Concerning existence of the Lyapunov sign-variable functions for linear systems of differential equations
Исследуются вопросы о существовании знакопеременных функций Ляпунова для линейных систем дифференциальных уравнений с переменными коэффициентами.
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| Datum: | 1992 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Russisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
1992
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/7800 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512760824070144 |
|---|---|
| author | Samoilenko , A. M. Kulik , V. L. Самойленко, А. М. Кулик, В. Л. |
| author_facet | Samoilenko , A. M. Kulik , V. L. Самойленко, А. М. Кулик, В. Л. |
| author_sort | Samoilenko , A. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-09-14T12:03:56Z |
| description | Исследуются вопросы о существовании знакопеременных функций Ляпунова для линейных систем дифференциальных уравнений с переменными коэффициентами. |
| first_indexed | 2026-03-24T03:33:55Z |
| format | Article |
| fulltext |
v nK s1 1.92s
A . M.. Ca~rnii i1c1mo, 'IJI .-Ropp. AI-I Y1,pmn1bl,
1· B . .JI. Kymm, p;-p cp113.-MaT. uay1; ( J-l IT-T MarnMannm AI-I Y1,paum,1, Rnen)
\. BOnpocy O cy1.u;ecTBOIJalHIIJ
aHa1.wnepeMeHHbIX (py111,.u;11ii Jlnnynona
~"l}l: JI'MHCMHbIX CHCTCl\:I ,[~HC}lcpepenu;1rnJibHblX ypannemdi
f .foc.~Cll.YHTTCH IJOnpOCbl O CYlllCCTIJOBUHIH! 3 HaKonepeMe! IH bl X 1py11KUHII Jhrny1rnIJa ,'\Jill JIHHCHHblll!
C HCTe M JlHcjxpepe HUH3JlbHbIX ypaB!-!CllHII C nepeMCHHbTMH 1rn•Kjxp11u11e 11TaMH •
.kocJii JVKYIOTbCH 11HT3HIIH npo icHyBa HHH 3HaK03MiHJrnX rj,yHKUiil JJ,rnyuona JlJHI -~iHiilirnX CH•
,c;re ~1 ,wtjJe p e nuiam, HHX pi llH51Hb i 3 3MiHHHMH KOetp iUiEHTaMH .
flycrb CHCTeMa JII-HieHHb!X .n.mpcpepeHUl-laJibHb!X ypaBHCHHH
(1}
•
n 1 + n2 = n, x1 = dx/dt,
C A M. CAM O l'!JIEHKO, B. JI. KYJIHK, 1992
00 ISSN 0041-6053. Jl,cp_ Mar. ,icypH., 1992. r. 44. M I
C Henpeph1BHOH
K03C\xjmm1eHTOB
CBOHCTBaMH.
H orpaim4e1rnoi1 Ha scei1 om R n x n-MepHoH MaTpnueif
A (t) = {Au (t)}7.i=1, A (t) E C0 (R), 06Jia,11,aeT cJie.n.y10W.HMH
I. Cyw.ecrnyeT n x n-MepttaH CHMMe-rp1141-1aH MaTpnua S1 (t) E C 1 (R)
(HenpepbIBHO .n.mpcpepeHunpyeMa H orpa1ml!e1-maH Ha R), y.n.osJiernopmow.aH
HepaBeHCTBY
(2)
2. CyruecrnyeT n 2 x n 2-Mep11aH CHMMerp11t11-iaH MaTp11ua S22 (t) E C1 (R),
HMelOIU3H CBOHCTBO
Tor.n.a Henocpe.D.CTBeHHO MO)K!-13 npoBepHTh, lJTO npOH3B0.D.HaH I<Ba,11,pan!l!HOH
cjJOpMbI VP (t, x) = p (S1 (t) x, x) + (S 22 (t) x2 , x2 ) B CHJIY rncTeMbI (I) np!f
)lOCTaT04HO 60JiblllHX 3Ha4eHHHX napaMeTpa p > 0 6y.n.eT TIOJIO)!{l,l'I'eJ]bHO on
p e)leJieHI-IOH:
O6o3Hatiaa
' W [SJ = S +SA+ A*S,
P 1 = diag {In,, 0}, P 2 = diag {0, In.},
S2 (t) = diag {O, S22 (t)},
HepaBeHCTBa (2), (3) 3arrnweM B s11,11,e
(W [S 1] x, x);;;,, II P 1x ![2 ,
(W [S2 ] P 2 x , P 2x) > II P 2 x /1 2 •
(4)
(5)
(6)
(7)
( )
T en epb npeJlTIOJIO)KHM, l!TO CHMMeTpHtIHaH MaTpnua S2 B HepaseIICTBe (8)
He HBJIHeTCH 6JIOCfl·JO-)lHa roHa.1b! IOI1, a HMeeT 6oJiee 06uu1 i'1 Bl!Jl S2 (t) =
= f s ;j (t) }7 t=l . B )la.llb!Ie1'.\1u e:11 y6e1tHMCH, '-ITO 3TO npeJl TI0JIO)KCI! Ht' lie C03j\aeT
OC06bIX TPY.J.tlJOCTeH .ll.JI}l Ll.0l(cl 3clTe.'lbCTBa Toro, 4T0 npOH3BO,'(l!;IH 1rna,1paTH4HOH
cpopMbI VP (t, x) = /J (S 1 (t) x , x) + (S 2 (t) x, x) B CHJI\' rncreMbI (I ) 6y.n.er
TI0JTOtKHTeJibll0 onpe,'1eJ1e1rnoii np11 ilOCTaTOlII-10 6o:: h l.1111 X 31Iall e IIHH X 11 a pa~1eTpa
p > 0. C ,11,pyroh CTopoHbI, npe2rnoJ1mKe1rne o 60J1ee o6meM B11.n.e ~laTJ)H J.U>i
S2 BeJleT K T0MY, '-ITO yu10BHe (3) Mm1rno 3aMeHI-nb 6oJiee cJia6hIM · '
(1S22 (t) + s22 '. I) A22 (t) + A;2 (t) s22 (t)I X 2 , Xz) + ([ s21 (t) A12 (t) +
+ A~2 (t) s;1 (t)J x2, x2):;;;,, 11 x2 l1 2.
PaCCMOTJ)HM 3TO Ha KOHKpeTI-IOM npmv1epe CHCTeMbl 1rnyx ypaBHe !lllH
'A. > 0 .
(9)
( 10) ' ,·
f1pOH3B0,UHaH KBan.paTl14HOH cj)OpMbl V = X1X2 B c1rny CHCTeMbl ( I 0) p;,ill 1a
' V = X~ + A (COS2 f) X~ > xr f103TOMY HepaBeHCTBO (2) BbIITOJIHHeTCH npH S 1 =
= ( 0 112 ) . Ol!eBHllHO, He cyw.ecTByeT HenpepbIBHO )ll1tj)Cjlepe 1mnpye~10»
1/2 0 - . ,
Ii orpaH11l!em1oi1 Ha R cpy1-1Ku1111 s (t), Ll.JIH KOTopoh 661 BhlllOJIIIHJJOCb 11 epase11c"rao
; (t) - 2s (t) sin t > I. CJien.oBaTeJibHO, HepaBeHCTBO (3) B paCCMclTp Me aeM0M
CJiy4ae He BbIITOJIHHeTCH, O.ll.1-IaKO HepaBeJ-ICTBO (9) BblITOJII-IHeTCH npH 5~2 (/) 1
-= - sin t, S 21 = 1/'A. : - cos t + 2 sin2t + 2 cos2 t ;;;,, I. TaKHM o6p a:,oM,
CHCTeMa ypaBHeI-IHH (10) HBJIHeTCH 3KCITOHeI-IUH3JibHO .ll.HXOTOMHlJHOH Ha BCeff
om R npH JII06oM 3J-IalleHHH napaMeTpa 'A.> 0 [I, c. 71.
Ooo6w.aH 1-1epaseI-IcTBa (7), (8) Ha cJiy4att npoH3BOJibHbIX Ma-rpHu P;~
npHXO.UHM K CJie.n.y,omeMy YTBep)K11.e1rn10.
T eo p e M a I. flycmb cyw,ec11Wy1om aee n x n-MepHbte cuM,1tempui1Hote
ISSN 0041-605:J. Jl,cp. Mar. :J1Cyp11.., 1992, T. 44, JI& l
.aampu4t>t S; (t) E C1 (R), i = I, 2, yooaJZemBopR1ow,ue HepaeencmeaM
(W[S1Jx,x)>IIM 1 (t}xll2 , (11)
(W [S2 [ M 2 (t) x, M 2 (t)x) > II M 2 (t}x 112. (12)
zce MampUl,{bt pa.3Mepa n x n M; (t) E C0 (R) 11Ate1om cAeoy10w,ue ceotlcmea:
M 1 (t)M 2 (t)=M 2 (t)M 1 (t) \ftER , (13)
det M (t} * 0 vt ER, M (t} = Ml (t) + M2 (t) (14)
• npu EJmo . .4t
II M-1 (t) II~ canst < oo.
Tozoa npott3800HGfl K8aopamu<mot1 q;opMbl
VP (t , x) = p (S1 (t) x, x) + (S2 (( x, x)
11 cuAy cucmeMbt (1) yooaAem aopRem Hepaaencmay
Vp (t, x)> a(p) II x 11 2 ,
zae a (p) > 0 npu oocmamo,uw 6011bwux sHa'ieNttflx napaMempa p:
P > II W f S2J !lo II M-1 116 + II W [S2J ll6 II M-I 116 II M 1!6-
( 15)
( 16)
(17)
Il o Ka 3 a T e JI b c T no. 3 anttCbIBaH npott3B0JJ.HYJO VP (t, x) H yl!HTbIBaH
«5o3Halle1me (4J If HepaBeI·ICTB0 (11) , HMeeM
VP (t, x) > p II M 1x 11 2 + (W [S2 ] x1 x). ( 18)
BTopoe c.1rnraeMoe 3anttrneM cJJe,ll.yIOlll.HM o6pa3oM:
(W [S2 ] (M 1 + M 2 ) M-1x , (M 1 + M 2 ) M-1x) = (W [S 2 ] M 1M-1x,
M 1M-1x) + 2 (W [S2 ] M 1M-1x, M 2M- 1x ) + (H? !S2 ] M 2M-1x, M 2M- 1x) =
=W1 +2W2 + W3 . (19)
YqHTbIBaH T0}K,ll.ecrno ( 13), HMeeM M; (t) M-1 (t) = M-1 (t} M 1 (t), i = 1, 2.
no3TOMY nepsoe H sropoe cJJa raeMhie B pane Herne ( 19) MO}KHO ouemnL TaK:
W1 = (W [S 2] M-1M 1x, M-1M 1x)>- II W [S 2 I llo II M-1M 1x 11 2 >
>- II W [S2] llo II M-11/6 II M1X 11 2,
(20)
W2 = (W [S 2 ] M-1M 1x, M-1M 2x) > - II W [S 2 ] llo II M-1M 1x 1111 M-1M 2x 11 >
>- II W fS2J llo II M-1 1!6 ll Mix JI II M2x 11-
TpeTbe CJiaraeMoe B paBeHCTBe (19), HCITOJ!b3YH HepaBeHCTBO (12), oueHHM CJie
«YIOIIJ.HM o6pa3oM:
Wa> II M2M- 1x 11 2 = II M- 1M2X 11 2 > 11 M llo2 II M2X 11 2• (21)
O6osHaqa51 w = II W [S2] llo, k = II M-1 /1 6. m = II Mf lo2 H yqHTbIBaH HepaneH
CTBa (20), (21), npo,11,omKaeM ouettKy (18):
' Vp(t, x)> (p -@k} II M 1x 11 2 -2@k ll M 1x II II M 2xl l + m II M 2x!12 • (22)
PaccMaTpHBaH COOTBeTCTBYIOlll.yIO npaBOH qacTH HepaBeHCTBa (22) KBa,ll.paTHlj
HYIO cpopMy (p - wll) ti - 2wktil2 + mtt Haii.n.eM ee Ha11MeHhWee aHaqeHHe Ha
e,nHHHl!HOH oKpy)KHOCTH t1 = cos <p, t2 = sin <p. 11MeeM
p-wk-m
(p - @k} cos21 <p - 2@k cos <p sin <p + m sin 2 cp = 2 cos 2<p -
I02 TSSN 0041-6053. YK.p. MaT. 3/CgpH., 1992, T. 44. JI I
- rok sin 2q, + p - ro2k + m > -v ( p - ~k - m r + ro2kl +
m(p -rok -ro2 .!!!:_)
p-rok+m .....__ m + 2 ::::? ------:,-------- = y (p) > o.
p-rok + m
Dpe 3TOM rrpe.nrronaraeM BhIIIOJIHem1e HepaBeHCTBa
k2
p>wk + ro 2 -.
m
laxHM o6pa30M, H3 oueHKH (22) CJie)],yeT
VP (t, x)> y(p)(il Mix 11 2 + II M 2x 11 2 );;;;,, y~p) (II Mix 11 + II M 2xll) ~
> y~) l!Mxll 2 > 211~~'llo llxi12 = 1'~:) 1i xl! 2•
TeopeMa AOKa3aHa.
(23)
3 a M e lj a H H e 1. EcJIH B YCJIOBHHX TeopeMhl l JlOTTOJIHHTeJibl-1O npe.uno
no>KHTb, l!TO I det S 1 (t)I ;;;;,, o - con s t > 0, To cncTeMa ypam1 e1-1HH (1) 6yAeT
3Kcno11e1-1u1-1aJibHO llHXOTOMH<-11-!0H 11a acen ocn R [2, 31.
3 a Mel[ a 1-1 H e 2. 0-rKa3 OT BblllOJIHeHHH TO)KlleCTBa (1 3) B ycJIOBHHX
'reOpeMbl I MO)KeT npHBeCTH K TOMY, l!TO He 6y11.eT cyw.ecTBOBaTb I<Ba,ll,paTHl!HaH
<popMa, KOTopaH HMeJia 6b! 3HaKoonpe,neJieIUIYIO npOH3BO)],l·IYIO B CHJIY CHCTeMbl
(1). llpOCJi ellHM 3TO Ha npHMepe CHCTeMbl .llBYX ypaBHeHHH
(24)
8.necb A(t) = . TTOJiaraH Si= HMeeM W[SiJ = SiA+ (-1 l) ( 0 - l)
-1 1 -1 2
+ A*Si = 2( 1 - 1). TTo3TOMY (W [S1] x, x) = 2 (x~ -2x1x2 + 4) = 2 x
-1 1
( l -1) X (x1 - x2 )2;;;;,, II M iX !1 2 , r.ne Mi = 0 0 . Tenepb 1-1enocpeACTBe1-1Ho rrpo-
BepHeM, l!TO npH s2 = (~ ~) BhIIIOJIHHeTCH 1-1epaBeHCTBO ((S2A + A*S2) M2X,
M 2x)>!IM 2xJi 2 , r.n.e M2 =(~ ~) · 3aMeTHM, l!TO MiM 2 =FM 2 M 1 • TaKHM
o6pa30M, Bee ycJIOBHH reopeMhl l, KpoMe (13), BblIIOJII-1511OTC51, I-1O npH 3TOM He
cymeCTsyer KBaapaTHl!HaH cpopMa, HMe10w.aH 31-1aKoonpe11.eJiem-1y10 npo113 ao.u
Hy10 B CHJIY CHCTeMbl (24). EcJIH 6bI TaKaH cpopMa cyw.ecTBOBaJia, TO ace pewe•
HH5l CHCTeMbl (24) Ha 6ecKOHel!HOCTH JIHOO 3KCIIOI-1e1-1u1-1aJibHO BO3 pacTaJII-!, JIH·
00 3aTyxaJIH K HYJIIO, HO 3TO He TaK: Xi (t) = C1t + C2, X2 (t) =Cit+ Cz + C1,
c1 - canst.
CJie.uyer OTMeTHTb, l!TO TO)KJlecTBO (13) )],JIH HeKOTOpb!X MaTpHU Mi MQ)l( (: J
He BhIIlOJIHHTbCH, HO TeM He MeHee CHCTeMa (1) 6y.neT 3KCTTOHeHuHaJibHO llHXO·
TOMI-Ill HOH Ha BCeH OCH R. B 3TOM MO)KHO y6e.UHTbC51, rrpoH3BO1lH 3aMeHy nepeMel-1·
HhIX x = Ny B ttepaBeHcrBe (8), Mi= Pi, M 2 = NP2 • TTpH 3TOM acer)la MO)K·
HO r1011.06paTh raKy10 MaTp1-1uy N, l!T06b1 M 1M 2 = PiNP2 * M 2M 1 = NP 2Pi.
T enepb npe)lf!OJIO)KHM, l!TO Hcxo11.Hy1O CHCTeMy JIHHeHHhl:X ypaaHemtii
MO)KHO 3amiCaTb B BHll.e
x1 = A 11 (t) x1 + A 12 (t) x2 + A13 (t) X3 ,
X 2 = A 21 (t) X 2 + A211 (t) x2 + A 23 (t) x3 ,
1 ISSN 0011-6Q5J. ~Kp. Mal. )/(,!jpH.., 1992. T. 44. M l
(25)
O603H3'iHM A2 = 32 23 ' X2 = 2 H upe,ZUI0J!0)KHM, 'ITO cymecTBYJOlt "(A A)" (x)
Aa2 Aaa X3
TJ)H CHMMeTpH<-11-ibre MaTpHUbI S1 (t), 52 (t), S3 :i (t) E C1 (R) pa3MepoB coorne-rcr:
BeHHo n X n, (n2 + n 3 ) X (n2 + n 3), n 3 X n 3 , y.n.oBJJeTBop51IOill,He ttepaBeHCJ>
BaM
( [S 1 (t) + S 1 (t) A (t) + A" (t) S 1 (t)] x, x) > II x 1 112.
( [Sz (t) + s2 (t) A2 (t) + A;s2 (t)] ;2, ;2) > I) X2112, (26'
([533 (t) + S33 (t) A33 t) + 11;3 (t) S3a (t)J X3, X3) > II X31! 2 .
Bo3HHKaeT B0IIpoc: cymeernyeT JlH IIpH 3THX ycJJ0BHHX KBa,ll.p3TI1'iH3H qmpMa,
HMeIOill,351 3H3K00IIpe.n.e.r1e1rnyIO IIp0H3B0.ll.HYIO B CHJJY ClICTeMbl (25)? Ecmi .n.a.
TO B KaK0M BH,D.e TaKyIO cpopMy M0)KH0 HCKaTb?
O6o3HatJaH S 2 = diag{0, S2 (t)}, S 3 = diag{0, S 33 (t)}, P 1 = diag{/n,, 0, O}~
P 2 =diag{0,In,,0}, P3 = diag{0,0,I11,}, uepaBe1-1crna (26) 3amnneM B BH,D.e
(W [S 1 ! x, x) > II P 1x !1 2 ,
(W [S 2 ] (P2 + P 3 ) x, (P2 + P 3 ) x);;;;,, JI P 2 x 112. (27)
(W [S 3] P 3x, P 3x);;;;,, II P,1x IJ2.
,UaJJee, rrpe.11II0J1ara51, 'ITO MaTpHUbI S 2 , S a B HepaBeHCTBax (27) He o65133TeJihHO
6JIO'iHO-,ll.HaroHaJlbHOro BH}la, a M3T()HI.I,bl Pi, P2, Boo6me r0B0pH, He rrpoeK
UH0HHbie, .n.oKa)KeM c.,1e.n.yror.uy10 TeopeMy.
T e Ope Ma 2. ll!JC!IIb C!J!ll,ecmay10m mpu CUAUtempU<lflbte AIOTnflU U,bt
pa3M.epa n X n S; (t) E C1 (R), i = D, yooa/lemsopmow,ue HepaaeHcmea.M
(W [S 1 ] x, x);;;;,, II M 1 (t) x 112 - (28)
(W [S 2 ] (M 2 (t) + Ma (t )) x, (M 2 (t) + M 3 (t)) x);;;;,, !I M 2 (t) x 11 2 , (29)
(30)
2,i)e n x n-AtepHbte Atampw{bt M ; (t) E C0 (R) wi,re10m c,~eoy,ow,ue cooucmoa:
Ml (t) IM 2 (t) + M3 (t)] = [M2 (t) + M3 (t)] M1 (t) ,
det M (t) =I= 0 Vt ER, 11 M - 1 Ct) II ~ const < oo,
M (t) = Mi(t) + M 2 (t) + M 3 (t).
(3 1)
T ozoa npOU3800HOfl. KBOO pamuimor1 q]Op Mbl
VP (t, x) = p2+" (S1 (t) x, x) + p (S2 (t) x, x) + (S 3 (t) x, x) (32)
8 CUily cucmntbl (25), cOe A, - npOU38UJ1 bHO (/JUKCUpOGOHHOe oer1cmaum eAbHOe
no.10J1cumeflbHoe 1tucflo , 6yoem rto;w J1cume,1bHO onpeoeAeHHor1 npu oocmamo"
Ho 00,·1blllUX 3HQt{,eHUfl.X napa,Hempa p > 0.
,1 0 r< a 3 a T e Jl b CT Bo. noKa)KeM, qTo rrpH .lWCT3T0'lH0 60JlbUTHX 3Ha
'I elll-1>1.\ !1apaMeTpa a> 0 ,D.Jl51 MaTpHUbJ crS 2 (t) + S 1 (t) 13b1no: 1ttne-ro1 1-1epa
BeHcTB0
(W laS 2 + SI] (M2 (t) + M 3 (t)) x, (M 2 (t) + M 3 (t)) x) >
> a (a) II (M 2 (t) + M 3 (t)) x 112, a (a) > 0. 33)
11crroJJb3YH I-IepaBeI-ICTBa (29), (30) , HMeeM
((oW[S2 ] + W [SaD (M 2 + M 3)x, (/11 2 + M 3)x) = o (W[S 2 J(M 2 + M~)x,
(M 2 + Ma) x) + (W [S 3 1 M 2 x, M 2x) + 2 (W iSaJ M 2 x, M 1x) +
+ (W [Sal Max, Max> > a II M 2 x 11 2 - II W !S3] llo II H2x ,I~ -
I04 · ISSN 0041-6053. lhcp. MaT. ,teypH., 1992, T. 44. M I
- 2 11 wrs 111 II M xii II M 11 + 11 M 11 2 :::;:,, a- II W[SsJ !lo-II WfSsl Iii
3 0 2 3x 3x ,:;c- a-llW[S3J!lo+l X
X (II M2X 11 2 + II M3X 11 2 ) .
Ta:trnM o6pa30M, HepaBeHCTBO (33) BbinO.'JI-IHeTCH rrpTT CJie.LJ.yIOW.eM 3Ha tieHTTH
ex. (o):
r.11.e napaMeTp o y.n.oaJieTBOpHeT 1-1epaBe1-1crny
a> II W [SsJ llo + II W [SsJ 115·
Tenepb o6o3HatiHM
~ l ~
Sil (t) = a(u) [uS 2 (t) + S3 (t)!. M 2 (t) = M 2 (t) + M3 (t)
(34)
(35)
H 33MenIM, l!TO Bb!TIOJIIHIIOTC51 ycJIOB!Hl Teop eMbl l, rne B Katiec·rne MaTpHU•
.S 2, /112 HCITOJib3YIOTC51 COOTBeTCTBellI-10 M3TfJH!lbl s a, 1\12· Ilo3TOMY npOTT 3 BOJJ.
H3 51 B c TTJiy cTTcTeMbl (25) 1rnanparnq1-1oi'l rjJOpMbI V,, (t, x) = 11 (S 1 (t) x , x) +
+ (S O (t) x, x) 6y.LI.eT ITOJIO)!rnTeJJb HO onpene.JJe1moh np11 nocTaTOtiHO 60JibWOM
sHa'-leHTTTT napaMeTpa 11:
1'] > II w [Sa] llo II M-l 11E + II w f.Snl 11 6 II M-l 116 II M !16- (36)
BeJITTLIHHa a (u), onpeneJrne~-ra,i paBel!crnoM (34), c-rpeM11TcJ1 K 1/2 rrpn u -+
-+ + oo. Ilo3TOMY II S0 (t)[l~u, a cMnoBare.:Jb 110 , II If/ IS"/ llo = CJ npH
0 -+ 00. T a 1rnM o6pa30M, rrpaBa51 '-1 3CTb nep,rneI-ICTBa (36) 9hBHBaJi e HTHa a
npH 60.r1b LUHX u. rioJ1a r a,1 u = p, 1'J = p 2 +\ rne A = c o n s t > 0, np11 l.lOCTa
To'-!1-10 60JibWTTX 3Ha'-leHHHX p, O'-leBHLlHO, 6 y n yT BbifJ OJ1!1 5!TbC51 u e paBeHCTBa
(35) 11 (36). 3-ro TT y6e)K]lae-r Hae B cnpaBe)lJITTBOCTl:I Teop eMbl.
3 a Me '-I a H TT e 3. cf>HKCHfJOBa1rnoe 'll:ICJIO A> 0 B KB3,ZlfJ3Tl:ILIHOH cj)OpMe
{32) MO}KHO Bb1fo1paTb rrpOH3BOJ1bHO, 1-1arrpHMep BCerna MO)KHO ITO.JJO)KHTb A= l.
5oJiee Toro, 3TY cpopMy MO)KHO Bb161:1paTb B BH,ue V,, (t,x) = pn.,+,,, , (Si(t)x,
x) + p 1· • (S 2 (t) x , x) + (S3 (t) x, x), rne A; - rrpott3BOJibHO cjmKCHpona1-rnb1e no
JIO)KHTeJibHbi e IlOCT051HHble.
B KaLJeCTBe npttMepa, 11.rr.rr1ocTpttpy1ow.ero npHMe1-1em1e TeopeMbI '.2, pac
C.40TpttM CllCTeMy ypaBHeI-mi'l
(37) ,
' x3 = (th t) X 1 + X2 + (th t) x3•
Henocpe.!ICTBeH!IO MO)KHO npoBepHTb, '-!TO npOH3B0,ZlHaH KBaJJ.paTH'I.HOli cpopMbl
V O = X 1X 2 + X 1X 3 + X 2X 3 (38)
B CHJIY CHCTeMhl (37) yJJ.OBJieTBOp51eT HepaBeHCTBY
' 2 2 2 2
V0 = X1 + (cos2 t ) X2 + (sin2 t) x 3 ;>, x1•
KpoMe Toro, npoH3BOJJ.Ha51 KBanpaTl:l'-IHOH cpopMbl Vo = X2X3 B CHJIY CHCTeMlil
• • 2
X2 = - (th t) X2, X3 = X2 + (th t) X3 HMeeT BHJJ. Vo = X2 H, HaKOHeU, npOH3BOJJ.-
H35l cj)yH1<u1:11:1 v = (th t) x~ B cttJiy ypaBHeHH51 x 3 = (th t) x 3 y,aoBJJernopHeT He
pane1-rcrny ~ ;>, iSxt i5 = canst> 0. TaKHM o6pa3oM, Ha OCHOBaHHl:I TeopeMhr 2 ·
ISSN 0041-6053. Y,cp. MaT. :JteypH.., 1992, ,. 44, M J 105-
''IIOlKHO yTBeplK)l.aTb, 'ITO KBa)l.paTHt!Ha51 cpopMa
VP (t, x) = p 3 (x1x 2 + x1x3 + X 2X 3) + px2x3 + (th t) x~ (39)
npH .UOCTaTO'lHO 6oJiblllHX 3Ha'leHH51X napaMeTpa p > 0 6yJ1.eT HMeTb ITOJIO)KH
TeJibHO onpe)l.eJieHHYIO npoH3B0Jl.HYIO B CHJIY CHCTeMbl (37). TiocKOJibKY KBa)l.
paTH'!HaH cpopMa (38) HBJIHeTC51 HeBblpO,K,!leHHOH, TO 11 cpopMa (39) TaKiKe 6yJ1.eT
HeBblpOIB:.UeHHOH npH 60JibUJHX 3Ha'leHHHX napaMeTpa p > 0, CJie)I.OBaTeJihHO,
CHCTeMa (37) HBJIHeTCH 3KCITOHeHIJ,HaJibHO .UHXOTOMH'lHOH Ha BCeH OCH R.
B 3aKJIIO'le1rne ccpopMyJIHpyeM yTBeP*.UeHHe, 0606ma1ow.ee TeopeMy 2.
T eo p e Ma 3. flycmb cyw,ecmayem m cuMMempu'lH.btX Mampu4 pa3Me
pa n x n S 1 (t) E C1 (R), i = l~ tw;)l(aaf/. us KOmopwc ya00/le11UJopRem npu
<tJCex x E Rn HepaseH-CmoaM
(W [S 1 ] x, x) > II M 1 (t) x 11 2 ,
m rn
(W [S2J ( L M; (t)) X, ( L M1 (t)) x) > II M 2 (t) x 112 ,
i = 2 i=2
(W [Sm- il (Mm-1 (t) + Mm (t)) X, (Mrn-1 (t) + Mm (t)) x) > II M rn-1 (t) X 112,
(W [Sml Mm (t) X, Mm (t) x) > II Mm (f) X 112,
,zae n x n-MepHbte MampUl{bt M 1 (t)EC0 (R), i = l, m, ma,cue, 'tmoux cyMMa
M (t) = Ml (t) + ... + Mm (t) o6Aaaaern CAeay,ow,u,HU ceoucmeaMu:
<le t M (t) =I= 0 V- t ER,
II M-1 (t) II ~ const < oo,
Ml (t) M (t) = M (t) Ml (t) yt ER.
·T02oa npOU3800Hafl ,ceaapamU'tf-iOU cpopMbt
V p (t, x) = p3m-2 (S1 (t) x, x) + p3rn-3 (S 2 (t) X, x) + ... + p3 (Sm-2 (t) X, x) +
+ P (Sm-1 (t) X, X) + (Sm (t) X, X)
·ti GUAY CUCmeMbl X = A (f) X 6yaem nOAO;J/CUmeAbHO onpeaeAeHHOU npu oocma
l1W'tHO 60AbutUX 3/-la'teHUflX napaMernpa p > 0.
l. MumponoAbCKutl IO. A., CaAtatlAeHKO A. M., Kl/AUK B. JI. YlccJiellOBaHHSI llHXOTOMHH mrneii
HbIX CHCTeM llHcfxl>epeuuHaJibHblX ypasueuHii c nOMOll.lbIO cj:>yHKUHii Jfanyuosa.- KHes: HayK.
llYMKa, 1990.- 272 C.
·2. CaMoilAeHKO A. M., KyAuK B. JI. 0 perym1pHblX ua sceii OCH JIHHefrnhlx CHCTeMax .11Hcp¢epeu
UHaJibHblX ypasueuHii / / llHcfxl>epeuu. ypasneHHll.- 1990.- 26, N2 8.- C. 1338- 1343.
3. BaAeee I(. r., <PuHux r. C. IlocTpoeuue cjJyHKUHii Jfanyuosa.- Kues: HayK. llYMKa, 1981.-
412 c,
IloJ1y<1euo 01,02,91
0092-2
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| id | umjimathkievua-article-7800 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus |
| last_indexed | 2026-03-24T03:33:55Z |
| publishDate | 1992 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/94/3fd387020b2fbc689aabeac10fda7c94.pdf |
| spelling | umjimathkievua-article-78002023-09-14T12:03:56Z Concerning existence of the Lyapunov sign-variable functions for linear systems of differential equations К вопросу о существовании знакопеременных функций Ляпунова для линейных систем дифференциальных уравнений Samoilenko , A. M. Kulik , V. L. Самойленко, А. М. Кулик, В. Л. Исследуются вопросы о существовании знакопеременных функций Ляпунова для линейных систем дифференциальных уравнений с переменными коэффициентами. Досліджуються питання про існування знакозмінних функцій Ляпунова для лінійних систем диференціальних рівнянь із змінними коефіцієнтами. Institute of Mathematics, NAS of Ukraine 1992-02-04 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7800 Ukrains’kyi Matematychnyi Zhurnal; Vol. 44 No. 1 (1992); 100-106 Український математичний журнал; Том 44 № 1 (1992); 100-106 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/7800/9468 Copyright (c) 1992 A. M. Samoilenko, V. L. Kulik |
| spellingShingle | Samoilenko , A. M. Kulik , V. L. Самойленко, А. М. Кулик, В. Л. Concerning existence of the Lyapunov sign-variable functions for linear systems of differential equations |
| title | Concerning existence of the Lyapunov sign-variable functions for linear systems of differential equations |
| title_alt | К вопросу о существовании знакопеременных функций Ляпунова для линейных систем дифференциальных уравнений |
| title_full | Concerning existence of the Lyapunov sign-variable functions for linear systems of differential equations |
| title_fullStr | Concerning existence of the Lyapunov sign-variable functions for linear systems of differential equations |
| title_full_unstemmed | Concerning existence of the Lyapunov sign-variable functions for linear systems of differential equations |
| title_short | Concerning existence of the Lyapunov sign-variable functions for linear systems of differential equations |
| title_sort | concerning existence of the lyapunov sign-variable functions for linear systems of differential equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7800 |
| work_keys_str_mv | AT samoilenkoam concerningexistenceofthelyapunovsignvariablefunctionsforlinearsystemsofdifferentialequations AT kulikvl concerningexistenceofthelyapunovsignvariablefunctionsforlinearsystemsofdifferentialequations AT samojlenkoam concerningexistenceofthelyapunovsignvariablefunctionsforlinearsystemsofdifferentialequations AT kulikvl concerningexistenceofthelyapunovsignvariablefunctionsforlinearsystemsofdifferentialequations AT samoilenkoam kvoprosuosuŝestvovaniiznakoperemennyhfunkcijlâpunovadlâlinejnyhsistemdifferencialʹnyhuravnenij AT kulikvl kvoprosuosuŝestvovaniiznakoperemennyhfunkcijlâpunovadlâlinejnyhsistemdifferencialʹnyhuravnenij AT samojlenkoam kvoprosuosuŝestvovaniiznakoperemennyhfunkcijlâpunovadlâlinejnyhsistemdifferencialʹnyhuravnenij AT kulikvl kvoprosuosuŝestvovaniiznakoperemennyhfunkcijlâpunovadlâlinejnyhsistemdifferencialʹnyhuravnenij |