1611689230-c82f95de5ae5b8e93bc9da7770e8f930 (826746), страница 6
Текст из файла (страница 6)
3) ñëåäóåò:deteòî åñòüe tAtA= e Tr (A)t 6= 0 ∀t ∈ R 1 ,- íåâûðîæäåííàÿ è ïîýòîìó(∃(e tA )−1 ,d(e tA )−1tA −1dt = −(e ) −1 · A,tA−1 (e ) t=0 = (IN ) = IN .t ∈ R 1,ïðè÷åì0(9 )Ôóíäàìåíòàëüíàÿ ìàòðèöà è ìàòðè÷íàÿ ýêñïîíåíòà òî æå âðåìÿ èç (13) ñëåäóåò0A(e tA )−1 = (e tA )−1 A,òî åñòü ñèëó(13 )(e tA )−1 ïåðåñòàíîâî÷íà ñ A.00(13 ) çàäà÷à Êîøè (9 ) ïåðåïèøåòñÿ òàê:(d(e tA )−1tA −11dt = −A(e ) , t ∈ R ,tA−1 (e ) t=0 = IN .Íî çàäà÷à Êîøèìàòðèöåé00(9 )00(9 )ÿâëÿåòñÿ ïðîñòî çàäà÷åé Êîøè (8) ñ−A:0Y (t) = −AY (t),Y (0) = IN ,t ∈ R 1;00(8 )òî åñòü (ñì. (11)):Y (t) = e−tA=∞Xk=0k(−1)·tkk!Ak .Ôóíäàìåíòàëüíàÿ ìàòðèöà è ìàòðè÷íàÿ ýêñïîíåíòàÈòàê, ìû äîêàçàëè çàìå÷àòåëüíóþ ôîðìóëó:(e tA )−1 = e −tA .(14)Ïðèìåð.Ìàòðèöà 2teY (t) =e 2tet0,detY (t)= −e 3t 6= 0,ÿâëÿåòñÿ ôóíäàìåíòàëüíîé ìàòðèöåé ðåøåíèé çàäà÷è Êîøèäëÿ ñèñòåìû 0yy = 1 = Ay ,y20ÌàòðèöàY (t)2e2t2e2tet0=10A=1102 2te2e 2t1et0,.0Y = AY .Ôóíäàìåíòàëüíàÿ ìàòðèöà è ìàòðè÷íàÿ ýêñïîíåíòàóäîâëåòâîðÿåò ìàòðè÷íîé ñèñòåìåÎáðàòíàÿ ìàòðèöàY−1=e −2te −t0e −tóäîâëåòâîðÿåò ñèñòåìå0(Y −1 ) = −Y −1 A.e tA(0):Ìàòðèöà−1B=YetAY (t)óìíîæåíèåì ñëåâà íà011−1ïîëó÷àåòñÿ èç 2te=e 2tet0Ðàíåå ìû äîêàçàëè, ÷òî= te(e tA )−1 = e −tA0èe 2t − e te 2t.||e tA || ≤ e |t|·||A||(ñì.(11), (14)).Ñëåäîâàòåëüíî:||e −tA || ≤ e |t|·||−A|| = e |t|·||A|| .Äàëåå, òàê êàêe tA · e −tA = IN||e tA || ≥è1||IN || = 1 ≤ ||e tA || · ||e −tA ||,òî≥ e −|t|·||A|| .Ôóíäàìåíòàëüíàÿ ìàòðèöà è ìàòðè÷íàÿ ýêñïîíåíòà||e −tA ||Èòàê, èìååì:e −|t|·||A|| ≤ ||e tA || ≤ e |t|·||A|| .(15)Çàìåòèì, ÷òî â îòëè÷èå îò ýêñïîíåíöèàëüíîé ôóíêöèè, âîîáùåãîâîðÿ,e t(A+B) 6= e tA · e tB .Ïðèìåð.A=1002,B=etA0100=eèet0A+B =,02tet(A+B)=, te0etB1102=.1t01e 2t − e te 2t,e t(A+B) 6= e tA e tB .Óïðàæíåíèÿ1.||A + B|| ≤ ||A|| + ||B||.Ôóíäàìåíòàëüíàÿ ìàòðèöà è ìàòðè÷íàÿ ýêñïîíåíòà2.Çàäà÷àdZ (s)ds= −Z (s) · A,Z (t) = IN ,0≤ s < t,íàçûâàåòñÿ ñîïðÿæåííîé ïî îòíîøåíèþ ê çàäà÷åÏîêàæèòå, ÷òîdY (t)dt= AY (t),Y (0) = IN .t > 0,Z (s) = Y (t) · Y −1 (s).3.
Ïîêàçàòü, ÷òîe t(A+B) = e tA · e tB⇔AB = BA.4.ÅñëèP(A) = Pk Ak + ... + P1 A + P0 ,Q(A) = Ql Al + ... + Q1 A + Q0 ,ïîëèíîìû îòAñ ïîñòîÿííûìè êîýôôèöåíòàìèe t(P+Q) = e tP · e tQ .Pi , Q j ,òî5. Âû÷èñëåíèå ìàòðè÷íîé ýêñïîíåíòû äëÿ íåêîòîðûõñïåöèàëüíûõ ìàòðèöÍà÷íåì ñî ñëó÷àÿ, êîãäàA- äèàãîíàëüíàÿ ìàòðèöà, òî åñòüτ1A = diag(τ1 , ... , τN ) = 0....0τNÏîñêîëüêó (ñì. 4)etA=∞ kXtk=0òî äëÿ äèàãîíàëüíîé ìàòðèöûAk!Ak ,(1)ïîëó÷àåì:∞ k∞ kXXt kt ke tA = diag(τ1 , ... ,τN ) = (e tτ1 , ... , e tτN ).k=0k!k=0k!Áåç òðóäà ìîæíî âû÷èñëèòü ìàòðè÷íóþ ýêñïîíåíòóe tAìàòðèö, êîòîðûå ïðèâîäÿòñÿ ê äèàãîíàëüíîé ôîðìå.Ïðèìåðû ìàòðè÷íîé ýêñïîíåíòûäëÿ òåõÒàêèìè ìàòðèöàìè ÿâëÿþòñÿ:à) ýðìèòîâû ìàòðèöûìàòðèöàU = U(A)A,òî åñòüA∗ = A.Òîãäà∃óíèòàðíàÿ(ñì.
1), òàêàÿ ÷òîA = U ∗ · diag(τ1 , ... , τN ) · U,τi = τi (A), i = 1,...,N- ñîáñòâåííûå çíà÷åíèÿá) íîðìàëüíûå ìàòðèöû ýòîì ñëó÷àå òîæå∃ïðèâîäÿùàÿ ìàòðèöóA,óíèòàðíàÿAA;AA∗ = A∗ A.ìàòðèöà U = U(A),òî åñòüê äèàãîíàëüíîìó âèäó:A = U ∗ · diag(τ1 , ... , τN ) · U,â) ìàòðèöûA,èìåþùèå ïðîñòîé ñïåêòð, òî åñòü íåêðàòíûåñîáñòâåííûå çíà÷åíèÿ τi = τi (A),∃T (A), detT 6= 0, òàêàÿ ÷òîi = 1,...,N . ýòîì ñëó÷àåA = T −1 · diag(τ1 , ...
, τN ) · T .Âî âñåõ ýòèõ ñëó÷àÿõ(e tA )0=Ae tA èëèe tAâû÷èñëÿåòñÿ òàê: òàê êàê0(e tA ) = U ∗ DUe tAÏðèìåðû ìàòðè÷íîé ýêñïîíåíòû0(e tA ) = T −1 DTe tA ,ãäåD = diag(τ1 , ... , τN ), òî åñòü ìàòðèöàZ (t) = Ue tA U ∗ (Z (t) = Te tA T −1 ) óäîâëåòâîðÿåòñëåäóþùåéçàäà÷å Êîøè:0Z (t) = DZ (t),Z (0) = IN .t ∈ R 1,Çíà÷èò (ñì. 4),Z (t) = e τ D = diag(e τ1 t , ... , e τN t )èe tA = U ∗ e tD U (T −1 e tD T ),÷òî è òðåáîâàëîñü äîêàçàòü.Äëÿ ïðîèçâîëüíûõ ìàòðèö äåëî îáñòîèò íåñêîëüêî ñëîæíåå.
Âòåîðèè ìàòðèöÒåîðåìà Øóðà∃îäíà âåñüìà ïîëåçíàÿ òåîðåìà, à èìåííîÏðèìåðû ìàòðè÷íîé ýêñïîíåíòûÅñëè äàíà ìàòðèöàA,∃òîóíèòàðíàÿ ìàòèöàU = U(A),ïðèâîäÿùàÿ A ê âåðõíåìó òðåóãîëüíîìó âèäó:A = U ∗ ∇U,ãäåτ1 p12 0 τ2∇=0 ...0pij , i = 1,...,N−1, j = 2,...,N0.........τN−1...0p1Np2N ,pN−1,N τN- íåêîòîðûå ïîñòîÿííûå.Èìåÿ â âèäó òåîðåìó Øóðà, ìîæíî ñðàçó ñ÷èòàòü, ÷òî âñèñòåìå0y = AyìàòðèöàA- âåðõíÿÿ òðåóãîëüíàÿ, èáî åñëèýòî íå òàê, òî äåëàÿ çàìåíó ïåðåìåííûõ z = Uy ïîëó÷àåì000y = Ay = U ∗ ∇Uy ⇒ Uy = ∇Uy ⇒ z = ∇z .Èòàê, âû÷èñëèì e tA , êîãäà A = ∇ - âåðõíÿÿ òðåóãîëüíàÿìàòðèöà.Ïðèìåðû ìàòðè÷íîé ýêñïîíåíòûÄëÿ ýòîãî íàì ïîíàäîáèòñÿ ôîðìóëà, äàþùàÿ ðåøåíèå çàäà÷èÊîøè äëÿ íåîäíîðîäíîãî óðàâíåíèÿ ïåðâîãî ïîðÿäêà (ñì.
1):ãäåa0y = ay + f (t), t ∈ R 1 ;y (0) = y0 , y0 ∈ R 1 (C 1 ),(2)- íåêîòîðàÿ ïîñòîÿííàÿ. Ýòà ôîðìóëà ëåãêî ìîæåò áûòüíàéäåíà.  ñàìîì äåëå, òàê êàêd −at{e y (t)} = e −at f (t),dty (t) = e at y0 =Ýëåìåíòûk -îãîZòîte a(t−s) f (s)ds.(3)0ñòîëáöà ìàòðèöûe tAíàõîäÿòñÿ êàê ðåøåíèåÏðèìåðû ìàòðè÷íîé ýêñïîíåíòûâåêòîðíîé ñèñòåìû ñ ñîîòâåòñòâóþùèìè íà÷àëüíûìè äàííûìè:0[k]1(y [k] (t)) = ∇y (t), t ∈ R ,0. ..
1y [k] (0) = ← koe ,0...ãäå y [k] (t)y1k (t) .. = . ,yN k (t)èëè, â ïîêîìïîíåíòíîé çàïèñè: 0Py1k (t) = τ1 y1k (t) + Ni=2 p1i yik (t), y1k (0) = 0; 0Py2k (t) = τ2 y2k (t) + Ni=3 p2i yik (t), y2k (0) = 0;...P 0ykk (t) = τk ykk (t) + Ni=k+1 pki yik (t), ykk (0) = 1,Ïðèìåðû ìàòðè÷íîé ýêñïîíåíòû(4)PN 0y(t)=τy(t)+k+1k+1,ki=k+2 pk+1,i yik (t),k+1,kyk+1,k (0) = 0; ...0yN−1,k (t) = τN−1 yN−1,k (t) + pN−1,N yN ,k (t),y 1,k (0) = 0; N−0yN ,k (t) = τN yN ,k (t), yN ,k (0) = 0.(5)Çàäà÷ó Êîøè ìû ðàçáèëè íà äâå ïîäçàäà÷è.
ßñíî, ÷òî çàäà÷à(5) äëÿ îïðåäåëåíèÿ ýëåìåíòîâñòîëáöà ìàòðèöûY (t) = e tAyk+1,k (t), ... , yN ,k (t) k -îãîìîæåò ðàññìàòðèâàòüñÿ îòäåëüíî,è ïîñêîëüêó íà÷àëüíûå äàííûå íóëåâûå, òî â ñèëó òåîðåìûåäèíñòâåííîñòè (ñì. 2):yi,k (t) ≡ 0,Ñëåäîâàòåëüíî, ìàòðèöài = k + 1, ... ,N .Y (t) = e tA(6)òîæå âåðõíÿÿ òðåóãîëüíàÿ,Ïðèìåðû ìàòðè÷íîé ýêñïîíåíòûêàê è ìàòðèöàA:Y (t) = e tAy11 y12 0 y22= 0 .....0...0.......yN−1,N−1y1Ny2N .yN−1,N yN,NÏîäçàäà÷à (4) ñ ó÷åòîì (6) ïåðåïèøåòñÿ òàê: 0Pk y1k (t) = τ1 y1k (t) + i=2 p1i yik (t),...0ykk (t) = τk ykk (t),y1k (0) = 0;0(4 )ykk (0) = 1.Ñ ïîìîùüþ ôîðìóëû (3) ìû ïîëó÷àåì ñëåäóþùèåðåêêóðåíòíûå ñîîòíîøåíèÿ äëÿ îïðåäåëåíèÿ ýëåìåíòîâj = 1, ... k k -îãî ñòîëáöà ìàòðèöû e tA :R t Pk yjk (t) = e τj t 0 { i=j+1 pji yik (s)}e −τj s ds,j = 1, ... k − 1;ykk (t) = e τk t .Ïðèìåðû ìàòðè÷íîé ýêñïîíåíòûyjk (t),(7) êà÷åñòâå ïðèìåðà ðàññìîòðèì ñëó÷àé, êîãäàk = 1, ...
N−1,à âñå îñòàëüíûåpk,k+1 = 1,pij = 0. ýòîì ñëó÷àå ñîîòíîøåíèÿ (7) ñèëüíî óïðîùàþòñÿ:Rt yjk (t) = e τj t 0 yj+1,k (s)e −τj s ds,j = 1, ..., k − 1;ykk (t) = e τk t . êà÷åñòâå äðóãîãî ïðèìåðà ðàññìîòðèì ñëó÷àé, êîãäà τj =0j = l, ..., k , l ≥ 1. Òîãäà èç ïîäçàäà÷è (4 ) ìîæíî âûäåëèòü0(7 )τ,çàäà÷ó äëÿ íàõîæäåíèÿ ýëåìåíòîâyl,k (t), yl+1,k (t), ... , ykk (t)k -îãî ñòîëáöà ìàòðèöû e tA (ïî-ïðåæíåìó, ñ÷èòàåì, ÷òîpk,k+1 = 1, k = 1,...,N−1, pij = 0 - âñå îñòàëüíûå): 0ylk (t) = τ ylk (t) + yl+1,k (t), ..ylk (0) = 0,.0yk−1,k (t) = τ yk−1,k (t) + ykk (t), 0ykk (t) = τ ykk (t), ykk (0) = 1.00yk−1,k (0) = 0,Ïðèìåðû ìàòðè÷íîé ýêñïîíåíòû(4 )Ïðèìåíÿÿ ê (400)0ôîðìóëó (7 ), ìû ïîñëåäîâàòåëüíî ïîëó÷àåì:ykk (t) = e τ t , yk−1,k (t) =t1!e τ t , ... , ylk (t) =t k−l(k−l)!eτt .Ñëåäñòâèåì ýòèõ ïðèìåðîâ ÿâëÿåòñÿ òîò ôàêò, ÷òî ìàòðè÷íàÿýêñïîíåíòàe tA ,ãäåA- æîðäàíîâà êëåòêàτA=0..0010.............0,1τíàõîäèòñÿ òàê:etA τte=0t1! e..τt...t N−1 τ t(N−1)! e........eτtÏðèìåðû ìàòðè÷íîé ýêñïîíåíòûÂåðíåìñÿ âíîâü ê îáùåìó ñëó÷àþ è ïîëó÷èì èç (7) íåêîòîðûåïîëåçíûå íåðàâåíñòâà. ñàìîì äåëå, èç (7) ñëåäóåò:RtPk |yjk (t)| ≤ i=j+1 |pji | · 0 |yik (s)| · |e τj (t−s) |ds,j = 1, ...
, k − 1,|ykk (t)| ≤ |e τk t |.(8)Äëÿ óïðîùåíèÿ íåðàâåíñòâ (8) äîêàæåì ñëåäóþùèé ôàêò: äëÿëþáîé ìàòðèöûB = (bij ), i, j = 1,...,Nèìååì:|bij | ≤ ||B||.(9)Äåéñòâèòåëüíî, ðàññìîòðèì âåêòîðy = Bx, y1 .. y = . ,yN x1 .. x = . ,xNïðè÷åìyi =NXj=1bij xj ,i = 1,...,N .Ïðèìåðû ìàòðè÷íîé ýêñïîíåíòûÒîãäà|yi | ≤ ||y || = ||Bx|| ≤ ||B|| · ||x||. 0Ïóñòü .. . x =1 ← j0 . .. .Ñëåäîâàòåëüíî,||x|| = 1èyi = bij0 ,0j0 = 1,...,N .Ïîýòîìó|bij0 | ≤ ||B||,i, j0 = 1,...,N ,÷òî è òðåáîâàëîñü.Ñ ó÷åòîì (9) èç (8) ïîëó÷àåì:RtPk |yjk ≤ ||A|| i=j+1 0 |yik (s)|e Λ(t−s) ds,j = 1, ..., k − 1,|ykk (t)| < e Λt .Ïðèìåðû ìàòðè÷íîé ýêñïîíåíòû0(8 )0Ïðè âûâîäå (8)ìû ïîëàãàëè, ÷òîτj = Reτj + Imτj ,èReτj ≤ Λ,ãäåΛ- íåêîòîðàÿ ïîñòîÿííàÿ.j = 1,...,N6. Êàíîíè÷åñêîå ïðåäñòàâëåíèå ìàòðè÷íîé ýêñïîíåíòûÎáùèå ðàññóæäåíèÿ îòíîñèòåëüíî âû÷èñëåíèÿ ìàòðè÷íîéýêñïîíåíòû e tA , òî åñòü óíäàìåíòàëüíîé ìàòðèöû ðåøåíèéçàäà÷è Êîøè ′1y = Ay ,t ∈ R ;(1)y (0) = y0 .Èç òåîðèè ìàòðèö èçâåñòíî, ÷òî ëþáàÿ ìàòðèöà A ìîæåò áûòüïðèâåäåíà ê êàíîíè÷åñêîé îðìå Æîðäàíà ñ ïîìîùüþíåêîòîðîé íåâûðîæäåííîé ìàòðèöû T = T (A), detT 6= 0:A= T −1 · A0 · T .(2) ïðåäñòàâëåíèè (2) ìàòðèöà A0 - êëåòî÷íîäèàãîíàëüíàÿ, óêîòîðîé íà äèàãîíàëè ñòîÿò ñòàíäàðòíûå æîðäàíîâû ÿùèêè.Ïðè ïðèâåäåíèè ìàòðèöû A ê æîðäàíîâîé îðìå A0 ìàòðèöàïåðåõîäà T îïðåäåëÿåòñÿ íå åäèíñòâåííûì îáðàçîì.Áîëåå òîãî, ñàì êàíîíè÷åñêèé âèä íå çàâèñèò íåïðåðûâíî îò AÏîñëåäíåå îçíà÷àåò âîò ÷òî: âîçüìåì â ïðåäñòàâëåíèè (2) â.Êàíîíè÷åñêîå ïðåäñòàâëåíèå ìàòðè÷íîé ýêñïîíåíòûêà÷åñòâå ïðèìåðà ñëåäóþùóþ ìàòðèöóA0τ1 0=0A00τ20......τ3 1 00 τ3 10 0 τ3τ4 10 τ40...τ5(3)è ðàññìîòðèì ñëåäóþùåå, çàâèñÿùåå îò ïàðàìåòðà ξ ñåìåéñòâîìàòðèöA+ ξ B = T −1 A0 T + ξ T −1 τ10..0.τ8=Êàíîíè÷åñêîå ïðåäñòàâëåíèå ìàòðè÷íîé ýêñïîíåíòûτ1 + ξτ2 + 2ξ−1 =T ...τ7 + 7ξτ 8 + 8ξT.Ïðè äîñòàòî÷íî ìàëûõ ξ ìîæíî ñäåëàòü âñå ñîáñòâåííûåçíà÷åíèÿ ìàòðèöû A + ξ · B ðàçëè÷íûìè è, ñëåäîâàòåëüíî, ïðèξ 6= 0 è äîñòàòî÷íî ìàëîì ξ ìàòðèöà A + ξ B äîëæíàïðèâîäèòñÿ ê ñëåäóþùåìó æîðäàíîâó âèäó:A+ ξ B = T̃ −1 diag(τ1 + ξ, ...
, τ8 + 8ξ)T̃ ,òî åñòü íàëèöî îòñóòñòâèå íåïðåðûâíîé çàâèñèìîñòè ïðèïðèâåäåíèè ìàòðèö ê æîðäàíîâîé îðìå.Òåì íå ìåíåå, åñëè ìàòðèöà A ïðèâåäåíà ê æîðäàíîâîé îðìåtA âû÷èñëÿåòñÿ òàê (ñì. 5):A0 , òî ìàòðè÷íàÿ ýêñïîíåíòà eetA = T −1 e tA0 T ,Êàíîíè÷åñêîå ïðåäñòàâëåíèå ìàòðè÷íîé ýêñïîíåíòûïðè÷åì e tA0 - áëî÷íîäèàãîíàëüíàÿ è ïî äèàãîíàëè ñòîÿòáëîêè, ÿâëÿþùèåñÿ ìàòðè÷íûìè ýêñïîíåíòàìè îò æîðäàíîâûõÿùèêîâ.Äîêàçàòåëüñòâî ïîñëåäíåãî àêòà î÷åâèäíî, òàê êàê åñëè(k ) ( 1),A0 = blok diag A0 , ...