 32 inclusion inclusion NOUN ijassa-1480 31 33 ( ( PUNCT ijassa-1480 31 34 2.1 2.1 NUM ijassa-1480 31 35 ) ) PUNCT ijassa-1480 31 36 using use VERB ijassa-1480 31 37 a a DET ijassa-1480 31 38 discrete discrete ADJ ijassa-1480 31 39 analogue analogue NOUN ijassa-1480 31 40 of of ADP ijassa-1480 31 41 the the DET ijassa-1480 31 42 direct direct ADJ ijassa-1480 31 43 lyapunov lyapunov NOUN ijassa-1480 31 44 method method NOUN ijassa-1480 31 45 . . PUNCT ijassa-1480 32 1 3 3 X ijassa-1480 32 2 . . X ijassa-1480 32 3 results result NOUN ijassa-1480 32 4 theorem theorem VERB ijassa-1480 32 5 3.1 3.1 NUM ijassa-1480 32 6 : : PUNCT ijassa-1480 32 7 the the DET ijassa-1480 32 8 following follow VERB ijassa-1480 32 9 conditions condition NOUN ijassa-1480 32 10 are be AUX ijassa-1480 32 11 equivalent equivalent ADJ ijassa-1480 32 12 : : PUNCT ijassa-1480 32 13 1 1 X ijassa-1480 32 14 . . X ijassa-1480 32 15 inclusion inclusion NOUN ijassa-1480 32 16 ( ( PUNCT ijassa-1480 32 17 2.1 2.1 NUM ijassa-1480 32 18 ) ) PUNCT ijassa-1480 32 19 is be AUX ijassa-1480 32 20 asymptotically asymptotically ADV ijassa-1480 32 21 stable stable ADJ ijassa-1480 32 22 . . PUNCT ijassa-1480 33 1 2 2 X ijassa-1480 33 2 . . X ijassa-1480 33 3 there there PRON ijassa-1480 33 4 exists exist VERB ijassa-1480 33 5 the the DET ijassa-1480 33 6 lyapunov lyapunov NOUN ijassa-1480 33 7 function function NOUN ijassa-1480 33 8 𝑣(𝑠 𝑣(𝑠 PROPN ijassa-1480 33 9 , , PUNCT ijassa-1480 33 10 π‘₯ π‘₯ NOUN ijassa-1480 33 11 ) ) PUNCT ijassa-1480 33 12 of of ADP ijassa-1480 33 13 the the DET ijassa-1480 33 14 quasiquadratic quasiquadratic ADJ ijassa-1480 33 15 form form NOUN ijassa-1480 33 16 𝑣(𝑠 𝑣(𝑠 NUM ijassa-1480 33 17 , , PUNCT ijassa-1480 33 18 π‘₯ π‘₯ NOUN ijassa-1480 33 19 ) ) PUNCT ijassa-1480 33 20 = = SYM ijassa-1480 33 21 π‘₯ π‘₯ DET ijassa-1480 33 22 ′𝐿(𝑠 ′𝐿(𝑠 NUM ijassa-1480 33 23 , , PUNCT ijassa-1480 33 24 π‘₯)π‘₯ π‘₯)π‘₯ ADJ ijassa-1480 33 25 , , PUNCT ijassa-1480 33 26 𝐿(𝑠 𝐿(𝑠 PRON ijassa-1480 33 27 , , PUNCT ijassa-1480 33 28 π‘₯ π‘₯ NOUN ijassa-1480 33 29 ) ) PUNCT ijassa-1480 33 30 = = SYM ijassa-1480 33 31 ( ( PUNCT ijassa-1480 33 32 𝑙 𝑙 X ijassa-1480 33 33 ( ( PUNCT ijassa-1480 33 34 𝑠 𝑠 PROPN ijassa-1480 33 35 , , PUNCT ijassa-1480 33 36 π‘₯ π‘₯ NOUN ijassa-1480 33 37 ) ) PUNCT ijassa-1480 33 38 ) ) PUNCT ijassa-1480 33 39 , , PUNCT ijassa-1480 33 40 , , PUNCT ijassa-1480 33 41 𝐿(𝑠 𝐿(𝑠 PROPN ijassa-1480 34 1 + + CCONJ ijassa-1480 34 2 𝑁 𝑁 PROPN ijassa-1480 34 3 , , PUNCT ijassa-1480 34 4 π‘₯ π‘₯ NOUN ijassa-1480 34 5 ) ) PUNCT ijassa-1480 34 6 = = SYM ijassa-1480 34 7 𝐿(𝑠 𝐿(𝑠 PROPN ijassa-1480 34 8 , , PUNCT ijassa-1480 34 9 π‘₯ π‘₯ NOUN ijassa-1480 34 10 ) ) PUNCT ijassa-1480 34 11 , , PUNCT ijassa-1480 34 12 𝐿′(𝑠 𝐿′(𝑠 ADV ijassa-1480 34 13 , , PUNCT ijassa-1480 34 14 π‘₯ π‘₯ NOUN ijassa-1480 34 15 ) ) PUNCT ijassa-1480 34 16 = = SYM ijassa-1480 35 1 𝐿(𝑠 𝐿(𝑠 PROPN ijassa-1480 35 2 , , PUNCT ijassa-1480 35 3 π‘₯ π‘₯ NOUN ijassa-1480 35 4 ) ) PUNCT ijassa-1480 35 5 = = SYM ijassa-1480 35 6 𝐿(𝑠 𝐿(𝑠 PROPN ijassa-1480 35 7 , , PUNCT ijassa-1480 35 8 πœ‡π‘₯ πœ‡π‘₯ NOUN ijassa-1480 35 9 ) ) PUNCT ijassa-1480 35 10 , , PUNCT ijassa-1480 35 11 π‘₯ π‘₯ PROPN ijassa-1480 35 12 β‰  β‰  PROPN ijassa-1480 35 13 0 0 NUM ijassa-1480 35 14 , , PUNCT ijassa-1480 35 15 πœ‡ πœ‡ ADP ijassa-1480 35 16 β‰  β‰  PROPN ijassa-1480 35 17 0 0 NUM ijassa-1480 35 18 , , PUNCT ijassa-1480 35 19 𝑣(𝑠 𝑣(𝑠 NUM ijassa-1480 35 20 , , PUNCT ijassa-1480 35 21 0 0 NUM ijassa-1480 35 22 ) ) PUNCT ijassa-1480 35 23 ≑ ≑ PROPN ijassa-1480 35 24 0 0 PUNCT ijassa-1480 35 25 ( ( PUNCT ijassa-1480 35 26 3.1 3.1 NUM ijassa-1480 35 27 ) ) PUNCT ijassa-1480 35 28 which which PRON ijassa-1480 35 29 is be AUX ijassa-1480 35 30 𝑁-periodic 𝑁-periodic ADJ ijassa-1480 35 31 in in ADP ijassa-1480 35 32 𝑠 𝑠 PROPN ijassa-1480 35 33 , , PUNCT ijassa-1480 35 34 homogeneous homogeneous ADJ ijassa-1480 35 35 ( ( PUNCT ijassa-1480 35 36 of of ADP ijassa-1480 35 37 second second ADJ ijassa-1480 35 38 order order NOUN ijassa-1480 35 39 ) ) PUNCT ijassa-1480 35 40 , , PUNCT ijassa-1480 35 41 strictly strictly ADV ijassa-1480 35 42 convex convex VERB ijassa-1480 35 43 in in ADP ijassa-1480 35 44 π‘₯ π‘₯ NOUN ijassa-1480 35 45 , , PUNCT ijassa-1480 35 46 and and CCONJ ijassa-1480 35 47 it it PRON ijassa-1480 35 48 satisfies satisfy VERB ijassa-1480 35 49 the the DET ijassa-1480 35 50 following follow VERB ijassa-1480 35 51 inequality inequality NOUN ijassa-1480 35 52 : : PUNCT ijassa-1480 35 53 π‘šπ‘Žπ‘₯ π‘šπ‘Žπ‘₯ ADJ ijassa-1480 35 54 ∈ ∈ NOUN ijassa-1480 35 55 ( ( PUNCT ijassa-1480 35 56 , , PUNCT ijassa-1480 35 57 ) ) PUNCT ijassa-1480 35 58 𝑣(𝑠 𝑣(𝑠 NUM ijassa-1480 35 59 , , PUNCT ijassa-1480 35 60 𝑦 𝑦 NOUN ijassa-1480 35 61 ) ) PUNCT ijassa-1480 35 62 ≀ ≀ NOUN ijassa-1480 35 63 πœƒπ‘£(𝑠 πœƒπ‘£(𝑠 PUNCT ijassa-1480 35 64 , , PUNCT ijassa-1480 35 65 π‘₯ π‘₯ NOUN ijassa-1480 35 66 ) ) PUNCT ijassa-1480 35 67 , , PUNCT ijassa-1480 35 68 π‘₯ π‘₯ PRON ijassa-1480 35 69 ∈ ∈ PROPN ijassa-1480 35 70 𝑅 𝑅 PROPN ijassa-1480 35 71 , , PUNCT ijassa-1480 35 72 𝑠 𝑠 PROPN ijassa-1480 35 73 = = SYM ijassa-1480 35 74 0,1 0,1 NUM ijassa-1480 35 75 , , PUNCT ijassa-1480 35 76 … … PUNCT ijassa-1480 35 77 ( ( PUNCT ijassa-1480 35 78 3.2 3.2 NUM ijassa-1480 35 79 ) ) PUNCT ijassa-1480 35 80 for for ADP ijassa-1480 35 81 some some DET ijassa-1480 35 82 πœƒ πœƒ NOUN ijassa-1480 35 83 ( ( PUNCT ijassa-1480 35 84 0 0 PUNCT ijassa-1480 35 85 < < X ijassa-1480 35 86 πœƒ πœƒ X ijassa-1480 35 87 < < X ijassa-1480 35 88 1 1 NUM ijassa-1480 35 89 ) ) PUNCT ijassa-1480 35 90 . . PUNCT ijassa-1480 36 1 in in ADP ijassa-1480 36 2 ( ( PUNCT ijassa-1480 36 3 3.1 3.1 NUM ijassa-1480 36 4 ) ) PUNCT ijassa-1480 36 5 the the DET ijassa-1480 36 6 prime prime NOUN ijassa-1480 36 7 means mean VERB ijassa-1480 36 8 the the DET ijassa-1480 36 9 transposition transposition NOUN ijassa-1480 36 10 operation operation NOUN ijassa-1480 36 11 . . PUNCT ijassa-1480 37 1 the the DET ijassa-1480 37 2 sufficiency sufficiency NOUN ijassa-1480 37 3 of of ADP ijassa-1480 37 4 the the DET ijassa-1480 37 5 conditions condition NOUN ijassa-1480 37 6 of of ADP ijassa-1480 37 7 theorem theorem ADJ ijassa-1480 37 8 3.1 3.1 NUM ijassa-1480 37 9 is be AUX ijassa-1480 37 10 established establish VERB ijassa-1480 37 11 ( ( PUNCT ijassa-1480 37 12 using use VERB ijassa-1480 37 13 the the DET ijassa-1480 37 14 lemma lemma PROPN ijassa-1480 37 15 given give VERB ijassa-1480 37 16 in in ADP ijassa-1480 37 17 [ [ X ijassa-1480 37 18 4 4 NUM ijassa-1480 37 19 ] ] PUNCT ijassa-1480 37 20 ) ) PUNCT ijassa-1480 37 21 by by ADP ijassa-1480 37 22 the the DET ijassa-1480 37 23 reasoning reasoning NOUN ijassa-1480 37 24 used use VERB ijassa-1480 37 25 in in ADP ijassa-1480 37 26 the the DET ijassa-1480 37 27 proof proof NOUN ijassa-1480 37 28 of of ADP ijassa-1480 37 29 exponential exponential ADJ ijassa-1480 37 30 stability stability NOUN ijassa-1480 37 31 of of ADP ijassa-1480 37 32 discrete discrete ADJ ijassa-1480 37 33 systems system NOUN ijassa-1480 37 34 in in ADP ijassa-1480 37 35 [ [ X ijassa-1480 37 36 2 2 NUM ijassa-1480 37 37 ] ] PUNCT ijassa-1480 37 38 . . PUNCT ijassa-1480 38 1 the the DET ijassa-1480 38 2 proof proof NOUN ijassa-1480 38 3 of of ADP ijassa-1480 38 4 the the DET ijassa-1480 38 5 necessity necessity NOUN ijassa-1480 38 6 follows follow VERB ijassa-1480 38 7 the the DET ijassa-1480 38 8 same same ADJ ijassa-1480 38 9 scheme scheme NOUN ijassa-1480 38 10 as as ADP ijassa-1480 38 11 the the DET ijassa-1480 38 12 proof proof NOUN ijassa-1480 38 13 of of ADP ijassa-1480 38 14 the the DET ijassa-1480 38 15 corresponding corresponding ADJ ijassa-1480 38 16 conditions condition NOUN ijassa-1480 38 17 of of ADP ijassa-1480 38 18 the the DET ijassa-1480 38 19 theorem theorem NOUN ijassa-1480 38 20 in in ADP ijassa-1480 38 21 [ [ X ijassa-1480 38 22 4 4 NUM ijassa-1480 38 23 ] ] PUNCT ijassa-1480 38 24 , , PUNCT ijassa-1480 38 25 where where SCONJ ijassa-1480 38 26 exponential exponential ADJ ijassa-1480 38 27 estimates estimate NOUN ijassa-1480 38 28 for for ADP ijassa-1480 38 29 solutions solution NOUN ijassa-1480 38 30 of of ADP ijassa-1480 38 31 inclusion inclusion NOUN ijassa-1480 38 32 ( ( PUNCT ijassa-1480 38 33 2.1 2.1 NUM ijassa-1480 38 34 ) ) PUNCT ijassa-1480 38 35 are be AUX ijassa-1480 38 36 used use VERB ijassa-1480 38 37 . . PUNCT ijassa-1480 39 1 theorem theorem VERB ijassa-1480 39 2 3.1 3.1 NUM ijassa-1480 39 3 is be AUX ijassa-1480 39 4 an an DET ijassa-1480 39 5 extension extension NOUN ijassa-1480 39 6 of of ADP ijassa-1480 39 7 the the DET ijassa-1480 39 8 classical classical ADJ ijassa-1480 39 9 theorem theorem NOUN ijassa-1480 39 10 for for ADP ijassa-1480 39 11 the the DET ijassa-1480 39 12 discrete discrete ADJ ijassa-1480 39 13 analogue analogue NOUN ijassa-1480 39 14 of of ADP ijassa-1480 39 15 direct direct ADJ ijassa-1480 39 16 lyapunov lyapunov NOUN ijassa-1480 39 17 method method NOUN ijassa-1480 39 18 in in ADP ijassa-1480 39 19 [ [ X ijassa-1480 39 20 2 2 NUM ijassa-1480 39 21 ] ] PUNCT ijassa-1480 39 22 to to ADP ijassa-1480 39 23 selector selector NOUN ijassa-1480 39 24 - - PUNCT ijassa-1480 39 25 linear linear NOUN ijassa-1480 39 26 periodic periodic ADJ ijassa-1480 39 27 difference difference NOUN ijassa-1480 39 28 inclusions inclusion NOUN ijassa-1480 39 29 ( ( PUNCT ijassa-1480 39 30 2.1 2.1 NUM ijassa-1480 39 31 ) ) PUNCT ijassa-1480 39 32 . . PUNCT ijassa-1480 40 1 the the DET ijassa-1480 40 2 reasoning reasoning NOUN ijassa-1480 40 3 used use VERB ijassa-1480 40 4 in in ADP ijassa-1480 40 5 the the DET ijassa-1480 40 6 proof proof NOUN ijassa-1480 40 7 of of ADP ijassa-1480 40 8 the the DET ijassa-1480 40 9 lemma lemma PROPN ijassa-1480 40 10 in in ADP ijassa-1480 40 11 [ [ PUNCT ijassa-1480 40 12 4 4 NUM ijassa-1480 40 13 ] ] PUNCT ijassa-1480 40 14 proves prove VERB ijassa-1480 40 15 that that SCONJ ijassa-1480 40 16 , , PUNCT ijassa-1480 40 17 under under ADP ijassa-1480 40 18 theorem theorem NOUN ijassa-1480 40 19 1 1 NUM ijassa-1480 40 20 , , PUNCT ijassa-1480 40 21 there there PRON ijassa-1480 40 22 exist exist VERB ijassa-1480 40 23 constants constant NOUN ijassa-1480 40 24 such such DET ijassa-1480 40 25 πœ† πœ† DET ijassa-1480 40 26 β‰₯ β‰₯ NOUN ijassa-1480 40 27 πœ† πœ† SYM ijassa-1480 40 28 > > X ijassa-1480 40 29 0 0 PUNCT ijassa-1480 41 1 that that SCONJ ijassa-1480 41 2 for for ADP ijassa-1480 41 3 all all DET ijassa-1480 41 4 𝑠 𝑠 PROPN ijassa-1480 41 5 β‰₯ β‰₯ NOUN ijassa-1480 41 6 0 0 NUM ijassa-1480 41 7 and and CCONJ ijassa-1480 41 8 π‘₯ π‘₯ DET ijassa-1480 41 9 ∈ ∈ PROPN ijassa-1480 41 10 𝑅 𝑅 PROPN ijassa-1480 41 11 , , PUNCT ijassa-1480 41 12 the the DET ijassa-1480 41 13 inequalities inequality NOUN ijassa-1480 41 14 πœ† πœ† PROPN ijassa-1480 41 15 β€–π‘₯β€– β€–π‘₯β€– PROPN ijassa-1480 41 16 ≀ ≀ PROPN ijassa-1480 41 17 𝑣(𝑠 𝑣(𝑠 NUM ijassa-1480 41 18 , , PUNCT ijassa-1480 41 19 π‘₯ π‘₯ NOUN ijassa-1480 41 20 ) ) PUNCT ijassa-1480 41 21 ≀ ≀ NOUN ijassa-1480 41 22 πœ† πœ† DET ijassa-1480 41 23 .β€–π‘₯β€– .β€–π‘₯β€– NOUN ijassa-1480 41 24 . . PUNCT ijassa-1480 42 1 ( ( PUNCT ijassa-1480 42 2 3.3 3.3 NUM ijassa-1480 42 3 ) ) PUNCT ijassa-1480 42 4 ( ( PUNCT ijassa-1480 42 5 3.3 3.3 NUM ijassa-1480 42 6 ) ) PUNCT ijassa-1480 42 7 implies imply VERB ijassa-1480 42 8 positive positive ADJ ijassa-1480 42 9 definiteness definiteness NOUN ijassa-1480 42 10 of of ADP ijassa-1480 42 11 the the DET ijassa-1480 42 12 function function NOUN ijassa-1480 42 13 𝑣(𝑠 𝑣(𝑠 PROPN ijassa-1480 42 14 , , PUNCT ijassa-1480 42 15 π‘₯ π‘₯ NOUN ijassa-1480 42 16 ) ) PUNCT ijassa-1480 42 17 . . PUNCT ijassa-1480 43 1 the the DET ijassa-1480 43 2 problem problem NOUN ijassa-1480 43 3 of of ADP ijassa-1480 43 4 constructing construct VERB ijassa-1480 43 5 the the DET ijassa-1480 43 6 lyapunov lyapunov ADJ ijassa-1480 43 7 function function NOUN ijassa-1480 43 8 is be AUX ijassa-1480 43 9 simplified simplify VERB ijassa-1480 43 10 if if SCONJ ijassa-1480 43 11 this this DET ijassa-1480 43 12 function function NOUN ijassa-1480 43 13 is be AUX ijassa-1480 43 14 selected select VERB ijassa-1480 43 15 from from ADP ijassa-1480 43 16 a a DET ijassa-1480 43 17 certain certain ADJ ijassa-1480 43 18 parametric parametric ADJ ijassa-1480 43 19 class class NOUN ijassa-1480 43 20 of of ADP ijassa-1480 43 21 functions function NOUN ijassa-1480 43 22 that that PRON ijassa-1480 43 23 depend depend VERB ijassa-1480 43 24 on on ADP ijassa-1480 43 25 a a DET ijassa-1480 43 26 finite finite ADJ ijassa-1480 43 27 number number NOUN ijassa-1480 43 28 of of ADP ijassa-1480 43 29 parameters parameter NOUN ijassa-1480 43 30 . . PUNCT ijassa-1480 44 1 the the DET ijassa-1480 44 2 class class NOUN ijassa-1480 44 3 of of ADP ijassa-1480 44 4 quasi quasi ADJ ijassa-1480 44 5 - - ADJ ijassa-1480 44 6 quadratic quadratic ADJ ijassa-1480 44 7 lyapunov lyapunov ADJ ijassa-1480 44 8 functions function NOUN ijassa-1480 44 9 ( ( PUNCT ijassa-1480 44 10 3.1 3.1 NUM ijassa-1480 44 11 ) ) PUNCT ijassa-1480 44 12 is be AUX ijassa-1480 44 13 not not PART ijassa-1480 44 14 parametric parametric ADJ ijassa-1480 44 15 . . PUNCT ijassa-1480 45 1 the the DET ijassa-1480 45 2 existence existence NOUN ijassa-1480 45 3 of of ADP ijassa-1480 45 4 a a DET ijassa-1480 45 5 parametric parametric ADJ ijassa-1480 45 6 class class NOUN ijassa-1480 45 7 of of ADP ijassa-1480 45 8 lyapunov lyapunov ADJ ijassa-1480 45 9 functions function NOUN ijassa-1480 45 10 , , PUNCT ijassa-1480 45 11 also also ADV ijassa-1480 45 12 defining define VERB ijassa-1480 45 13 necessary necessary ADJ ijassa-1480 45 14 and and CCONJ ijassa-1480 45 15 sufficient sufficient ADJ ijassa-1480 45 16 conditions condition NOUN ijassa-1480 45 17 for for ADP ijassa-1480 45 18 the the DET ijassa-1480 45 19 asymptotic asymptotic ADJ ijassa-1480 45 20 stability stability NOUN ijassa-1480 45 21 of of ADP ijassa-1480 45 22 inclusion inclusion NOUN ijassa-1480 45 23 ( ( PUNCT ijassa-1480 45 24 3.1 3.1 NUM ijassa-1480 45 25 ) ) PUNCT ijassa-1480 45 26 , , PUNCT ijassa-1480 45 27 can can AUX ijassa-1480 45 28 be be AUX ijassa-1480 45 29 formulated formulate VERB ijassa-1480 45 30 as as ADP ijassa-1480 45 31 a a DET ijassa-1480 45 32 theorem theorem VERB ijassa-1480 45 33 . . PUNCT ijassa-1480 45 34 theorem theorem ADJ ijassa-1480 45 35 3.2 3.2 NUM ijassa-1480 45 36 : : PUNCT ijassa-1480 46 1 inclusion inclusion NOUN ijassa-1480 46 2 ( ( PUNCT ijassa-1480 46 3 2.1 2.1 NUM ijassa-1480 46 4 ) ) PUNCT ijassa-1480 46 5 is be AUX ijassa-1480 46 6 asymptotically asymptotically ADV ijassa-1480 46 7 stable stable ADJ ijassa-1480 46 8 iff iff NOUN ijassa-1480 46 9 , , PUNCT ijassa-1480 46 10 for for ADP ijassa-1480 46 11 some some DET ijassa-1480 46 12 integer integer NOUN ijassa-1480 46 13 𝑀 𝑀 PROPN ijassa-1480 46 14 β‰₯ β‰₯ NUM ijassa-1480 46 15 𝑛 𝑛 PROPN ijassa-1480 46 16 , , PUNCT ijassa-1480 46 17 there there PRON ijassa-1480 46 18 exists exist VERB ijassa-1480 46 19 a a DET ijassa-1480 46 20 periodic periodic NOUN ijassa-1480 46 21 in in ADP ijassa-1480 46 22 𝑠 𝑠 PROPN ijassa-1480 46 23 ( ( PUNCT ijassa-1480 46 24 of of ADP ijassa-1480 46 25 period period NOUN ijassa-1480 46 26 𝑁 𝑁 PROPN ijassa-1480 46 27 ) ) PUNCT ijassa-1480 46 28 , , PUNCT ijassa-1480 46 29 piecewise piecewise NOUN ijassa-1480 46 30 - - PUNCT ijassa-1480 46 31 quadratic quadratic ADJ ijassa-1480 46 32 lyapunov lyapunov NOUN ijassa-1480 46 33 function function NOUN ijassa-1480 46 34 luapunov luapunov NOUN ijassa-1480 46 35 functions function NOUN ijassa-1480 46 36 for for ADP ijassa-1480 46 37 periodic periodic ADJ ijassa-1480 46 38 selector selector NOUN ijassa-1480 46 39 - - PUNCT ijassa-1480 46 40 linear linear NOUN ijassa-1480 46 41 difference difference NOUN ijassa-1480 46 42 inclusions inclusion NOUN ijassa-1480 46 43 189 189 NUM ijassa-1480 46 44 copyright copyright NOUN ijassa-1480 46 45 Β© Β© PROPN ijassa-1480 46 46 2024 2024 NUM ijassa-1480 46 47 assa assa NOUN ijassa-1480 46 48 . . PUNCT ijassa-1480 47 1 adv adv PROPN ijassa-1480 47 2 . . PUNCT ijassa-1480 48 1 in in ADP ijassa-1480 48 2 systems system NOUN ijassa-1480 48 3 science science NOUN ijassa-1480 48 4 and and CCONJ ijassa-1480 48 5 appl appl NOUN ijassa-1480 48 6 . . PUNCT ijassa-1480 49 1 ( ( PUNCT ijassa-1480 49 2 2024 2024 NUM ijassa-1480 49 3 ) ) PUNCT ijassa-1480 49 4 𝑣 𝑣 PROPN ijassa-1480 49 5 ( ( PUNCT ijassa-1480 49 6 𝑠 𝑠 PROPN ijassa-1480 49 7 , , PUNCT ijassa-1480 49 8 π‘₯ π‘₯ NOUN ijassa-1480 49 9 ) ) PUNCT ijassa-1480 49 10 = = VERB ijassa-1480 49 11 π‘šπ‘Žπ‘₯ π‘šπ‘Žπ‘₯ NOUN ijassa-1480 49 12 𝑙 𝑙 NOUN ijassa-1480 49 13 ( ( PUNCT ijassa-1480 49 14 𝑠 𝑠 NOUN ijassa-1480 49 15 ) ) PUNCT ijassa-1480 49 16 , , PUNCT ijassa-1480 49 17 π‘₯⟩ π‘₯⟩ INTJ ijassa-1480 49 18 , , PUNCT ijassa-1480 49 19 ( ( PUNCT ijassa-1480 49 20 𝑙 𝑙 X ijassa-1480 49 21 ( ( PUNCT ijassa-1480 49 22 𝑠 𝑠 PROPN ijassa-1480 49 23 + + CCONJ ijassa-1480 49 24 𝑁 𝑁 PROPN ijassa-1480 49 25 ) ) PUNCT ijassa-1480 49 26 = = SYM ijassa-1480 49 27 𝑙 𝑙 PROPN ijassa-1480 49 28 ( ( PUNCT ijassa-1480 49 29 𝑠 𝑠 NOUN ijassa-1480 49 30 ) ) PUNCT ijassa-1480 49 31 , , PUNCT ijassa-1480 49 32 ( ( PUNCT ijassa-1480 49 33 3.4 3.4 NUM ijassa-1480 49 34 ) ) PUNCT ijassa-1480 49 35 for for ADP ijassa-1480 49 36 which which PRON ijassa-1480 49 37 the the DET ijassa-1480 49 38 inequality inequality NOUN ijassa-1480 49 39 ( ( PUNCT ijassa-1480 49 40 3.2 3.2 NUM ijassa-1480 49 41 ) ) PUNCT ijassa-1480 49 42 is be AUX ijassa-1480 49 43 satisfied satisfied ADJ ijassa-1480 49 44 for for ADP ijassa-1480 49 45 all all PRON ijassa-1480 49 46 𝑠 𝑠 PROPN ijassa-1480 49 47 β‰₯ β‰₯ NOUN ijassa-1480 49 48 0 0 NUM ijassa-1480 49 49 and and CCONJ ijassa-1480 49 50 π‘₯ π‘₯ DET ijassa-1480 49 51 ∈ ∈ PROPN ijassa-1480 49 52 𝑅 𝑅 PROPN ijassa-1480 49 53 , , PUNCT ijassa-1480 49 54 and and CCONJ ijassa-1480 49 55 the the DET ijassa-1480 49 56 𝑛-dimensional 𝑛-dimensional ADJ ijassa-1480 49 57 periodic periodic ADJ ijassa-1480 49 58 vectors vector NOUN ijassa-1480 49 59 ( ( PUNCT ijassa-1480 49 60 𝑙 𝑙 X ijassa-1480 49 61 ( ( PUNCT ijassa-1480 49 62 𝑠 𝑠 NOUN ijassa-1480 49 63 ) ) PUNCT ijassa-1480 49 64 ( ( PUNCT ijassa-1480 49 65 𝑙 𝑙 X ijassa-1480 49 66 ( ( PUNCT ijassa-1480 49 67 𝑠 𝑠 PROPN ijassa-1480 49 68 + + CCONJ ijassa-1480 49 69 𝑁 𝑁 PROPN ijassa-1480 49 70 ) ) PUNCT ijassa-1480 49 71 = = SYM ijassa-1480 49 72 𝑙 𝑙 PROPN ijassa-1480 49 73 ( ( PUNCT ijassa-1480 49 74 𝑠 𝑠 NOUN ijassa-1480 49 75 ) ) PUNCT ijassa-1480 49 76 ) ) PUNCT ijassa-1480 49 77 , , PUNCT ijassa-1480 49 78 𝑗 𝑗 NOUN ijassa-1480 49 79 = = SYM ijassa-1480 49 80 1 1 NUM ijassa-1480 49 81 , , PUNCT ijassa-1480 49 82 𝑀 𝑀 PROPN ijassa-1480 49 83 satisfy satisfy VERB ijassa-1480 49 84 the the DET ijassa-1480 49 85 condition condition NOUN ijassa-1480 49 86 π‘Ÿπ‘Žπ‘›π‘˜πΏ(𝑠 π‘Ÿπ‘Žπ‘›π‘˜πΏ(𝑠 PROPN ijassa-1480 49 87 ) ) PUNCT ijassa-1480 50 1 = = SYM ijassa-1480 50 2 𝑛 𝑛 PROPN ijassa-1480 50 3 ≀ ≀ PROPN ijassa-1480 50 4 𝑀 𝑀 PROPN ijassa-1480 50 5 , , PUNCT ijassa-1480 50 6 𝐿(𝑠 𝐿(𝑠 PRON ijassa-1480 50 7 ) ) PUNCT ijassa-1480 50 8 = = SYM ijassa-1480 51 1 𝑙 𝑙 PROPN ijassa-1480 51 2 ( ( PUNCT ijassa-1480 51 3 𝑠 𝑠 NOUN ijassa-1480 51 4 ) ) PUNCT ijassa-1480 51 5 , , PUNCT ijassa-1480 51 6 … … PUNCT ijassa-1480 51 7 , , PUNCT ijassa-1480 51 8 𝑙 𝑙 X ijassa-1480 51 9 ( ( PUNCT ijassa-1480 51 10 𝑠 𝑠 PROPN ijassa-1480 51 11 ) ) PUNCT ijassa-1480 51 12 , , PUNCT ijassa-1480 51 13 𝑠 𝑠 PROPN ijassa-1480 51 14 = = SYM ijassa-1480 51 15 0,1 0,1 NUM ijassa-1480 51 16 , , PUNCT ijassa-1480 51 17 … … PUNCT ijassa-1480 51 18 ( ( PUNCT ijassa-1480 51 19 3.5 3.5 NUM ijassa-1480 51 20 ) ) PUNCT ijassa-1480 51 21 ( ( PUNCT ijassa-1480 51 22 i.e. i.e. X ijassa-1480 51 23 , , PUNCT ijassa-1480 51 24 periodic periodic ADJ ijassa-1480 51 25 ( ( PUNCT ijassa-1480 51 26 𝑛 𝑛 PROPN ijassa-1480 51 27 Γ— Γ— PROPN ijassa-1480 51 28 𝑀 𝑀 PROPN ijassa-1480 51 29 ) ) PUNCT ijassa-1480 51 30 matrix matrix NOUN ijassa-1480 52 1 𝐿(𝑠)(𝐿(𝑠 𝐿(𝑠)(𝐿(𝑠 PROPN ijassa-1480 52 2 + + CCONJ ijassa-1480 52 3 𝑁 𝑁 PROPN ijassa-1480 52 4 ) ) PUNCT ijassa-1480 52 5 = = NUM ijassa-1480 52 6 𝐿(𝑠 𝐿(𝑠 NUM ijassa-1480 52 7 ) ) PUNCT ijassa-1480 52 8 ) ) PUNCT ijassa-1480 52 9 has have VERB ijassa-1480 52 10 a a DET ijassa-1480 52 11 maximum maximum ADJ ijassa-1480 52 12 rank rank NOUN ijassa-1480 52 13 for for ADP ijassa-1480 52 14 all all DET ijassa-1480 52 15 𝑠 𝑠 PROPN ijassa-1480 52 16 β‰₯ β‰₯ NOUN ijassa-1480 52 17 0 0 NUM ijassa-1480 52 18 ) ) PUNCT ijassa-1480 52 19 . . PUNCT ijassa-1480 53 1 we we PRON ijassa-1480 53 2 denote denote VERB ijassa-1480 53 3 by by ADP ijassa-1480 53 4 βŸ¨β‹… βŸ¨β‹… PROPN ijassa-1480 53 5 , , PUNCT ijassa-1480 53 6 β‹…βŸ© β‹…βŸ© NUM ijassa-1480 53 7 a a DET ijassa-1480 53 8 scalar scalar ADJ ijassa-1480 53 9 product product NOUN ijassa-1480 53 10 of of ADP ijassa-1480 53 11 vectors vector NOUN ijassa-1480 53 12 . . PUNCT ijassa-1480 54 1 the the DET ijassa-1480 54 2 set set NOUN ijassa-1480 54 3 of of ADP ijassa-1480 54 4 lyapunov lyapunov ADJ ijassa-1480 54 5 functions function NOUN ijassa-1480 54 6 ( ( PUNCT ijassa-1480 54 7 3.4 3.4 NUM ijassa-1480 54 8 ) ) PUNCT ijassa-1480 54 9 forms form VERB ijassa-1480 54 10 a a DET ijassa-1480 54 11 parametric parametric ADJ ijassa-1480 54 12 class class NOUN ijassa-1480 54 13 . . PUNCT ijassa-1480 55 1 the the DET ijassa-1480 55 2 parameters parameter NOUN ijassa-1480 55 3 defining define VERB ijassa-1480 55 4 this this DET ijassa-1480 55 5 class class NOUN ijassa-1480 55 6 are be AUX ijassa-1480 55 7 the the DET ijassa-1480 55 8 components component NOUN ijassa-1480 55 9 of of ADP ijassa-1480 55 10 periodic periodic ADJ ijassa-1480 55 11 vectors vector NOUN ijassa-1480 55 12 𝑙 𝑙 X ijassa-1480 55 13 ( ( PUNCT ijassa-1480 55 14 𝑠 𝑠 NOUN ijassa-1480 55 15 ) ) PUNCT ijassa-1480 55 16 ( ( PUNCT ijassa-1480 55 17 𝑗 𝑗 NOUN ijassa-1480 55 18 = = SYM ijassa-1480 55 19 1 1 NUM ijassa-1480 55 20 , , PUNCT ijassa-1480 55 21 𝑀 𝑀 PROPN ijassa-1480 55 22 , , PUNCT ijassa-1480 55 23 𝑠 𝑠 PROPN ijassa-1480 55 24 β‰₯ β‰₯ NOUN ijassa-1480 55 25 0 0 NUM ijassa-1480 55 26 ) ) PUNCT ijassa-1480 55 27 and and CCONJ ijassa-1480 55 28 the the DET ijassa-1480 55 29 integer integer PROPN ijassa-1480 55 30 𝑀 𝑀 PROPN ijassa-1480 55 31 β‰₯ β‰₯ X ijassa-1480 55 32 𝑛. 𝑛. NOUN ijassa-1480 55 33 if if SCONJ ijassa-1480 55 34 the the DET ijassa-1480 55 35 rank rank NOUN ijassa-1480 55 36 condition condition NOUN ijassa-1480 55 37 ( ( PUNCT ijassa-1480 55 38 3.5 3.5 NUM ijassa-1480 55 39 ) ) PUNCT ijassa-1480 55 40 is be AUX ijassa-1480 55 41 satisfied satisfied ADJ ijassa-1480 55 42 , , PUNCT ijassa-1480 55 43 the the DET ijassa-1480 55 44 function function NOUN ijassa-1480 55 45 𝑣 𝑣 X ijassa-1480 55 46 ( ( PUNCT ijassa-1480 55 47 𝑠 𝑠 PROPN ijassa-1480 55 48 , , PUNCT ijassa-1480 55 49 π‘₯ π‘₯ NOUN ijassa-1480 55 50 ) ) PUNCT ijassa-1480 55 51 is be AUX ijassa-1480 55 52 positively positively ADV ijassa-1480 55 53 defined define VERB ijassa-1480 55 54 in in ADP ijassa-1480 55 55 𝑅 𝑅 PROPN ijassa-1480 55 56 and and CCONJ ijassa-1480 55 57 its its PRON ijassa-1480 55 58 level level NOUN ijassa-1480 55 59 surfaces surface NOUN ijassa-1480 55 60 at at ADP ijassa-1480 55 61 any any DET ijassa-1480 55 62 fixed fix VERB ijassa-1480 55 63 one one NOUN ijassa-1480 55 64 𝑠 𝑠 PRON ijassa-1480 55 65 β‰₯ β‰₯ NOUN ijassa-1480 55 66 0 0 NUM ijassa-1480 55 67 are be AUX ijassa-1480 55 68 centrally centrally ADV ijassa-1480 55 69 symmetric symmetric ADJ ijassa-1480 55 70 convex convex NOUN ijassa-1480 55 71 polyhedrons polyhedron NOUN ijassa-1480 55 72 . . PUNCT ijassa-1480 56 1 the the DET ijassa-1480 56 2 vectors vector NOUN ijassa-1480 56 3 𝑙 𝑙 X ijassa-1480 56 4 ( ( PUNCT ijassa-1480 56 5 𝑠 𝑠 PROPN ijassa-1480 56 6 ) ) PUNCT ijassa-1480 56 7 , , PUNCT ijassa-1480 56 8 𝑗 𝑗 NOUN ijassa-1480 56 9 = = SYM ijassa-1480 56 10 1 1 NUM ijassa-1480 56 11 , , PUNCT ijassa-1480 56 12 𝑀 𝑀 PROPN ijassa-1480 56 13 defining define VERB ijassa-1480 56 14 the the DET ijassa-1480 56 15 norms norm NOUN ijassa-1480 56 16 to to ADP ijassa-1480 56 17 their their PRON ijassa-1480 56 18 faces face NOUN ijassa-1480 56 19 . . PUNCT ijassa-1480 57 1 proof proof NOUN ijassa-1480 57 2 . . PUNCT ijassa-1480 58 1 sufficiency sufficiency NOUN ijassa-1480 58 2 . . PUNCT ijassa-1480 59 1 the the DET ijassa-1480 59 2 sufficiency sufficiency NOUN ijassa-1480 59 3 of of ADP ijassa-1480 59 4 the the DET ijassa-1480 59 5 conditions condition NOUN ijassa-1480 59 6 of of ADP ijassa-1480 59 7 theorem theorem ADJ ijassa-1480 59 8 3.2 3.2 NUM ijassa-1480 59 9 is be AUX ijassa-1480 59 10 established establish VERB ijassa-1480 59 11 according accord VERB ijassa-1480 59 12 to to ADP ijassa-1480 59 13 the the DET ijassa-1480 59 14 standard standard ADJ ijassa-1480 59 15 scheme scheme NOUN ijassa-1480 59 16 of of ADP ijassa-1480 59 17 the the DET ijassa-1480 59 18 proof proof NOUN ijassa-1480 59 19 of of ADP ijassa-1480 59 20 exponential exponential ADJ ijassa-1480 59 21 stability stability NOUN ijassa-1480 59 22 in in ADP ijassa-1480 59 23 [ [ X ijassa-1480 59 24 2 2 NUM ijassa-1480 59 25 ] ] PUNCT ijassa-1480 59 26 using use VERB ijassa-1480 59 27 inequality inequality NOUN ijassa-1480 59 28 ( ( PUNCT ijassa-1480 59 29 3.2 3.2 NUM ijassa-1480 59 30 ) ) PUNCT ijassa-1480 59 31 and and CCONJ ijassa-1480 59 32 estimates estimate VERB ijassa-1480 59 33 πœ† πœ† X ijassa-1480 59 34 β€–π‘₯β€– β€–π‘₯β€– PROPN ijassa-1480 59 35 ≀ ≀ PROPN ijassa-1480 59 36 𝑣 𝑣 PRON ijassa-1480 59 37 ( ( PUNCT ijassa-1480 59 38 𝑠 𝑠 PROPN ijassa-1480 59 39 , , PUNCT ijassa-1480 59 40 π‘₯ π‘₯ NOUN ijassa-1480 59 41 ) ) PUNCT ijassa-1480 59 42 ≀ ≀ NOUN ijassa-1480 59 43 πœ† πœ† DET ijassa-1480 59 44 β€–π‘₯β€– β€–π‘₯β€– PROPN ijassa-1480 59 45 , , PUNCT ijassa-1480 59 46 πœ† πœ† DET ijassa-1480 59 47 β‰₯ β‰₯ NOUN ijassa-1480 59 48 πœ† πœ† SYM ijassa-1480 59 49 > > X ijassa-1480 59 50 0 0 PUNCT ijassa-1480 59 51 for for ADP ijassa-1480 59 52 function function NOUN ijassa-1480 59 53 ( ( PUNCT ijassa-1480 59 54 3.4 3.4 NUM ijassa-1480 59 55 ) ) PUNCT ijassa-1480 59 56 under under ADP ijassa-1480 59 57 condition condition NOUN ijassa-1480 59 58 ( ( PUNCT ijassa-1480 59 59 3.5 3.5 NUM ijassa-1480 59 60 ) ) PUNCT ijassa-1480 59 61 . . PUNCT ijassa-1480 60 1 necessity necessity NOUN ijassa-1480 60 2 . . PUNCT ijassa-1480 61 1 it it PRON ijassa-1480 61 2 follows follow VERB ijassa-1480 61 3 from from ADP ijassa-1480 61 4 theorem theorem ADJ ijassa-1480 61 5 3.1 3.1 NUM ijassa-1480 61 6 that that PRON ijassa-1480 61 7 for for ADP ijassa-1480 61 8 inclusion inclusion NOUN ijassa-1480 61 9 ( ( PUNCT ijassa-1480 61 10 2.1 2.1 NUM ijassa-1480 61 11 ) ) PUNCT ijassa-1480 61 12 there there PRON ijassa-1480 61 13 exists exist VERB ijassa-1480 61 14 a a DET ijassa-1480 61 15 periodic periodic NOUN ijassa-1480 61 16 in in ADP ijassa-1480 61 17 𝑠 𝑠 PROPN ijassa-1480 62 1 lyapunov lyapunov NOUN ijassa-1480 62 2 function function VERB ijassa-1480 62 3 𝑣(𝑠 𝑣(𝑠 PROPN ijassa-1480 62 4 , , PUNCT ijassa-1480 62 5 π‘₯ π‘₯ NOUN ijassa-1480 62 6 ) ) PUNCT ijassa-1480 62 7 of of ADP ijassa-1480 62 8 the the DET ijassa-1480 62 9 quasiquadratic quasiquadratic ADJ ijassa-1480 62 10 form form NOUN ijassa-1480 62 11 satisfying satisfy VERB ijassa-1480 62 12 the the DET ijassa-1480 62 13 conditions condition NOUN ijassa-1480 62 14 of of ADP ijassa-1480 62 15 theorem theorem NOUN ijassa-1480 62 16 3.1 3.1 NUM ijassa-1480 62 17 . . PUNCT ijassa-1480 63 1 consider consider VERB ijassa-1480 63 2 centrally centrally ADV ijassa-1480 63 3 symmetric symmetric ADJ ijassa-1480 63 4 convex convex NOUN ijassa-1480 63 5 bodies body NOUN ijassa-1480 63 6 𝑃 𝑃 VERB ijassa-1480 63 7 ( ( PUNCT ijassa-1480 63 8 𝑠 𝑠 NOUN ijassa-1480 63 9 ) ) PUNCT ijassa-1480 63 10 = = PRON ijassa-1480 63 11 { { PUNCT ijassa-1480 63 12 π‘₯ π‘₯ NOUN ijassa-1480 63 13 : : PUNCT ijassa-1480 63 14 𝑣(𝑠 𝑣(𝑠 NUM ijassa-1480 63 15 , , PUNCT ijassa-1480 63 16 π‘₯ π‘₯ NOUN ijassa-1480 63 17 ) ) PUNCT ijassa-1480 63 18 ≀ ≀ NUM ijassa-1480 63 19 1 1 NUM ijassa-1480 63 20 } } PUNCT ijassa-1480 63 21 , , PUNCT ijassa-1480 63 22 𝑃 𝑃 PROPN ijassa-1480 63 23 ( ( PUNCT ijassa-1480 63 24 𝑠 𝑠 NOUN ijassa-1480 63 25 ) ) PUNCT ijassa-1480 63 26 = = PRON ijassa-1480 63 27 { { PUNCT ijassa-1480 63 28 π‘₯ π‘₯ NOUN ijassa-1480 63 29 : : PUNCT ijassa-1480 63 30 𝑣(𝑠 𝑣(𝑠 NUM ijassa-1480 63 31 , , PUNCT ijassa-1480 63 32 π‘₯ π‘₯ NOUN ijassa-1480 63 33 ) ) PUNCT ijassa-1480 63 34 ≀ ≀ NOUN ijassa-1480 64 1 π‘Ÿ π‘Ÿ NOUN ijassa-1480 64 2 } } PUNCT ijassa-1480 64 3 , , PUNCT ijassa-1480 64 4 𝑠 𝑠 PROPN ijassa-1480 64 5 = = SYM ijassa-1480 64 6 0,1 0,1 NUM ijassa-1480 64 7 , , PUNCT ijassa-1480 64 8 . . PUNCT ijassa-1480 64 9 . . PUNCT ijassa-1480 65 1 . . PUNCT ijassa-1480 66 1 , , PUNCT ijassa-1480 66 2 0 0 PUNCT ijassa-1480 66 3 < < X ijassa-1480 66 4 π‘Ÿ π‘Ÿ X ijassa-1480 66 5 < < X ijassa-1480 66 6 1 1 X ijassa-1480 66 7 . . PUNCT ijassa-1480 67 1 let let VERB ijassa-1480 67 2 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ijassa-1480 67 3 𝐴 𝐴 PROPN ijassa-1480 67 4 be be AUX ijassa-1480 67 5 the the DET ijassa-1480 67 6 set set NOUN ijassa-1480 67 7 of of ADP ijassa-1480 67 8 interior interior ADJ ijassa-1480 67 9 points point NOUN ijassa-1480 67 10 of of ADP ijassa-1480 67 11 the the DET ijassa-1480 67 12 set set NOUN ijassa-1480 67 13 𝐴. 𝐴. PROPN ijassa-1480 67 14 since since SCONJ ijassa-1480 67 15 0 0 NUM ijassa-1480 67 16 < < X ijassa-1480 67 17 π‘Ÿ π‘Ÿ X ijassa-1480 67 18 < < X ijassa-1480 67 19 1 1 NUM ijassa-1480 67 20 , , PUNCT ijassa-1480 67 21 then then ADV ijassa-1480 67 22 𝑃 𝑃 PROPN ijassa-1480 67 23 ( ( PUNCT ijassa-1480 67 24 𝑠 𝑠 NOUN ijassa-1480 67 25 ) ) PUNCT ijassa-1480 68 1 βŠ‚ βŠ‚ PROPN ijassa-1480 68 2 𝑖𝑛𝑑 𝑖𝑛𝑑 PUNCT ijassa-1480 69 1 𝑃 𝑃 PROPN ijassa-1480 69 2 ( ( PUNCT ijassa-1480 69 3 𝑠 𝑠 NOUN ijassa-1480 69 4 ) ) PUNCT ijassa-1480 69 5 . . PUNCT ijassa-1480 70 1 it it PRON ijassa-1480 70 2 follows follow VERB ijassa-1480 70 3 from from ADP ijassa-1480 70 4 theorem theorem NOUN ijassa-1480 70 5 20.4 20.4 NUM ijassa-1480 70 6 in in ADP ijassa-1480 70 7 [ [ X ijassa-1480 70 8 6 6 NUM ijassa-1480 70 9 ] ] PUNCT ijassa-1480 70 10 that that SCONJ ijassa-1480 70 11 there there PRON ijassa-1480 70 12 exists exist VERB ijassa-1480 70 13 a a DET ijassa-1480 70 14 centrally centrally ADV ijassa-1480 70 15 symmetric symmetric ADJ ijassa-1480 70 16 convex convex NOUN ijassa-1480 70 17 polyhedron polyhedron NOUN ijassa-1480 70 18 𝐷 𝐷 PROPN ijassa-1480 70 19 ( ( PUNCT ijassa-1480 70 20 𝑠 𝑠 NOUN ijassa-1480 70 21 ) ) PUNCT ijassa-1480 70 22 , , PUNCT ijassa-1480 71 1 that that SCONJ ijassa-1480 71 2 the the DET ijassa-1480 71 3 relations relation NOUN ijassa-1480 71 4 𝑃 𝑃 PROPN ijassa-1480 71 5 ( ( PUNCT ijassa-1480 71 6 𝑠 𝑠 NOUN ijassa-1480 71 7 ) ) PUNCT ijassa-1480 71 8 βŠ‚ βŠ‚ PROPN ijassa-1480 71 9 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ijassa-1480 71 10 𝐷 𝐷 PROPN ijassa-1480 71 11 ( ( PUNCT ijassa-1480 71 12 𝑠 𝑠 NOUN ijassa-1480 71 13 ) ) PUNCT ijassa-1480 71 14 βŠ‚ βŠ‚ PROPN ijassa-1480 71 15 𝐷 𝐷 PROPN ijassa-1480 71 16 ( ( PUNCT ijassa-1480 71 17 𝑠 𝑠 NOUN ijassa-1480 71 18 ) ) PUNCT ijassa-1480 72 1 βŠ‚ βŠ‚ PROPN ijassa-1480 72 2 𝑖𝑛𝑑 𝑖𝑛𝑑 PUNCT ijassa-1480 73 1 𝑃 𝑃 PROPN ijassa-1480 73 2 ( ( PUNCT ijassa-1480 73 3 𝑠 𝑠 NOUN ijassa-1480 73 4 ) ) PUNCT ijassa-1480 73 5 βŠ‚ βŠ‚ PROPN ijassa-1480 73 6 𝑃 𝑃 PROPN ijassa-1480 73 7 ( ( PUNCT ijassa-1480 73 8 𝑠 𝑠 NOUN ijassa-1480 73 9 ) ) PUNCT ijassa-1480 73 10 . . PUNCT ijassa-1480 74 1 ( ( PUNCT ijassa-1480 74 2 3.6 3.6 NUM ijassa-1480 74 3 ) ) PUNCT ijassa-1480 74 4 let let VERB ijassa-1480 74 5 2𝑀 2𝑀 PROPN ijassa-1480 74 6 be be AUX ijassa-1480 74 7 the the DET ijassa-1480 74 8 number number NOUN ijassa-1480 74 9 of of ADP ijassa-1480 74 10 faces face NOUN ijassa-1480 74 11 of of ADP ijassa-1480 74 12 the the DET ijassa-1480 74 13 polyhedron polyhedron NOUN ijassa-1480 74 14 𝐷 𝐷 PROPN ijassa-1480 74 15 ( ( PUNCT ijassa-1480 74 16 𝑠 𝑠 PROPN ijassa-1480 74 17 ) ) PUNCT ijassa-1480 74 18 , , PUNCT ijassa-1480 74 19 then then ADV ijassa-1480 74 20 there there PRON ijassa-1480 74 21 are be VERB ijassa-1480 74 22 𝑀 𝑀 PROPN ijassa-1480 74 23 pairs pair NOUN ijassa-1480 74 24 of of ADP ijassa-1480 74 25 centrally centrally ADV ijassa-1480 74 26 symmetric symmetric ADJ ijassa-1480 74 27 faces face NOUN ijassa-1480 74 28 . . PUNCT ijassa-1480 75 1 let let VERB ijassa-1480 75 2 π‘Ÿ π‘Ÿ PRON ijassa-1480 75 3 > > X ijassa-1480 75 4 0 0 PUNCT ijassa-1480 75 5 be be AUX ijassa-1480 75 6 the the DET ijassa-1480 75 7 distance distance NOUN ijassa-1480 75 8 from from ADP ijassa-1480 75 9 a a DET ijassa-1480 75 10 point point NOUN ijassa-1480 75 11 π‘₯ π‘₯ NOUN ijassa-1480 75 12 = = SYM ijassa-1480 75 13 0 0 NUM ijassa-1480 75 14 to to ADP ijassa-1480 75 15 the the DET ijassa-1480 75 16 faces face NOUN ijassa-1480 75 17 of of ADP ijassa-1480 75 18 the𝑗 the𝑗 NOUN ijassa-1480 75 19 pair pair NOUN ijassa-1480 75 20 , , PUNCT ijassa-1480 75 21 and and CCONJ ijassa-1480 75 22 let let VERB ijassa-1480 75 23 ±𝑛 ±𝑛 PROPN ijassa-1480 75 24 ( ( PUNCT ijassa-1480 75 25 𝑛 𝑛 NOUN ijassa-1480 75 26 = = SYM ijassa-1480 75 27 1 1 NUM ijassa-1480 75 28 ) ) PUNCT ijassa-1480 75 29 , , PUNCT ijassa-1480 75 30 𝑗 𝑗 NOUN ijassa-1480 75 31 = = SYM ijassa-1480 75 32 1 1 NUM ijassa-1480 75 33 , , PUNCT ijassa-1480 75 34 𝑀 𝑀 PROPN ijassa-1480 75 35 be be VERB ijassa-1480 75 36 the the DET ijassa-1480 75 37 unit unit NOUN ijassa-1480 75 38 external external ADJ ijassa-1480 75 39 normals normal NOUN ijassa-1480 75 40 to to ADP ijassa-1480 75 41 the the DET ijassa-1480 75 42 faces face NOUN ijassa-1480 75 43 of of ADP ijassa-1480 75 44 this this DET ijassa-1480 75 45 pair pair NOUN ijassa-1480 75 46 . . PUNCT ijassa-1480 76 1 in in ADP ijassa-1480 76 2 order order NOUN ijassa-1480 76 3 the the DET ijassa-1480 76 4 surface surface NOUN ijassa-1480 76 5 of of ADP ijassa-1480 76 6 the the DET ijassa-1480 76 7 polyhedron polyhedron NOUN ijassa-1480 76 8 𝐷 𝐷 PROPN ijassa-1480 76 9 ( ( PUNCT ijassa-1480 76 10 𝑠 𝑠 NOUN ijassa-1480 76 11 ) ) PUNCT ijassa-1480 76 12 to to PART ijassa-1480 76 13 be be AUX ijassa-1480 76 14 the the DET ijassa-1480 76 15 surface surface NOUN ijassa-1480 76 16 of of ADP ijassa-1480 76 17 the the DET ijassa-1480 76 18 level level NOUN ijassa-1480 77 1 𝛺 𝛺 PROPN ijassa-1480 77 2 ( ( PUNCT ijassa-1480 77 3 𝑠 𝑠 NOUN ijassa-1480 77 4 ) ) PUNCT ijassa-1480 77 5 = = PRON ijassa-1480 77 6 { { PUNCT ijassa-1480 77 7 π‘₯ π‘₯ NOUN ijassa-1480 77 8 : : PUNCT ijassa-1480 77 9 𝑣 𝑣 X ijassa-1480 77 10 ( ( PUNCT ijassa-1480 77 11 𝑙(𝑠 𝑙(𝑠 NOUN ijassa-1480 77 12 ) ) PUNCT ijassa-1480 77 13 , , PUNCT ijassa-1480 77 14 π‘₯ π‘₯ X ijassa-1480 77 15 ) ) PUNCT ijassa-1480 77 16 = = SYM ijassa-1480 77 17 1 1 X ijassa-1480 77 18 } } PUNCT ijassa-1480 77 19 for for ADP ijassa-1480 77 20 function function NOUN ijassa-1480 77 21 𝑣 𝑣 X ijassa-1480 77 22 ( ( PUNCT ijassa-1480 77 23 𝑙(𝑠 𝑙(𝑠 NOUN ijassa-1480 77 24 ) ) PUNCT ijassa-1480 77 25 , , PUNCT ijassa-1480 77 26 π‘₯ π‘₯ NOUN ijassa-1480 77 27 ) ) PUNCT ijassa-1480 77 28 ( ( PUNCT ijassa-1480 77 29 3.4 3.4 NUM ijassa-1480 77 30 ) ) PUNCT ijassa-1480 77 31 , , PUNCT ijassa-1480 77 32 choose choose VERB ijassa-1480 77 33 𝑙 𝑙 PROPN ijassa-1480 77 34 ( ( PUNCT ijassa-1480 77 35 𝑠 𝑠 NOUN ijassa-1480 77 36 ) ) PUNCT ijassa-1480 77 37 = = NOUN ijassa-1480 77 38 π‘Ÿ π‘Ÿ NOUN ijassa-1480 77 39 𝑛 𝑛 NOUN ijassa-1480 77 40 , , PUNCT ijassa-1480 77 41 𝑗 𝑗 NOUN ijassa-1480 77 42 = = SYM ijassa-1480 77 43 1 1 NUM ijassa-1480 77 44 , , PUNCT ijassa-1480 77 45 𝑀 𝑀 PROPN ijassa-1480 77 46 , , PUNCT ijassa-1480 77 47 𝑠 𝑠 PROPN ijassa-1480 77 48 = = SYM ijassa-1480 77 49 0,1 0,1 NUM ijassa-1480 77 50 , , PUNCT ijassa-1480 77 51 . . PUNCT ijassa-1480 78 1 .. .. PUNCT ijassa-1480 79 1 then then ADV ijassa-1480 79 2 the the DET ijassa-1480 79 3 vectors vector NOUN ijassa-1480 79 4 𝑙 𝑙 X ijassa-1480 79 5 ( ( PUNCT ijassa-1480 79 6 𝑠 𝑠 PROPN ijassa-1480 79 7 ) ) PUNCT ijassa-1480 79 8 , , PUNCT ijassa-1480 79 9 𝑗 𝑗 NOUN ijassa-1480 79 10 = = SYM ijassa-1480 79 11 1 1 NUM ijassa-1480 79 12 , , PUNCT ijassa-1480 79 13 𝑀 𝑀 PROPN ijassa-1480 79 14 , , PUNCT ijassa-1480 79 15 𝑠 𝑠 PROPN ijassa-1480 79 16 = = SYM ijassa-1480 79 17 0,1 0,1 NUM ijassa-1480 79 18 , , PUNCT ijassa-1480 79 19 . . PUNCT ijassa-1480 79 20 .. .. PUNCT ijassa-1480 80 1 satisfy satisfy VERB ijassa-1480 80 2 rank rank NOUN ijassa-1480 80 3 condition condition NOUN ijassa-1480 80 4 ( ( PUNCT ijassa-1480 80 5 3.5 3.5 NUM ijassa-1480 80 6 ) ) PUNCT ijassa-1480 80 7 , , PUNCT ijassa-1480 80 8 because because SCONJ ijassa-1480 80 9 otherwise otherwise ADV ijassa-1480 80 10 the the DET ijassa-1480 80 11 surfaces surface NOUN ijassa-1480 80 12 of of ADP ijassa-1480 80 13 the the DET ijassa-1480 80 14 level level NOUN ijassa-1480 80 15 function function NOUN ijassa-1480 80 16 𝑣 𝑣 X ijassa-1480 80 17 ( ( PUNCT ijassa-1480 80 18 𝑙(𝑠 𝑙(𝑠 NOUN ijassa-1480 80 19 ) ) PUNCT ijassa-1480 80 20 , , PUNCT ijassa-1480 80 21 π‘₯ π‘₯ X ijassa-1480 80 22 ) ) PUNCT ijassa-1480 80 23 will will AUX ijassa-1480 80 24 not not PART ijassa-1480 80 25 be be AUX ijassa-1480 80 26 bounded bound VERB ijassa-1480 80 27 . . PUNCT ijassa-1480 81 1 let let VERB ijassa-1480 81 2 𝛿(𝐾 𝛿(𝐾 ADJ ijassa-1480 81 3 , , PUNCT ijassa-1480 81 4 𝑧 𝑧 NOUN ijassa-1480 81 5 ) ) PUNCT ijassa-1480 81 6 = = VERB ijassa-1480 81 7 π‘šπ‘Žπ‘₯ π‘šπ‘Žπ‘₯ ADJ ijassa-1480 81 8 ∈ ∈ NOUN ijassa-1480 81 9 ( ( PUNCT ijassa-1480 81 10 𝑧 𝑧 PROPN ijassa-1480 81 11 , , PUNCT ijassa-1480 81 12 π‘₯ π‘₯ NOUN ijassa-1480 81 13 ) ) PUNCT ijassa-1480 81 14 , , PUNCT ijassa-1480 81 15 𝑧 𝑧 PROPN ijassa-1480 81 16 ∈ ∈ PROPN ijassa-1480 81 17 𝑅 𝑅 PROPN ijassa-1480 81 18 be be VERB ijassa-1480 81 19 the the DET ijassa-1480 81 20 reference reference NOUN ijassa-1480 81 21 function function NOUN ijassa-1480 81 22 of of ADP ijassa-1480 81 23 compact compact ADJ ijassa-1480 81 24 𝐾 𝐾 PROPN ijassa-1480 81 25 βŠ‚ βŠ‚ PROPN ijassa-1480 81 26 𝑅 𝑅 PROPN ijassa-1480 81 27 ( ( PUNCT ijassa-1480 81 28 see see VERB ijassa-1480 81 29 [ [ X ijassa-1480 81 30 6 6 NUM ijassa-1480 81 31 ] ] NUM ijassa-1480 81 32 ) ) PUNCT ijassa-1480 81 33 . . PUNCT ijassa-1480 82 1 since since SCONJ ijassa-1480 82 2 𝛿(𝑃 𝛿(𝑃 NOUN ijassa-1480 82 3 ( ( PUNCT ijassa-1480 82 4 𝑠 𝑠 NOUN ijassa-1480 82 5 ) ) PUNCT ijassa-1480 82 6 , , PUNCT ijassa-1480 82 7 𝑧 𝑧 X ijassa-1480 82 8 ) ) PUNCT ijassa-1480 82 9 = = SYM ijassa-1480 82 10 βˆšπ‘Ÿπ›Ώ(𝑃 βˆšπ‘Ÿπ›Ώ(𝑃 X ijassa-1480 82 11 ( ( PUNCT ijassa-1480 82 12 𝑠 𝑠 NOUN ijassa-1480 82 13 ) ) PUNCT ijassa-1480 82 14 , , PUNCT ijassa-1480 82 15 𝑧 𝑧 X ijassa-1480 82 16 ) ) PUNCT ijassa-1480 82 17 , , PUNCT ijassa-1480 82 18 it it PRON ijassa-1480 82 19 follows follow VERB ijassa-1480 82 20 from from ADP ijassa-1480 82 21 ( ( PUNCT ijassa-1480 82 22 3.6 3.6 NUM ijassa-1480 82 23 ) ) PUNCT ijassa-1480 82 24 and and CCONJ ijassa-1480 82 25 property property NOUN ijassa-1480 82 26 9 9 NUM ijassa-1480 82 27 in in ADP ijassa-1480 82 28 [ [ X ijassa-1480 82 29 1 1 X ijassa-1480 82 30 ] ] X ijassa-1480 82 31 βˆšπ‘Ÿπ›Ώ(𝑃 βˆšπ‘Ÿπ›Ώ(𝑃 PROPN ijassa-1480 82 32 ( ( PUNCT ijassa-1480 82 33 𝑠 𝑠 NOUN ijassa-1480 82 34 ) ) PUNCT ijassa-1480 82 35 , , PUNCT ijassa-1480 82 36 𝑧 𝑧 X ijassa-1480 82 37 ) ) PUNCT ijassa-1480 82 38 < < X ijassa-1480 82 39 𝛿(𝐷 𝛿(𝐷 NOUN ijassa-1480 82 40 ( ( PUNCT ijassa-1480 82 41 𝑠),z 𝑠),z NOUN ijassa-1480 82 42 ) ) PUNCT ijassa-1480 82 43 < < X ijassa-1480 82 44 𝛿(𝑃 𝛿(𝑃 PUNCT ijassa-1480 82 45 ( ( PUNCT ijassa-1480 82 46 𝑠),z 𝑠),z NOUN ijassa-1480 82 47 ) ) PUNCT ijassa-1480 82 48 , , PUNCT ijassa-1480 82 49 z z NOUN ijassa-1480 82 50 β‰  β‰  PROPN ijassa-1480 82 51 0 0 NUM ijassa-1480 82 52 . . PUNCT ijassa-1480 83 1 ( ( PUNCT ijassa-1480 83 2 3.7 3.7 NUM ijassa-1480 83 3 ) ) PUNCT ijassa-1480 83 4 let let VERB ijassa-1480 83 5 us we PRON ijassa-1480 83 6 show show VERB ijassa-1480 83 7 that that SCONJ ijassa-1480 83 8 there there PRON ijassa-1480 83 9 exists exist VERB ijassa-1480 83 10 a a DET ijassa-1480 83 11 number number NOUN ijassa-1480 83 12 π‘Ÿ π‘Ÿ NOUN ijassa-1480 83 13 ( ( PUNCT ijassa-1480 83 14 π‘Ÿ π‘Ÿ X ijassa-1480 83 15 < < X ijassa-1480 83 16 π‘Ÿ π‘Ÿ X ijassa-1480 83 17 < < X ijassa-1480 83 18 1 1 NUM ijassa-1480 83 19 ) ) PUNCT ijassa-1480 83 20 , , PUNCT ijassa-1480 83 21 such such ADJ ijassa-1480 83 22 that that SCONJ ijassa-1480 83 23 the the DET ijassa-1480 83 24 inequality inequality NOUN ijassa-1480 83 25 βˆšπ‘Ÿπ›Ώ(𝑃 βˆšπ‘Ÿπ›Ώ(𝑃 PROPN ijassa-1480 83 26 ( ( PUNCT ijassa-1480 83 27 𝑠 𝑠 NOUN ijassa-1480 83 28 ) ) PUNCT ijassa-1480 83 29 , , PUNCT ijassa-1480 83 30 𝑧 𝑧 X ijassa-1480 83 31 ) ) PUNCT ijassa-1480 83 32 < < X ijassa-1480 83 33 π‘Ÿ π‘Ÿ PRON ijassa-1480 83 34 𝛿(𝐷 𝛿(𝐷 NOUN ijassa-1480 83 35 ( ( PUNCT ijassa-1480 83 36 𝑠),z 𝑠),z NOUN ijassa-1480 83 37 ) ) PUNCT ijassa-1480 83 38 , , PUNCT ijassa-1480 83 39 z z NOUN ijassa-1480 83 40 β‰  β‰  PROPN ijassa-1480 83 41 0 0 NUM ijassa-1480 83 42 . . PUNCT ijassa-1480 84 1 ( ( PUNCT ijassa-1480 84 2 3.8 3.8 NUM ijassa-1480 84 3 ) ) PUNCT ijassa-1480 84 4 suppose suppose VERB ijassa-1480 84 5 that that SCONJ ijassa-1480 84 6 inequality inequality NOUN ijassa-1480 84 7 ( ( PUNCT ijassa-1480 84 8 3.8 3.8 NUM ijassa-1480 84 9 ) ) PUNCT ijassa-1480 84 10 is be AUX ijassa-1480 84 11 not not PART ijassa-1480 84 12 satisfied satisfied ADJ ijassa-1480 84 13 , , PUNCT ijassa-1480 84 14 i.e. i.e. X ijassa-1480 84 15 , , PUNCT ijassa-1480 84 16 for for ADP ijassa-1480 84 17 any any DET ijassa-1480 84 18 π‘Ÿ π‘Ÿ NOUN ijassa-1480 84 19 ( ( PUNCT ijassa-1480 84 20 π‘Ÿ π‘Ÿ X ijassa-1480 84 21 < < X ijassa-1480 84 22 π‘Ÿ π‘Ÿ X ijassa-1480 84 23 < < X ijassa-1480 84 24 1 1 NUM ijassa-1480 84 25 ) ) PUNCT ijassa-1480 84 26 βˆšπ‘Ÿπ›Ώ(𝑃 βˆšπ‘Ÿπ›Ώ(𝑃 PROPN ijassa-1480 84 27 ( ( PUNCT ijassa-1480 84 28 𝑠 𝑠 NOUN ijassa-1480 84 29 ) ) PUNCT ijassa-1480 84 30 , , PUNCT ijassa-1480 84 31 𝑧 𝑧 X ijassa-1480 84 32 ) ) PUNCT ijassa-1480 84 33 β‰₯ β‰₯ NOUN ijassa-1480 84 34 π‘Ÿ π‘Ÿ NOUN ijassa-1480 84 35 𝛿(𝐷 𝛿(𝐷 PROPN ijassa-1480 84 36 ( ( PUNCT ijassa-1480 84 37 𝑠),z 𝑠),z NOUN ijassa-1480 84 38 ) ) PUNCT ijassa-1480 84 39 or or CCONJ ijassa-1480 84 40 π‘Ÿ π‘Ÿ PRON ijassa-1480 84 41 π‘Ÿ π‘Ÿ NOUN ijassa-1480 84 42 𝛿(𝑃 𝛿(𝑃 X ijassa-1480 84 43 ( ( PUNCT ijassa-1480 84 44 𝑠 𝑠 NOUN ijassa-1480 84 45 ) ) PUNCT ijassa-1480 84 46 , , PUNCT ijassa-1480 84 47 𝑧 𝑧 X ijassa-1480 84 48 ) ) PUNCT ijassa-1480 84 49 β‰₯ β‰₯ NOUN ijassa-1480 84 50 𝛿(𝐷 𝛿(𝐷 NOUN ijassa-1480 84 51 ( ( PUNCT ijassa-1480 84 52 𝑠),z 𝑠),z NOUN ijassa-1480 84 53 ) ) PUNCT ijassa-1480 84 54 , , PUNCT ijassa-1480 84 55 z z NOUN ijassa-1480 84 56 β‰  β‰  PROPN ijassa-1480 84 57 0 0 NUM ijassa-1480 84 58 , , PUNCT ijassa-1480 84 59 that that PRON ijassa-1480 84 60 contradicts contradict VERB ijassa-1480 84 61 inequality inequality NOUN ijassa-1480 84 62 ( ( PUNCT ijassa-1480 84 63 3.7 3.7 NUM ijassa-1480 84 64 ) ) PUNCT ijassa-1480 84 65 . . PUNCT ijassa-1480 85 1 function function NOUN ijassa-1480 85 2 βˆšπ‘Ÿ βˆšπ‘Ÿ NOUN ijassa-1480 85 3 𝛿(𝐷 𝛿(𝐷 PROPN ijassa-1480 85 4 ( ( PUNCT ijassa-1480 85 5 𝑠),z 𝑠),z NOUN ijassa-1480 85 6 ) ) PUNCT ijassa-1480 85 7 = = NUM ijassa-1480 85 8 𝛿(𝐷 𝛿(𝐷 NOUN ijassa-1480 85 9 , , PUNCT ijassa-1480 85 10 z z NOUN ijassa-1480 85 11 ) ) PUNCT ijassa-1480 85 12 is be AUX ijassa-1480 85 13 the the DET ijassa-1480 85 14 reference reference NOUN ijassa-1480 85 15 function function NOUN ijassa-1480 85 16 of of ADP ijassa-1480 85 17 the the DET ijassa-1480 85 18 polyhedron polyhedron NOUN ijassa-1480 85 19 𝐷 𝐷 PROPN ijassa-1480 85 20 ( ( PUNCT ijassa-1480 85 21 𝑠 𝑠 NOUN ijassa-1480 85 22 ) ) PUNCT ijassa-1480 85 23 = = PRON ijassa-1480 85 24 { { PUNCT ijassa-1480 85 25 π‘₯ π‘₯ NOUN ijassa-1480 85 26 : : PUNCT ijassa-1480 85 27 𝑣 𝑣 X ijassa-1480 85 28 ( ( PUNCT ijassa-1480 85 29 𝑙(𝑠 𝑙(𝑠 NOUN ijassa-1480 85 30 ) ) PUNCT ijassa-1480 85 31 , , PUNCT ijassa-1480 85 32 π‘₯ π‘₯ NOUN ijassa-1480 85 33 ) ) PUNCT ijassa-1480 85 34 ≀ ≀ NOUN ijassa-1480 85 35 π‘Ÿ π‘Ÿ NOUN ijassa-1480 85 36 } } PUNCT ijassa-1480 85 37 , , PUNCT ijassa-1480 85 38 which which PRON ijassa-1480 85 39 is be AUX ijassa-1480 85 40 similar similar ADJ ijassa-1480 85 41 to to ADP ijassa-1480 85 42 polyhedron polyhedron PROPN ijassa-1480 85 43 𝐷 𝐷 PROPN ijassa-1480 85 44 ( ( PUNCT ijassa-1480 85 45 𝑠 𝑠 NOUN ijassa-1480 85 46 ) ) PUNCT ijassa-1480 85 47 = = PRON ijassa-1480 85 48 { { PUNCT ijassa-1480 85 49 π‘₯ π‘₯ NOUN ijassa-1480 85 50 : : PUNCT ijassa-1480 85 51 𝑣 𝑣 X ijassa-1480 85 52 ( ( PUNCT ijassa-1480 85 53 𝑙(𝑠 𝑙(𝑠 NOUN ijassa-1480 85 54 ) ) PUNCT ijassa-1480 85 55 , , PUNCT ijassa-1480 85 56 π‘₯ π‘₯ NOUN ijassa-1480 85 57 ) ) PUNCT ijassa-1480 85 58 ≀ ≀ NUM ijassa-1480 85 59 1 1 NUM ijassa-1480 85 60 } } PUNCT ijassa-1480 85 61 . . PUNCT ijassa-1480 86 1 therefore therefore ADV ijassa-1480 86 2 ( ( PUNCT ijassa-1480 86 3 3.8 3.8 NUM ijassa-1480 86 4 ) ) PUNCT ijassa-1480 86 5 is be AUX ijassa-1480 86 6 equivalent equivalent ADJ ijassa-1480 86 7 to to ADP ijassa-1480 86 8 the the DET ijassa-1480 86 9 relation relation NOUN ijassa-1480 86 10 𝑃 𝑃 NOUN ijassa-1480 86 11 ( ( PUNCT ijassa-1480 86 12 𝑠 𝑠 NOUN ijassa-1480 86 13 ) ) PUNCT ijassa-1480 86 14 βŠ‚ βŠ‚ PROPN ijassa-1480 86 15 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ijassa-1480 86 16 𝐷 𝐷 PROPN ijassa-1480 86 17 ( ( PUNCT ijassa-1480 86 18 𝑠 𝑠 NOUN ijassa-1480 86 19 ) ) PUNCT ijassa-1480 86 20 βŠ‚ βŠ‚ PROPN ijassa-1480 86 21 𝐷 𝐷 PROPN ijassa-1480 86 22 ( ( PUNCT ijassa-1480 86 23 𝑠 𝑠 NOUN ijassa-1480 86 24 ) ) PUNCT ijassa-1480 86 25 . . PUNCT ijassa-1480 87 1 ( ( PUNCT ijassa-1480 87 2 3.9 3.9 NUM ijassa-1480 87 3 ) ) SYM ijassa-1480 87 4 190 190 NUM ijassa-1480 87 5 m. m. NOUN ijassa-1480 87 6 morozov morozov NOUN ijassa-1480 87 7 copyright copyright NOUN ijassa-1480 88 1 Β© Β© PROPN ijassa-1480 88 2 2024 2024 NUM ijassa-1480 88 3 assa assa PROPN ijassa-1480 88 4 adv adv PROPN ijassa-1480 88 5 . . PUNCT ijassa-1480 89 1 in in ADP ijassa-1480 89 2 systems system NOUN ijassa-1480 89 3 science science NOUN ijassa-1480 89 4 and and CCONJ ijassa-1480 89 5 appl appl NOUN ijassa-1480 89 6 . . PUNCT ijassa-1480 90 1 ( ( PUNCT ijassa-1480 90 2 2024 2024 NUM ijassa-1480 90 3 ) ) PUNCT ijassa-1480 90 4 it it PRON ijassa-1480 90 5 follows follow VERB ijassa-1480 90 6 from from ADP ijassa-1480 90 7 inequality inequality NOUN ijassa-1480 90 8 ( ( PUNCT ijassa-1480 90 9 3.2 3.2 NUM ijassa-1480 90 10 ) ) PUNCT ijassa-1480 90 11 for for ADP ijassa-1480 90 12 the the DET ijassa-1480 90 13 function function NOUN ijassa-1480 90 14 𝑣(𝑠 𝑣(𝑠 PROPN ijassa-1480 90 15 , , PUNCT ijassa-1480 90 16 π‘₯ π‘₯ NOUN ijassa-1480 90 17 ) ) PUNCT ijassa-1480 90 18 that that SCONJ ijassa-1480 90 19 𝐹(𝑃 𝐹(𝑃 X ijassa-1480 90 20 ( ( PUNCT ijassa-1480 90 21 𝑠 𝑠 NOUN ijassa-1480 90 22 ) ) PUNCT ijassa-1480 90 23 ) ) PUNCT ijassa-1480 91 1 βŠ‚ βŠ‚ PROPN ijassa-1480 91 2 𝑃 𝑃 PROPN ijassa-1480 91 3 ( ( PUNCT ijassa-1480 91 4 𝑠 𝑠 NOUN ijassa-1480 91 5 ) ) PUNCT ijassa-1480 91 6 , , PUNCT ijassa-1480 91 7 where where SCONJ ijassa-1480 91 8 𝐹(𝑃 𝐹(𝑃 X ijassa-1480 91 9 ( ( PUNCT ijassa-1480 91 10 𝑠 𝑠 NOUN ijassa-1480 91 11 ) ) PUNCT ijassa-1480 91 12 ) ) PUNCT ijassa-1480 92 1 = = PUNCT ijassa-1480 92 2 βˆͺ βˆͺ ADP ijassa-1480 92 3 ∈ ∈ PROPN ijassa-1480 92 4 ( ( PUNCT ijassa-1480 92 5 ) ) PUNCT ijassa-1480 92 6 𝐹(𝑠 𝐹(𝑠 NUM ijassa-1480 92 7 , , PUNCT ijassa-1480 92 8 π‘₯ π‘₯ NOUN ijassa-1480 92 9 ) ) PUNCT ijassa-1480 92 10 . . PUNCT ijassa-1480 93 1 ( ( PUNCT ijassa-1480 93 2 3.6 3.6 NUM ijassa-1480 93 3 ) ) PUNCT ijassa-1480 93 4 implies imply VERB ijassa-1480 93 5 that that SCONJ ijassa-1480 93 6 𝐷 𝐷 PROPN ijassa-1480 93 7 ( ( PUNCT ijassa-1480 93 8 𝑠 𝑠 NOUN ijassa-1480 93 9 ) ) PUNCT ijassa-1480 94 1 βŠ‚ βŠ‚ PROPN ijassa-1480 94 2 𝑃 𝑃 PROPN ijassa-1480 94 3 ( ( PUNCT ijassa-1480 94 4 𝑠 𝑠 NOUN ijassa-1480 94 5 ) ) PUNCT ijassa-1480 94 6 . . PUNCT ijassa-1480 95 1 therefore therefore ADV ijassa-1480 95 2 𝐹 𝐹 PROPN ijassa-1480 95 3 𝐷 𝐷 PROPN ijassa-1480 95 4 ( ( PUNCT ijassa-1480 95 5 𝑠 𝑠 NOUN ijassa-1480 95 6 ) ) PUNCT ijassa-1480 96 1 βŠ‚ βŠ‚ PRON ijassa-1480 96 2 𝐹 𝐹 PROPN ijassa-1480 96 3 𝑃 𝑃 PROPN ijassa-1480 96 4 ( ( PUNCT ijassa-1480 96 5 𝑠 𝑠 NOUN ijassa-1480 96 6 ) ) PUNCT ijassa-1480 96 7 βŠ‚ βŠ‚ PROPN ijassa-1480 97 1 𝑃 𝑃 PROPN ijassa-1480 97 2 ( ( PUNCT ijassa-1480 97 3 𝑠 𝑠 NOUN ijassa-1480 97 4 ) ) PUNCT ijassa-1480 97 5 . . PUNCT ijassa-1480 98 1 ( ( PUNCT ijassa-1480 98 2 3.10 3.10 NUM ijassa-1480 98 3 ) ) PUNCT ijassa-1480 98 4 it it PRON ijassa-1480 98 5 follows follow VERB ijassa-1480 98 6 from from ADP ijassa-1480 98 7 ( ( PUNCT ijassa-1480 98 8 3.9 3.9 NUM ijassa-1480 98 9 ) ) PUNCT ijassa-1480 98 10 and and CCONJ ijassa-1480 98 11 ( ( PUNCT ijassa-1480 98 12 3.10 3.10 NUM ijassa-1480 98 13 ) ) PUNCT ijassa-1480 98 14 𝐹(𝐷 𝐹(𝐷 X ijassa-1480 98 15 ( ( PUNCT ijassa-1480 98 16 𝑠 𝑠 NOUN ijassa-1480 98 17 ) ) PUNCT ijassa-1480 98 18 ) ) PUNCT ijassa-1480 99 1 βŠ‚ βŠ‚ PROPN ijassa-1480 99 2 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ijassa-1480 99 3 𝐷 𝐷 PROPN ijassa-1480 99 4 ( ( PUNCT ijassa-1480 99 5 𝑠 𝑠 NOUN ijassa-1480 99 6 ) ) PUNCT ijassa-1480 99 7 and and CCONJ ijassa-1480 99 8 therefore therefore ADV ijassa-1480 99 9 the the DET ijassa-1480 99 10 condition condition NOUN ijassa-1480 99 11 π‘šπ‘Žπ‘₯ π‘šπ‘Žπ‘₯ ADJ ijassa-1480 99 12 ∈ ∈ NOUN ijassa-1480 99 13 ( ( PUNCT ijassa-1480 99 14 , , PUNCT ijassa-1480 99 15 ) ) PUNCT ijassa-1480 99 16 𝑣 𝑣 X ijassa-1480 99 17 ( ( PUNCT ijassa-1480 99 18 𝑙(𝑠 𝑙(𝑠 NOUN ijassa-1480 99 19 ) ) PUNCT ijassa-1480 99 20 , , PUNCT ijassa-1480 99 21 𝑦 𝑦 X ijassa-1480 99 22 ) ) PUNCT ijassa-1480 99 23 < < X ijassa-1480 99 24 π‘Ÿ π‘Ÿ NOUN ijassa-1480 99 25 , , PUNCT ijassa-1480 99 26 π‘₯ π‘₯ PRON ijassa-1480 99 27 ∈ ∈ PROPN ijassa-1480 99 28 𝐷 𝐷 PROPN ijassa-1480 99 29 ( ( PUNCT ijassa-1480 99 30 𝑠 𝑠 PROPN ijassa-1480 99 31 ) ) PUNCT ijassa-1480 99 32 , , PUNCT ijassa-1480 99 33 𝑠 𝑠 PROPN ijassa-1480 99 34 = = SYM ijassa-1480 99 35 0,1 0,1 NUM ijassa-1480 99 36 , , PUNCT ijassa-1480 99 37 … … PUNCT ijassa-1480 99 38 ( ( PUNCT ijassa-1480 99 39 3.11 3.11 NUM ijassa-1480 99 40 ) ) PUNCT ijassa-1480 99 41 since since SCONJ ijassa-1480 99 42 𝑣 𝑣 X ijassa-1480 99 43 ( ( PUNCT ijassa-1480 99 44 𝑙(𝑠 𝑙(𝑠 NOUN ijassa-1480 99 45 ) ) PUNCT ijassa-1480 99 46 , , PUNCT ijassa-1480 99 47 πœ‡π‘₯ πœ‡π‘₯ NOUN ijassa-1480 99 48 ) ) PUNCT ijassa-1480 99 49 = = PUNCT ijassa-1480 99 50 πœ‡ πœ‡ ADP ijassa-1480 99 51 𝑣 𝑣 X ijassa-1480 99 52 ( ( PUNCT ijassa-1480 99 53 𝑙(𝑠 𝑙(𝑠 NOUN ijassa-1480 99 54 ) ) PUNCT ijassa-1480 99 55 , , PUNCT ijassa-1480 99 56 π‘₯ π‘₯ NOUN ijassa-1480 99 57 ) ) PUNCT ijassa-1480 99 58 and and CCONJ ijassa-1480 99 59 𝐹(𝑠 𝐹(𝑠 NUM ijassa-1480 99 60 , , PUNCT ijassa-1480 99 61 πœ‡π‘₯ πœ‡π‘₯ NOUN ijassa-1480 99 62 ) ) PUNCT ijassa-1480 99 63 = = SYM ijassa-1480 100 1 πœ‡πΉ(𝑠 πœ‡πΉ(𝑠 PROPN ijassa-1480 100 2 , , PUNCT ijassa-1480 100 3 π‘₯ π‘₯ NOUN ijassa-1480 100 4 ) ) PUNCT ijassa-1480 100 5 for for ADP ijassa-1480 100 6 all all DET ijassa-1480 100 7 𝑠 𝑠 PROPN ijassa-1480 100 8 β‰₯ β‰₯ NOUN ijassa-1480 100 9 0 0 NUM ijassa-1480 100 10 , , PUNCT ijassa-1480 100 11 π‘₯ π‘₯ DET ijassa-1480 100 12 ∈ ∈ PROPN ijassa-1480 100 13 𝑅 𝑅 PROPN ijassa-1480 100 14 and and CCONJ ijassa-1480 100 15 πœ‡ πœ‡ ADP ijassa-1480 100 16 ∈ ∈ PROPN ijassa-1480 100 17 𝑅 𝑅 PROPN ijassa-1480 100 18 , , PUNCT ijassa-1480 100 19 then then ADV ijassa-1480 100 20 , , PUNCT ijassa-1480 100 21 for for ADP ijassa-1480 100 22 the the DET ijassa-1480 100 23 above above ADV ijassa-1480 100 24 - - PUNCT ijassa-1480 100 25 constructed construct VERB ijassa-1480 100 26 function function NOUN ijassa-1480 100 27 𝑣 𝑣 X ijassa-1480 100 28 ( ( PUNCT ijassa-1480 100 29 𝑙(𝑠 𝑙(𝑠 NOUN ijassa-1480 100 30 ) ) PUNCT ijassa-1480 100 31 , , PUNCT ijassa-1480 100 32 π‘₯ π‘₯ NOUN ijassa-1480 100 33 ) ) PUNCT ijassa-1480 100 34 from from ADP ijassa-1480 100 35 ( ( PUNCT ijassa-1480 100 36 3.11 3.11 NUM ijassa-1480 100 37 ) ) PUNCT ijassa-1480 100 38 , , PUNCT ijassa-1480 100 39 the the DET ijassa-1480 100 40 inequality inequality NOUN ijassa-1480 100 41 π‘šπ‘Žπ‘₯ π‘šπ‘Žπ‘₯ VERB ijassa-1480 100 42 ∈ ∈ NOUN ijassa-1480 100 43 ( ( PUNCT ijassa-1480 100 44 , , PUNCT ijassa-1480 100 45 ) ) PUNCT ijassa-1480 100 46 𝑣 𝑣 X ijassa-1480 100 47 ( ( PUNCT ijassa-1480 100 48 𝑙(𝑠 𝑙(𝑠 NOUN ijassa-1480 100 49 ) ) PUNCT ijassa-1480 100 50 , , PUNCT ijassa-1480 100 51 𝑦 𝑦 NOUN ijassa-1480 100 52 ) ) PUNCT ijassa-1480 100 53 ≀ ≀ NOUN ijassa-1480 100 54 π‘Ÿ π‘Ÿ PRON ijassa-1480 100 55 𝑣 𝑣 X ijassa-1480 100 56 ( ( PUNCT ijassa-1480 100 57 𝑙(𝑠 𝑙(𝑠 NOUN ijassa-1480 100 58 ) ) PUNCT ijassa-1480 100 59 , , PUNCT ijassa-1480 100 60 π‘₯ π‘₯ NOUN ijassa-1480 100 61 ) ) PUNCT ijassa-1480 100 62 , , PUNCT ijassa-1480 100 63 0 0 PUNCT ijassa-1480 100 64 < < X ijassa-1480 100 65 π‘Ÿ π‘Ÿ X ijassa-1480 100 66 < < X ijassa-1480 100 67 1 1 NUM ijassa-1480 100 68 , , PUNCT ijassa-1480 100 69 π‘₯ π‘₯ DET ijassa-1480 100 70 ∈ ∈ PROPN ijassa-1480 100 71 𝑅 𝑅 PROPN ijassa-1480 100 72 , , PUNCT ijassa-1480 100 73 𝑠 𝑠 PROPN ijassa-1480 100 74 = = SYM ijassa-1480 100 75 0,1 0,1 NUM ijassa-1480 100 76 , , PUNCT ijassa-1480 100 77 … … PUNCT ijassa-1480 100 78 ( ( PUNCT ijassa-1480 100 79 3.12 3.12 NUM ijassa-1480 100 80 ) ) PUNCT ijassa-1480 100 81 of of ADP ijassa-1480 100 82 the the DET ijassa-1480 100 83 form form NOUN ijassa-1480 100 84 ( ( PUNCT ijassa-1480 100 85 3.2 3.2 NUM ijassa-1480 100 86 ) ) PUNCT ijassa-1480 100 87 . . PUNCT ijassa-1480 101 1 this this PRON ijassa-1480 101 2 completes complete VERB ijassa-1480 101 3 the the DET ijassa-1480 101 4 proof proof NOUN ijassa-1480 101 5 of of ADP ijassa-1480 101 6 theorem theorem NOUN ijassa-1480 101 7 3.2 3.2 NUM ijassa-1480 101 8 . . PUNCT ijassa-1480 102 1 ο‚  ο‚  PRON ijassa-1480 102 2 consider consider VERB ijassa-1480 102 3 piecewise piecewise NOUN ijassa-1480 102 4 linear linear VERB ijassa-1480 102 5 lyapunov lyapunov NOUN ijassa-1480 102 6 functions function NOUN ijassa-1480 102 7 𝑉 𝑉 PROPN ijassa-1480 102 8 ( ( PUNCT ijassa-1480 102 9 𝑠 𝑠 PROPN ijassa-1480 102 10 , , PUNCT ijassa-1480 102 11 π‘₯ π‘₯ NOUN ijassa-1480 102 12 ) ) PUNCT ijassa-1480 102 13 = = VERB ijassa-1480 102 14 π‘šπ‘Žπ‘₯ π‘šπ‘Žπ‘₯ NOUN ijassa-1480 102 15 𝑙 𝑙 NOUN ijassa-1480 102 16 ( ( PUNCT ijassa-1480 102 17 𝑠 𝑠 NOUN ijassa-1480 102 18 ) ) PUNCT ijassa-1480 102 19 , , PUNCT ijassa-1480 102 20 π‘₯⟩ π‘₯⟩ X ijassa-1480 102 21 . . PUNCT ijassa-1480 103 1 ( ( PUNCT ijassa-1480 103 2 3.13 3.13 NUM ijassa-1480 103 3 ) ) PUNCT ijassa-1480 103 4 a a DET ijassa-1480 103 5 corollary corollary NOUN ijassa-1480 103 6 of of ADP ijassa-1480 103 7 theorem theorem ADJ ijassa-1480 103 8 3.2 3.2 NUM ijassa-1480 103 9 is be AUX ijassa-1480 103 10 theorem theorem VERB ijassa-1480 103 11 3.3 3.3 NUM ijassa-1480 103 12 : : PUNCT ijassa-1480 103 13 inclusion inclusion NOUN ijassa-1480 103 14 ( ( PUNCT ijassa-1480 103 15 2.1 2.1 NUM ijassa-1480 103 16 ) ) PUNCT ijassa-1480 103 17 is be AUX ijassa-1480 103 18 asymptotically asymptotically ADV ijassa-1480 103 19 stable stable ADJ ijassa-1480 103 20 iff iff NOUN ijassa-1480 103 21 , , PUNCT ijassa-1480 103 22 for for ADP ijassa-1480 103 23 some some DET ijassa-1480 103 24 integer integer NOUN ijassa-1480 103 25 𝑀 𝑀 PROPN ijassa-1480 103 26 β‰₯ β‰₯ NOUN ijassa-1480 103 27 𝑛 𝑛 ADP ijassa-1480 103 28 there there ADV ijassa-1480 103 29 exists exist VERB ijassa-1480 103 30 a a DET ijassa-1480 103 31 periodic periodic NOUN ijassa-1480 103 32 on on ADP ijassa-1480 103 33 𝑠 𝑠 PROPN ijassa-1480 103 34 ( ( PUNCT ijassa-1480 103 35 period period NOUN ijassa-1480 103 36 𝑁 𝑁 PROPN ijassa-1480 103 37 ) ) PUNCT ijassa-1480 103 38 , , PUNCT ijassa-1480 103 39 piecewise piecewise NOUN ijassa-1480 103 40 - - PUNCT ijassa-1480 103 41 linear linear NOUN ijassa-1480 103 42 lyapunov lyapunov NOUN ijassa-1480 103 43 function function NOUN ijassa-1480 103 44 𝑉 𝑉 PROPN ijassa-1480 103 45 ( ( PUNCT ijassa-1480 103 46 𝑠 𝑠 PROPN ijassa-1480 103 47 , , PUNCT ijassa-1480 103 48 π‘₯ π‘₯ NOUN ijassa-1480 103 49 ) ) PUNCT ijassa-1480 103 50 ( ( PUNCT ijassa-1480 103 51 3.13 3.13 NUM ijassa-1480 103 52 ) ) PUNCT ijassa-1480 103 53 , , PUNCT ijassa-1480 103 54 satisfying satisfy VERB ijassa-1480 103 55 condition condition NOUN ijassa-1480 103 56 ( ( PUNCT ijassa-1480 103 57 3.5 3.5 NUM ijassa-1480 103 58 ) ) PUNCT ijassa-1480 103 59 and and CCONJ ijassa-1480 103 60 inequality inequality NOUN ijassa-1480 103 61 ( ( PUNCT ijassa-1480 103 62 3.2 3.2 NUM ijassa-1480 103 63 ) ) PUNCT ijassa-1480 103 64 for for ADP ijassa-1480 103 65 all all DET ijassa-1480 103 66 π‘₯ π‘₯ DET ijassa-1480 103 67 ∈ ∈ PROPN ijassa-1480 103 68 𝑅 𝑅 PROPN ijassa-1480 103 69 , , PUNCT ijassa-1480 103 70 𝑠 𝑠 PROPN ijassa-1480 103 71 β‰₯ β‰₯ NOUN ijassa-1480 103 72 0 0 NUM ijassa-1480 103 73 . . PUNCT ijassa-1480 104 1 proof proof NOUN ijassa-1480 104 2 . . PUNCT ijassa-1480 105 1 since since SCONJ ijassa-1480 105 2 𝑣 𝑣 PRON ijassa-1480 105 3 ( ( PUNCT ijassa-1480 105 4 𝑠 𝑠 PROPN ijassa-1480 105 5 , , PUNCT ijassa-1480 105 6 π‘₯ π‘₯ NOUN ijassa-1480 105 7 ) ) PUNCT ijassa-1480 105 8 = = VERB ijassa-1480 105 9 π‘šπ‘Žπ‘₯ π‘šπ‘Žπ‘₯ ADJ ijassa-1480 105 10 βŸ¨π‘™ βŸ¨π‘™ X ijassa-1480 105 11 ( ( PUNCT ijassa-1480 105 12 𝑠 𝑠 NOUN ijassa-1480 105 13 ) ) PUNCT ijassa-1480 105 14 , , PUNCT ijassa-1480 105 15 π‘₯⟩ π‘₯⟩ PRON ijassa-1480 105 16 = = PUNCT ijassa-1480 105 17 π‘šπ‘Žπ‘₯ π‘šπ‘Žπ‘₯ NOUN ijassa-1480 105 18 𝑙 𝑙 NOUN ijassa-1480 105 19 ( ( PUNCT ijassa-1480 105 20 𝑠 𝑠 NOUN ijassa-1480 105 21 ) ) PUNCT ijassa-1480 105 22 , , PUNCT ijassa-1480 105 23 π‘₯⟩ π‘₯⟩ PUNCT ijassa-1480 105 24 = = SYM ijassa-1480 105 25 𝑉 𝑉 PROPN ijassa-1480 105 26 ( ( PUNCT ijassa-1480 105 27 𝑠 𝑠 PROPN ijassa-1480 105 28 , , PUNCT ijassa-1480 105 29 π‘₯ π‘₯ NOUN ijassa-1480 105 30 ) ) PUNCT ijassa-1480 105 31 , , PUNCT ijassa-1480 105 32 then then ADV ijassa-1480 105 33 each each DET ijassa-1480 105 34 function function NOUN ijassa-1480 105 35 𝑣 𝑣 X ijassa-1480 105 36 ( ( PUNCT ijassa-1480 105 37 𝑠 𝑠 PROPN ijassa-1480 105 38 , , PUNCT ijassa-1480 105 39 π‘₯ π‘₯ NOUN ijassa-1480 105 40 ) ) PUNCT ijassa-1480 105 41 can can AUX ijassa-1480 105 42 be be AUX ijassa-1480 105 43 matched match VERB ijassa-1480 105 44 with with ADP ijassa-1480 105 45 function function NOUN ijassa-1480 105 46 𝑉 𝑉 PROPN ijassa-1480 105 47 ( ( PUNCT ijassa-1480 105 48 𝑠 𝑠 PROPN ijassa-1480 105 49 , , PUNCT ijassa-1480 105 50 π‘₯ π‘₯ NOUN ijassa-1480 105 51 ) ) PUNCT ijassa-1480 105 52 = = SYM ijassa-1480 105 53 𝑣 𝑣 X ijassa-1480 105 54 ( ( PUNCT ijassa-1480 105 55 s s X ijassa-1480 105 56 , , PUNCT ijassa-1480 105 57 x x NOUN ijassa-1480 105 58 ) ) PUNCT ijassa-1480 105 59 . . PUNCT ijassa-1480 106 1 in in ADP ijassa-1480 106 2 the the DET ijassa-1480 106 3 same same ADJ ijassa-1480 106 4 way way NOUN ijassa-1480 106 5 each each DET ijassa-1480 106 6 function function NOUN ijassa-1480 106 7 𝑉 𝑉 PROPN ijassa-1480 106 8 ( ( PUNCT ijassa-1480 106 9 𝑠 𝑠 PROPN ijassa-1480 106 10 , , PUNCT ijassa-1480 106 11 π‘₯ π‘₯ NOUN ijassa-1480 106 12 ) ) PUNCT ijassa-1480 106 13 ( ( PUNCT ijassa-1480 106 14 3.13 3.13 NUM ijassa-1480 106 15 ) ) PUNCT ijassa-1480 106 16 can can AUX ijassa-1480 106 17 be be AUX ijassa-1480 106 18 associated associate VERB ijassa-1480 106 19 with with ADP ijassa-1480 106 20 function function NOUN ijassa-1480 106 21 𝑣 𝑣 X ijassa-1480 106 22 ( ( PUNCT ijassa-1480 106 23 𝑠 𝑠 PROPN ijassa-1480 106 24 , , PUNCT ijassa-1480 106 25 π‘₯ π‘₯ NOUN ijassa-1480 106 26 ) ) PUNCT ijassa-1480 106 27 = = SYM ijassa-1480 106 28 𝑉 𝑉 PROPN ijassa-1480 106 29 ( ( PUNCT ijassa-1480 106 30 𝑠 𝑠 PROPN ijassa-1480 106 31 , , PUNCT ijassa-1480 106 32 π‘₯ π‘₯ NOUN ijassa-1480 106 33 ) ) PUNCT ijassa-1480 106 34 . . PUNCT ijassa-1480 107 1 from from ADP ijassa-1480 107 2 this this PRON ijassa-1480 107 3 and and CCONJ ijassa-1480 107 4 theorem theorem VERB ijassa-1480 107 5 3.2 3.2 NUM ijassa-1480 107 6 follows follow VERB ijassa-1480 107 7 the the DET ijassa-1480 107 8 validity validity NOUN ijassa-1480 107 9 of of ADP ijassa-1480 107 10 theorem theorem ADJ ijassa-1480 107 11 3.3 3.3 NUM ijassa-1480 107 12 . . PUNCT ijassa-1480 108 1 ο‚  ο‚  PRON ijassa-1480 108 2 4 4 NUM ijassa-1480 108 3 . . PUNCT ijassa-1480 108 4 example example NOUN ijassa-1480 108 5 consider consider VERB ijassa-1480 108 6 the the DET ijassa-1480 108 7 torsional torsional ADJ ijassa-1480 108 8 vibrations vibration NOUN ijassa-1480 108 9 of of ADP ijassa-1480 108 10 the the DET ijassa-1480 108 11 crankshafts crankshaft NOUN ijassa-1480 108 12 of of ADP ijassa-1480 108 13 a a DET ijassa-1480 108 14 single single ADJ ijassa-1480 108 15 - - PUNCT ijassa-1480 108 16 cylinder cylinder NOUN ijassa-1480 108 17 engine engine NOUN ijassa-1480 108 18 with with ADP ijassa-1480 108 19 a a DET ijassa-1480 108 20 flywheel flywheel NOUN ijassa-1480 108 21 with with ADP ijassa-1480 108 22 allowance allowance NOUN ijassa-1480 108 23 for for ADP ijassa-1480 108 24 the the DET ijassa-1480 108 25 inertia inertia NOUN ijassa-1480 108 26 of of ADP ijassa-1480 108 27 connecting connect VERB ijassa-1480 108 28 rods rod NOUN ijassa-1480 108 29 and and CCONJ ijassa-1480 108 30 pistons piston NOUN ijassa-1480 108 31 . . PUNCT ijassa-1480 109 1 let let VERB ijassa-1480 109 2 us we PRON ijassa-1480 109 3 assume assume VERB ijassa-1480 109 4 that that SCONJ ijassa-1480 109 5 the the DET ijassa-1480 109 6 mass mass NOUN ijassa-1480 109 7 of of ADP ijassa-1480 109 8 the the DET ijassa-1480 109 9 flywheel flywheel NOUN ijassa-1480 109 10 is be AUX ijassa-1480 109 11 sufficiently sufficiently ADV ijassa-1480 109 12 large large ADJ ijassa-1480 109 13 , , PUNCT ijassa-1480 109 14 so so SCONJ ijassa-1480 109 15 that that SCONJ ijassa-1480 109 16 the the DET ijassa-1480 109 17 rotation rotation NOUN ijassa-1480 109 18 of of ADP ijassa-1480 109 19 the the DET ijassa-1480 109 20 shaft shaft NOUN ijassa-1480 109 21 can can AUX ijassa-1480 109 22 be be AUX ijassa-1480 109 23 considered consider VERB ijassa-1480 109 24 uniform uniform ADJ ijassa-1480 109 25 . . PUNCT ijassa-1480 110 1 let let VERB ijassa-1480 110 2 πœ” πœ” PRON ijassa-1480 110 3 angular angular ADJ ijassa-1480 110 4 velocity velocity NOUN ijassa-1480 110 5 of of ADP ijassa-1480 110 6 the the DET ijassa-1480 110 7 flywheel flywheel NOUN ijassa-1480 110 8 , , PUNCT ijassa-1480 110 9 𝑐 𝑐 X ijassa-1480 110 10 torsional torsional ADJ ijassa-1480 110 11 stiffness stiffness ADJ ijassa-1480 110 12 coefficient coefficient NOUN ijassa-1480 110 13 of of ADP ijassa-1480 110 14 the the DET ijassa-1480 110 15 shaft shaft NOUN ijassa-1480 110 16 . . PUNCT ijassa-1480 111 1 in in ADP ijassa-1480 111 2 [ [ X ijassa-1480 111 3 3 3 X ijassa-1480 111 4 ] ] X ijassa-1480 111 5 it it PRON ijassa-1480 111 6 is be AUX ijassa-1480 111 7 shown show VERB ijassa-1480 111 8 that that SCONJ ijassa-1480 111 9 vibrations vibration NOUN ijassa-1480 111 10 are be AUX ijassa-1480 111 11 described describe VERB ijassa-1480 111 12 by by ADP ijassa-1480 111 13 second second ADJ ijassa-1480 111 14 order order NOUN ijassa-1480 111 15 differential differential NOUN ijassa-1480 111 16 equation equation NOUN ijassa-1480 111 17 𝑑 𝑑 PROPN ijassa-1480 111 18 π‘ž π‘ž X ijassa-1480 111 19 𝑑𝑑 𝑑𝑑 ADP ijassa-1480 111 20 + + PROPN ijassa-1480 111 21 𝑐 𝑐 NOUN ijassa-1480 111 22 πœ” πœ” ADP ijassa-1480 111 23 𝑝(𝑑)π‘ž 𝑝(𝑑)π‘ž NOUN ijassa-1480 111 24 = = SYM ijassa-1480 111 25 0 0 NUM ijassa-1480 111 26 , , PUNCT ijassa-1480 111 27 ( ( PUNCT ijassa-1480 111 28 4.14 4.14 NUM ijassa-1480 111 29 ) ) PUNCT ijassa-1480 111 30 where where SCONJ ijassa-1480 111 31 π‘ž π‘ž PROPN ijassa-1480 111 32 – – PUNCT ijassa-1480 111 33 generalized generalized ADJ ijassa-1480 111 34 coordinate coordinate NOUN ijassa-1480 111 35 associated associate VERB ijassa-1480 111 36 with with ADP ijassa-1480 111 37 the the DET ijassa-1480 111 38 angle angle NOUN ijassa-1480 111 39 of of ADP ijassa-1480 111 40 rotation rotation NOUN ijassa-1480 111 41 of of ADP ijassa-1480 111 42 the the DET ijassa-1480 111 43 crank crank NOUN ijassa-1480 111 44 , , PUNCT ijassa-1480 111 45 and and CCONJ ijassa-1480 111 46 periodic periodic ADJ ijassa-1480 111 47 function function NOUN ijassa-1480 111 48 𝑝(𝑑 𝑝(𝑑 PROPN ijassa-1480 111 49 ) ) PUNCT ijassa-1480 111 50 of of ADP ijassa-1480 111 51 the the DET ijassa-1480 111 52 period period NOUN ijassa-1480 111 53 2πœ‹can 2πœ‹can NUM ijassa-1480 111 54 be be AUX ijassa-1480 111 55 determined determine VERB ijassa-1480 111 56 by by ADP ijassa-1480 111 57 the the DET ijassa-1480 111 58 kinetic kinetic ADJ ijassa-1480 111 59 energy energy NOUN ijassa-1480 111 60 of of ADP ijassa-1480 111 61 the the DET ijassa-1480 111 62 crank crank NOUN ijassa-1480 111 63 together together ADV ijassa-1480 111 64 with with ADP ijassa-1480 111 65 the the DET ijassa-1480 111 66 associated associated ADJ ijassa-1480 111 67 moving move VERB ijassa-1480 111 68 masses masse NOUN ijassa-1480 111 69 ( ( PUNCT ijassa-1480 111 70 connecting connect VERB ijassa-1480 111 71 rod rod NOUN ijassa-1480 111 72 and and CCONJ ijassa-1480 111 73 piston piston NOUN ijassa-1480 111 74 ) ) PUNCT ijassa-1480 111 75 . . PUNCT ijassa-1480 112 1 equation equation NOUN ijassa-1480 112 2 ( ( PUNCT ijassa-1480 112 3 4.1 4.1 NUM ijassa-1480 112 4 ) ) PUNCT ijassa-1480 112 5 is be AUX ijassa-1480 112 6 a a DET ijassa-1480 112 7 special special ADJ ijassa-1480 112 8 case case NOUN ijassa-1480 112 9 of of ADP ijassa-1480 112 10 an an DET ijassa-1480 112 11 equation equation NOUN ijassa-1480 112 12 of of ADP ijassa-1480 112 13 more more ADJ ijassa-1480 112 14 general general ADJ ijassa-1480 112 15 form form NOUN ijassa-1480 112 16 𝑑 𝑑 X ijassa-1480 112 17 𝑧 𝑧 NOUN ijassa-1480 112 18 𝑑𝑑 𝑑𝑑 ADP ijassa-1480 112 19 + + NOUN ijassa-1480 112 20 𝑏𝑓(𝑑)𝑧 𝑏𝑓(𝑑)𝑧 PROPN ijassa-1480 112 21 = = SYM ijassa-1480 112 22 0 0 NUM ijassa-1480 112 23 , , PUNCT ijassa-1480 112 24 ( ( PUNCT ijassa-1480 112 25 4.2 4.2 X ijassa-1480 112 26 ) ) PUNCT ijassa-1480 112 27 luapunov luapunov NOUN ijassa-1480 112 28 functions function NOUN ijassa-1480 112 29 for for ADP ijassa-1480 112 30 periodic periodic ADJ ijassa-1480 112 31 selector selector NOUN ijassa-1480 112 32 - - PUNCT ijassa-1480 112 33 linear linear NOUN ijassa-1480 112 34 difference difference NOUN ijassa-1480 112 35 inclusions inclusion NOUN ijassa-1480 112 36 191 191 NUM ijassa-1480 112 37 copyright copyright NOUN ijassa-1480 112 38 Β© Β© PROPN ijassa-1480 112 39 2024 2024 NUM ijassa-1480 112 40 assa assa NOUN ijassa-1480 112 41 . . PUNCT ijassa-1480 113 1 adv adv PROPN ijassa-1480 113 2 . . PUNCT ijassa-1480 114 1 in in ADP ijassa-1480 114 2 systems system NOUN ijassa-1480 114 3 science science NOUN ijassa-1480 114 4 and and CCONJ ijassa-1480 114 5 appl appl NOUN ijassa-1480 114 6 . . PUNCT ijassa-1480 115 1 ( ( PUNCT ijassa-1480 115 2 2024 2024 NUM ijassa-1480 115 3 ) ) PUNCT ijassa-1480 115 4 where where SCONJ ijassa-1480 115 5 𝑓(𝑑 𝑓(𝑑 NOUN ijassa-1480 115 6 ) ) PUNCT ijassa-1480 115 7 is be AUX ijassa-1480 115 8 a a DET ijassa-1480 115 9 periodic periodic ADJ ijassa-1480 115 10 function function NOUN ijassa-1480 115 11 of of ADP ijassa-1480 115 12 time time NOUN ijassa-1480 115 13 ( ( PUNCT ijassa-1480 115 14 with with ADP ijassa-1480 115 15 period period NOUN ijassa-1480 115 16 𝑇 𝑇 PROPN ijassa-1480 115 17 > > X ijassa-1480 115 18 0 0 NUM ijassa-1480 115 19 ) ) PUNCT ijassa-1480 115 20 , , PUNCT ijassa-1480 115 21 and and CCONJ ijassa-1480 115 22 𝑏 𝑏 DET ijassa-1480 115 23 ∈ ∈ PROPN ijassa-1480 115 24 𝐼 𝐼 PROPN ijassa-1480 115 25 , , PUNCT ijassa-1480 115 26 𝐼 𝐼 PROPN ijassa-1480 115 27 = = PUNCT ijassa-1480 116 1 [ [ X ijassa-1480 116 2 𝑏 𝑏 NOUN ijassa-1480 116 3 , , PUNCT ijassa-1480 116 4 𝑏 𝑏 PROPN ijassa-1480 116 5 ] ] PUNCT ijassa-1480 116 6 is be AUX ijassa-1480 116 7 a a DET ijassa-1480 116 8 certain certain ADJ ijassa-1480 116 9 parameter parameter NOUN ijassa-1480 116 10 . . PUNCT ijassa-1480 117 1 let let VERB ijassa-1480 117 2 us we PRON ijassa-1480 117 3 introduce introduce VERB ijassa-1480 117 4 notation notation NOUN ijassa-1480 117 5 π‘₯ π‘₯ NOUN ijassa-1480 117 6 = = SYM ijassa-1480 117 7 𝑧 𝑧 PROPN ijassa-1480 117 8 , , PUNCT ijassa-1480 117 9 π‘₯ π‘₯ PROPN ijassa-1480 117 10 = = SYM ijassa-1480 117 11 = = X ijassa-1480 117 12 , , PUNCT ijassa-1480 117 13 π‘₯ π‘₯ NOUN ijassa-1480 117 14 = = PUNCT ijassa-1480 118 1 π‘₯ π‘₯ PART ijassa-1480 118 2 π‘₯ π‘₯ X ijassa-1480 118 3 , , PUNCT ijassa-1480 118 4 𝐴(𝑑 𝐴(𝑑 PROPN ijassa-1480 118 5 , , PUNCT ijassa-1480 118 6 π‘Ž π‘Ž NOUN ijassa-1480 118 7 , , PUNCT ijassa-1480 118 8 𝑏 𝑏 NOUN ijassa-1480 118 9 ) ) PUNCT ijassa-1480 118 10 = = SYM ijassa-1480 118 11 0 0 NUM ijassa-1480 118 12 1 1 NUM ijassa-1480 118 13 βˆ’π‘π‘“(𝑑 βˆ’π‘π‘“(𝑑 NUM ijassa-1480 118 14 ) ) PUNCT ijassa-1480 118 15 0 0 PUNCT ijassa-1480 118 16 . . PUNCT ijassa-1480 119 1 equation equation NOUN ijassa-1480 119 2 ( ( PUNCT ijassa-1480 119 3 4.2 4.2 NUM ijassa-1480 119 4 ) ) PUNCT ijassa-1480 119 5 will will AUX ijassa-1480 119 6 take take VERB ijassa-1480 119 7 the the DET ijassa-1480 119 8 form form NOUN ijassa-1480 119 9 of of ADP ijassa-1480 119 10 second second ADJ ijassa-1480 119 11 order order NOUN ijassa-1480 119 12 system system NOUN ijassa-1480 119 13 𝑑π‘₯ 𝑑π‘₯ VERB ijassa-1480 119 14 𝑑𝑑 𝑑𝑑 ADP ijassa-1480 119 15 = = SYM ijassa-1480 119 16 𝐴(𝑑 𝐴(𝑑 NOUN ijassa-1480 119 17 , , PUNCT ijassa-1480 119 18 𝑏)π‘₯ 𝑏)π‘₯ NOUN ijassa-1480 119 19 , , PUNCT ijassa-1480 119 20 𝐴(𝑑 𝐴(𝑑 X ijassa-1480 119 21 + + CCONJ ijassa-1480 119 22 𝑇 𝑇 PROPN ijassa-1480 119 23 ) ) PUNCT ijassa-1480 119 24 ≑ ≑ PROPN ijassa-1480 119 25 𝐴(𝑑 𝐴(𝑑 PROPN ijassa-1480 119 26 ) ) PUNCT ijassa-1480 119 27 , , PUNCT ijassa-1480 119 28 𝑏 𝑏 PROPN ijassa-1480 119 29 ∈ ∈ PROPN ijassa-1480 119 30 𝐼 𝐼 PROPN ijassa-1480 119 31 , , PUNCT ijassa-1480 119 32 𝑑 𝑑 X ijassa-1480 119 33 β‰₯ β‰₯ NOUN ijassa-1480 119 34 0 0 NUM ijassa-1480 119 35 , , PUNCT ijassa-1480 119 36 𝑇 𝑇 PROPN ijassa-1480 119 37 > > X ijassa-1480 119 38 0 0 NUM ijassa-1480 119 39 , , PUNCT ijassa-1480 119 40 π‘₯ π‘₯ DET ijassa-1480 119 41 ∈ ∈ PROPN ijassa-1480 119 42 𝑅 𝑅 PROPN ijassa-1480 119 43 . . PUNCT ijassa-1480 120 1 ( ( PUNCT ijassa-1480 120 2 4.3 4.3 NUM ijassa-1480 120 3 ) ) PUNCT ijassa-1480 120 4 consider consider VERB ijassa-1480 120 5 discrete discrete ADJ ijassa-1480 120 6 analogue analogue NOUN ijassa-1480 120 7 of of ADP ijassa-1480 120 8 system system NOUN ijassa-1480 120 9 ( ( PUNCT ijassa-1480 120 10 4.3 4.3 NUM ijassa-1480 120 11 ) ) PUNCT ijassa-1480 120 12 π‘₯(𝑠 π‘₯(𝑠 NOUN ijassa-1480 120 13 + + CCONJ ijassa-1480 120 14 1 1 X ijassa-1480 120 15 ) ) PUNCT ijassa-1480 120 16 = = SYM ijassa-1480 120 17 𝐴(𝑠 𝐴(𝑠 NOUN ijassa-1480 120 18 , , PUNCT ijassa-1480 120 19 𝑏)π‘₯(𝑠 𝑏)π‘₯(𝑠 PROPN ijassa-1480 120 20 ) ) PUNCT ijassa-1480 120 21 , , PUNCT ijassa-1480 120 22 𝐴(𝑠 𝐴(𝑠 PUNCT ijassa-1480 120 23 + + CCONJ ijassa-1480 120 24 𝑀 𝑀 PROPN ijassa-1480 120 25 ) ) PUNCT ijassa-1480 120 26 ≑ ≑ PROPN ijassa-1480 120 27 𝐴(𝑠 𝐴(𝑠 PROPN ijassa-1480 120 28 ) ) PUNCT ijassa-1480 120 29 , , PUNCT ijassa-1480 120 30 𝑏 𝑏 PROPN ijassa-1480 120 31 ∈ ∈ PROPN ijassa-1480 120 32 𝐼 𝐼 PROPN ijassa-1480 120 33 , , PUNCT ijassa-1480 120 34 𝑀 𝑀 PROPN ijassa-1480 120 35 ∈ ∈ PROPN ijassa-1480 120 36 𝑁 𝑁 PROPN ijassa-1480 120 37 , , PUNCT ijassa-1480 120 38 π‘₯ π‘₯ PRON ijassa-1480 120 39 ∈ ∈ PROPN ijassa-1480 120 40 𝑅 𝑅 PROPN ijassa-1480 120 41 , , PUNCT ijassa-1480 120 42 ( ( PUNCT ijassa-1480 120 43 4.4 4.4 NUM ijassa-1480 120 44 ) ) PUNCT ijassa-1480 120 45 where where SCONJ ijassa-1480 120 46 𝑠 𝑠 PROPN ijassa-1480 120 47 = = SYM ijassa-1480 120 48 0,1 0,1 NUM ijassa-1480 120 49 , , PUNCT ijassa-1480 120 50 . . PUNCT ijassa-1480 121 1 .. .. PUNCT ijassa-1480 121 2 is be AUX ijassa-1480 121 3 discrete discrete ADJ ijassa-1480 121 4 time time NOUN ijassa-1480 121 5 . . PUNCT ijassa-1480 122 1 system system NOUN ijassa-1480 122 2 ( ( PUNCT ijassa-1480 122 3 4.4 4.4 NUM ijassa-1480 122 4 ) ) PUNCT ijassa-1480 122 5 is be AUX ijassa-1480 122 6 equivalent equivalent ADJ ijassa-1480 122 7 to to ADP ijassa-1480 122 8 periodic periodic ADJ ijassa-1480 122 9 selector selector NOUN ijassa-1480 122 10 - - PUNCT ijassa-1480 122 11 linear linear NOUN ijassa-1480 122 12 difference difference NOUN ijassa-1480 122 13 inclusion inclusion NOUN ijassa-1480 122 14 ( ( PUNCT ijassa-1480 122 15 2.1 2.1 NUM ijassa-1480 122 16 ) ) PUNCT ijassa-1480 122 17 , , PUNCT ijassa-1480 122 18 where where SCONJ ijassa-1480 122 19 the the DET ijassa-1480 122 20 multivalued multivalued ADJ ijassa-1480 122 21 function function NOUN ijassa-1480 122 22 𝐹(𝑠 𝐹(𝑠 NUM ijassa-1480 122 23 , , PUNCT ijassa-1480 122 24 π‘₯ π‘₯ NOUN ijassa-1480 122 25 ) ) PUNCT ijassa-1480 122 26 ( ( PUNCT ijassa-1480 122 27 𝐹(𝑠 𝐹(𝑠 PROPN ijassa-1480 122 28 + + NUM ijassa-1480 122 29 𝑀 𝑀 PROPN ijassa-1480 122 30 , , PUNCT ijassa-1480 122 31 π‘₯ π‘₯ NOUN ijassa-1480 122 32 ) ) PUNCT ijassa-1480 122 33 ≑ ≑ PROPN ijassa-1480 122 34 𝐹(𝑠 𝐹(𝑠 NUM ijassa-1480 122 35 , , PUNCT ijassa-1480 122 36 π‘₯ π‘₯ NOUN ijassa-1480 122 37 ) ) PUNCT ijassa-1480 122 38 ) ) PUNCT ijassa-1480 122 39 is be AUX ijassa-1480 122 40 defined define VERB ijassa-1480 122 41 at at ADP ijassa-1480 122 42 each each DET ijassa-1480 122 43 point point NOUN ijassa-1480 122 44 ( ( PUNCT ijassa-1480 122 45 𝑠 𝑠 PROPN ijassa-1480 122 46 , , PUNCT ijassa-1480 122 47 π‘₯ π‘₯ NOUN ijassa-1480 122 48 ) ) PUNCT ijassa-1480 122 49 , , PUNCT ijassa-1480 122 50 π‘₯ π‘₯ DET ijassa-1480 122 51 ∈ ∈ PROPN ijassa-1480 122 52 𝑅 𝑅 PROPN ijassa-1480 122 53 by by ADP ijassa-1480 122 54 the the DET ijassa-1480 122 55 relation relation NOUN ijassa-1480 122 56 𝐹(𝑠 𝐹(𝑠 NUM ijassa-1480 122 57 , , PUNCT ijassa-1480 122 58 π‘₯ π‘₯ NOUN ijassa-1480 122 59 ) ) PUNCT ijassa-1480 122 60 = = SYM ijassa-1480 123 1 { { PUNCT ijassa-1480 123 2 𝑦 𝑦 NOUN ijassa-1480 123 3 : : PUNCT ijassa-1480 123 4 𝑦 𝑦 NOUN ijassa-1480 123 5 = = SYM ijassa-1480 123 6 𝐴(𝑠 𝐴(𝑠 NOUN ijassa-1480 123 7 , , PUNCT ijassa-1480 123 8 𝑏)π‘₯ 𝑏)π‘₯ NOUN ijassa-1480 123 9 , , PUNCT ijassa-1480 123 10 𝑏 𝑏 PRON ijassa-1480 123 11 ∈ ∈ PROPN ijassa-1480 123 12 𝐼 𝐼 PROPN ijassa-1480 123 13 } } PUNCT ijassa-1480 123 14 . . PUNCT ijassa-1480 124 1 5 5 X ijassa-1480 124 2 . . X ijassa-1480 124 3 conclusion conclusion NOUN ijassa-1480 124 4 for for ADP ijassa-1480 124 5 periodic periodic ADJ ijassa-1480 124 6 selector selector NOUN ijassa-1480 124 7 - - PUNCT ijassa-1480 124 8 linear linear NOUN ijassa-1480 124 9 difference difference NOUN ijassa-1480 124 10 inclusion inclusion NOUN ijassa-1480 124 11 ( ( PUNCT ijassa-1480 124 12 2.1 2.1 NUM ijassa-1480 124 13 ) ) PUNCT ijassa-1480 124 14 asymptotic asymptotic ADJ ijassa-1480 124 15 stability stability NOUN ijassa-1480 124 16 criteria criterion NOUN ijassa-1480 124 17 were be AUX ijassa-1480 124 18 obtained obtain VERB ijassa-1480 124 19 . . PUNCT ijassa-1480 125 1 they they PRON ijassa-1480 125 2 use use VERB ijassa-1480 125 3 quasi quasi ADJ ijassa-1480 125 4 - - ADJ ijassa-1480 125 5 quadratic quadratic ADJ ijassa-1480 125 6 , , PUNCT ijassa-1480 125 7 piecewise piecewise NOUN ijassa-1480 125 8 quadratic quadratic ADJ ijassa-1480 125 9 and and CCONJ ijassa-1480 125 10 piecewise piecewise PROPN ijassa-1480 125 11 linear linear PROPN ijassa-1480 125 12 lyapunov lyapunov NOUN ijassa-1480 125 13 functions function NOUN ijassa-1480 125 14 . . PUNCT ijassa-1480 126 1 the the DET ijassa-1480 126 2 example example NOUN ijassa-1480 126 3 of of ADP ijassa-1480 126 4 technical technical ADJ ijassa-1480 126 5 problem problem NOUN ijassa-1480 126 6 leading lead VERB ijassa-1480 126 7 to to ADP ijassa-1480 126 8 consideration consideration NOUN ijassa-1480 126 9 of of ADP ijassa-1480 126 10 periodic periodic ADJ ijassa-1480 126 11 differential differential NOUN ijassa-1480 126 12 and and CCONJ ijassa-1480 126 13 difference difference NOUN ijassa-1480 126 14 inclusions inclusion NOUN ijassa-1480 126 15 is be AUX ijassa-1480 126 16 given give VERB ijassa-1480 126 17 . . PUNCT ijassa-1480 127 1 the the DET ijassa-1480 127 2 obtained obtain VERB ijassa-1480 127 3 results result NOUN ijassa-1480 127 4 can can AUX ijassa-1480 127 5 be be AUX ijassa-1480 127 6 used use VERB ijassa-1480 127 7 in in ADP ijassa-1480 127 8 the the DET ijassa-1480 127 9 study study NOUN ijassa-1480 127 10 of of ADP ijassa-1480 127 11 the the DET ijassa-1480 127 12 stability stability NOUN ijassa-1480 127 13 of of ADP ijassa-1480 127 14 control control NOUN ijassa-1480 127 15 systems system NOUN ijassa-1480 127 16 with with ADP ijassa-1480 127 17 periodic periodic ADJ ijassa-1480 127 18 parameters parameter NOUN ijassa-1480 127 19 . . PUNCT ijassa-1480 128 1 in in ADP ijassa-1480 128 2 particular particular ADJ ijassa-1480 128 3 , , PUNCT ijassa-1480 128 4 such such ADJ ijassa-1480 128 5 systems system NOUN ijassa-1480 128 6 are be AUX ijassa-1480 128 7 the the DET ijassa-1480 128 8 tracking tracking NOUN ijassa-1480 128 9 systems system NOUN ijassa-1480 128 10 , , PUNCT ijassa-1480 128 11 which which PRON ijassa-1480 128 12 elements element NOUN ijassa-1480 128 13 operate operate VERB ijassa-1480 128 14 on on ADP ijassa-1480 128 15 alternating alternate VERB ijassa-1480 128 16 current current ADJ ijassa-1480 128 17 and and CCONJ ijassa-1480 128 18 the the DET ijassa-1480 128 19 control control NOUN ijassa-1480 128 20 systems system NOUN ijassa-1480 128 21 with with ADP ijassa-1480 128 22 pulse pulse NOUN ijassa-1480 128 23 - - PUNCT ijassa-1480 128 24 amplitude amplitude NOUN ijassa-1480 128 25 modulation modulation NOUN ijassa-1480 128 26 . . PUNCT ijassa-1480 129 1 the the DET ijassa-1480 129 2 extracted extract VERB ijassa-1480 129 3 in in ADP ijassa-1480 129 4 theorems theorem NOUN ijassa-1480 129 5 3.2 3.2 NUM ijassa-1480 129 6 , , PUNCT ijassa-1480 129 7 3.3 3.3 NUM ijassa-1480 129 8 piecewise piecewise NOUN ijassa-1480 129 9 - - PUNCT ijassa-1480 129 10 quadratic quadratic ADJ ijassa-1480 129 11 and and CCONJ ijassa-1480 129 12 piecewise piecewise NOUN ijassa-1480 129 13 - - PUNCT ijassa-1480 129 14 linear linear NOUN ijassa-1480 129 15 lyapunov lyapunov ADJ ijassa-1480 129 16 functions function NOUN ijassa-1480 129 17 of of ADP ijassa-1480 129 18 the the DET ijassa-1480 129 19 form form NOUN ijassa-1480 129 20 ( ( PUNCT ijassa-1480 129 21 3.4 3.4 NUM ijassa-1480 129 22 ) ) PUNCT ijassa-1480 129 23 , , PUNCT ijassa-1480 129 24 ( ( PUNCT ijassa-1480 129 25 3.13 3.13 NUM ijassa-1480 129 26 ) ) PUNCT ijassa-1480 129 27 establish establish VERB ijassa-1480 129 28 necessary necessary ADJ ijassa-1480 129 29 and and CCONJ ijassa-1480 129 30 sufficient sufficient ADJ ijassa-1480 129 31 conditions condition NOUN ijassa-1480 129 32 of of ADP ijassa-1480 129 33 asymptotic asymptotic ADJ ijassa-1480 129 34 stability stability NOUN ijassa-1480 129 35 for for ADP ijassa-1480 129 36 inclusion inclusion NOUN ijassa-1480 129 37 ( ( PUNCT ijassa-1480 129 38 2.1 2.1 NUM ijassa-1480 129 39 ) ) PUNCT ijassa-1480 129 40 . . PUNCT ijassa-1480 130 1 these these DET ijassa-1480 130 2 functions function NOUN ijassa-1480 130 3 can can AUX ijassa-1480 130 4 be be AUX ijassa-1480 130 5 used use VERB ijassa-1480 130 6 when when SCONJ ijassa-1480 130 7 developing develop VERB ijassa-1480 130 8 numerical numerical ADJ ijassa-1480 130 9 methods method NOUN ijassa-1480 130 10 of of ADP ijassa-1480 130 11 stability stability NOUN ijassa-1480 130 12 analysis analysis NOUN ijassa-1480 130 13 for for ADP ijassa-1480 130 14 systems system NOUN ijassa-1480 130 15 equivalent equivalent ADJ ijassa-1480 130 16 to to ADP ijassa-1480 130 17 difference difference NOUN ijassa-1480 130 18 inclusion inclusion NOUN ijassa-1480 130 19 ( ( PUNCT ijassa-1480 130 20 2.1 2.1 NUM ijassa-1480 130 21 ) ) PUNCT ijassa-1480 130 22 . . PUNCT ijassa-1480 131 1 references reference NOUN ijassa-1480 131 2 1 1 NUM ijassa-1480 131 3 . . PUNCT ijassa-1480 132 1 blagodatskikh blagodatskikh PROPN ijassa-1480 132 2 , , PUNCT ijassa-1480 132 3 v. v. PROPN ijassa-1480 132 4 i. i. PROPN ijassa-1480 132 5 & & CCONJ ijassa-1480 132 6 filippov filippov PROPN ijassa-1480 132 7 , , PUNCT ijassa-1480 132 8 a. a. PROPN ijassa-1480 132 9 f. f. PROPN ijassa-1480 132 10 ( ( PUNCT ijassa-1480 132 11 1986 1986 NUM ijassa-1480 132 12 ) ) PUNCT ijassa-1480 132 13 . . PUNCT ijassa-1480 133 1 differential differential ADJ ijassa-1480 133 2 inclusions inclusion NOUN ijassa-1480 133 3 and and CCONJ ijassa-1480 133 4 optimal optimal ADJ ijassa-1480 133 5 control control NOUN ijassa-1480 133 6 , , PUNCT ijassa-1480 133 7 proc proc PROPN ijassa-1480 133 8 . . PUNCT ijassa-1480 133 9 steklov steklov PROPN ijassa-1480 133 10 inst inst PROPN ijassa-1480 133 11 . . PUNCT ijassa-1480 133 12 math math NOUN ijassa-1480 133 13 . . PUNCT ijassa-1480 133 14 , , PUNCT ijassa-1480 133 15 169 169 NUM ijassa-1480 133 16 , , PUNCT ijassa-1480 133 17 199–259 199–259 NUM ijassa-1480 133 18 . . PUNCT ijassa-1480 134 1 2 2 NUM ijassa-1480 134 2 . . X ijassa-1480 134 3 halanay halanay PROPN ijassa-1480 134 4 , , PUNCT ijassa-1480 134 5 a. a. PROPN ijassa-1480 134 6 & & CCONJ ijassa-1480 134 7 wexler wexler PROPN ijassa-1480 134 8 , , PUNCT ijassa-1480 134 9 d. d. PROPN ijassa-1480 134 10 ( ( PUNCT ijassa-1480 134 11 1968 1968 NUM ijassa-1480 134 12 ) ) PUNCT ijassa-1480 134 13 . . PUNCT ijassa-1480 135 1 theoria theoria PROPN ijassa-1480 135 2 calitativa calitativa PROPN ijassa-1480 135 3 a a DET ijassa-1480 135 4 sistemelor sistemelor NOUN ijassa-1480 135 5 cu cu PROPN ijassa-1480 135 6 impulsuri impulsuri NOUN ijassa-1480 136 1 [ [ X ijassa-1480 136 2 qualitative qualitative ADJ ijassa-1480 136 3 theory theory NOUN ijassa-1480 136 4 of of ADP ijassa-1480 136 5 pulse pulse NOUN ijassa-1480 136 6 systems system NOUN ijassa-1480 136 7 ] ] PUNCT ijassa-1480 136 8 . . PUNCT ijassa-1480 137 1 bucharest bucharest PROPN ijassa-1480 137 2 , , PUNCT ijassa-1480 137 3 romania romania PROPN ijassa-1480 137 4 : : PUNCT ijassa-1480 137 5 editura editura PROPN ijassa-1480 137 6 academiei academiei PROPN ijassa-1480 137 7 republicii republicii PROPN ijassa-1480 137 8 socialiste socialiste PROPN ijassa-1480 137 9 romania romania PROPN ijassa-1480 137 10 , , PUNCT ijassa-1480 138 1 [ [ X ijassa-1480 138 2 in in ADP ijassa-1480 138 3 romanian romanian NOUN ijassa-1480 138 4 ] ] X ijassa-1480 138 5 . . PUNCT ijassa-1480 139 1 3 3 X ijassa-1480 139 2 . . X ijassa-1480 139 3 malkin malkin PROPN ijassa-1480 139 4 , , PUNCT ijassa-1480 139 5 i. i. PROPN ijassa-1480 139 6 g. g. PROPN ijassa-1480 139 7 ( ( PUNCT ijassa-1480 139 8 1952 1952 NUM ijassa-1480 139 9 ) ) PUNCT ijassa-1480 139 10 . . PUNCT ijassa-1480 140 1 theory theory NOUN ijassa-1480 140 2 of of ADP ijassa-1480 140 3 stability stability NOUN ijassa-1480 140 4 of of ADP ijassa-1480 140 5 motion motion NOUN ijassa-1480 140 6 . . PUNCT ijassa-1480 141 1 washington washington PROPN ijassa-1480 141 2 , , PUNCT ijassa-1480 141 3 dc dc PROPN ijassa-1480 141 4 : : PUNCT ijassa-1480 141 5 us us PROPN ijassa-1480 141 6 atomic atomic ADJ ijassa-1480 141 7 energy energy NOUN ijassa-1480 141 8 comission comission NOUN ijassa-1480 141 9 . . PUNCT ijassa-1480 142 1 4 4 X ijassa-1480 142 2 . . X ijassa-1480 142 3 morozov morozov NOUN ijassa-1480 142 4 , , PUNCT ijassa-1480 142 5 m. m. NOUN ijassa-1480 142 6 v. v. PROPN ijassa-1480 142 7 ( ( PUNCT ijassa-1480 142 8 2021 2021 NUM ijassa-1480 142 9 ) ) PUNCT ijassa-1480 142 10 . . PUNCT ijassa-1480 143 1 a a DET ijassa-1480 143 2 criterion criterion NOUN ijassa-1480 143 3 for for ADP ijassa-1480 143 4 the the DET ijassa-1480 143 5 asymptotic asymptotic ADJ ijassa-1480 143 6 stability stability NOUN ijassa-1480 143 7 of of ADP ijassa-1480 143 8 a a DET ijassa-1480 143 9 periodic periodic ADJ ijassa-1480 143 10 selectorlinear selectorlinear NOUN ijassa-1480 143 11 differential differential NOUN ijassa-1480 143 12 inclusion inclusion NOUN ijassa-1480 143 13 , , PUNCT ijassa-1480 143 14 automation automation NOUN ijassa-1480 143 15 & & CCONJ ijassa-1480 143 16 remote remote ADJ ijassa-1480 143 17 control control NOUN ijassa-1480 143 18 , , PUNCT ijassa-1480 143 19 82(1 82(1 NOUN ijassa-1480 143 20 ) ) PUNCT ijassa-1480 143 21 , , PUNCT ijassa-1480 143 22 63–72 63–72 NUM ijassa-1480 143 23 , , PUNCT ijassa-1480 143 24 https://doi.org/10.1134/s0005117921010045 https://doi.org/10.1134/s0005117921010045 NUM ijassa-1480 143 25 5 5 NUM ijassa-1480 143 26 . . PUNCT ijassa-1480 143 27 morozov morozov NOUN ijassa-1480 143 28 , , PUNCT ijassa-1480 143 29 m. m. NOUN ijassa-1480 143 30 v. v. PROPN ijassa-1480 143 31 ( ( PUNCT ijassa-1480 143 32 2022 2022 NUM ijassa-1480 143 33 ) ) PUNCT ijassa-1480 143 34 . . PUNCT ijassa-1480 144 1 on on ADP ijassa-1480 144 2 the the DET ijassa-1480 144 3 stability stability NOUN ijassa-1480 144 4 of of ADP ijassa-1480 144 5 periodic periodic ADJ ijassa-1480 144 6 difference difference NOUN ijassa-1480 144 7 inclusions inclusion NOUN ijassa-1480 144 8 , , PUNCT ijassa-1480 144 9 advances advance NOUN ijassa-1480 144 10 in in ADP ijassa-1480 144 11 systems system NOUN ijassa-1480 144 12 science science NOUN ijassa-1480 144 13 and and CCONJ ijassa-1480 144 14 applications application NOUN ijassa-1480 144 15 , , PUNCT ijassa-1480 144 16 22(1 22(1 NUM ijassa-1480 144 17 ) ) PUNCT ijassa-1480 144 18 , , PUNCT ijassa-1480 144 19 167–175 167–175 NUM ijassa-1480 144 20 . . PUNCT ijassa-1480 145 1 6 6 NUM ijassa-1480 145 2 . . X ijassa-1480 145 3 rockafellar rockafellar ADJ ijassa-1480 145 4 , , PUNCT ijassa-1480 145 5 r. r. PROPN ijassa-1480 145 6 t. t. PROPN ijassa-1480 145 7 ( ( PUNCT ijassa-1480 145 8 1970 1970 NUM ijassa-1480 145 9 ) ) PUNCT ijassa-1480 145 10 . . PUNCT ijassa-1480 146 1 convex convex VERB ijassa-1480 146 2 analysis analysis NOUN ijassa-1480 146 3 . . PUNCT ijassa-1480 147 1 princeton princeton PROPN ijassa-1480 147 2 , , PUNCT ijassa-1480 147 3 nj nj PROPN ijassa-1480 147 4 : : PUNCT ijassa-1480 147 5 princeton princeton PROPN ijassa-1480 147 6 university university PROPN ijassa-1480 147 7 press press NOUN ijassa-1480 147 8 . . PUNCT