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1):Y (t ) = A · Y′è äëÿ íåå ïîñòàâèòü çàäà÷ó Êîøè: ′Y = AY ,Y (0) = Y0 ,t ∈ R1(9)Ïðîñòðàíñòâî è óíäàìåíòàëüíàÿ ñèñòåìà ðåøåíèéãäå Y0 - ïðîèçâîëüíàÿ íåâûðîæäåííàÿ ìàòðèöà (õîòÿ,åñòåñòâåííî, çàäà÷ó Êîøè (9) ìîæíî ðàññìàòðèâàòü ñ ëþáîéìàòðèöåé Y0 ).Îïðåäåëåíèå 1. Ìàòðèöà Y (t ) íàçûâàåòñÿ óíäàìåíòàëüíîéìàòðèöåé ðåøåíèé çàäà÷è Êîøè (1).Äëÿ çàäà÷è Êîøè (9) èìååò ìåñòî îäíîçíà÷íàÿ ðàçðåøèìîñòü(êîððåêòíîñòü, ñì. 2): äëÿ ëþáîé ìàòðèöû Y0 , detY0 6= 0∃Y (t ), ýëåìåíòû êîòîðîé ÿâëÿþòñÿ íåïðåðûâíûìè èíåïðåðûâíîäèåðåíöèðóåìûìè óíêöèÿìè, îïðåäåëåííûìè∀t ∈ R , è òàêàÿ, ÷òîY (t ) = AY (t ), Y (0) = Y0 .′Ýòèìè óñëîâèÿìè Y (t ) îïðåäåëåíà îäíîçíà÷íî.Äîêàæåì ñëåäóþùóþ îðìóëó: íà ðåøåíèÿõ çàäà÷è Êîøè (9)âûïîëíåíî ñîîòíîøåíèå△(t ) = △0 · exp {Tr (A) · t },(10)Ïðîñòðàíñòâî è óíäàìåíòàëüíàÿ ñèñòåìà ðåøåíèéãäå △(t ) =detY (t ), △0 =detY0 ,Tr (A) =NXk =1akk- ñëåä ìàòðèöû A.Êñòàòè, èç (10) ñëåäóåò óòâåðæäåíèå î òîì, ÷òî ìàòðèöà Y (t ),ñîñòàâëåííàÿ èç N ëèíåéíîíåçàâèñèìûõ ðåøåíèé çàäà÷èÊîøè (7), íåâûðîæäåííàÿ.Âåðíåìñÿ ê (10).
Èìååì:y11 . . . y 1N .... .. NX′′′ △ (t ) =det yk 1 . . . yk N . .... k =1 .. yN 1 . . . yNNÒàê êàêykl′=NXj =1akj · yjl (t ), l = 1,...,N,òîÏðîñòðàíñòâî è óíäàìåíòàëüíàÿ ñèñòåìà ðåøåíèéy11... .. . ′det yk 1 .. ....yN 1...y 1Ny11 .... .. NX′yk N = akj · det yj 1 ....
j =1 .. yN 1yNN.........y 1N.. . yj N =.. . yNN= akk · detY (t ) = akk · △(t ).Ñëåäîâàòåëüíî,△ (t ) = △(t ) · Tr (A),′òî åñòü (èç 1)△(t ) = △(0) · e Tr (A)t = △0 · e Tr (A)t ,÷òî è òðåáîâàëîñü! çàêëþ÷åíèå îòìåòèì, ÷òî, ïîñêîëüêó âñå ðåøåíèÿ çàäà÷èÊîøè (7) îáðàçóþò ëèíåéíîå ïðîñòðàíñòâî ðàçìåðíîñòè N , òîëþáîå ðåøåíèå ýòîé çàäà÷è - ëèíåéíàÿ êîìáèíàöèÿ ñòîëáöîâÏðîñòðàíñòâî è óíäàìåíòàëüíàÿ ñèñòåìà ðåøåíèéìàòðèöûY (t ):y (t ) =ãäåCNXi =1Ci y [i ](t ) = Y (t ) · C ,- ïðîèçâîëüíûé âåêòîðC1.
C = .. .CN òî æå âðåìÿ ëåãêî âûðàçèòü âåêòîðy0 = Y (0)Còî åñòüC÷åðåçy0 := Y0 C ,C = Y0−1 · y0 .Ôîðìóëà (11) ïðåäñòàâëÿåò ñîáîé îáùåå ðåøåíèå ëèíåéíîé′ñèñòåìû Y = AY .(11)Ãëàâà I. Ëèíåéíûå ñèñòåìû ñ ïîñòîÿííûìèêîýôôèöèåíòàìè. Ëèíåéíîå óðàâíåíèå ñ ïîñòîÿííûìèêîýôôèöèåíòàìè âûñîêîãî ïîðÿäêà. Çàäà÷à Êîøè.Çàäà÷à Êîøè äëÿ ëèíåéíûõ ñèñòåì ñ ïåðåìåííûìèêîýôôèöèåíòàìè. Àïðèîðíûå îöåíêè1.
Ïðåäâàðèòåëüíûå ñâåäåíèÿÎïðåäåëåíèå 1. Îáûêíîâåííûì äèôôåðåíöèàëüíûìóðàâíåíèåì n-îãî ïîðÿäêà íàçûâàåòñÿ ñîîòíîøåíèå âèäà0F (t, y , y , ..., y (n) ) = 0.(1)Ðåøåíèåì óðàâíåíèÿ (1) íà èíòåðâàëåôóíêöèÿy = ϕ(t),îïðåäåëåííàÿ íà(a, b) íàçûâàåòñÿ(a, b) âìåñòå ñî ñâîèìèïðîèçâîäíûìè äî n-îãî ïîðÿäêà âêëþ÷èòåëüíî è òàêàÿ, ÷òîïîäñòàíîâêà ôóíêöèèòîæäåñòâî äëÿtèçy = ϕ(t)(a, b).â (1) ïðåâðàùàåò åãî âÇàìå÷àíèå 1. Âñþäó (åñëè íå îãîâîðåíî îñîáî) ïîäïîíèìàåìêîíå÷íûéèíòåðâàë.Ïðåäâàðèòåëüíûå ñâåäåíèÿ(a, b)Ïðîñòåéøèå ïðèìåðû ÎÄÓ - ëèíåéíûå óðàâíåíèÿ ñïîñòîÿííûìè êîýôôèöåíòàìè, ò.å. êîãäà â (1) ôóíêöèÿëèíåéíà ïîF0y , y , ..., y (n) :0Ly = y (n) + a1 y (n−1) + ...
+ an−1 y + an y = 0,ïðè÷åìa1 , ..., an(2)- íåêîòîðûå ïîñòîÿííûå (âåùåñòâåííûå èëèêîìïëåêñíûå).(2) - ëèíåéíîå îäíîðîäíîå ÎÄÓ ïîðÿäêà n ñ ïîñòîÿííûìèêîýôôèöåíòàìè.Çàìå÷àíèå 2. ×åðåçLâ (2) îáîçíà÷åí äèôôåðåíöèàëüíûéîïåðàòîðL=äåéñòâóþùèé íàd n−1ddn+ a1 n−1 + ... + an−1 + an ,ndtdtdtôóíêöèþ y = y (t).Åñëè âìåñòî (2) ðàññìîòðåòü óðàâíåíèå ñïðàâîé ÷àñòüþLy = f (t),ãäåf = f (t)- èçâåñòíàÿ ôóíêöèÿ îò t , òî (4) - ëèíåéíîåíåîäíîðîäíîå óðàâíåíèå.(3)Ïðåäâàðèòåëüíûå ñâåäåíèÿ(4)Ïðèìåð 1.Íàéòè òàêèå êðèâûå íà ïëîñêîñòè(t, y ),÷òîáû tg óãëà íàêëîíàêàñàòåëüíîé (ïî îòíîøåíèþ ê ïîëóîñè Ît) â ëþáîé òî÷êå ýòèõêðèâûõ ðàâíÿëñÿ îðäèíàòåyýòîé òî÷êè, óìíîæåííîé íàíåêîòîðîå âåùåñòâåííîå ÷èñëîa(ñì. Ðèñ.)yy = y(t)at0Ðèñ.Ïðåäâàðèòåëüíûå ñâåäåíèÿ0C00= y ⇒ y = ay ⇒ (e −at y ) = 0 ⇔ e −at y = C ,Òàê êàê tg αãäå- ïðîèçâîëüíàÿ âåùåñòâåííàÿ êîíñòàíòà⇒ y = y (t) = Ce at(5)- óðàâíåíèå ñåìåéñòâà êðèâûõ.Îïðåäåëåíèå 2.
Ðåøåíèå äèôôåðåíöèàëüíîãî óðàâíåíèÿn-îãîïîðÿäêà (1), çàâèñÿùåå îò n ïðîèçâîëüíûõ ïîñòîÿííûõCi , i=1, ..., n,y = ϕ(t, C1 , ..., Cn ),íàçûâàåòñÿîáùèì ðåøåíèåìýòîãî óðàâíåíèÿ.Òàêèì îáðàçîì, ñîîòíîøåíèå (5) çàäàåò îáùåå ðåøåíèåóðàâíåíèÿ0y = ay .Îïðåäåëåíèå 3. Çàäà÷åé Êîøè äëÿ óðàâíåíèÿ (1) íàçûâàåòñÿçàäà÷à î íàõîæäåíèè òàê íàçûâàåìîãîy = ϕ(t)÷àñòíîãî ðåøåíèÿóðàâíåíèÿ (1), óäîâëåòâîðÿþùåãîíà÷àëüíûìt = t0 , t0 ∈ (a, b):0ϕ(t0 ) = ϕ0 , ϕ (t0 ) = ϕ1 , ϕ(n−1) (t0 ) = ϕn−1 ,ãäå ϕ0 , ..., ϕn−1 - íåêîòîðûå çàäàííûå ïîñòîÿííûå.óñëîâèÿìïðèÏðåäâàðèòåëüíûå ñâåäåíèÿÄëÿ óðàâíåíèÿãäåy0y 0 = ay çàäà÷à Êîøè ôîðìóëèðóåòñÿ 0y = ay ,t ∈ (a, b);y (t0 ) = y0 , t0 ∈ (a, b),òàê:(6)- íåêîòîðàÿ çàäàííàÿ ïîñòîÿííàÿ. Çíàÿ ôîðìóëó îáùåãîðåøåíèÿ óðàâíåíèÿy 0 = ay(ò.å.
(5)) ìîæíî ðåøèòü çàäà÷óÊîøè (6).Èìååì:y0 = Ce at0 ⇒ C = y0 e −at ⇒èñêîìîå ðåøåíèå òàêîâî:y = y0 e a(t−t0 ) .(7)Ãåîìåòðè÷åñêèé ñìûñë ðåøåíèÿ çàäà÷è Êîøè (6): èç âñåõêðèâûõ, îïèñûâàåìûõ ôîðìóëîé (5) íàäî âûáðàòü òàêóþ,êîòîðàÿ ïðîõîäèò ÷åðåç çàäàííóþ òî÷êó(t0 , y0 )íà ïëîñêîñòè.Çàìå÷àíèå 3. Ðåøåíèå çàäà÷è Êîøè (6) îïðåäåëåíî ïðè âñåõt ∈ R n , ∀t0 ∈ R 1 , y0 ∈ R 1 .Áóäåì ðàññìàòðèâàòü è ñèñòåìû. Îãðàíè÷åìñÿ ïîêà òîëüêîïðèìåðàìè, íå äàâàÿ ñòðîãîãî îïðåäåëåíèÿ ñèñòåìû.Ïðåäâàðèòåëüíûå ñâåäåíèÿÏðèìåð 2.y1 0 = −y2 ,0y2 = y1 .(8)Çäåñüy1,2 = y1,2 (t) - íåèçâåñòíûå ôóíêöèè.îáîçíà÷åíèÿ:y1 (t)y (t) =- âåêòîð-ôóíêöèÿ.y2 (t)ÂâåäåìÒîãäà ñèñòåìó (8) ìîæíî ïåðåïèñàòü òàê: (âåêòîðíûé âèä)0y (t) =y1y20⇒dydt= y (t) ⇒0y =Ay , ãäå 0 −1A=100y10y2=0−110 y1,y2èëè- ìàòðèöà. îáùåì ñëó÷àå ñèñòåìó ëèíåéíûõ óðàâíåíèé ñ ïîñòîÿííûìèêîýôôèöåíòàìè äëÿNíåèçâåñòíûõ ôóíêöèéy1 (t), ..., yN (t)çàïèøåì â âåêòîðíîì âèäå:0y =dy= Ay ,dtÏðåäâàðèòåëüíûå ñâåäåíèÿ(9)ãäåy =aijy1 (t)...a11.
. . a1N ... , A = (aij ) = ... , i,j=1,...,N,yN (t)aN 1 . . . aNN- ýëåìåíòû ìàòðèöûA,ïîñòîÿííûå âåùåñòâåííûå (èëèêîìïëåêñíûå) ÷èñëà.Ïîêîìïîíåíòíàÿ çàïèñü ñèñòåìû (9) ñîñòîèò èçNóðàâíåíèéNXdyi=aij yij , i = 1,...,N .dt0(9 )j=1Çàìåòèì, ÷òî ñèñòåìà (9) - îäíîðîäíàÿ, ñèñòåìà æådyi= Ay + f (t),dtãäåf (t) = f1 (t)...00(9 )- çàäàííàÿ âåêòîð-ôóíêöèÿ, íåîäíîðîäíàÿ.fN (t)Ïðåäâàðèòåëüíûå ñâåäåíèÿÍàêîíåö, íàðÿäó ñ (9) áóäåì ðàññìàòðèâàòü òàê íàçûâàåìûåìàòðè÷íûå óðàâíåíèÿdY= AY ,dtãäåy11.
. . y1N(10).. Y = (yij ) = .... ; i, j =1,...,N,yN 1 . . . yNNïðè ýòîì, ïî îïðåäåëåíèþdY=dt0y11...00. . . y1N...0.yN 1 . . . yNNÏîêîìïîíåíòíàÿ çàïèñü ñèñòåìû (10) òàêîâà:NXdyir=aij yjr ,dtj, r = 1, ..., Nj=1Ïðåäâàðèòåëüíûå ñâåäåíèÿ0(10 )Èç (100)ñëåäóåò, ÷òî íà ñàìîì äåëå (10) ìîæíî ïåðåïèñàòü ââèäå âåêòîðíîé ñèñòåìû. Äåéñòâèòåëüíî, îáîçíà÷èì ÷åðåçy [k] , k=1,...,Nñëåäóþùèå âåêòîð-ôóíêöèèy11y1N y [1] = ... , ... , y [N] = ...
.y1NyNNÒîãäà èç (10dy [k]dt0)ñëåäóåò, ÷òî= Ay [k] , k=1,...,N èëè [1] yAd . . . =dt0y [N]0...y [1 ] .. . ,Ay [N]00(10 )òî åñòü ìû ïîëó÷èëè ñèñòåìó âèäà (9).Ñâåäåíèÿ èç òåîðèè ìàòðèö.R N - N -ìåðíîå âåùåñòâåííîå åâêëèäîâîN -ìåðíîå êîìïëåêñíîå ïðîñòðàíñòâî.ïðîñòðàíñòâî,Ïðåäâàðèòåëüíûå ñâåäåíèÿCN-y =y1 (t)...,yN (t)ïðè÷åì∀t ∈ R 1 y (t) ∈ R NÊàê èçâåñòíî, äëÿ âåêòîðîâ èçRN(èëè(èëèCN)C N ).ìîæíî ââåñòèäëèíó (íîðìó):vu NpuX|yi |2||y || = (y , y ) = t∀t ∈ R 1 .i=1Çäåñü(y , x) =NXyi xii=1-ñêàëÿðíîå ïðîèçâåäåíèåâåêòîðîâ, y1x1 .. .. y = . , x = . .yNxNÏðåäâàðèòåëüíûå ñâåäåíèÿ(11), ∈ CN .Âàðèàíò äëÿ x yÑêàëÿðíîå ïðîèçâåäåíèå îáëàäàåò ñâîéñòâàìè:1)(y , x)= (x, y ),α ∈ C 1,2)(αy , x) = α(y , x),3)(y , αx) = α(y , x),d ∈ C 1,4)||y || = 0⇔ y = 0.Íåðàâåíñòâà Êóðàíòà:λmin (B)||y ||2 ≤ (By , y ) ≤ λmax (B)||y ||2 ,ãäåB = B∗(12)- ýðìèòîâà ìàòðèöà,λmin (B), λmax (B) - íàèìåíüøååB .
Íàïîìíèì, ÷òîåñëè B = (bij ), òî B ∗ = (b̄ji ), i, j = 1, ..., N . Êðîìå òîãî, âñåñîáñòâåííûå ÷èñëà ýðìèòîâîé ìàòðèöû B âåùåñòâåííûå.è íàèáîëüøåå ñîáñòâåííûå ÷èñëà ìàòðèöûÄîêàæåì (12).∃óíèòàðíîå ïðåîáðàçîâàíèåU (U −1 = U ∗ )òàêîå, ÷òîB = U ∗ DU,D = diag (λ1 , ..., λN ), λi = λi (B), i =1,...,N - ñîáñòâåííûåìàòðèöû B , ïðè÷åì λ1 = λmin (B), λN = λmax (B).Ïðåäâàðèòåëüíûå ñâåäåíèÿ÷èñëàÒîãäà(By , y ) = (U ∗ DUy , y ) = (DUy , Uy ) = λ1 |z1 |2 + ...++λN |zN |2 ,ãäå z1 .. z = . = Dy .zNÑ äðóãîé ñòîðîíû2λ1 ||z|| ≤NXλi |zi |2 ≤ λN ||z||2 .(13)i=1Òàê êàê||z||2 = (z, z) = (Uy , Uy ) = (U ∗ Uy , y ) = (y , y ) = ||y ||2 ,òî èç (13) ñëåäóåò (12).y ∈ R N èëè (C N )A = (aij ), i,j=1,...N .Êðîìå íîðìû âåêòîðàíîðìó ìàòðèöûìîæíî ââåñòè òàêæå èÏðåäâàðèòåëüíûå ñâåäåíèÿÍàèáîëåå óïîòðåáèòåëüíûìè ÿâëÿþòñÿåâêëèäîâàîïåðàòîðíàÿ íîðìàèíîðìà (ïîñëåäíÿÿ = ôðîáåíèóñîâà íîðìà).Îïðåäåëåíèå 4.
Îïåðàòîðíîé íîðìîé ìàòðèöûAíàçûâàþòâåëè÷èíó||Ay ||=||A|| = supy 6=0 ||y ||ssupy 6=0(Ay , Ay ).(y , y )(14)Ïðåîáðàçóåì ïðàâóþ ÷àñòü (14):=Çàìåòèì, ÷òîA∗ Armaxýðìèòîâà èmax(A∗ Ay , y ).||y ||=1A∗ A ≥ 0.(A∗ Ay , y ) = λmax (A∗ A) ≥ 0.||y ||=1Òàêèì îáðàçîì,||A|| =pλmax (A∗ A) ≥ 0.Ïðåäâàðèòåëüíûå ñâåäåíèÿ(15) îòëè÷èå îò îïåðàòîðíîé íîðìû ìàòðèöûíîðìà ìàòðèöûAAôðîáåíèóñîâàââîäèòñÿ òàê:vuXu N||A||E = t|aij |2 .(16)i,j=1Ïîêàæåì, ÷òî||A|| ≤ ||A||E .(17)Ñíà÷àëà âñïîìíèì íåðàâåíñòâî Áóíÿêîâñêîãî - Øâàðöà:|(x, y )|2 ≤ (x, x) · (y , y ).(18)Âåðíåìñÿ ê (17)vvu N NuXNNXuX Xu N X2||Ay || = t (aij yj ) ≤ t (|aij |2 )(|yj |2 ) =i=1 j=1= ||y || · ||A||Ei=1 j=1⇒||Ay ||≤ ||A||E||y ||j=1⇒Ïðåäâàðèòåëüíûå ñâåäåíèÿ(17) äîêàçàíî.
Ïîëåçíîå íåðàâåíñòâî:||A · B|| ≤ ||A|| · |B|| ñàìîì äåëå:s||A · B|| =rmax(ABy , ABy )=(y , y )y 6=0sup(A∗ ABy , By ) ≤||y ||=1≤rmax[λmax (A∗ A) · (By , By )] = ||A|| · ||B||.||y ||=1(19)2. Ðàçðåøèìîñòü çàäà÷è Êîøè äëÿ îäíîðîäíûõëèíåéíûõ ñèñòåì ñ ïîñòîÿííûìè êîýôôèöèåíòàìèÈçó÷àåìy 0 = Ay ,(1)îäíîðîäíóþ ñèñòåìó ëèíåéíûõ äèôôåðåíöèàëüíûõ óðàâíåíèé ñïîñòîÿííûìè êîýôôèöèåíòàìè.Ïîä çàäà÷åé Êîøè äëÿ (1) áóäåì ïîíèìàòü ñëåäóþùóþ çàäà÷óíàõîæäåíèÿ íåèçâåñòíîé âåêòîð-ôóíêöèè y = y (t): 0y = Ay , t ∈ R 1 ;(2)y (t0 ) = y0 , t0 ∈ R 1 ,ãäå y0 ∈ R N (èëè C N ) - íåêîòîðûé çàäàííûé âåêòîð, t0 çíà÷åíèå íåçàâèñèìîé ïåðåìåííîé t , ïðè êîòîðîì çàäàþòñÿíà÷àëüíûå óñëîâèÿ. Êðîìå òîãî, â îòëè÷èå îò çàäà÷è Êîøè(6) èç 1, ìû ðàññìàòðèâàåì çàäà÷ó Êîøè íà âñåéâåùåñòâåííîé îñè R 1 .Çàìåòèì, ÷òî íà÷àëüíûå óñëîâèÿ ìîæíî ðàññìàòðèâàòü òîëüêîïðè t = 0.
Ïîëîæèì τ = t − t0 , z(τ ) = y (τ + t0 ), òîãäà (1) è (2)Ðàçðåøèìîñòü çàäà÷è Êîøèïðåîáðàçóþòñÿ òàê:ddτ z(τ )= Az(τ ),z(0) = y0 .0(2 )0Âîçâðàùàÿñü â (2 ) ê ñòàðûì îáîçíà÷åíèÿì, ïîëó÷èì çàäà÷ó(2) ïðè t0 = 0.Êëàññè÷åñêîå îïðåäåëåíèå êîððåêòíîñòè çàäà÷è Êîøè(2)Îïðåäåëåíèå 1. Çàäà÷à Êîøè (2) íàçûâàåòñÿ êîððåêòíîé, åñëè1) åå ðåøåíèå ∃ ∀y0 ∈ R N (èëè C N );2) åå ðåøåíèå åäèíñòâåííî äëÿ çàäàííîãî âåêòîðà y0 ;3) åå ðåøåíèå íåïðåðûâíî çàâèñèò îò âåêòîðà y0 , ò.å. îòíà÷àëüíûõ äàííûõ.Ðàçîáüåì ïðîöåññ èññëåäîâàíèÿ çàäà÷è Êîøè íà íåñêîëüêîýòàïîâ.I) Äîïóñòèì, ÷òî çàäà÷à Êîøè (2) íà èíòåðâàëå (−T , T ), T > 0- íåêîòîðàÿ êîíñòàíòà èìååò íåïðåðûâíîå è íåïðåðûâíî äèôôåðåíöèðóåìîå ðåøåíèå y = y (t) (ò.å. êàæäàÿ êîìïîíåíòàâåêòîð-ôóíêöèè y = y (t) îáëàäàåò ýòèìè ñâîéñòâàìè).Ðàçðåøèìîñòü çàäà÷è ÊîøèÒàê êàê y = y (t) - ðåøåíèå çàäà÷è Êîøè (2), òî â ñèëóñèñòåìû ïîëó÷èì:dd00(y (t), y (t)) = ||y (t)||2 = (y (t), y (t))+dtdt0+(y (t), y (t)) = (Ay (t), y (t)) + (A∗ y (t), y (t)) == (By (t), y (t)) ≤ M+ ||y (t)||2 .Çäåñü B = A + A∗ = B ∗ , M+ = λmax (B).Ïóñòü ñíà÷àëà 0 ≤ t < T .
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