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, A0ãäåè(i )A0, i = 1, ... , k - æîðäàíîâû ÿùèêè, òî(k ) l l1 lA0 = blok diag (A0 ) , ... , (A0 ) ,tA0 =e∞ lXtA0l= 0,1,2,...(1)(k ) = blok diag e tA0 , ... , e tA0 .l!l =0Íàïîìíèì òàêæå, ÷òî åñëèτ 1...B0 = 0 . . .0..0.τ...01τÊàíîíè÷åñêîå ïðåäñòàâëåíèå ìàòðè÷íîé ýêñïîíåíòûæîðäàíîâà êëåòêà ïîðÿäêà τtetB0 = eNt1! e..τe 4t.t N −1...(N −1)! e..eτt.τtçàäàåòñÿ (3), òîA0eτt....0Åñëè, òî (ñì. 5):tA0 = blok diagt1! eτ4 tτ te 4eτ1 tτe 3, e τ2 t , 00tt1! eτ3 tτe 30tt 2 e τ3 t 2!t τ3 t ,1! eτ te 3, e τ5 t .0Åñëè òåïåðü âíîâü âåðíóòüñÿ ê çàäà÷å Êîøè ′1y = Ay ,t ∈ R ;y (0) = y0 ,(4)Êàíîíè÷åñêîå ïðåäñòàâëåíèå ìàòðè÷íîé ýêñïîíåíòûòî âçÿâ â êà÷åñòâå ïðèìåðà ìàòðèöó A = T −1 A0 T ñ ìàòðèöåéA0 âèäà (3), ìîæíî çàêëþ÷èòü, ÷òî ïðîèçâîëüíîå ðåøåíèå (4)òàêîâî, ÷òî ëþáàÿ êîìïîíåíòà âåêòîðà - ðåøåíèÿ y = y (t ) ëèíåéíàÿ êîìáèíàöèÿ âûðàæåíèéeτ1 t,eτ2 t, e τ3 t ,teτ3 t,2 τ3t et,eτ4 t,teτ4 t,eτ5.Òàêîâà æå ñèòóàöèÿ è â îáùåì ñëó÷àå.
Ëþáàÿ êîìïîíåíòàâåêòîðà - ðåøåíèÿ çàäà÷è Êîøè (4) - ëèíåéíàÿ êîìáèíàöèÿâûðàæåíèé t lj e τj t , τj = τj (A), j = 1,...,N - ñîáñòâåííûå çíà÷åíèÿA, ñòåïåíü lj íå ïðåâîñõîäèò ÷èñëà kj − 1, kj - êðàòíîñòü τj . ñâÿçè ñ ïðèâåäåíèåì ìàòðèöû A ê æîðäàíîâîé îðìå A0ìîæíî ïðåäëîæèòü è òàêóþ ïðîöåäóðó íàõîæäåíèÿ ðåøåíèéçàäà÷è Êîøè (1).Ïóñòü τ0 = τ0 (△) - íåêîòîðîå ñîáñòâåííîå çíà÷åíèå ìàòðèöû Aè ïóñòü íàì óäàëîñü íàéòè îòâå÷àþùèé ýòîìó ñîáñòâåííîìóçíà÷åíèþ ñîáñòâåííûé âåêòîð y0 è ïðèñîåäèíåííûå âåêòîðàÊàíîíè÷åñêîå ïðåäñòàâëåíèå ìàòðè÷íîé ýêñïîíåíòûy1, ... , yr òàêèå, ÷òî:Ay0Ay1= τ0 ,= τ0 y1 + y0 ,...Ayr = τ0 yr + yr −1Íàïîìíèì, ÷òî âåêòîðà y0 , y1 , ... , yr - ëèíåéíî-íåçàâèñèìû.Ñêîíñòðóèðóåì ñëåäóþùèå àãðåãàòû:τ te 0 · y0 ,t τ0 t y ,τ te 0 · y1 +01! et e τ0 t y + t e τ0 t yτ te 0 y +21!12!0è ò.ä.Íåòðóäíî ïðîâåðèòü, ÷òî ñêîíñòðóèðîâàííûå òàê âåêòîðà′ÿâëÿþòñÿ ðåøåíèåì ñèñòåìû y = Ay , à íà÷àëüíûå äàííûå ïðèt = 0 ýòèõ ðåøåíèé áóäóò ëèíåéíî - íåçàâèñèìû.Óïðàæíåíèÿ.1) Ïîêàçàòü, ÷òî ðåøåíèå ìàòðè÷íîãî óðàâíåíèÿdXdt= XB ,X(0) = CÊàíîíè÷åñêîå ïðåäñòàâëåíèå ìàòðè÷íîé ýêñïîíåíòûäàåòñÿ îðìóëîé X = Ce Bt .2) Ïîêàçàòü, ÷òî ðåøåíèå ìàòðè÷íîãî óðàâíåíèÿdXdt= AX + XB ,X(0) = Cäàåòñÿ îðìóëîé X = e tA · C · e tB .Óêàçàíèå 1.Âûïèñàííàÿ ñèñòåìà ìîæåò áûòü ñâåäåíà ê âåêòîðíîìóÊàíîíè÷åñêîå ïðåäñòàâëåíèå ìàòðè÷íîé ýêñïîíåíòûóðàâíåíèþyy′x11= Gy , ãäå ..
. x1N x21 .. . = x2N , .. . xN 1 .. . G11 .. . 1N = A ⊗ IN + IN ⊗ B T , y (0) = ... = g , N 1 .. . NNNNxïðè÷åì X = (xij ), C = (ij ), i , j = 1,...,N .Ñëåäîâàòåëüíî, çàäà÷à Êîøèy′= Gy ,y(0) = g ,Êàíîíè÷åñêîå ïðåäñòàâëåíèå ìàòðè÷íîé ýêñïîíåíòûà âìåñòå ñ íåé è èñõîäíàÿ çàäà÷à èìååò åäèíñòâåííîå ðåøåíèå( ) = e tA Y (t ),X tãäåY(t ) - íåêîòîðàÿ ìàòðèöà, ïîäëåæàùàÿ îïðåäåëåíèþ.7. Ôóíäàìåíòàëüíûå ñèñòåìû ðåøåíèé çàäà÷è Êîøèäëÿ îäíîãî ëèíåéíîãî óðàâíåíèÿ ñ ïîñòîÿííûìèêîýôôèöèåíòàìèÅùå ðàç âåðíåìñÿ ê ëèíåéíîìó óðàâíåíèþ ñ ïîñòîÿííûìèêîýôôèöèåíòàìè:0Lx = x (N) + a1 x (N−1) + ... + aN−1 x + aN x = 0.(1)Ïîëó÷èì äëÿ (1) íåêîòîðûå ÷àñòíûå ðåøåíèÿ, êîòîðûå áóäåìèñêàòü â âèäåx = eτt ,(2)ãäå τ - íåêîòîðûå ïîñòîÿííûå.Èìååì äëÿ τ :PN (τ ) = τ N + a1 τ N−1 + ... + aN−1 τ + aN = 0.(3)Ïîëèíîì PN (τ ) íàçûâàåòñÿ õàðàêòåðèñòè÷åñêèì ïîëèíîìîìäëÿ äèôôåðåíöèàëüíîãî óðàâíåíèÿ (1).1) Åñëè õàðàêòåðèñòè÷åñêèé ïîëèíîì PN (τ ) èìååò ðàçëè÷íûåêîðíè τj , j = 1...N , òî ÷àñòíûå ðåøåíèÿxj = e τj t ,j = 1...N(4)Ôóíäàìåíòàëüíûå ñèñòåìû ðåøåíèé çàäà÷è Êîøèîáðàçóþò ôóíäàìåíòàëüíóþ ñèñòåìó ðåøåíèé çàäà÷è Êîøè äëÿ(1). ñàìîì äåëå, ôóíäàìåíòàëüíàÿ ìàòðèöà ðåøåíèé:x1...xN0 x0...xN 1Φ(t) = .= ..x1(N−1) .
. .e τ1 tτ1 e τ1 t...=τ1N−1 e τ1 txN(N−1)e τNτ N e τN t ,N−1 τN tτN e.........à detΦ(t) = W (t):PNW = e(i=1 τi )tdet 1τ1.........τ1N−1 . . .1τN =τNN−1Ôóíäàìåíòàëüíûå ñèñòåìû ðåøåíèé çàäà÷è ÊîøèPN= e(i=1 τi )tY(τi − τj ).i>j2) Åñëè ñðåäè êîðíåé τj åñòü êðàòíûå, òî ôóíäàìåíòàëüíóþñèñòåìó ðåøåíèé áóäåì ñòðîèòü ïî-äðóãîìó. ÏóñòüPN (τ ) = (τ − τ1 )k1 (τ − τ2 )k2 ...(τ − τp )kp ,Pãäå ki - êðàòíîñòü τi , pi=1 ki = N .Çàìåòèì, ÷òîL=d N−1dddN+a+ ... aN−1 + aN = PN ( ),1dt Ndt N−1dtdtïðè÷åì (1) ìîæíî ïåðåïèñàòü â îäíîì èç ñëåäóþùèõ âèäîâ:Lx = PN (ddd)x = ( − τ1 )k1 ...( − τp )kp x(t) = 0.dtdtdtÈëè(dddd− τ2 )k2 ( − τ3 )k3 ...( − τp )kp ( − τ1 )k1 x(t) = 0.dtdtdt Ôóíäàìåíòàëüíûådt ñèñòåìû ðåøåíèé çàäà÷è ÊîøèÈëè(dddd− τ1 )k1 ( − τ3 )k3 ...( − τp )kp ( − τ2 )k2 x(t) = 0.dtdtdtdtè ò.ä.Ñëåäîâàòåëüíî, ëþáîå ðåøåíèå êàæäîãî èç óðàâíåíèé(d− τ1 )k1 x(t) = 0,dt...d− τp )kp x(t) = 0dtÿâëÿåòñÿ ðåøåíèåì è óðàâíåíèÿ (1).Ëåãêî ïîíÿòü, ÷òî óðàâíåíèå(d− τ )k x(t) = 0,dtèìååò ñëåäóþùèå ÷àñòíûå ðåøåíèÿ:(eτt ,t1!e τ t , ...
,k ≥1t k−1 τ te .(k − 1)!Ôóíäàìåíòàëüíûå ñèñòåìû ðåøåíèé çàäà÷è ÊîøèÇàìå÷àíèå.Äîêàçàòåëüñòâî ýòîãî ôàêòà îñíîâàíî íà ñëåäóþùåé ôîðìóëå(ñìîòðè óïðàæíåíèå 1 ê ýòîìó ïàðàãðàôó)(dt l−1 τ tt l−m−1− τ )m [e ]=eτt ,dt(l − 1)!(l − m − 1)!(5)ãäå l − m − 1 ≥ 0, l − 1 ≥ 1. Çàìåòèì, ÷òî(d− τ )e τ t = 0.dtÓ÷èòûâàÿ ýòî, ìîæíî óòâåðæäàòü, ÷òî (1) èìååò ñëåäóþùèéíàáîð ðåøåíèék −1x1 (t) = e τ1 t , x2 (t) = 1t! e τ1 t , ...
xk1 (t) = (kt 11−1)! e τ1 t ,k −1xk1 +1 (t) = e τ2 t , ... xk1 +k2 (t) = (kt 22−1)! e τ2 t ,... xN (t) = t kp −1 e τp t .(kp −1))!(6)Ôóíäàìåíòàëüíûå ñèñòåìû ðåøåíèé çàäà÷è ÊîøèÌîæíî ïîêàçàòü, ÷òî ñîâîêóïíîñòè ÷àñòíûõ ðåøåíèé (6)îáðàçóåò ôóíäàìåíòàëüíóþ ñèñòåìó ðåøåíèé (òî åñòü îíèëèíåéíî íåçàâèñèìû - óïðàæíåíèå 2).Èòàê, ðàçîáðàíû äâà ñëó÷àÿ ïîñòðîåíèÿ Ôóíäàìåòàëüíîéñèñòåìû ðåøåíèé óðàâíåíèÿ (1):Ïåðâûé - õàðàêòåðèñòè÷åñêèé ïîëèíîì íå èìååò êðàòíûõêîðíåé;âòîðîé - â ñëó÷àå êðàòíûõ êîðíåé.Îäíàêî, åñëè êîýôôèöèåíòû ai , i = 1,...,N (1) çàâèñÿò îò êàêèõ ëèáî ïàðàìåòðîâ, òî ïðè èçìåíåíèè ýòèõ ïàðàìåòðîâ êîðíèõàðàêòåðèñòè÷åñêîãî óðàâíåíèÿ, áóäó÷è, ñêàæåì, ïðîñòûìè,ìîãóò ñáëèæàòüñÿ è ñòàíîâèòüñÿ êðàòíûìè, à çàòåì ñíîâàðàñõîäèòüñÿ è ò.ä.Ïîýòîìó, äëÿ ïðàêòè÷åñêèõ íóæä ñëåäóåòîòûñêàòü òàêóþ ôóíäàìåíòàëüíóþ ñèñòåìó ðåøåíèé, êîòîðàÿîáñëóæèâàëà áû è ñëó÷àé ïðîñòûõ, è ñëó÷àé êðàòíûõ êîðíåéõàðàêòåðèñòè÷åñêîãî ïîëèíîìà.
Ìû ñåé÷àñ ïðèñòóïèì êïîñòðîåíèþ òàêîé ôóíäàìåíòàëüíîé ñèñòåìû ðåøíèéóðàâíåíèÿ (1).Ôóíäàìåíòàëüíûå ñèñòåìû ðåøåíèé çàäà÷è ÊîøèÐàçëîæèìPN (τ ) = (τ − τN )...(τ − τ1 ),ãäå τi , i = 1...N - êîðíè õàðàêòåðèñòè÷åñêîãî ïîëèíîìà (ìîãóòáûòü è êðàòíûå) (3),(ddd− τN )...( − τ2 )( − τ1 )x = 0.dtdtdtÑâåäåì ýòî óðàâíåíèå ê ñèñòåìå ëèíåéíûõ óðàâíåíèé (íå òàê,êàê â 3):Ââåäåì îáîçíà÷åíèÿ:y 1 = x = P0 (y2 = (d)x,dtP0 (d) = 1,dtdddd− τ2 )y1 = P1 ( )x, P1 ( ) =− τ1 ,dtdtdtdt...ddyN = ( − τN−1 )yN−1 = PN−1 ( )x,dtdtÔóíäàìåíòàëüíûå ñèñòåìû ðåøåíèé çàäà÷è ÊîøèN−1Y ddPN−1 ( ) =( − τi ),dtdti=1ïðè÷åì óðàâíåíèå (1) ýêâèâàëåíòíî óðàâíåíèþ äëÿ yN :(d= τN ) = 0.dt èòîãå äëÿ âåêòîðà y : 1y1 −τ1 y = ... = ...yNN−1(−1)01τ1 . .
.......τN−1x= T ... x (N−1)x0x...00 ·= (N−2) x1(N−1)xÔóíäàìåíòàëüíûå ñèñòåìû ðåøåíèé çàäà÷è Êîøèìû ïîëó÷àåì âåêòîðíóþ ñèñòåìó:dy = Ay ,dtτ1 1τ2A=0 ...0 ...1......0...1τN.τN−1 5 ìû óæå ðàçîáðàëè âîïðîñ î íàõîæäåíèè ìàòðè÷íîéýêñïîíåíòû îò òàêîé ìàòðèöû. Ìàòðè÷íàÿ ýêñïîíåíòà â ýòîìñëó÷àå èìååò âèä âåðõíåé òðåóãîëüíîé ìàòðèöû:e tAy11 y12 0 y22 = Y (t) = yij (t) = 0 ...0 ............yN−1,N−1y1Ny2N ,yN−1,N yNNÔóíäàìåíòàëüíûå ñèñòåìû ðåøåíèé çàäà÷è Êîøèi, j = 1,...,N , ãäå τ Rt e i · 0 yi+1,j (s) · e τi s ds,yij (t) =e τi t , i = j,0,i > j.i < j,Ëåãêî âèäåòü, ÷òî ýëåìåíòû ïåðâîé ñòðîêè ìàòðèöû e tA :x(t) = y11 (t), ...
, xN (t) = y1N (t)(7)ÿâëÿþòñÿ ðåøåíèåì íàøåãî óðàâíåíèÿ, òî åñòüPN (ddd)y1k (t) = ( − τN )...( − τ1 )y1k (t) = 0.dtdtdtÏîêàæåì òåïåðü, ÷òî âðîíñêèàíx1... ..W (t) = det .x1(N−1) . . .xN.. 6 0,. =xN(N−1)Ôóíäàìåíòàëüíûå ñèñòåìû ðåøåíèé çàäà÷è Êîøèòî åñòü (7) ñîñòàâëÿþò ôóíäàìåíòàëüíóþ ñèñòåìó ðåøåíèéóðàâíåíèÿ (1). ñàìîì äåëå, ïîñêîëüêóxky1k y [k] (t) = ... = T ... , òîyN kxk(N−1)Y (t) = T · Φ(t) èdetY (t) = e Tr (A)t = detT · detΦ(t) = detΦ(t) 6= 0,÷òî è òðåáîâàëîñü äîêàçàòü.Ñêîíñòðóèðîâîííîé ôóíäàìåíàëüíîé ñèñòåìîé ðåøåíèé (1),íåïðåðûâíî çàâèñÿùåé îò τ1 , τ2 , ..., τN , óäîáíî ïîëüçîâàòüñÿ âòåîðåòè÷åñêèõ èññëåäîâàíèÿõ.
Ïðè ðåøåíèè æå óðàâíåíèÿ (1)íåâûñîêîãî ïîðÿäêà óäîáíî ïîëüçîâàòüñÿ ïðîñòåéøèìèôóíäàìåíòàëüíûìè ñèñòåìàìè ðåøåíèé, ðàññìîòðåííûìè âíà÷àëå ïàðàãðàôà.Ôóíäàìåíòàëüíûå ñèñòåìû ðåøåíèé çàäà÷è ÊîøèÍàêîíåö, åñëè êîýôôèöèåíòû A â ñèñòåìåy = Ayêîìïëåêñíûå, òî åñòü A = B + iC , òîd(u + iv ) = (B + iC )(u + iv ),dtýêâèâàëåíòíîddt uB=vC−CBy = u + iv uvñèñòåìå ñ âåùåñòâåííûìè êîýôôèöèåíòàìè.8.Ñèñòåìà íåîäíîðîäíûõ ëèíåéíûõóðàâíåíèé ñ ïîñòîÿííûìè êîýôôèöèåíòàìè.Òåîðåìà î íåïðåðûâíîé çàâèñè-ìîñòè ðåøåíèÿ îò ïàðàìåòðàÐàññìàòðèâàåòñÿ ñëåäóþùàÿ çàäà÷à Êîøè:(0y = Ay + f (t),y(0) = y0 ∈ C Nt ∈ [−T, T ], 0 < T < ∞,(èëè RN ).(1)Çäåñüf (t) = f1(t)..fN (t)−âåêòîð-ôóíêöèÿ ïðàâûõ ÷àñòåé, fi(t), i =1,...N - íåïðåðûâíûå ôóíêöèè îò t íà [−T, T ],A - êâàäðàòíàÿ ìàòðèöà ïîðÿäêà N ñ ïîñòîÿííûìè êîýôôèöèåíòàìè,y = y(t) = y1(t)..yN (t)-âåêòîð èñêîìûõ ôóíêöèé,y0 = y10(t)..yN 0(t)-âåêòîð íà÷àëüíûõ äàííûõ.Ïðåäïîëîæèì, ÷òî çàäà÷à Êîøè (1) èìååòíà [−T, T ] íåïðåðûâíîå è íåïðåðûâíî - äèôôåðåíöèðóåìîå ðåøåíèå y = y(t).Óìíîæèì ñèñòåìó (1) ñëåâà íà e−tA:0e−tAy = e−tAAy + e−tAf (t).Òàê êàêe−tAA = Ae−tA,è0(e−tA) = −A(e−tA),òî00−tA−tA−tAey −eAy = {ey} = e−tAf (t).ÒîãäàZ td0Èëèds{e−sAy(s)}ds =Z te−sAf (s)ds.0e−tAy(t) − e−0·ty(0) =Z te−sAf (s)ds.0Äàëååy(t) = etAy0 +Z t0e(t−s)Af (s)ds.(2)Èòàê, åñëè ðåøåíèå çàäà÷è Êîøè (1) ñóùåñòâóåò, òî îíî îïðåäåëÿåòñÿ ôîðìóëîé (2).Îáðàòíî, íåïîñðåäñòâåííîé ïðîâåðêîé ëåãêî óñòàíîâèòü, ÷òî âåêòîð - ôóíêöèÿy = y(t), çàäàâàåìàÿ ôîðìóëîé (2), óäîâëå0òâîðÿåò óðàâíåíèÿì y = Ay + f (t) è y(0) =y0.Äîêàæåì åäèíñòâåííîñòü ðåøåíèÿ (2) çàäà÷è Êîøè (1).Îò ïðîòèâíîãî: ïóñòüdy I.II dt = Ay I,II + f (t), y I,II (0) = y .t ∈ [−T, T ],0Òîãäà äëÿy = y I (t) − y II (t)(0y = Ay, t ∈ [−T, T ],y(0) = y0,ðåøåíèåì êîòîðîé ÿâëÿåòñÿ y(t), ñïðàâåäëèâî, ÷òî y(t) ≡ 0, òî åñòü yI (t) ≡ yII (t) íà[−T, T ].Ñ ïîìîùüþ (2) ïîëó÷èì è "àïðèîðíóþ" îöåíêó äëÿ ðåøåíèÿ çàäà÷è Êîøè (2).Èç (2) ñëåäóåò:||y(t)|| = ||etAy0+Z te(t−s)Af (s)ds|| ≤ ||etA||·||y0||+0+||Z te(t−s)Af (s)ds|| ≤ e|t|·||A||||y0||+0Z t(t−s)A||e|| · ||f (s)||ds ≤ e|t|·||A||||y0||++0Z t|t−s|·||A||+eds · M0 =0e|t|·||A|| − 1|t|·||A||=e||y0|| + M0,||A||ãäåM0 =max ||f (s)||.s∈[−T,T ]Òî åñòüe|t|·||A||||y(t)|| ≤ e||y0|| + M0|t|·||A|| − 1||A||.(3)Çàìåòèì, ÷òî (3) ñïðàâåäëèâî è ïðè A = 0,|t|·||A||ïðè ýòîì e ||A||−1 íóæíî çàìåíèòü íà |t| (ñì.óïðàæíåíèå 2 ê ýòîìó ïàðàãðàôó).Íåðàâåíñòâî (3) ìîæíî îãðóáèòü òàê:||y(t)|| ≤ C1(T, M ),(3 )0ãäåC1(T, M ) = (1 + M )eT M − 1,M = max(M0, ||y0||, ||A||).Ïðè âûâîäåâåíñòâîì:0(3 )ìû âîñïîëüçîâàëèñü íåðà-e|t|·||A||| − 1eT M − 1≤||A||Mïðè|t| ≤ T , ||A|| ≤ M(óïðàæíåíèå 3 ê 8).Çàìå÷àíèå.Åñëè íà÷àëüíûå äàííûå â (1) çàäàþòñÿ ïðèt = t0, t0 ∈ (−T, T ),ïèøóòñÿ òàê:òî îöåíêè (3), (3 ) ïåðå0|t−t0 |·||A|| − 1e||y(t)|| ≤ e|t−t0|·||A||||y0|| + M0||A||(4)è||y(t)|| ≤ C2(T, M ),ãäå(4 )0C2(T, M ) = (1 + M )e2T M − 1.Ïðåäïîëîæèì, ÷òî ìû èìååì äâå çàäà÷è Êîøè: ñ yI è yII .Ïóñòü4(t) = y I (t) − y II (t),ω(t) = f I (t) − f II (t),δ = y0I (t) − y0II (t),Λ = AI − AII ,4y II = y II (tI ) − y II (tII ),0(y I − y II ) = AI y I − AI y II + AI y II − AII y II .Òîãäà(04 (t) = AI 4(t) + Λy II (t) + ω(t), t ∈ [−T, T ],4(tI ) = −4y II + δ, tI,II ∈ (−T, T ).È èñïîëüçóÿ îöåíêè (4),(4 ),0||4(t)|| ≤ e2T ·||A||||4(tI )|| +nïîëó÷àåìmax ||ω(s)||+s∈[−T,T ]o e2T ||AI || − 1+||Λ|| · max ||y II (s) ·s∈[−T,T ]||AI ||≤non2TMII≤e· ||4y || + ||δ|| + max ||ω(s)||+s∈[−T,T ]+C2(T, M ) · ||Λ||ãäåM = maxo e2T M − 1MM0I,II , ||y0I,II ||, ||AI,II ||,.×òîáû çàâåðøèòü âûâîä íóæíîé îöåíêè, íàìîñòàëîñü òîëüêî îöåíèòü àãðåãàò ||4yII ||.Ïîñêîëüêó4y II =òîZ tIdtII dsy II (s)ds =Z tItII{AII y II (s)+f II (s)}ds, Z tIII{M C2(T, M ) + M }ds =||4y || ≤ tII= C3(t, M ) · |tI − tII |.Èòàê, îêîí÷àòåëüíî ïîëó÷àåì:||4(t)|| = ||y I (t) − y II (t)|| ≤ K(T, M ) · {|tI − tII |++||y0I −y0II ||+||AI −AII ||+ max ||f I (s)−f II (s)||}.s∈[−T,T ]Çäåñü(5)nK(T, M ) = max e2T M , e2T M C3(T, M ),e2T M − 1 e2T M − 1 oC2(T, M ),.MMÑëåäîâàòåëüíî, ìû ïîêàçàëè, ÷òî ðåøåíèåçàäà÷è Êîøè (1) íåïðåðûâíî çàâèñèò îò íà÷àëüíûõ äàííûõ, êîýôôèöèåíòîâ ìàòðèöûA,ïðàâîé ÷àñòè, â òîì ñìûñëå, ÷òî ìàëîåèçìåíåíèå ïîñëåäíèõ âåäåò ê ìàëîìó èçìåíåíèþ ñàìîãî ðåøåíèÿ ⇒ òåîðåìà î íåïðåðûâíîé çàâèñèìîñòè ðåøåíèÿ çàäà÷è Êîøèîò ïàðàìåòðà, âõîäÿùåãî â êîýôôèöèåíòûìàòðèöû A, ïðàâóþ ÷àñòü, íà÷àëüíûå äàííûå.Èòàê, ïóñòü ìû èìååì ñëåäóþùóþ çàäà÷óÊîøè:0y = A(µ)y + f (t, µ), t ∈ [−T, T ], µ ∈ [−L, L],0 < T, L < ∞,(6)y(t0) = y0(µ), t0 = t0(µ), t0 ∈ (−T, T ), µ ∈ [−L, L].Äàëåå áóäåì ñ÷èòàòü, ÷òî1.f (t, µ) - íåïðåðûâíàÿâåêòîð - ôóíêöèÿ âîáëàñòè Ω̄ = {(t, µ) |t| ≤ T, |µ| ≤ L}.2.Ìàòðèöà A(µ) èìååò íåïðåðûâíûå ïî µ êîýôôèöèåíòû íà [−L, L].3.Ôóíêöèÿ t0 = t0(µ) íåïðåðûâíà ïî µ íà[−L, L].4.Âåêòîð y0(µ) èìååò íåïðåðûâíûå ïî µ êîìïîíåíòû íà [−L, L].Ñ ó÷åòîì 1-4 ìîæíî óòâåðæäàòü, ÷òî ñóùåñòâóåò ìîäóëü íåïðåðûâíîñòè ω(ξ), òî åñòüòàêàÿ ôóíêöèÿ, ÷òî ω(ξ) > 0 ïðè ξ > 0,ω(ξ) → +0 ïðè ξ → +0 è||f (t, µ1) − f (t, µ2)|| ≤ ω(|µ1 − µ2|),||y0(µ1) − y0(µ2)|| ≤ ω(|µ1 − µ2|),||t0(µ1) − t0(µ2)|| ≤ ω(|µ1 − µ2|),âåçäå µ ∈ [−L, L].Ñ äðóãîé ñòîðîíû, ñóùåñòâóåò ïîñòîÿííàÿM , òàêàÿ ÷òî||A(µ1)−A(µ2)|| ≤ ω(|µ1−µ2|),||A(µ)||, ||y0(µ)||, ||f (t, µ)|| ≤ M,µ ∈ [−L, L],t ∈ [−T, T ].Áîëåå òîãî, îáîçíà÷àÿ ÷åðåçy I (t) = y(t, µ1),y II = y(t, µ2),AI = A(µ1),AII = A(µ2),y0I = y0(µ1),y0II = y0(µ2),tI = t0(µ1),tII = t0(µ2),ìû ïðèõîäèì ê ñèòóàöèè, îïèñàííîé âûøåè, ñëåäîâàòåëüíî, âîñïîëüçóåìñÿ íåðàâåíñòâîì (5), êîòîðîå ìû ïåðåïèøåì òàê:||y(t, µ1)−y(t, µ2)|| ≤ max ||y(t, µ1)−y(t, µ2)|| ≤t∈[−T,T ]≤ K(T, M )·{|t0(µ1)−t0(µ2)|+||y0(µ1)−y0(µ2)||++||A(µ1)−A(µ2||+ max ||f (s, µ1)−f (s, µ2)||} ≤s∈[−T,T ]≤ 4K(T, M )ω(|µ1 − µ2|).Ïîñëåäíåå îçíà÷àåò, ÷òî ðåøåíèå çàäà÷è Êîøè (6) íåïðåðûâíî çàâèñèò îò µ ïðè óñëîâèè, ÷òî âûïîëíåíû 1-4.(ñóòü òåîðåìû î íåïðåðûâíîé çàâèñèìîñòèîò ïàðàìåòðà µ).Çàìåòèì òàêæå, ÷òî ïîñêîëüêó||y(t1, µ) − y(t2, µ)|| ≤ C3(T, M ) · |t1 − t2| ≤≤ K(T, M ) · |t1 − t2|ïðè t1,2 ∈ [−T, T ], µ ∈ [−L, L], òî ñïðàâåäëèâàè áîëåå îáùàÿ îöåíêà:||y(t1, µ1)−y(t2, µ2)|| ≤ K(T, M ){|t1−t2|+4ω(|µ1−µ2|)},t1,2 ∈ [−T, T ],µ1,2 ∈ [−L, L].Ïîñëåäíåå íåðàâåíñòâî ïîêàçûâàåò, ÷òî âåêòîð - ôóíêöèÿ y = y(t, µ), ÿâëÿþùàÿñÿ ðåøåíèåì çàäà÷è Êîøè (6), ÿâëÿåòñÿ íåïðåðûâíîé ôóíêöèåé ïî t, µ â Ω̄.Òàê êàêdy(t, µ) = A(µ)y(t, µ) + f (t, µ),dt0òî yt(t, µ) òîæå íåïðåðûâíà ïî ñîïîêóïíîñòèïåðåìåííûõ â Ω̄.Íàêîíåö, âñå âûøåïåðå÷èñëåííîå ïî÷òè äîñëîâíî ìîæíî ïîâòîðèòü è äëÿ ñëó÷àÿ, êîãäà µ - âåêòîðíûé ïàðàìåòð, òî åñòü µ =(µ1, ..., µm).Óïðàæíåíèÿ.1.