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⇒∂2y = t22 y ⇒∂t 20022 ∂xx 2 ∂by.b=1∂x∂b∂x ∂b b=1 .= 0,.Ðåøåíèå:t33−13t .Ãëàâà III. Êðàåâûå çàäà÷è äëÿ ëèíåéíûõ ñèñòåì ïåðâîãîïîðÿäêà è ëèíåéíûõ óðàâíåíèé âûñîêîãî ïîðÿäêà13. Êðàåâûå çàäà÷è äëÿ ëèíåéíûõ óðàâíåíèé I ïîðÿäêà.Ìàòðèöà Ãðèíà. Ñîáñòâåííûå çíà÷åíèÿ0y = A(t)y + f (t),t ∈ [a, b].(1)Èíîãäà A = A(t, λ), ãäå λ - ïàðàìåòð (âîîáùå ãîâîðÿ,êîìïëåêñíûé). 9 ðàññìàòðèâàëàñü çàäà÷à Êîøè:0y (t) = A(t)y (t) + f (t), |t| ≤ T , 0 < T < ∞,y (0) = y0 ∈ C N (èëè R N ).Ãðàíè÷íûå óñëîâèÿ íà êîíöàõ a, b (÷èñëî óñëîâèé = N ïîðÿäêó ñèñòåìû (1))PNt=a:1 lij yj (a) = li , i = 1,...,N−M ,Pj=Nt=b:j=1 rij yj (a) = ri , i = 1,...,M .(2)Êðàåâûå çàäà÷è ëèíåéíûõ ñèñòåì 1îãî ïîðÿäêàÏåðåïèøåì êðàåâûå óñëîâèÿ (2) â äðóãîì âèäå:Ly (a) = l, L = (lij ), i = 1,...,N−M , j = 1,...,N ;Ry (a) = r , R = (rij ), i = 1,...,M , j = 1,...,N .0(2 ) 9 íàéäåíî îáùåå ðåøåíèå çàäà÷è Êîøè:y (t) = Y (t)Y −1 (0)y0 +tZY (t)Y −1 (t)f (τ )dτ ,0(3)ãäå Y (t) - ôóíäàìåíòàëüíàÿ ìàòðèöà ðåøåíèé óðàâíåíèÿ0y (t) = A(t)y (t).Åñëè íà÷àëüíûå äàííûå çàäàíû ïðè t = t0 , òî ðàâåíñòâî (3)ïðåîáðàçóåòñÿ â ñëåäóþùåå:y (t) = Y (t)Y −1 (t0 )y0 +ZtY (t)Y −1 (t)f (τ )dτ .0(3 )t0Íàéäåì ðåøåíèå (1), (2).Êðàåâûå çàäà÷è ëèíåéíûõ ñèñòåì 1îãî ïîðÿäêà0Ïîëàãàÿ â (3 ) t = b , t0 = a, y0 = y (a), ïîëó÷èìy (b) = Y (b)Y−1Z(a)y (a) +bY (b)Y −1 (τ )f (τ )dτ =a= Y (b)Y −1 (a)y (a) + g .
(4)0Òîãäà èç (2 ) èìååì:Ly (a) = l,RY (b)Y −1 (a)y (a) = r − Rg ,èëèLlRY (b)Y −1 (a) y (a) = r − Rg .|{z}| {z }(5)ϕK̃Ïåðåïèøåì (5) òàê. Ââåäåì âåêòîð z = Y −1 (a)y (a), òî åñòüy (a) = Y (a)z .Òîãäà èç (5) ïîëó÷àåì ñèñòåìó ëèíåéíûõ óðàâíåíèé:Kz = ϕ,0(5 )Êðàåâûå çàäà÷è ëèíåéíûõ ñèñòåì 1îãî ïîðÿäêàK=LY (a).RY (b)0Óðàâíåíèå (5 ) îäíîçíà÷íî ðàçðåøèìî äëÿ ëþáîãî ϕ ⇔detK 6= 0.(6)Çàìåòèì, ÷òî âñå ñêàçàííîå íå çàâèñèò îò âèäàôóíäàìåíòàëüíîé ìàòðèöû. ñàìîì äåëå, ïóñòüỸ (t) = Y (t)B, (detB 6= 0), LỸ (a)LY (a)BKỸ === KY B.RY (b)BR Ỹ (b)Âìåñòî detK ââåäåì ïàðàìåòð4=detK,detY (a)⇒4Y = 4Ỹ .Êðàåâûå çàäà÷è ëèíåéíûõ ñèñòåì 1îãî ïîðÿäêà(èç 3) d y10 −1y1f (t)=+ 11 0y2f2 (t)dt y2Ïðèìåð 1.íà [a, b].r11 y1 (b) + r12 y2 (b) = r1 ,l11 y1 (a) + l12 y2 (a) = l1 .Ôóíäàìåíòàëüíàÿ ìàòðèöà:cos(t − a) − sin(t − a)Y (t) =.sin(t − a) cos(t − a)4 = (l11 r12 − l12 r11 ) cos(b − a) − (l11 r11 + l12 r12 ) sin(b − a).Çíà÷èò, ñôîðìóëèðîâàííàÿ êðàåâàÿ çàäà÷à îäíîçíà÷íîðàçðåøèìà äëÿ ëþáûõ f1 (t), f2 (t), åñëètg(b − a) 6=l11 r12 − l12 r11.l11 r11 + l12 r12Êðàåâûå çàäà÷è ëèíåéíûõ ñèñòåì 1îãî ïîðÿäêàÇàìå÷àíèå 1.Ïðåäïîëàãàåòñÿ, ÷òî cos(b − a) 6= 0.
Åñëè cos(b − a) = 0, òîíåîáõîäèìîå óñëîâèå ðàçðåøèìîñòè âûãëÿäèò ñëåäóþùèìîáðàçîì: l11 r11 + l12 r22 6= 0.Ïóñòü ãðàíè÷íûå óñëîâèÿ (2) îäíîðîäíû, òî åñòü l = 0, r = 0.Ìàòðèöû ÃðèíàÏóñòü G (t, t0 ) ìàòðèöà ñî ñâîéñòâàìè:G (t, t0 ) = G0 (t, t0 ) + G1 (t, t0 ),G0 (t, t0 ) =0, a ≤ t < t0 ≤ b;Y (t)Y −1 (t0 ), a ≤ t0 < t ≤ b;(7)(8)G1 (t, t0 ) = Y (t)B,(ñìîòðè Ðèñ. 1)Êðàåâûå çàäà÷è ëèíåéíûõ ñèñòåì 1îãî ïîðÿäêàxba0abtÐèñ. 1âûáîð "ïîñòîÿííîé" ìàòðèöû B áóäåò ñäåëàí ïîçæå.ßñíî, ÷òî G0 (t, t0 ) íå çàâèñèò îò âûáîðà Y (t).Êðàåâûå çàäà÷è ëèíåéíûõ ñèñòåì 1îãî ïîðÿäêàÄàëåå íà èíòåðâàëàõ (a, t0 ), (t0 , b)dG0 (t, t0 ) = A(t)G0 (t, t0 ),dt(9)à ïðè t = t0 ïîëó÷àåì ñêà÷îê:G0 (t0 + 0, t0 ) − G0 (t0 − 0, t0 ) = IN .Îïðåäåëèì B:LG1 (a, t0 ) = −LG0 (a, t0 ) = 0,RG1 (b, t0 ) = −RG0 (b, t0 ) = −RY (b)Y −1 (t0 ),èëè(10)(11)0.−RY (b)Y −1 (t0KB =Åñëè 4 =6 0, òî B îäíîçíà÷íî íàõîäèòñÿ. ñèëó îïðåäåëåíèÿ, G1 (t, t0 ) - íåïðåðûâíî - äèôôåðåíöèðóåìàíà (a, b) ïî t , t0 .Óñëîâèÿ íà G (t, t0 ):Êðàåâûå çàäà÷è ëèíåéíûõ ñèñòåì 1îãî ïîðÿäêàdG (t, t0 ) = A(t)G (t, t0 ) íà (a, t0 ), (t0 , b);dtG (t0 + 0, t0 ) − G (t0 − 0, t0 ) = IN ,LG (a, t0 ) = 0,RG (b, t0 ) = 0.(I)(II)(III)Ìîæíî äîêàçàòü, ÷òî åñëè 4 =6 0, òî (I), (II), (III) îïðåäåëÿþòG (t, t0 ) îäíîçíà÷íî (óïðàæíåíèå).Îïðåäåëåíèå 1.Ìàòðèöà G (t, t0 ) ïîðÿäêà N ñ ãðàíè÷íûìè óñëîâèÿìè (I), (II),0(III) íàçûâàåòñÿ ìàòðèöåé Ãðèíà äëÿ y = A(t)y .Ïóñòü A = A(t, λ), R = R(λ), L = L(λ).
Çíà÷åíèÿ λ, ïðèêîòîðûõ 4(λ) = 0 - ñîáñòâåííûå çíà÷åíèÿ êðàåâîé çàäà÷è.Åñëè 4(λ0 ) = 0, òî ñèñòåìàK (λ0 )z = 0èìååò íåïðåðûâíîå ðåøåíèå.Êðàåâûå çàäà÷è ëèíåéíûõ ñèñòåì 1îãî ïîðÿäêàÏîëîæèìy (a) = Y (a, λ0 )z,ãäå Y (t, λ0 ) - ôóíäàìåíòàëüíàÿ ìàòðèöà ðåøåíèé. Òîãäàôóíêöèÿy (t) = Y (t, λ0 )Y −1 (a, λ0 )y (a)(12)(13)óäîâëåòâîðÿåò ñèñòåìå0y (t) = A(t, λ0 )y (t)è îäíîðîäíûì êðàåâûì óñëîâèÿìL(λ0 )y (a) = 0,R(λ0 )y (b) = 0.Èòàê, åñëè 4(λ0 ) = 0, òî îäíîðîäíàÿ ñèñòåìà ñ îäíîðîäíûìèêðàåâûìè óñëîâèÿìè èìååò íåíóëåâîå ðåøåíèå y (t) (ñìîòðèôîðìóëó (13)).ßñíî, ÷òî òàêèå ðåøåíèÿ îáðàçóþò ëèíåéíîå ïðîñòðàíñòâîN ≥ dim ≥ 1.Êðàåâûå çàäà÷è ëèíåéíûõ ñèñòåì 1îãî ïîðÿäêàÏóñòü çàäà÷à (1), (2) ñ îäíîðîäíûìè êðàåâûìè óñëîâèÿìèòàêîâà, ÷òî 4 =6 0, òî åñòü äîïóñêàåò ïîñòðîåíèå ìàòðèöûÃðèíà G (t, t0 ). Òîãäà ðåøåíèå ìîæíî çàïèñàòü òàê:Z by (t) =G (t, t0 )f (t0 )dt0 .(14)a0Ïðîâåðèì, ÷òî (14) äàåò ðåøåíèå y = A(t)y + f (t) íà (a, b).
Âñàìîì äåëå,Z bZ t0y (t) = {G (t, t0 )f (t0 )dt0 +G (t, t0 )f (t0 )dt0 } =atZ t=Gt f (t, t0 )dt + G (t + 0, t)f (t)+aZ b+Gt f (t, t0 )dt − G (t − 0, t)f (t).0tÄàëåå:ZLy (a) = LbG (a, t0 )f (t0 )dt0 =aÊðàåâûå çàäà÷è ëèíåéíûõ ñèñòåì 1îãî ïîðÿäêàZ=b[LG (a, t0 )]f (t0 )dt0 = 0.aRy (b) = 0.Ïóñòü L(λ), R(λ), A(t, λ) - àíàëèòè÷åñêèå ôóíêöèèêîìïëåêñíîãî ïàðàìåòðà λ, îïðåäåëåííûå äëÿ ëþáîãî λ 6= ∞,òî åñòü îíè öåëûå àíàëèòè÷åñêèå îò λ. Òîãäà 4(λ) - öåëàÿàíàëèòè÷åñêàÿ îò λ. ñàìîì äåëå, ôóíäàìåíòàëüíàÿ ìàòðèöà ðåøåíèé Y (t, λ):dY (t, λ) = A(t, λ)Y (t, λ),dtY (a, λ) = IN .Ðåøåíèå çàäà÷è Êîøè äèôôåðåíöèðóåìî ïî λ, òàê êàêêîýôôèöèåíòû îáëàäàþò ñâîéñòâîì äèôôåðåíöèðóåìîñòè.Íàïðèìåð,A(t, λ) = a0 (t) + λA1 (t) + ... + λk Ak (λ),R(λ) = R0 + λR1 + ... + λm Rm ,L(λ) = L0 + λL1 + ...
+ λn Ln ,Êðàåâûå çàäà÷è ëèíåéíûõ ñèñòåì 1îãî ïîðÿäêàAj (t), j = 0, ..., k - íåïðåðûâíû. ⇒ 4(λ) - öåëàÿ àíàëèòè÷åñêàÿ.Îòìåòèì îäíî âàæíîå ñâîéñòâî, êàñàþùååñÿ ðàñïîëîæåíèÿíóëåé ôóíêöèè 4(λ).Åñëè 4(λ) 6≡ 0, òî êîðíè 4(λ) = 0 ðàñïîëîæåíû äèñêðåòíî íàïëîñêîñòè, íå èìåþò íè îäíîé êîíå÷íîé ïðåäåëüíîé òî÷êè. ñàìîì äåëå, â ïðîòèâíîì ñëó÷àå èç òåîðåìû åäèíñòâåííîñòèñëåäóåò, ÷òî 4 ≡ 0.14. Îãðàíè÷åííûå ðåøåíèÿ ëèíåéíûõ íåîäíîðîäíûõñèñòåì ñ ïîñòîÿííûìè êîýèöèåíòàìè. Êðàåâûåóñëîâèÿ, óäîâëåòâîðÿþùèå óñëîâèþ ËîïàòèíñêîãîÄàíà ñèñòåìà′y= Ay + f (t ),ñ ïîñòîÿííîé ìàòðèöåéA.Ïóñòü âñå åå ñîáñòâåííûå ÷èñëàReτj (A)Ïóñòüf(t )6= 0,t∈ R1τj (A), j = 1,...,Nj(1)òàêîâû, ÷òî= 1,...,N .(2)- íåïðåðûâíàÿ âåêòîð - óíêöèÿ, ïðè÷åìkf (t )k ≤ F < ∞,t∈ R 1.Îêàçûâàåòñÿ, â ýòîì ñëó÷àå ñóùåñòâóåò è åäèíñòâåííî ðåøåíèåky (t )k ≤ Q < ∞,Qt∈ R 1.- onst.Äîêàæåì åäèíñòâåííîñòü ðåøåíèÿ çàäà÷è (1), (2).Îãðàíè÷åííûå ðåøåíèÿ.
Êðàåâûå óñëîâèÿIy (t )I,IIky (t )k ≤ QI ,II ≤ ∞.Ïóñòü ñóùåñòâóþòèyII (t ),Òîãäà ðàçíîñòü äâóõ ðåøåíèé( ) = y I (t ) − y II (t )y tóäîâëåòâîðÿåò óñëîâèÿì:Ïîêàæåì, ÷òî′= Ay ,ky (t )k ≤ QI + QII = Q < ∞.y( )=0y täëÿ ëþáîãî(3)t.Äîêàæåì ðÿä ëåìì.Ëåììà 1.
(åëüàíäà - Øèëîâà)Åñëè Reτj (A)≤ −σ (σ > 0), j = 1,...,N ,ke tA k ≤ M (M= M ( kAσ k , N )òî ïðèt≥ 0:σkAk, N )e − 2 t ,σ- íåêîòîðàÿ ïîñòîÿííàÿ.Äîêàçàòåëüñòâî.Îãðàíè÷åííûå ðåøåíèÿ. Êðàåâûå óñëîâèÿ(4) ñèëóìîæíî ñ÷èòàòü, ÷òî òåîðåìû Øóðà, τ10A = 0p12τ20........p1p2.0NNτN.Äàëåå (ñì. 5 è 1, îðìóëó (7)),ãäåY (t ) = y11y120y220000........Ny2,N −1y1, −1.......NNy −1, −10Ny2N y1,yN −1,N NNyïðè÷åì çíàåì (5) ðåêêóðåíòíóþ îðìóëó äëÿ ýëåìåíòîâÊðîìå òîãî, åñëè Reτj (A)≤ Λ, j = 1,...,N ,òîRtPk |yjk (t )| ≤ kAk i =j +1 0 |yik (s )|e Λ(t −s ) ds ,j = 1,...k − 1,k = 1,...,N ;|ykk (t )| ≤ e Λt , k = 1,...,N .yij .Îãðàíè÷åííûå ðåøåíèÿ. Êðàåâûå óñëîâèÿ(5)Èç (5) ïîëó÷àåì:|ykk (t )| ≤ e Λt ,|yk −1,k (t )| ≤ kAk|yk −2,k (t )| ≤ kAkè òàê äàëååãäåjk (x )Pt1!eΛt+t1!(kAkt )22!eeΛt,Λt|yjk (t )| ≤ Pjk (kAkt )e Λt ,- ïîëèíîì ñòåïåíèk−jîòíîñèòåëüíîxñïîñòîÿííûìè êîýèöèåíòàìè.ÏîýòîìóvuXNqX Xu N XΛtA[ ]2 e Λt e 2 t .ke k ≤ t[Pij (kAkt )]2 e 2Λt =i =1 j =1Òàê êàêΛ = −σ (σ > 0) ⇒qX XvuXNu N X...
≤ max t[ ]2 e −σt =t ≥0i =1 j =1Îãðàíè÷åííûå ðåøåíèÿ. Êðàåâûå óñëîâèÿvuXNu N XkAk 2 −skAkt= max[Pij (s )] e= M(, N) ⇒s ≥0σσi =1 j =iσke tA k ≤ M ( )e − 2 t ,÷òî è òðåáîâàëîñü.Ëåììà 2.Åñëè óAâñå ñîáñòâåííûå ÷èñëà≥ σ > 0, j = 1,...,N , òî(t ) = Ay óäîâëåòâîðÿåò ïðèτj (A)òàêîâû, ÷òîReτj (A)ëþáîå ðåøåíèåyt′ky (t )k ≥Òàê êàê Reτj (−A)≥01M()≤ −σ (σ > 0),σe 2( )y tóðàâíåíèÿíåðàâåíñòâó:t ky (0)k.òî ïî ëåììå 1 èìååì:σkAk, N )e − 2 t .σ−tA y (t ), òîy (0) = eke t (−A) k ≤ M (Òàê êàê( ) = e tA y (0)y tèëèky (0)k ≤ ke −tA kky (t )k ≤ M ( )e − 2 t ky (t )k.σÎãðàíè÷åííûå ðåøåíèÿ. Êðàåâûå óñëîâèÿ(6)Ëåììà 3.Åñëè Reτjy′= Ay≤ −σ , j = 1,...,N ,òî ëþáîå ðåøåíèåóäîâëåòâîðÿåò ïðèky (t )k ≥t≤01M()( )y tóðàâíåíèÿíåðàâåíñòâóe− σ2 tky (0)k.Äîêàçàòåëüñòâî ñëåäóåò èç ëåììû 2: çàìåíà(7)τ = −t .Äîêàçàòåëüñòâî îñíîâíîãî óòâåðæäåíèÿ.àçáåðåì 2 ñëó÷àÿ:1) Âñå ñîáñòâåííûå ÷èñëà ëåæàò ïî îäíó ñòîðîíó îò ìíèìîéîñè: Reτj (A)≥σ>0(áåç îãðàíè÷åíèÿ îáùíîñòè).Ïî ëåììå 2 èìååì:ky (0)k ≤ M ( )e − 2 t ky (t )k ≤ M ( )Qe − 2 tσσ∀t ≥ 0⇒ y (0) ≡ 0.2) Ñîáñòâåííûå çíà÷åíèÿ ëåæàò êàê ñëåâà, òàê è ñïðàâà îíìíèìîé îñè, ïðè÷åì |Reτj (A)|≥ σ > 0, j = 1,...,N .Îãðàíè÷åííûå ðåøåíèÿ.
Êðàåâûå óñëîâèÿ ýòîì ñëó÷àåAAA+èN++ N− = N .A−ïðèâîäèñÿ ê æîðäàíîâîé îðìå:=T−1A+00A−- êâàäðàòíûå, ïîðÿäêîâA+=T,detTN+ , N− .6= 0,ïðè÷åìτ1τ210τ2τ3100τ3100τ3...,Îãðàíè÷åííûå ðåøåíèÿ. Êðàåâûå óñëîâèÿA−=τ l +100A−=A+00A−′z+zτ l +11τ l +1..èìåþò ïîëîæèòåëüíûå- îòðèöàòåëüíûå. Reλj (A+ )= 1,...,N+; Reλk (A− ) ≤ −σ ,ÒîãäàÏóñòüz = Ty . ′00A+jz1..Âñå ñîáñòâåííûå çíà÷åíèÿâåùåñòâåííûå ÷àñòè,τlk= 1,...,N−.≥ σ,èëè= A+ z+ ,′z−= A− z− ,z=z+z−.Äàëååkz+ k ≤ kT kQ ,Ñëåäîâàòåëüíî,( )≡0y tkz− k ≤ kT kQ .è åäèíñòâåííîñòü ðåøåíèÿ çàäà÷è (1),(2) äîêàçàíà.Îãðàíè÷åííûå ðåøåíèÿ. Êðàåâûå óñëîâèÿÒåïåðü ïîêàæåì ñóùåñòâîâàíèå îãðàíè÷åííîãî ðåøåíèÿ.Äëÿ ýòîãî ââåäåì â ðàññìîòðåíèå êâàäðàòíóþ ìàòðèöó ïîðÿäêà( )N , ìàòðèöó ðèíà G täëÿ ñèñòåìû y′= Ayñ ïîìîùüþàêñèîì:1)2)kG (t )k îãðàíè÷åíà ∀t ∈ R 1 ;G (t ) íåïðåðûâíà âñþäà, êðîìåG3) ïðèt6= 0G′t= 0,ãäå(+0) − G (−0) = IN ;(t ) = AG (t ).Çàìåòèì, ÷òî ñ ïîìîùüþ 1), 2), 3) ìàòðèöàîäíîçíà÷íî â ïðåäïîëîæåíèè, ÷òîA( )G tîïðåäåëÿåòñÿíå èìååò ÷èñòî ìíèìûõñîáñòâåííûõ çíà÷åíèé. ñàìîì äåëå: ïî ïðåäûäóùåìó( ) = GI (t ) − GII (t )G tÏðåäñòàâèì ìàòðèöóAA⇒( ) ≡ 0.G tâ âèäå:=T−1A+00A−T,Îãðàíè÷åííûå ðåøåíèÿ.