Asymptotic analysis of a singularly perturbed linear system with a singular limiting beam of matrices
С помощью методов возмущений и диаграмм Ньютона проводится асимптотический анализ общего решения линейной сингулярно возмущенной системы обыкновенных дифференциальных уравнений в случае вырождения матрицы при производной при стремлении малого параметра к нулю.  редполагается, что пучок...
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| Date: | 1992 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | Russian |
| Published: |
Institute of Mathematics, NAS of Ukraine
1992
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/7801 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512763205386240 |
|---|---|
| author | Yakovets , V. P. Яковец, В. П. |
| author_facet | Yakovets , V. P. Яковец, В. П. |
| author_sort | Yakovets , V. P. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-09-14T12:03:56Z |
| description | С помощью методов возмущений и диаграмм Ньютона проводится асимптотический анализ общего решения линейной сингулярно возмущенной системы обыкновенных дифференциальных уравнений в случае вырождения матрицы при производной при стремлении малого параметра к нулю.  редполагается, что пучок предельных матриц системы сингулярный и имеет по одному минимальному индексу для строк и столбцов. |
| first_indexed | 2026-03-24T03:33:57Z |
| format | Article |
| fulltext |
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n pH'leM B 0<pO+ 1 = 0. llepe3 e" 3;1ecb o6o3 I-1a 11e11 se1u-op, k -H Koop,1lJIIWTa KO
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108 JSSN 0041-6053. Y Kp. AtaT. :m:ypH., 1992, T. 44, M J
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u = ['J..H (A8 ) B - H () .. 0 ) f] u + <p (A0).
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ISSN 0041-6053. Y,cp. MaT. aicypH.., 1992, T. 44, M 1 109 ,
BHM BbipameHHe L (A, e) B BHlle qx>pMaJibHoro pa3JIO)KeI-l!rn
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TO
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.....,/
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,;...; .....,/
k = I k = I
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k k- j
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k-j
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i= O
Tor.n;a corJiac110 (21)
~ oo k oo 00 00 oo
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t::aiJ j-::a l k= l ,- .,......., J l,•-= i 1, _..., ,
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HL = i: i: Vk+i,i [J../J + f Vko·
U>rJI8CHO (22)
i = I k=O k=I
k
Vk+J./ = L (- I) k+iehiu ((HB);+i' (Hr\_;) ;!Jr, (24)
i=O
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B (t, e) H r (t, e). rio,ucTaBHB 3TH pa3JIO)l{eHHH B (24) H crpynnHpOBaB qJieH&
npH O,llHHaKOBblX CTeneHHX e, noJiyqHM
oo k ~
Vk+i ,i = LL (- 1/+'e:+htPl+i,k-dHB; Hr; ;tJ', (25)
<= 0 i=O
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,neHtt fr i MHO)KHTeJieii: BH,lla HBk, HBk, ... , HBk,· H j MHO)l{HTeJJeH Bh,Ua Hr. l
1 2 1
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2 /
(corJiaCHO pa3JIO)KeHHHM (2), (14) HH,UeKCbl k, - Ha6opbl ueJibIX HeoTpHUaTeJib-
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Vllo = i: (- Ile•P~ (Hr), (26)
s-0
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Hr ;2 , ... , Hr ill C HaTypaJibHbIMH HH,lleKC8MH, CYMM8 KOTOpblX paBHa S. f1epe
rpynnHp0B8B CJiaraeMbie B (25), OKOH"'laTeJJbHO IlO,llyqHM
oo [s/ h] ~
Vk+i,i = L L (- o*'s5Pt+1:,.-dHB; Hr)[Ji (21)
s= O i=O
((a] - ueJiaH qaCTb qHcJJa a).
rio,ucTaBHB (26), (27) B (23) H npHHHB BO BHHMaHHe, qTo Bb1pa)l{eHHe
Pt;7:k-t (H B; Hr) HMeeT CMbICJJ TOJib1<0 npH k- i ~ s - h;, T. e. np11 i ~
~ k ~ s + i - hi , a Bb1pm1<et111e P'J, (Hr) - np11 I ~ k ~ s, 6y,ueM HMeTb
~ oo oo [s/h] s+i-hi _
HL =LL L I (- I/+kP:+7~k-i (HB; Hr);!} ' p/Je•+
i= ! s=O i= O 1,=i
00 S
+LL (- I)kP1(Hr)e5 •
.s= lk= I
CneJiaB nepecTaHoBKY HH.UeKcoB, .UJIH onepaTOpa L (1., e) noJiyq11M cJie,ny~
paaJiome ~rne:
~ oo 00 [s/1!] s-hi
L('A, e) =LL L L (- IlN+Z'.1(HB; Hr);/)i ['Ak] es +
k= I s=O i= O i= O
00 S
+LL (- 1)1P j (Hr)e•,
s= l j = l
ISSN 0041-6053. YKp. Mar. :HCypn., 1992, T. 44, M 1
pi (Hf) TeM, qTQ B HHX nepBbie MHO)KHTeJIH BO Bcex CJiaraeMbIX He CO,!I.ep)K3T H.
OTCIO,ll,a IIOJiyciaeM CJie,ny1ow.He cpopMy.TJbI ,ll,JI51 K03CpcjmutteHTOB pa3JIO)KE7
turn ( 19):
Los= L (- l)i(P i(Hf)cp,1jJ), S= 1,2, ... ,
j=J
[s/h l s-h i
(29)
(30}
Lks fA k] = LI <- l i{Pti=Z\{HB;Hf) cp ,1jJ)2Ji[Ak], {31)
i=O i=O
k, s = 1, 2, ... .
4 . I1 c c JI e ,!I. o B a H H e y p a B H e H H H p a 3 B e T B JI e H 11 51 • llmi
o npe1IeJ1 e u1rn <JlYHKUHH ' -o (t) H BJ,l)J:3 pa3JIO)Kel-! H51 cpyHKUHH A (t, e) no E HCnOJT b-
3yeM MeT0ll n11arp aMM I-lbIOTOI-Ia [ 15], 110CT3BHB B COOTBeTCTBHe Ka)KllOMY 0T
JTHllf-lOM Y o T 11 yJI51 1<03cjicjmu11e1ny L,,., ypaBiremrn paaneT BJieHH51 (18) TOLIKY
(k; s) B rrp HMoyroJibHOH CHCTeMe JW0j)llI!HaT xOy. flpH 3TOM , noCKOJlbKY Hae HH
Tepecy,or MaJibre pewe1-11rn y paB1-1er-1~151 pa 3BeTBJTeHHH , 6y,!I.eM paccMaTpHBaTb
TOJibK0 yob!B3 !0LIJ,H e y<IaCTKH CO0TBeTCTB YI0l.llHX m-iarpaMM . 1Vrn 3Toro pacCMOT
pHM He KOTOp ble cno(1CT Ba K03lpcjrnuHel-!TOB ypanr-I e HH51 paaneTBJieHH51 (1 8) .
y q HTblB351 (12), H3 (29) HMe eM
L1io (AJ = 0, /? = 0, 1, ... . (32)
C.r:1enoBaTeJ1bHO, Ha OCH a6CUHCC B CO0TBeTCTBYIOW.H X ,ll.HarpaMMax ,ll.JlH ypaBHemrn
( 18) TOLfKH OTCYTCTBYI0T ,
JI e M M a . CnpaseoAuao paseHc11180
s
c/ Los (1. 0 )/iJ1ci = Id ) (- l / (Pt; (H B; 1-Jf) cp, 'ljJ), k, s = 1, 2, ... . (33)
,.;..J
1I O I{ a 3 a T e JI b C T B O paBe HCTBa (33) npoBe,ll.eM MeT0,ll.OM MaTeMaTHLfeC0
K OH mrnyKUHH cHaciaJJa no s, a aaTeM no k.
IloKa}KeM , cirn
iJLos O,o)liJA.o = ~ (- 1 i (PL (H B; Hr) cp, 1jl), s = 1, 2, ... . (34)
i=O
T a K KaK cor JiaCH0 (30) Loi = -(r I, cp, 1jl), TO, HCIIOJ1b3Y51 COOTI-IOW e HH51 (8),
{ 15), 6yneM HM eTb
I
iJ L01liJ'A() = ((Bl -1\HBo - B0Hr1) <p, 'ljJ) = L (- li <PL (f-JB; HI') cp, 1jl),
i=tl
Clle,!I.onaTeJJbHO, npH s = l cpopMyJJa (33) cnp aBe)lJJ HBa . 1Ionycn1M, ciTo oHa
<:npaBe)IJI H Ba nptt s < k. Torna, nocK0JJbKY
TO
k-t
iJLo,k-1 ~ ~ ~ k
iJ1. + Lo,k- iHB, + B,HLo,k-i = ,£.J (- llP1,? (HB; Hf),
0 ___ i=O
i=l,k-1.
Y'1HTblB351 (35). (8), (9) H 0tJeBH)l,HOe paBeHCTB0
k+I-j
P1 (Hr) = ~ PJ:::t (I-Ir) Hr,,
i=I
(35)
I 12 ,i 'rrSSN 0041-6053, Y,w Mar. rllCYPH, , 1992, r, 44. M 1
tiy .n.eM HMeTb
k 1,+ t-J
L0,. = -(f,.<p, ,p) + L L (- Ii (Pt=f (Hf) Hf;(P , 'Ii)= - (f1/P, 'lj)) +
i =2 i= l
k-1 k-i k-l ~
+ L L (- l);+l (PJ-i (Hr) Hf;q>, 'Ii)= - (r,,cp, ,p) - L (Lo,k-iHl';cp, 'lj>),
l= l j= l i=l
k- l ~
~ (( olo.k-i iJLoklo')..o = (B1i<r>, 'll)- ((f1iHBo + BoHrh) q>, '¢) - ,t... -~ +
i = l
k-l k-t
= (B,,cp , '¢)- ((f,,HBo + B.Hfh) (J), cp)- L L (- l)jPt;' (HB; Hf) X
i = l j=O
k -1 k-i k-1 k-i
x Hf;q>, 'Ii)+ L L (- 1/ (PJ-i (Hf) ff B;rp, '¢)- L L (- If (PJ-\Hf) x
i=I i = l i= I j=l
X HI';H B 0rp , 'lj)) = (B1<<P, 'IJ)-((f,,HB0 + B 0Hfh) <p, 'ii)+
k~I k- j k-2 k-j-1
+ L L c- 1/+ 1 (Pt°"/ (HB; Hr) Hr,<p, 'Ii)+ L L c- 1);+ 1 cPJ+i' (Hf) x
j= l i = I j=O i= l
k-1 k- l
X H B ;<p, ,P) + L (- 1);+1 (PJ+i(Hf) H B.cp, 'Ii)- L (P~-1(H B) Hr;(f), 'Ii)=
i = I l=I
k-2 k-j
= (B,,<p, 'll)- ((fhff Bo+ BoHr,.) <p, 'Ii)+ L (- l)i+I [L (Pf-:i' (H B; Hr) X
j = I i , I
k-j-t k -1
X Hf;rp, 'Ii) + L (PJ+t (Hf) H B;<p, t)] - L (Pf-i (Hf) H B;cp, 'lJ)-
l =O i=l
k-1
- L (P/- 1 (H B) Hf;cp, 'Ii)+ (- 1)1' [(P: (Hf) H B 0<p, 'P) +
i=I
◊TC!Olla, rrpHHHMaH BO BHHMaHHe, lffQ
k-j k-j-1
I: Pti'<HB; Hf)Hr, + L PJ+f (Hf}HB, = Pti+1 (HB; Hr)1
i=l i=O
·OTKylla CJienyeT, 'ITO cpopMyJia (33) cnpaBe,ll.JIHBa rrpH s = ll, a 3Ha'IHT, OHa
•cnpaBe.n.mrna rrpH JII060M k.
JSSN 0041-6053. Yttp. Mar. xypH., 1992, r. 44, M l 8-1·812 113
lloKa)KeM Tenepb tj)OpMyJiy
0~ i: (- l)i (Pti (H B; /-Ir) cp, 'lji) =(fl+ 1) ± (- 1/ (P1,+ 1,i (HB; Hf) rp, '\j)).
o i=O j=O
(36)
Ee cnpase.QJIHBOCTb npH fl = 0 .noKa3aHa BbJUie. D.onycTHM, lJTO 01-1a cnpaBe.U.
JIHBa flf)J,f fl < r. BocnOJib30B3BWHCb COOTHOWeHHeM
s-j s-i+I
P',,, i<HB; Hr) = 1 Pt-=-.\.j(HB; Hr)HB; + L Pt-:/-1(HB ; Hf)Hri
i=O i = l
II BBe)]JI .llJlfl COI<paw,e1m51 33 flHCH o603H3l!eHHe
_L (- 1/Pt ,j(HB; Hr)= Rh.s•
i=O
6y).IeM HMeTb
, s-i s s-i+I
R,. = L ~)- 1/P:=L(HB; Hf)HB i + L L (- 1/P;-:/-i(HB; Hf)Hf;=
i=O i=O j= I i= I
s
= L Rr-1, s- iHB; - L Rr,s-tHr;.
i = O i=I
Toma , YIIHTbIBaH (8), (9), noJiyqaeM
s
O~o (Rrs(J), 'IJ) = L (( oRra{:s-i + Rr-1, s- tHBo+ BoHR&- 1,s-i)HB;,(p ,'IJ)+
i = U
s '
+ L (Rr-1 ,s-iHBiHBoCP,'IJ)- L(( 0~~:-i +Rr. s-JiBo+BijHR,, s--1) X
i=O i = I
X Hf;(J), ,ii)- f (Rr, ,- JlfJIBocp , 'ljJ) + f (R , ,s-iHB;ep, 1jJ) =
i= I i= I
:,. 5 ~
=r (( ~ Rr,s-;HB1- _L Rr+l,s-Jif1)<r, 'lji) +(( I R r-1,s-iHBi -
i= O i=l i = ll
s s s
- L Rr, s-tHf;)HBo(J), ,ii)+ (( I Rr,s-diB; - L Rr+l,s-iHr;)(J), '\j)) =
l = l i=l i=l
s
= r (Rr+l, s(JJ, 1jJ) + (( L Rr, s-tHB;- L Rr+l.s-iHf1) cp, 'ljl) =
i=O i=I
oTI<yna cJie.n,yeT, l!TO tj:JopMyJia (36) cnpaBe.llJIHBa npn fl = r. Cne.noBaTeJJhIIO,
oHa cnpaBe.n,mrna npH JI1060M k. 113 tj)OPMYJibI (36) J1en <o noJ1ytJaeM paBeHCTBO
(33). J1eMMa .noKa3aHa.
C yIIeTOM (33) q>opMyJia (31) 11peo6pa3yeTCSI K BHAY
[s/hl
Lits p/] = L -(/l_+_I -:i),.,..! - k,s= 1, 2, ... , (37 )
i = U
OTKy.n.a, B qaOTHOCTH, CJie.nyeT, lJTO np11 S ~ h Bblpa)Kemfe L,,_. He co.n,ep)l<HT
U4 ,ISSN 0041-6053. Y,cp. Mar. 31eypn., 1992, r. 44, M I
orreparnpa ,nmjxpepeHLLHpoBa HHH .0:
L .= ---i-- akL,os, s= D , k= l ,2 , ....
n 1?! OAo
(38)
nocKOJibKY K03cjxjJHUHel-ITbl Ll< s ypaBHeHH5! p a3BeTBJi eHH 51 (1 8) 3 3BHC51T
OT Ao, TO OT Ao 6y,neT 33BHCeTb H CTp y ,nypa COOTBeTCTBYIOll.lHX )U1arpaMM H bIO
TOHa . P acCMOTpHM Bblj)3.t!i.e~Il1e Lot (),o) - Y4HTb!BaH (4), (6) , (10), (11 ) , (15),
HMeeM
p . .- Q - · .-+ A~01,h (B0HB 0 cp (A0), tp(t,0)) = A0 ( L A~B 1 (GA 0 /-' cp, )~ 'A6 (G*A~/- 11jl)-
i=O j=O
f) ,_ Q _, 11
- ( ~ A1 A (GA )p-i ,_, , i (G*A* )Q-i ) - - ~., M O n--,
.:_. 0 1 0 (p, _2,. /1,Q O Cp - .:_. il /1,Q •
(39)
i=O j=O i= O
r,ne
CHMBOJIOM PLi (GA; GB) 3,/1,eCh o6o3Ha lfeHo Bb1pa,1<eH 11e, KOTopoe cocTaB
JrneTcH no TaKOMY )Ke npaBHJiy , lJTO 11 BBe)lem1oe BbIUJe Bb1pm1<e1rne Pf,i(HB ;
Hr), c TeM JIHll.lh ovrn ,rneM, lJTO B )lammM Bb1pa)KeHirn 11c no.r1b3y10-rcH 1<03cll
qmu11ettTb1 pa3,1m1,eH 1111
A (t, c:) = L c:k A,, (t) , B (t, e) = L c-"B" (t)
k~O k ~ !
11 MaTpnua G BMeCTO H. T o me OTHOCHTCH 11 K 0603 Ha t1 eumo Pl (GA). 3)lecb
YlfTeHo TaKme c JieJLy1oruee J1er1<0 npoBep HeMoe cuof'1cTBo MaTpHUbI A 0 : (GA 0)i =
= 0 np11 i > m a x {p + 1, q}, (A 0G) i = 0 np11 j > m a x {p, q + I}.
T a KHM o6pa 3oM, L01 (1,0) npenCTa BJ15leT co6of'1 Ml!Orolwei-1 OTHOCHTeJJhHO
"-o, CTeneHb KOToporo He npeBh1maeT n. nycTb A0 (t) - 1<0p e1-1b ypam1e m rn
L 01 (A0 ) = 0 ( 4 1)
KJJ3THOCTH r ~ n . n onyCTHM, tITO OH YJLOB,1eTBOJJHeT y CJIOBH!O
L0 2 (A0 ) = ((l\H (A0 ) r 1 - r 2 ) cp (A0 ), 1j., (A0 )) =f= 0 V t E [t 0 ; T]. ( 42)
Tor)la corJiacHo (38)
(43)
H COOTBeTCTBYIOLU3 5l n11arpaMMa 1-l b !OTOHa rrp encTaBJI HeT COOOH 0Tpe30K , co
emmmoI.IlHH T04KH (0; 2) H (r; I) . noc KOJ1b KY TaHreHC yrna H3KJlOHa 3TOro OT
p e3 Ka !{ OTJJHUaTeJibHOM Y n a npaBJi e H1110 OCH a6cu11cc paBe !-l 1/r, TO cor:1aC HO
MeTOllY n 11 a rp aMM 1-l b IOTOHa 1151 cj1 y mmm1 A (t, e) H BeI<TOJJ u (t, e) 6 y ny-r pa3 -
Jia r a T bC51 B rjJOpMaJ[bf-Ibl e JJ5lllbl no CT e n e ll H M µ = e l /r H T a 101 x pa3JJO:>Kel·IHi:'i MO}l(
HO TIOCTJJOHTb poBHO r. flpM 3TOM nepBbir1 K03cjxjmuHel·IT A1 (t) pa3JIO.t!{e!U!H
A (t, e) y )l_OBJieTBOpHeT ypaB He l-lHIO
L02 (11.0 ) + Lr1 (A0 ) A; = 0, (44)
OTKyna H3XOllHM r pa3JJHLJHblX OTJlfll!HbIX OT HYJ!fl cf)YHKUl-111 "-1 (t):
Ai{f) = ' VI z::g:; /exp[! (arg(- t:~~:;)+2n(j-l))]. l=G.
(45)
nocJJenyromne K03cjx p11u11eHTbl pa3JJO:>KeHHH A (t, c:) onpe)leJIHM He rrocpe,U
CTBer11-10 113 yp a B1-1emrn pa,lBeTBJJeHHH (18), HCITOJ!b3YH MeTO)l Heonpe)leJieHHbiX
JSSN 0041-6053. :t,w Mai. 3/CypH,., 1992, T, 44, M I 115
x o 3'.jxp HUHe1nos . LlJrn 9Tor o noncTaBHM cj1opMaJib 1-JbIH pHJl
"A (t, e) = I µ k11," (t) (46)
k = I
B ypaBHe 1-rne (18). Crpy nnHpOB3 B YJi e Hbl n pH OJlHH3KOBbIX CTe rreHHX f::l, IlOJiy
•rnM
oo [i/ r] i-rs
L ( 11, , e) = \.---, ~ '""' L [Pi-rs (11,) ·1 µ 1 = 0 L L...a .;..J hs k ,
i= O s= O k = O
r ne CHMBOJIOM P!, (11,) 0003HatJe11a cyMMa BCeB03M0)KHblX n pOH3Bene1rn11 h !1,' I-IQ
)KJ,fTe.'leH 11, , , ' 11, i , , .•. ' 11, ,,, C Ha TypaJibHbl MH HH)leI<CaMI I . c yMMa KOTOp b! X paBHa
;, a o nepaTOpb! f/)j B COCTaBe L hs neik TBYIOT Ha K3)K)lOe CJiaraeMoe Bb!p3 )i(e
mrn Pt-rs 0-) B OTLl eJi b HOCTH 110 T3KOMY )i(e npaBHJJy' Kell( Ha , / B qJo pMyJ1e
(3 i ). CTp u 3TOM no onpene:1e Hmo 110.r1a r aeM L0 8 1P~ ( 11,)] = L08 , s = 0 , l, ... .
ripHpaB HHB3 H 3necb 1<03qJqJHUHeJ-ITbl npH o n111-1aKOBbIX CTeneHHX µ, IlOJiy
'IBeM CHCTeMy ypaBHeHHH
li/ r ] i-r,
"' '""' L !Pi-rs (11,)] - 0 L, ,:;..i hs k - , i= O, I, ...• (47)
s= O i<=f'
Y •!! ITb!BclH (32), (43) , npHxonHM K BbIBOLlY, lJTO npH i < 2r ypaBJ-JeHHH (47)
Sl RJlHIOTCH TO)i<necTB3MH, a npH i = 2r IlOJiyYHM y p a BI-JeHJ-Ie (44), H3 KOTOporo
o npeneJI HeTcH 11, 1 (t) . llon y cTHM, lJTO c!)YH KUHH 11,; (t ) H3Becrn b1 npH i < m. T o r
JJ.a nm1 onpen enem rn 11, m (t) HCITOJib3 y eM yp aBHe HH e (47) npu i = 2r + m - 1,
KOTOpoe C ytleTOM (32), (43) 3 311HCbIBaeTC5l B BHJJ,e
r"A~-l"Amlr1 + gm (t) = 0,
r2r+ m-11
r 2r+ m-1-rr
gm (t) = L I Lhs w f+m-l-PS (A)]+
<=2 k=O
- H3BeCTHoe Bb!pameHHe, B KOTOpOM
p;.+ m- l ( A), KOTOpaH He coneplliHT '-m·
'1>YHKUH5l Am ( t):
p~+rn-1 ("A) - Ta YaCTb Bb;pamemm
0TC]Oll3 OJl,HOBH3YHO or::peneJl5leTC5l
"A (t) - - gm (t)
m - r"A~-llri ("Ao)
(48)
Jl;rn orrpeneJieHHH K03 cj1<p HUHeHTOB pa3JI0)i(el-JH51 BeKTOpa U (t, e) BOCITOJJb-
8YeMCH cpopMyJJof1 (17) . YLJHTbIBaH (20), (21) , noJJyYaeM
00 00
U (t, e) = cp (A0 ) + H (11,0 ) Lcp ( 11,0 ) = cp + L L H Lhs [A kl e 5cp,
S=O k= O
~
rne Lk• - Ko9cp¢Hu11e1-1Tb1 pasJiomeHHH (28) ((L"8cp, 'ljl) = L ,..).
TToncrnBHB c,on a noCTpoeH1-1oe pasJimKe1me ( 46) JlJIH '>., (t . e) H crpynm1per
BaB YJieHbl npH OL!HI-l3KOBblX CTenem1x µ, 6yneM HMeTb
oo oo [i/r] i-rs _
U (t; e) = I µ'u1 (t) = (j) + I I I H (,•.o) Lhs w~-r,s (11,)I cpµi,
i=O i= ! s=O k=O
'1 6 ISSN 0041-6053. YKp. MaT. (JICypH., 1992, T. 44. M I
OTKyJJ.a noJiyqaeM CJ1e}.1,y1ow.11e cj)OpMyJibl }.1,Jl51 K03 cjx )rnuH e IITOB pa3JIO)!{eHH51 BeK
Top a u (t, e) no CTene 1rnM µ:
I i / rJ i-r. ~
u0 (t)=<p(A0), u,(t)= LL H(A0)L,.. fP~-rs (A,lj(p, i= 1,2, .... (49)
S= O k = O
J1e rKO y6eJJ.HTbC51 , 4TO TT 0CTpoe HHbie T 3 1<HM o6pa30M pa3JJO)!{eHH51 }.1,JIH A (t, e}
H U (t , e) YJl.0BJieTBO p5110T CHCTeMe (1) B CMbI CJie paBe I-ICTBa c!)OpMaJibHbIX p51)].0B.
B p e 3 yJibTaTe Jl.0 K a 3 a1-1a cJie;1y10 1u,H1 Teop eMa.
T eo p e Ma l. E c11u ypasHei-1.ue (41) u,weem KopeHb A0 (t) KpamHocm u r,
OAfl Konw p ow fi 1JtnOAHflemcfl yc11oaue (42) , m o cuc111e,11a (I) UAteem r rpopAtallb
H',tX <tQCll2HbtX peuteHllU 81./0Q (12), eiJe c/J!JHKlfllfl A (t, e) u aeKmop 1.l (t, e) n p eiJ
Cr/Ul(//lfl!OtnCfl (J 8UOe pa3/lOJ/CeHllU n o cm eneHflA·t ~l = f, l / r ' KOJcjxjJu4ueHtnbl KOmo
p blX onpeoe/lfl!OtnCJl C no,1to114btO p e,cyp peHl71HbtX c/JOf)At!Jll ( 45)' ( 48)' ( 49).
OrMeTHM, 4TO ecJIH .'laHHbIH 1<ope 11b A0 (t) ypaBI1e1-rn51 (41) 11 e yJJ.on.1eTBO
p HeT YCJ10BHIO (42) (TO>KJ_(ecrne1-11-10 n o t), TO 1-1eo6XO,ll.IIMO npo nep llTb , YJl0B, IeT
BOp51eT Jll-1 OH ycJIOBHIO Lo; (Ao) =f= 0 npH HeK0TOp0M i > 2. EcJIH )1.Jlfl HeKOTO
poro m Loi ("-o) = 0 npH i < m, 1-1 0
(50)
TO y6b!B310W.Hl'i yqacT0K ,QH a rp a MMbl H b IOTOJ-l a J.lJI51 3T0ro l<O()Hfl 6yJJ.eT HMeTb
6oJiee CJIO>Kl-l blr1 BHLI. H MO)KeT C0CT0 51Tb 113 I-I eCK0JlbKHX :1ne1-1beB, TaK KaK Ha
,QHarpaMMe M0 ryT 1-13X0L1.HTbC5l T0 LJ KH npHMbIX y = s, s = 2, m - 1, eCJI H COOT
B e-rc·rny1ou.1,11 e 1<03:jicp11u11e1-rrb1 L,,., ft = 1, r - I, o-r.1111LJ HbI OT 1-1y.riH . B 3 TOM
CJiyciae [1 5 ] pa3 JIOiKe HHH ll.JI51 ), (t, e) H 1.l (t, e) GynyT CT()0HT bCH 0 T,l.l.eJJbl-1 0 ,l.l.JIH
Ka>Knoro 3Be 1-1 a .u.11arpaMMbI. TToKa3 a TeJ1 H eren e 1m napaMeTpa £ , no K0T0J)blM
CJieny eT Been, 3 T H pa3JJO)KeHH51 , paBIIHI0TCH Ta 11re 1-1caM yrJI0B H3KJI0Ha C00T
BeTCTBYIOU(HX :rne HbeB I{ 0TpHUaTeJibH0My 1ianpaBJie 1-11-110 OCH a6CUHCC . KoJIH
q ecTB0 p a 3 JIHlll-I bl X pa3 JIO)KeHHH B 3T0M cJi yLJae 6y,L1.e-r p a BI-10 Kp a nJ0CTH r nan
H0ro 1<op1rn Ao (t), fl 0CK0Jl bKY 3TOH K() 3 TH0CTH paB H a c yMMa )lJl HH npoe 1m .11ii
Bcex 3 Be HbeB }.lHarpaMMbl 1-1a 0Cb a6cm1cc. Cne,QyeT 0,l.1.1-131<0 HMeTb B BHJ.lY, ll'fO
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H3 aJire6paH4eCKHX ypan11e1 rntt, KaK npH Jl0I<a3aTe.'lbCTBe TeopeMbl I, a H3
JJ.Hcj:>:pep e 1-1u11a JibHbIX, rI0CK0JlbKY cor.riac H0 (37) I<03cj)cj:>HUH e l-!Tbl L,<S [Ak] npa
fl > h Mory-r cone p ,1<aTb one pa-rop nmj):j:le p e 11u11posaa1151 2J .
5. P e Ill e 11 11 H B To p o H r p y n n bl. k13 np0Bene1rnoro Bbillle a1-1aJIH-
3a CJien ye-r , 41'0 ecJIH ypaBHe 1-rne (41) HMeeT n KOpHe tt (c ytJeT0M HX KpaTI-!OCTH),
Ka>KJ.lblH 113 K0TOpb!X YJ.l0BJiernopHeT ycJI0BHIO (42) HJIH (50) npH HeK0T0pOM m
(CBOHM J.lJIH K3>K,l.l.0ro Kop,rn), TO M0>KH0 fl0CTp0HTb 11 pa3 JIHLJI-Ib!X JIHHettHO He -
33BHCHMbIX cj:>opMaJJbHbIX pemeHHH CHCTeMbl (I), K0T0()ble C0CT3BHT ee o6w.ee
cpopMaJibHOe p e m e 1rne.
YpaB1-1e1-rne (4 1) HMeeT n KOpHeH, eCJIH K03cjx)mu11e1-1T npH Ai B HeM 0TJIHlieH
OT HYJIH, T. e.
{5 1)
JlerKO y6e,l.l.HTbCH , LJT0 9T0 ycJI0BHe 06ecneLJ1maeT Heoco6eHHOCTb MaTpHUbl
B (t, s) np11 Jl.0CTaTOLJH0 MaJibIX e > 0, nocK0JibKY ecJIH MaTp11uy B (t , e) p ac
cMaTpHBaTb Kai< rIOJIHI-IOMH3Jlbl-I b lH ny40K MaTpHU 0TI-I0CHTeJlbl-10 e, TO 113 Bbr
IT0JIHeJ-IHH ycJI0BHH (5 1) CJienyeT ero peryJIHpl-l0CTb [17].
.llonycTHM, 'ITO ycJI0BHe (5 1) T0>K,l.l.eCTBeI-JJ-10 He BblIT0JIHHeTCH H, KpoMe
K09:jxpHUHeHTa Mo1 (t), B ypaBHeHHH (41) paBHbl 1-IYJIIO Bee K03cj)cjmuHeHTbl
Af il (t) npH i < s. Torna 3T0 ypan1-1e1-111e HMeeT 11 - S KOpHei:i H , CJie.il.0BaTe.'I bHO,
pemeHH H BH}.la (12) M0>KH0 rI0CT()0HTb T0JibK0 n - s. B 3T0M CJ1yc1ae B03HHKaer
rrpo6Ji eMa Il0HCKa 0CTaJibI-lblX pewe HHH CHCTeMbl (1). B Jl3Jlb!-IettWeM peweHHff
BHJla (1 2) 6yneM I-133blB3Tb pewe1-JHHMH nepBOH rpynnhI, a HCK0MbJe 1-1e)].0CTaIO
m11 e peweHHH - peweHHHMH BTOpOH rpynnbJ.
CneAyH l9-l l J, 3TH pewe1-111H 6yJJ.eM crpoHTb B BHJl.e
I
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ll_Z
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noCTynHM TaJrnM )Ke o6pa30M , KaK H n p n IICCJi e,UOBaJ-JHH peweHHH BH.Lla (1 2) .
no:rcTaBJ!51H (52) B CHCTeMy (1) , IlOJiytlaeM 3cl) l3'1Y O B03My1ue1-1H H Hy.'lenoro
COOCTBE'I IHOfO 311a 11e1IH 51 If COOTBeTCTBy101uero co6crnem-10ro BeJnopa JIHHei1-
HOro onepaTopa Bn OTflOCllTe.Yl bHO JIHl·Iett1-JOro 011ep aropa Ao:
IB 0 + B (t, e)]v (t, 1:-) = s(t, c,)l((t, e)v(t, e),
r .ue
B . (t) = B. (t) , s = 1, 2, ... ;
(54 )
K s (t) = As(!) - B ,-liD, s = 0, I , ... . (55 )
YpaBHeHHe p a3BeTBJieHHH ,U.'IH 3TOH 33.11.3 1-111 HMCCT BH.Ll
K aK II B npe.ub1JJ.y1UeM cJ1yqae, n pe.ucTaBHM onepaTop1me Bbipa}Keune B BHJJ.e
cjJOpl\laJ!bHOfO pa3JJO}Ketnrn
M (s, e) = L Mhs lsk] e5
, (57)
k,s;;.o
r )le k - cyMMapHaH CTeneJ-lb cjiyHKUHH £, co.uep>KatUeHCH B 3aIIHCH oneparnp
HOJ1 cj)yHKUHH M hs [1;kJ. l1MeeM
GM = i (1;GJ( - GB)".
k= I
H cnoJib3YH cooTHomeHHe
II npoBe,UH HHJJ.YKUHIO no /,, noJiytraeM cJ1eny1muy 10 cpopMyJiy:
1, s-1 k-s
(sGK - GB)k = (- 1/ (GBl + 2 ) L (- l) i+s-J-ke1"a ((GB)1;
s= I i= O ;-~o
(59)
r ,ue 3aBTTCHMOCTb a ((GB);; (GB)j; (GK)s-1-i r1;•- 11 OT £ onpene.1rneTC5! no ra
KOM )' npaBn,1y. Ka)l<i!OMY cJiaraeMoMy onepaTopHoro Bbipa>KeHHH o ((GB);;
(GH )i : (GK)s- 1- ;) orne11aeT npo11s1:1e.ne1 ]1,fe i MHO)l<HTeJieH D 's , KOTOf'hTe cooT
BeTcTBy10-r 1111-1mKHTeJ15JM B , 11 s - I - i MHm1rnTe.nett s, cooTBeTcTBYIDlUHX
MHO)l{HTCJIHM /(. IlopH.LlOK pa3MemeHH5l MHO)l<HTeJICH D1; II £ TaKOB }Ke, Kai( H
rrop nnoK pa3Mem emrn Ml·JO)!{HTeJreii B II /( B COOTBeTCTBYIOll.leM CJJ a raeMOM.
Dp,-1 3TOM onepaTop nwjxj1epe1-11111pona1-1H 5I D ,uei'\c'rnyeT Ha nee Bbipa}KeHHe, Ha
xo;umteecH cnpana nepe;i HIIM. I-lanpHMep,
~ a ((GB)1 ; (GB)1 ; (G/()1) ls,2 J = GBGKGBD1; 2 + GBGBGJ(D~ 2 +
+ GBGBGK.D'¢, 2 + GBGKGB's,D's, + GKGBGB's,D's, + GKGBGB1;D's,.
no,ncTaBHB Bblpa>Ken1-1 e (59) B (58) II npone,nH TclKHe )!,e Bb!KJ13,i.1,KH, KaK H nptt
HCCJie,UoBa HHH onep aTO pl--!OfO BhIJ) 3)!(eHlrn L (!,. , E) , J(JJH K03cj;cjH,JUHeHTOB ypaB
Hemrn pa3BeTBJiemrn (56) noJI'.)"ll-!M cJ1e,ny1011LHe cj:JopMy.1b1:
M oo= 0,
118
(60 )
(61)
ISSN 0041-6053, YKp. Mar. :JJCypn., 1992, r. 44, N! 1
X (GB; GK; GB) lsk- ll sGKr,P1=t-i,-i, (GB);,~), k, s = 1, 2, . .. , (62)
rJI.e 33Bl-l.CHMOCTb onep a TOpHOH cpyHKUHH P{:k-1-i,j, (GB, GK, GB) [sk- ll OT S
on p ene.rrneTC51 TIO Ta1<0My )Ke np aBHJJY, CITO H B cj)OpMyJJe (59). CooTBeTCTBYIOI.I.l,He
nt,Ipa>I<eHH51 fsk- l] COCTaB.'IHIOTC51 11.JlH K cDi(,LLO fO CJJaraeMOfO B OT,LLeJJbHOCT11;
l lO!)HJlOK pa3Melll,eHlrn MHO,I<HTeJlett Ds 11 £ COOTBeTCTByeT TI0p511l.Ky pa3M eIUe
llH51 M HO)KHTe.r1eii B;, K; H tte 3aBHCl1T OT 11x HH.lle1<coB. Hanp11Me p,
P l. 1.1 (GB; GI(; GB) [s 2 1 = B0GK0GB1Ds 2 + B0GB1GK0Ds2 + KoGBoGB1sDs +
+ K0GB1GB 0sDs + BpB0GK0Ds 2 + B 1GK0GB0sDs.
TTpoaHaJJH3 HpyeM n oJJyYeH Hble cjJOpMyJJbl. TTpm-rnr.rnH BO BHHMa HHe (11),
33 KJIIOYaeM, 'ITO
M ,,0 = 0, I? = 0, 1, .... (63)
C.rie11.0 BaTeJ1bHO, Ha 11.11ar p a MMax HblOTOHa , COOTBeTCTBY!Olll.HX ypam1eHH!O p a3 -
BeTBJi e Hlrn (56), I<aK 11 B npe.llblll.YIUeM cJJyYae, OTCYTCTBYIOT TOYKH OCH a6CUHCC.
l13 cfxlPMYJJbI (62) HMeeM
M,,11sk1 = ccP1 (GK) - Pl. 1 cGK; GB>);,~> sk -
- 81,f, (P9,k-2 (GB; GK) GKo;• ~) [s''- 1]1;, I? = 1, 2, ...
(B1, 11 -CHMBOJJ K,pmreKepa). TaK KaK GB0 =B~B 0 =diag{l, ••• ,1,0}, TO
HC110Jlb3Y51 COOTHOWeHH51 ( 11 ), HeTpy 11.1-IO TIOK333Tb, 'ITO
KpoMe Toro , PL (GK; GB) co11.ep,1rnT TOJibKO /(0 = A0 , a see c .rrnraeMbre B
coc-rase Pk (GK), conep)KaIUHe B0D, paBHbI Hymo, nocKOJlbKY ue HMeIOT MHO·
)KHTeJ1e[1, 33BHC511Ul1X OT t. CJJe,LLoBaTeJibHO,
C paBHHBaH (64) c (39), (40), npttxo11.HM I< BbIBO.llY, l!TO K03cp<frnuHeHTbl Mn
ypaB1-1e1urn pa :rnernJieHHH (56) coBna.LJ.aioT c K03clxj)HUHeIITaMH np11 "''J-1 B ypas
He1rn11 ( 4 1).
3ToT BbIBOll. ,LLaeT peweHHe Bonpoca O fIOCTpoeHHH s peweHHH CHCTeMbl (1),
.LlOTIOJlIUIIOI.I.l,HX y,1<e fIOCTpoe 1-1Hble n - S p e w e HHH nepBOH rpynnbl. JlonycTHM.
'-ITO
(65)
Torna, ecJIH
M 11 (t) =0, M.1 (tJ,,bO VfE[t,; TJ, i = l, s-1, (66)
TO COOTBeTCTBy101.uaH 1l,Ha rp a MMa I-lblOTOHa ll.J151 ypaBHeHH51 pa3BeTBJieHH51
(56) rr p e11cTas11HeT co6ott oTpe3oK, coe.LJ.11Hm01UHH TOYKH (O; 2) H (s; I), Tattre1-1c
yrJi a H3KJIOH3 KOTOpo ro K OTPHU3TeJibl-lOMY HanpasJieHmO OCH a6CUHCC paBeH
1/s. C 1e110BaTeJ1bHO, cpyHI<UHH S (t, e ) H BeKTOp V (t, e) pa3JJaratOTC51 B PH1l.bl
no c-re ll t" HHM V = e115, 1-1 TaKHX pa3JIO)KeHHH 6y11.eT p aBHO s . TTpH 3TOM n epBbIH
K03.jx ji11w1e HT s1 (t) p a3JI0,1,e 1-11-151 .LJ.JIH s (t, e) y noBJieTBOpHeT ypasueHHIO
(67)
ISSN 0041-6053. :l,cp. MaT. 31Cypn., 1992, r. 44, M l 119
OTKy,11.a B CHJIY YCJIOBHH (65), (66) Haii.neM s pa3JIH'IH61X OTJIH'IHbIX OT HYJIH cpyHK·
U,HH S1 (t):
S1 (t) = s v·1 ::: I exp [ ~ (arg (- ::: ) + 2n (j- 1))1, l = [s. (68)
IlJIH onpelleJieHHH nocJieJJ,y1ow.Hx K03clxlrnuHeHTOB pa3JimKeHHH s (t, e).
KaK H npH llOCTpoeHHH pemeHHH nepBOH rpynnbI, BOCfIO.llb3YeMCH Henocpell
CTBemro ypaBHe1rneM pa3BeTBJieHHH (56), npHMernrn MeTO,l.l Heonpe,l.leJieHHblX
K03:jxjrnuHeHTOB. TTo.ncTaBHB cpopMaJibHbIH pHll
00
s (t, e) = L vks,, (t) (69)
k= I
B ypaa1-1em1e (56) H crpyrrnHpOBaB B HeM K03CpqrnuHeHTbl rrpH O.ll.HHaKOBblX cTe
rre1rnx v, a 3aTeM 11pHpaBH5lB HX 11ymo, noJiycIHM CHCTeMy ypaBHeHHH
[k/s] k-si L L Mn [PJ-si (s)l = 0, k = 0, 1, ... , (70)
i = O j=O
r .ne orreparop M ii .neiicrayer OTlleJibHO 1-1a Ka,K,11.oe cJiaraeMoe B cocTaBe
pJ-s' (S) TaKHM )I<e o6pa30M, KaK H Ha Si, He HapymaH YCTaHOBJieHI-IOro pa3-
MeLUeIHIH MHO)KHTeJie11 £1, l;~, ... . TTpH 3TOM llO orrpe1leJ1e1rn10 rroJiaraeM
M 0 ; [Pg (l;)l = M 0 ;, i = 0, 1, ... .
Cornac1-10 (65), (66), rrpH k < 2s ypaaHeHHH (70) Hi3JJ51IOTCH TO,K1lecrna,n1,
a rrpH k = 2s noJJyqaeM ypaaHeHHe (67). PewaH 3TH ypaaHemrn rrp1-1 fl > 2s,
HaX0,11.HM JII06b1e K03CjxjrnuHeI-ITbl pa3JJO)KeHHH £ (t , e). TaI<, ecJJH K03CjxjJHUHeHTbl
l;h (t) H3BeCTHbl rrpH k < m, TO , ITOJIO)KHB B (70) k = 2s + m - 1, HaH)leM
t (t) = - fm (!) (71)
<om s1;1-1MS1
[ 2s+m-1]
s 2s+m-1-sl
fm (i) = L L Mn [P7'+m-l-si (l;)] +
i=2 i=O
s+m-1 + L Mil [P j+m-1 (l;)] + p;+m-1 (l;) M s l
/=s+ l
- H3BeCTHoe Bblpa,Ke1me, B KOTOpOM p:+m-l (l;)- Ta 'IaCTb p :+m-l (s), KOTO
paH He COllep,KHT £m·
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TOpa V (t, i;) no crerreHHM V BOCITOJib3YeMCH cpopMyJIOH
v(t, e) = <p + GM (1;, e) <p.
TTo,11.craBHB CI0,11.a pa3JIO>KeHHH (57), (69) 11 crpymrnpoBaB Ko3tjxpmi:HeHTbl npH
O,ll.HH3KOBbIX CTerreHHX rrapaMeTpa v, ITOJIY'IHM CJ1e,11.y1ow.He cpopMyJibl,
[/</S I k-si _ _
Vg = cp, Uk (t) = I L GM}i [PJ-sl (s)] <p, k = I , 2, ••• • (72)
i=O j= O
B pe3yJihTaTe noJiytIHM cJienymw.y10 TeopeMy.
T e o p e M a 2. E c1tu 8btno1t1-uuomrn yc1toauH (65), (66) , mo cucmeJ,w ypae
n e1-1,w1 ( 1) uMeem s cpopMILllbHbt.x 1tacm1-tb1x pewe1-tuii auoa (52), zoe cpy1-tKqtm
l; (t, e) u eeKmop v (t , e) npeiJcmas1tR1omcR pa31to?1Ce1-tufl.MU no cmenen R.M v =
= i;l/s , KO,HprjJuu,ueH!llbl KO!llOpblX onpeiJe1t5l/O!llC5l C nOMOUl,b/O peKyppeH!llHblX
qJOPMYA (68) , (7 1), (72).
6. H e K o To p bl e B bl Bo .ll bl. BbmOJIHeHHe ycJIOBHH TeopeM I 11 2
U03BOJIHeT . .uo~po11Tb pOBHO n 'laCTHblX cpopMaJl bHblX pemeHHH CHCTeMbl (1),
120, JSSN 0041-6053. ~11,p. MaT. ~ypn., 1992, T. 44, M I
'-!aCTb 113 KOTOpbIX HMeeT BH,ll. (12), a BTopaH '-1 8CTb - (52). Mern,ll.aMH [6] MQ}KHO
noKa3aTb, l[TO 3TH p eweHHH JIHl!errno He3aBHCl1Mbl 11, CJie,ll.OBaTeJibHO , ,LJ.a!OT
o 6 w.ee cpopMaJibHOe pewe1111 e !.l3l·!IIOH CHCTeMbl.
3aMeTHM, l!TQ rJiaBI-IbIM ycJIOBHeM, o6ecrre'-111BlllHM cyrrteCTBOBaI-IHe s pe111 e
IIl1l1 arnpou rpynnhr, HBHJIOCb ycJToB~Ie (65). EeJIH c yw.eernyeT Ta KOH 1-10Mep
k, tJTO Mui (t) = 0 np11 i < k, 110 M 0,. (t) =I= 0 yt E lt0 ; Tl, TO np11 BblflO.TIHe111111
yeJIOBHH (66) TaK,Ke 6y;1eT CTpOHTbCH s p eme,mH BTOpOH rpyrrnbl. T O.!lbKO miar
p aMMa HblOTOHa 6y.neT HMeTb 60.11ee eJIO}KHblH B11.ll. H np1-1 k > h KO:xjx trnuHel!Tbl
p a3JIO,KeHHH ,LJ.JIH cj)yHKUH H ~ (t, i::) MoryT onpe.neJJHTbeH He 113 a.11re6paw1ee1rnx ,
a 113 JLHcpcj)epe11u11 aJ1bHb1x ypas11e1-11111.
KaK y,Ke OTMe 1JaJIOeb, OTJIHl!He OT tty.'IH XOTH 6bl 0,11.HOro 113 K03Cjx jrnu11eHTOB
M o,, (t) o6eenC'll!BaeT 1-1eoeo6e 111-1oeTb MaTpl'. l.l bl B (t, e) np11 ,~oeTaT04HO Ma.TlblX
e > 0, '-!TO 11 fl03 BOJ151eT noeTpOHTb n JIHH eHHO He3 aBHeHMblX peweHHH eHcTeMbl
(1). EeJJH oKa, KeTeH, 'ITO Bee M 0 i (t) == 0, TO MaTp11ua B (t, e) 6y,ll.eT Bb1pmK,LieH
Hoi-i H rrp11 BblfIOJlI-JeJ-IHH yeJJOBHH (66) peweHHH BTOJ)OH rpynrrbl 6y,ll.eT MeHbWe,
'-!eM S.
KoJiw1eeTBo :JTH X perneH11H onpe.neJIHeTeH noBe,ne~rneM Ko3 cj)ctmu11e1-n0B
Mn, j = 1, s- l, i = 2, 3, ... . A .HMem-10, eeJJH Mii = 0 npu j = 1, m,
m < s, i = 2, 3, . . . ' TO Ha eooTBeTC'rBYIOW.e l1 .n11arpaMMe H b!OTOI-Ia 6y,nyT
OTCYTCTBOBaTb TO'--IKl1 npHMblX X = i, i < m 11, 3 Ha'IHT, pe1.ue1mi1 6yJ.teT s - m.
Beero }Ke B 3 TOM CJ1y 1rne 6y,11.e-r n - m p eweHHH eHCTeMbl (!) BMeeTe e pellleHHHMlf
nepsott rpynrrh1.
EeM1 OKa, KeTCH, LITO L 01 U·-0) = 0 rrpu mo6hIX A0 , TO eornae110 (38) see
Ln (Ao) = 0 H Ha KOOJ))lHHaTHOH ITJIOeKOeTH , eOOTBeTeTBy1om,ei1 yp aBuernuo
pa3BeTBJieHHH (18), 6yAyT OTeyTeTBOBaTb TO'-!Kl-1 npm.mtt y = l . B :>TOM e,1y
'-!ae cj)yHKUHH A0 (t) 6y,ll.eT onpe.UeJIHTbCH 1-13 ypaBHeI-rnH L 02 (A0) = 0, KOTOpoe,
Ka K 1-1 ypas1-1eH11e (4 1), 6y.ue-r npe,ncTaBJ151Tb eo6oii u eJJoe aJire6pa1-1,1ee1<0e ypaB
He1-111e n-ii c-renerrn, rrp11t1eM Ko3cp(jmu1-1eHTaMH rrpH A~-i 6ynyT Bb1p a}l(eI-111H
M 2 i (t). Eem-1 H L 02 (A0 ) = 0 rrp1-1 mo6b1X A0 , TO .ll.JJH orrpe,LLeJiemrn A0 (t) 1-1enOJih
syeTeH ypaBHe,rne Loa (A0) = 0, 1-1 T. ,11. .
HaKoHeu, ecJI H oKa)KeTeH, tJTO Bee L 08 (A0) = 0, TO eornaeHo (37) 1-1 Bee
Lhs (A0) = 0. B 3TOM eJiytiae pernemrn nepBoH rpyrrrrbI 6y)lyT OTeyTeTBOB8Tb .
noeKOJlbKY K03cpcpl-lU11eHTbl M "' ypaB1-1e 1-11rn pa3BeTBJiemrn (56) eBH3aHbl C
LIL,, TO OHH TaK}Ke 6y,nyT paBHbl 1-1ymo, a 3 1-Ia'-IHT, 6yAyT OTeyTeTBOB3Tb H pe
rnemrn BTOj)OH rpyrrrrbl. MO}Kl-!0 ITOKa3aTb, '-ITO OTJil-llJHe OT HYJIH XOTH 6bl O)lHOro
K03(pcj}11UHeHT3 Loi (Ao) o6ecne'-IHBaeT peryJIHj)HOCTb CHeTeMbl (1) [ 18, 19], lJTO
l--1 rapaHTHpyeT c yrn.eeTBOBaHHe ee o6mero peme1-11151 B BH,ll.e JllJHeHHOli KOM6HIIa
Ul--ll--l k ~ n JJl1HeHHO J-Ie3aa1-1e11MbIX '--laeTHbIX pewe1-11-111. EeJJ1-1 }Ke Bee L 0 ; = 0,
TO e1-1eTeMa ( 1) e1rnryJJHpHa 1-1 ee 0611iee pewe1-11-1e 1-1Mee-r n pyryJO CTPYKTYPY·
B 3aKJIJO'-leH11e OTMeTHM, lJTO, 1-ICIT0Jlb3YH MeTO,llbl pa6oT [ 1, 6], MO}KHO
,ll.OKa33Tb, '--ITO Bee cpopM3JlbHbie peweu1151 e1-1creMbl (1), CTJ)OHll.J,HecH no orr1-1 ea1-1-
HOMY BblllJe aJJrOpl-lTMY' rrp11 onpe,ll.eJieHHblX yCJIOBI-IHX 11MeIOT acHMilTOT!-1'-leeKHit
xapaKTep B CMb)CJle [20].
I. <Pe/.l{eltKO C. <P., IJJKUAb /-I. {1., HUKOAeltKO JI. JI.. ACHMilTQTHqecKHe MeT0J{bl B reopHH JIHHeH
HblX 11Hcjxj>epeHUHaJ11,tt1,1x ypairne ,rniL- KHeB : HayK. llYMKa, 1966.- 252 c .
2. 8(1.9()8 8 . AcHMilTOTHqecKHe pa3JlQ)Ke HH51 pewe1-mH o6blKHOBeHHblX 1!HcjxpepeHUH3JlbHblX.
ypaBH e ttHH.- M. : MHp, 1968.- 464 c.
3. W Ki/UJ M. I . A.cHM ilTOTHqHi MeT0l!H B 11mj)epe1miaJibH HX piBHSIHHSIX.- KHIB : Bmua WK. ,
1971. - 226 C.
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| id | umjimathkievua-article-7801 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus |
| last_indexed | 2026-03-24T03:33:57Z |
| publishDate | 1992 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/7d/1be3ce801f243f7f8ab1c4f4c5c4297d.pdf |
| spelling | umjimathkievua-article-78012023-09-14T12:03:56Z Asymptotic analysis of a singularly perturbed linear system with a singular limiting beam of matrices Асимптотический анализ сингулярно возмущенной линейной системы с сингулярным предельным пучком матриц Yakovets , V. P. Яковец, В. П. С помощью методов возмущений и диаграмм Ньютона проводится асимптотический анализ общего решения линейной сингулярно возмущенной системы обыкновенных дифференциальных уравнений в случае вырождения матрицы при производной при стремлении малого параметра к нулю. &nbsp;редполагается, что пучок предельных матриц системы сингулярный и имеет по одному минимальному индексу для строк и столбцов. За допомогою методів збурень та діаграм Ньютона проводиться асимптотичний аналіз загального розв’язку лінійної сингулярно збуреної системи звичайних диференціальних рів-нянь у випадку виродження матриці при похідній, коли малий параметр прямує до нуля. Передбачається, що пучок граничних матриць системи сингулярний і має по одному мінімальному індексу для рядків і стовпців. Institute of Mathematics, NAS of Ukraine 1992-02-04 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7801 Ukrains’kyi Matematychnyi Zhurnal; Vol. 44 No. 1 (1992); 106-122 Український математичний журнал; Том 44 № 1 (1992); 106-122 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/7801/9469 Copyright (c) 1992 V. P. Yakovets |
| spellingShingle | Yakovets , V. P. Яковец, В. П. Asymptotic analysis of a singularly perturbed linear system with a singular limiting beam of matrices |
| title | Asymptotic analysis of a singularly perturbed linear system with a singular limiting beam of matrices |
| title_alt | Асимптотический анализ сингулярно возмущенной линейной системы с сингулярным предельным пучком матриц |
| title_full | Asymptotic analysis of a singularly perturbed linear system with a singular limiting beam of matrices |
| title_fullStr | Asymptotic analysis of a singularly perturbed linear system with a singular limiting beam of matrices |
| title_full_unstemmed | Asymptotic analysis of a singularly perturbed linear system with a singular limiting beam of matrices |
| title_short | Asymptotic analysis of a singularly perturbed linear system with a singular limiting beam of matrices |
| title_sort | asymptotic analysis of a singularly perturbed linear system with a singular limiting beam of matrices |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7801 |
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