Диссертация (Эллипсоидальные методы в решении задач достижимости и синтеза управлений для систем с запаздыванием), страница 10

Ïðèt ∈ [h − δ, h + δ]äàííîå âûðàæåíèå ðàâíîìåðíî îãðàíè÷åíî. Âûáèðàÿ èçíà÷àëüíîíî ìàëûì ìîæíî ñäåëàòü âûâîä, ÷òîZhA1(τ )ym (τ )dτ =0Zh02A1(τ )Im(τ )dτ + o2 , ko2k < εÓ÷èòûâàÿ óñëîâèÿ (3.13), ïîëó÷àåìZh02(t)dt =A1 (t)ImZh0A1(t)Z0−hx0(τ )c(t, τ )dτ dtδäîñòàòî÷-84äåm m−kmkh (k − 1)h m ( h t)e− h t , τ ∈ [− , −],h (m − k)!mmc(t, τ ) =0, ïðè îñòàëüíûõ τÌåíÿåì ìåñòàìè ïîðÿäîê èíòåãðèðîâàíèÿ, ïîëó÷àåì:Z0x0(τ )−hZ0A1(t)x0(τ )c(t, τ )dtdτ−hÎöåíèâàÿ âíóòðåííèé èíòåãðàë, àíàëîãè÷íî (3.21) ïîëó÷àåì:Z0A1 (t)x0(τ )c(t, τ )dt = A1 (τ + h)x(τ ) + o1 (τ )−h ñëó÷àå åñëèäît.t < h,âìåñòî èíòåãðàëà îò0äîhïîÿâèòñÿ èíòåãðàë îò0Âñå îñòàëüíûå ðàññóæäåíèÿ àíàëîãè÷íû.Òàêèì îáðàçîì, èìååì äëÿ àïïðîêñèìèðóþùåé óíêöèè èíòåãðàëüíîå óðàâíåíèå.y0 (t) = y0(0) +ZtA0(τ )y0(τ )dτ +0+ZtZt0A1(τ )(y0(τ − h) + o1 (τ ))dτ +B(τ )u(τ )dτ + o(t, δ)0äåy0 (τ − h) = x0(τ − h),ïðèt ∈ (0, h)Êîòîðîå îòëè÷àåòñÿ îò èñõîäíîãî óðàâíåíèÿ íà âåëè÷èíó íå ïðåâîñõîäÿùóþëþáóþ íàïåðåä çàäàííóþ ïðè ñòðåìëåíèèmê áåñêîíå÷íîñòè.

Èç ÷åãî ìîæíîñäåëàòü âûâîä, (íàïðèìåð èñïîëüçóÿ ëåììó ðîíóîëëà-Áåëëìàíà) ÷òîðàâíîìåðíî ñõîäèòñÿ êx(t) .y0(t)853.3Àïïðîêñèìàöèÿ èñõîäíîé ñèñòåìû ìåòîäîìïðÿìûõ äëÿ ñëó÷àÿ ïîñòîÿííûõ êîýèöèåíòîâàññìîòðèì ëèíåéíóþ óïðàâëÿåìóþ ñèñòåìó ñ çàïàçäûâàíèåì.ẋ(t) = Ax(t − h) + Bu(t), t ∈ [0, t1](3.24)x(0) = x0, x(τ ) = x0(τ ), τ ∈ [−h, 0).(3.25)àññìîòðèì åå àïïðîêñèìàöèþ:ẏ0(t) = Aym (t) + Bu(t),ẏ1 (t) =mh (y0 (t)− y1 (t)),(3.26)...ẏm (t) =ãäåmh (ym−1 (t)yi (t) ∈ Rn , i = 0, 1, ..., m .− ym (t)),Ñ íà÷àëüíûìè óñëîâèÿìè:y0 (0) = y00 = x0, yi (t0) = yi0 =mh(−i+1)h/mZx0(τ )dτ, i = 1, 2, ..., m.(3.27)−ih/mÎáîçíà÷èì çàX(p) , Y0(p) , Y1(p) ,..., Ym (p) , U (p)çîâàíèÿ Ëàïëàñà îò ñîîòâåòñòâóþùèõ óíêöèéèçîáðàæåíèÿ ïðåîáðà-x(t) , y0 (t) , y1 (t) ,..., ym (t) ,86u(t) :X(p) =Y0 (p) =Y1 (p) =+∞R0+∞R0+∞Re−pτ x(τ )dτ,e−pτ y0 (τ )dτ,e−pτ y1 (τ )dτ,0(3.28)...Ym (p) =U (p) =+∞Re−pτ ym (τ )dτ,0+∞Re−pτ u(τ )dτ.0Çàìåòèì, ÷òî ðåøåíèÿ ñèñòåì (3.24), (3.25) è (3.26), (3.27) ðàñòóò íå áûñòðåå ýêñïîíåíòû.

Ñëåäîâàòåëüíî, ñóùåñòâóåò òàêîå ïîëîæèòåëüíîå ÷èñëî÷òî ïðèp > P0P0 ,ñïðàâåäëèâû ñëåäóþùèå âûðàæåíèÿlim e−pτ x(τ ) = 0,τ →∞lim e−pτ y0(τ ) = 0,τ →∞lim e−pτ y1(τ ) = 0,τ →∞(3.29)...lim e−pτ ym (τ ) = 0τ →∞lim e−pτ u(τ ) = 0τ →∞Ïðèìåíèì ïðåîáðàçîâàíèå Ëàïëàñà ê îáåèì ñèñòåìàì (ó÷èòûâàÿ (3.29)).Z+∞pX(p) = x0 + e−phAe−pτ x0(τ − h)dτ e−ph A + U (p).(3.30)0pY0(p) = AYm(p) + BU (p),pY1(p) = y10 + mh (Y0 (p) − Y1 (p)),...0(Ym−1(p) − Ym (p)),pYm (p) = ym+mh(3.31)87Ïðîâåäÿ ýëåìåíòàðíûå ïðåîáðàçîâàíèÿ, ïîëó÷àåì ñëåäóþùèå âûðàæåíèÿ:X(p) = (pI − e−phA)−1(x0 +Z0e−p(τ +h) Ax0(τ )dτ + U (p)).(3.32)−h=pI −m mhm m( )(p+ h )A−1 Y0 (p) =0ymy00 + A p+m +hp0ymYm (p) =+p+ mhÇàìåòèì, ÷òî()m−1+ ...y10( mh)(p+ mh )m+ U (p) .(3.33)Y0(p) mhmp+ hy10m +hm 20y10 mY(p)y20hh+m +2m 2+ hp+ mp+hhmm0 m m−10y1 hY0(p) mym−1 hh + + ...m mm mm 2p+p+p+ hhhY1 (p) =Y2 (p) =m0ym−1hm 2p+ hp+m mhm+mh(3.34)mphlim= e−ph .= lim 1 −m→∞ pm→∞ph + m!m0 m m−100yym−1ymh+ ...

1 h m m =m +2p+ hp+ hp+ mh! mm m−1mh00hhh ym+ ym−1+ ...y10m−1mmp+ h mp+ hp+ mh(3.35)(3.36) ñèëó (3.27), (3.35) ìîæíî óòâåðæäàòü î ñëåäóþùåé ñõîäèìîñòè:limmhm→∞p+mhhm00ym+ ym−1=Z0mhp+−p(τ +h)emh + ...y10x0(τ )dτpm m−1hm−1+mh!=(3.37)−hÑîîòâåòñòâåííî ñõîäÿòñÿ äðóã ê äðóãó âûðàæåíèÿ (3.32), (3.33) èçîáðàæåíèéX(p)èY0(p)883.4åãóëÿðèçàöèÿ çàäà÷è ñèíòåçààññìîòðèì ëèíåéíóþ óïðàâëÿåìóþ ñèñòåìó ñ çàïàçäûâàíèåì (1.4)-(2.1)[t0, t1 ] , t1 > t0 + h .íà îòðåçêåñòâîM(·)èç ïðîñòðàíñòâà0C ∈ int M(·) .Çàèêñèðóåì öåëåâîå îãðàíè÷åííîå ìíîæå-C[−h, 0] . Íå îãðàíè÷èâàÿ îáùíîñòè, ñ÷èòàåì, ÷òîÇàèêñèðóåì ðàâíîìåðíî îãðàíè÷åííîå ìíîæåñòâî óïðàâëå-íèé âèäà (1.5).

Íå îãðàíè÷èâàÿ îáùíîñòè, ñ÷èòàåì, ÷òîÔèêñèðóåìε0 ∈ int P (τ ) .- òðåáóåìóþ òî÷íîñòü ïîïàäàíèÿ íà öåëåâîå ìíîæåñòâî. Ôèê-K1 -ñèðóåì ïàðàìåòð ðåãóëÿðèçàöèèìàêñèìàëüíî äîïóñòèìóþ íîðìó íà-÷àëüíîé óíêöèè.Ñîãëàñíî òåîðåìå 30 ñóùåñòâóåò ÷èñëîïðîêñèìàöèèε/2 .Äëÿ ýòîãîmm,îáåñïå÷èâàþùåå òî÷íîñòü àï-ñòðîèì ñèñòåìó (3.12) ñ îãðàíè÷åíèåì íàïðàâîì êîíöå:y(t1 ) ∈ Mm.ÌíîæåñòâîâåêòîðàMmñòðîèòñÿ ïî ìíîæåñòâóy ∈ Mm ,M(·) òàêèì îáðàçîì, ÷òî äëÿ ëþáîãîïîñòðîåííàÿ èç íåãî ëîìàíàÿíà ðàññòîÿíèè îò ìíîæåñòâàM(·)(3.38)ỹ(·) ∈ C[−h, 0]â ïðîñòðàíñòâåC[−h, 0]d(ỹ(·), M(·)) ≤ ε/2.Ïîñëå ýòîãî ñòðîèì ìíîæåñòâî ðàçðåøèìîñòèîíàëüíî óìåíüøàÿ öåëåâîå ìíîæåñòâîMmáóäåò ëåæàòüíå áîëåå ÷åìε/2 :(3.39)Wm[t]ñèñòåìû.

Ïðîïîðöè-è ìíîæåñòâî óïðàâëåíèé, â ñèëóîãðàíè÷åííîñòè ìíîæåñòâà ðàçðåøèìîñòè ëèíåéíîé ñèñòåìû îáûêíîâåííûõäèåðåíöèàëüíûõ óðàâíåíèé, ìîæíî äîáèòüñÿ óñëîâèÿWm [t0] ≤ K1 .Âìå-ñòî ýòîãî ìîæíî âçÿòü â êà÷åñòâå èñêîìîãî ìíîæåñòà ïåðåñå÷åíèå ìíîæåñòâàðàçðåøèìîñòè ñ øàðîì ðàäèóñàK1 , òåì ñàìûì ãàðàíòèðóÿ âûïîëíåíèÿ îãðà-íè÷åíèÿ íà ìàêñèìàëüíóþ íîðìó íà÷àëüíîé óíêöèè.89Òàêèì îáðàçîì, åñëè äâèãàòüñÿ èç òàêîãî ìíîæåñòâà ðàçðåøèìîñòè â ñèëó ñèñòåìû (3.12) ïðè ëþáîì óïðàâëåíèè, óäîâëåòâîðÿþùåì îãðàíè÷åíèþ,òî â ìîìåíòt1ñîîòâåòñòâóþùàÿ ëîìàíàÿỹáóäåò àïïðîêñèìèðîâàòü ðåàëü-íîå ðåøåíèå ñèñòåìû (1.4) ñ ëþáûì íà÷àëüíûì óñëîâèåì, óäîâëåòâîðÿþùèì(3.13)) ñ òî÷íîñòüþìíîæåñòâîMmε/2 . À ïîñòðîåííûé ñèíòåç äëÿ îáûêíîâåííîé ñèñòåìû íàáóäåò îáåñïå÷èâàòü ïîïàäàíèå íà ìíîæåñòâîñèñòåìû (1.4) ñ òî÷íîñòüþ äîε.M(·)ðåøåíèÿëàâà 4Óïðàâëåíèåàïïðîêñèìèðóþùåéñèñòåìîé îáûêíîâåííûõäèåðåíöèàëüíûõóðàâíåíèé äàííîé ãëàâå ðàññìîòðåíû ìåòîäû óïðàâëåíèÿ ñèñòåìîé îáûêíîâåííûõäèåðåíöèàëüíûõ óðàâíåíèé, àïïðîêñèìèðóþùåé ñèñòåìó ñ çàïàçäûâàíèåì.

Îñíîâíûå ìåòîäû è âûðàæåíèÿ, èñïîëüçóåìûå â äàííîé ãëàâå, îïóáëèêîâàíû â ðàáîòå [62℄.90914.1Ìåòîä äèíàìè÷åñêîãî ïðîãðàììèðîâàíèÿÎïðåäåëèâ ïîðÿäîêmäëÿ àïïðîêñèìàöèè ñèñòåìû ñ çàïàçäûâàíèåì ñè-ñòåìîé îáûêíîâåííûõ äèåðåíöèàëüíûõ óðàâíåíèé ìåòîäîì ïðÿìûõ ïîëó÷àåì ñëåäóþùóþ ñèñòåìó.ẋ(t) = A(t)x(t) + B0 (t)u(t)äå ìàòðèöûA(t) ∈ R(m+1)n×(m+1)n ,àB(t) ∈ R(m+1)n×n(4.1)îïðåäåëÿþòñÿ ñëåäó-þùèì îáðàçîìΘ A0(t) Θ m h I −mI ΘhmA(t) =  ΘI −mhhI ...ΘΘΘãäåΘèIìåðíîñòè... ΘA1 (t) ... ΘΘ ... ΘΘ ... .. mm... h I − h I,(4.2)B(t) Θ ,B0(t) =  ... Θ(4.3)- ñîîòâåòñòâåííî íóëåâàÿ è åäèíè÷íàÿ êâàäðàòíûå ìàòðèöû ðàç-n × n.Îãðàíè÷åíèå íà óïðàâëåíèå îñòàþòñÿ ïðåæäíèìè (3.11).Öåëåâîå ìíîæåñòâîMñòðîèòñÿ ñîãëàñíî (3.39).Äëÿ ðåøåíèÿ çàäà÷è óïðàâëåíèÿ ìîæíî âîñïîëüçîâàòüñÿ ìåòîäîì äèíàìè÷åñêîãî ïðîãðàììèðîâàíèÿ.92Äëÿ ýòîãî ââîäèòñÿ óíêöèÿ öåíûV (t, x) = min d2(x(t1), M).(4.4)uÊîòîðóþ ìîæíî àíàëèòè÷åñêè âûïèñàòü èñïîëüçóÿ àïïàðàò âûïóêëîãî àíàëèçà [56℄V (t, x) = max{hX(t1 , t)x, li −lZt1tρ (−B0′ (τ )X ′ (t1 , τ )l| P (τ )) dτ −(4.5)−ρ (l| M) − 1/4 hl, li}Äàííàÿ óíêöèÿ öåíû óäîâëåòâîðÿåò óðàâíåíèþ àìèëüòîíà-ßêîáè-Áåëëìàíà∂V (t, x)∂V (t, x+ min, A(t)x + B0(t)u∂tu∈P (t)∂x(4.6)V (t1, x)t = d2 (x(t1), M)(4.7)Òðåáóåìûé ñèíòåç óïðàâëåíèÿ çäåñü ñîñòîèò èç ìèíèìèçàòîðîâU (t, x) = Arg minu∈P (t)∂V (t, x)B0′ (t),u∂xuâ (4.6).Äëÿ óïðîùåíèÿ âû÷èñëåíèé çàìåòèì, ÷òî óíêöèÿ öåíû ìîæåò áûòü âûðàæåíà ÷åðåç ìíîæåñòâî ðàçðåøèìîñòèW [t] .Ïðèìåíÿÿ ìåòîäû âûïóêëîãîàíàëèçà, ïîëó÷àåì [56℄:V (t, x) = d2 (X(t1, t)x, X(t1, t)W [t]).äåX(t1 , t)(4.8)óíäàìåíòàëüíàÿ ìàòðèöà ñèñòåìû (4.1):∂X(t, τ )= A(t)X(t, τ ),∂tX(τ, τ ) = I.1V (t, x) = max{hX ′ (t1 , t)l, xi − ρ (X ′ (t1 , t)l| W [t]) − hl, li} =l41′′= max{hl, xi − ρ (l| W [t]) − hX (t, t1)l, X (t, t1)li}l4(4.9)93 ñèëó ñèëüíîé âûïóêëîñòè ìàêñèìèçàòîð åäèíñòâåííûé, ïîýòîìó ìîæíî âûðàçèòü ïîëíóþ ïðîèçâîäíóþ óíêöèè öåíû, èñïîëüçóÿ òåîðåìó î äèåðåíöèðîâàíèè óíêöèè ìàêñèìóìà [9℄.

Ïóñòül0åäèíñòâåííûé ìàêñèìèçàòîð ââûðàæåíèè (4.9), òîãäà ïðîèçâîäíàÿÏîýòîìó ddV (t, x(t)) 0= l , A(t)x(t) + B0(t)u(t) − ρ l0 W [t] −dtdtd 1′ 00X(t, t1) l , X(t, t1)l .−dt 4U (t, x) = Arg min B0′ (t)l0, u .(4.10)u∈P (t)Çàìåòèì, ÷òî ñòðàòåãèÿ óïðàâëåíèÿU (t, x)ÿâëÿåòñÿ ìíîãîçíà÷íûì îòîáðà-æåíèåì, ïîýòîìó, óðàâíåíèå (4.1) ïðåâðàùàåòñÿ â äèåðåíöèàëüíîå âêëþ÷åíèåẋ(τ ) ∈ A(τ )x(τ ) + B0 (τ )U (t, x(τ )),τ ∈ [t0 , t1],(4.11)Íî ïîñêîëüêó ðàçìåðíîñòü ñèñòåìû âåëèêà, äàííûå âûðàæåíèÿ, íåñìîòðÿ íà ñâîé ÿâíûé âèä, îáëàäàþò áîëüøîé âû÷èñëèòåëüíîé ñëîæíîñòüþ. Íîåñëè çàìåíèòü òî÷íîå ìíîæåñòâîîöåíêóW[ t]W [t]íà åãî âíóòðåííþþ ýëëèïñîèäàëüíóþòî âûðàæåíèÿ ñóùåñòâåííî óïðîñòÿòñÿ.Âûâîä óðàâíåíèé ýëëèïñîèäàëüíîé àïïðîêñèìàöèè [56, 57℄, îñíîâàí íàëþöèîííîì óðàâíåíèèäëÿ ìíîæåñòâà ðàçðåøèìîñòè è åãî àïïðîêñèìàöèé:lim σ −1h+ (W [t − σ], (I − σA)W [t] − σB(t)P (t)) = 0,σ→0+ñ êîíå÷íûì óñëîâèåìBr (0)}ýâî-W [t1] = M .Çäåñü(4.12)h+ (X, Y ) = min{r > 0|X ⊆ Y + ïîëóìåòðèêà Õàóñäîðà â ïðîñòðàíñòâå êîìïàêòíûõ ïîäìíîæåñòâR(m+1)n .

Íàèáîëüøåå ïî âêëþ÷åíèþ ðåøåíèå (4.12) ñîâïàäàåò ñî ìíîæåñòâîìðàçðåøèìîñòèW [t] .Äëÿ öåëåé ñèíòåçà óïðàâëåíèé âàæíî ñëåäóþùåå ñâîéñòâî ðåøåíèé ýâîëþöèîííîãî óðàâíåíèÿ [56℄:94Òåîðåìà 31.(4.12),Ïóñòü Z[t] òàêîå ðåøåíèå ýâîëþöèîííîãî óðàâíåíèÿ÷òî óíêöèÿ ρ (l| Z[t]) äèåðåíöèðóåìà ïî t äëÿ ëþáîãî l ∈ R(m+1)n .Òîãäà óíêöèÿZ(t, x) = d2(X(t1 , t)x, X(t1, t)Z[t])óäîâëåòâîðÿåò äèåðåíöèàëüíîìó íåðàâåíñòâódZ(t, x(t))= Zt + min hZx , A(t)x + ui ≤ 0.u∈P (t)dtu∈P (t)min(4.13)Êàê ñëåäñòâèå (4.13) âûïîëíÿåòñÿ ñëåäóþùåå ñâîéñòâî ïîçèöèîííîãî óïðàâëåíèÿ, îïðåäåë¼ííîãî êàê ìíîæåñòâî ìèíèìèçàòîðîâ â (4.13):UZ (t, x) = Arg minhZx , B(t)ui.(4.14)u∈P (t)Åñëè íà÷àëüíàÿ òî÷êàx(t0)(4.11) íàõîäèòñÿ âíóòðèòðóáêå.òðàåêòîðèè äèåðåíöèàëüíîãî âêëþ÷åíèÿZ[t0] , òî îñòàëüíàÿ ÷àñòü òðàåêòîðèèòàêæå ëåæèò âZ[t0] .

Ïîñëåäíåå âåðíî, ïîñêîëüêó ðàññòîÿíèå îò x(t) äî Z[t] ÿâëÿåòñÿíåâîçðàñòàþùåé óíêöèåé.Ñëåäîâàòåëüíî, åñëè âíóòðåííÿÿ àïïðîêñèìàöèÿ ìíîæåñòâà ðàçðåøèìîñòè ÿâëÿåòñÿ ðåøåíèåì ýâîëþöèîííîãî óðàâíåíèÿ, òî ïîçèöèîííàÿ ñòðàòåãèÿ(4.14) ðåøàåò çàäà÷ó öåëåâîãî óïðàâëåíèÿ íà ìíîæåñòâî÷àëüíûõ ñîñòîÿíèé èçZ[t0] . Ïðè ýòîì óïðàâëåíèå ìîæåòîðìóëàì (4.9)(4.10) ñ çàìåíîéW [t]íàZ[t] .äëÿ âñåõ íà-áûòü âû÷èñëåíî ïîÒàêóþ ñòðàòåãèþ óïðàâëåíèÿìîæíî èíòåðïðåòèðîâàòü êàê ïðèöåëèâàíèå íà òðóáêó4.2MZ[t] .Ýëëèïñîèäàëüíûé ñèíòåçÍàëîæèì íà ìíîæåñòâî óïðàâëåíèÿ è öåëåâîå ìíîæåñòâî ýëëèïñîèäàëüíûåîãðàíè÷åíèÿ.u(t) ∈ E(p(t), P (t))(4.15)95M = E(x1, X1 ).(4.16) ýòîì ñëó÷àå ìíîæåñòâî ðàçðåøèìîñòè ïðåäñòàâèìî â âèäå [57℄[W [t] ={E(x−(t), X−(t))}(4.17)kl0 k=1ẋ−(t) = A(t)x−(t) + B0(t)p(t),(4.18)x−(t1) = x1;X− (t) = Q∗ (t)′Q∗(t),Q̇∗(t) = Q(t)A(t) − S(t)[B0(t)P (t)B0′ (t)]1/2,1/2Q (t1) = X11/2S(t)(B0(t)P (t)B0′ (t))1/2l∗(t) = λ(t)X− l∗(t),(4.19)∗Äèåðåíöèàëüíîå óðàâíåíèå äëÿ óíêöèèS T (t)S(t) = I.X− (t)ïðèìåò âèäẊ− (t) = AX−(t) + X− (t)AT +1/21/2+X− (t)S(t)(B0(t)P (t)B0′ (t))1/2 + (B0(t)P (t)B0′ (t))1/2S ′ (t)X− (t),X− (t1 ) = 0.(4.20)Ïîëó÷åííàÿ òàêèì îáðàçîì ýëëèïñîèäàëüíàÿ àïïðîêñèìàöèÿ êàñàåòñÿ ìíîæåñòâà ðàçðåøèìîñòè âäîëü õîðîøèõ íàïðàâëåíèél∗(t) = X ′ (t0 , t)l0 :ρ (l∗(t)| E(x−(t), X−(t))) = ρ (l∗(t)| W [t]) .(4.21)Ïðè ýòîì óïðàâëåíèå ìîæåò áûòü âû÷èñëåíî ïî îðìóëàì (4.9)(4.10) ñçàìåíîéW [t]íàE(x−(t), X−(t)) ..Çàìåòèì, ÷òî ïðè êàæäîì èêñèðîâàííîìãî ìíîæåñòâàíàïðàâëåíèéW [t]l0ïðîèñõîäèò êàñàíèå òî÷íî-è âíóòðåííåé ýëëèïñîèäàëüíîé îöåíêè âäîëü õîðîøèõl∗(t) = X ′ (t0, t)l0 .ṼE (t, x) = max{hl, xi − ρ (l| E(x−(t), X−(t))) −l1hl, li}4(4.22)96â êà÷åñòâå íà÷àëüíîãîxâçÿòü òî÷êó êàñàíèÿ òî÷íîãî ìíîæåñòâà è âíóòðåí-íåãî ýëëèïñîèäà. ñëó÷àå ýëëèïñîèäàëüíîé àïïðîêñèìàöèè ìàêñèìèçàòîðl0â îðìóëå(4.9), íåîáõîäèìûé äëÿ âû÷èñëåíèÿ óïðàâëåíèÿ, ìîæåò áûòü íàéäåí êàêl0 = 2λ(X−(t) + λF (t))−1(x(t) − x− (t)),′(4.23)F (t) = X (t, t1)X(t, t1),ãäåλ åäèíñòâåííûé íåîòðèöàòåëüíûé êîðåíü óðàâíåíèÿhX− (t) + λF (t))−1(x(t) − x−(t)), X−(t)(X−(t) + λF (t))−1(x(t) − x− (t))i = 1,(4.24)èëèl0 = 0 ,åñëè (4.24) íå èìååò íåîòðèöàòåëüíûõ ðåøåíèé.Íèæå ïðèâåäåíû ãðàè÷åñêèå èëëþñòðàöèè.Ýëëèïñîèäàëüíûå îöåíêè ìíîæåñòâà ðàçðåøèìîñòè:Óïðàâëåíèå è îòêëîíåíèå íîðìû ðåøåíèÿ îò íà÷àëà êîîðäèíàò:97Çàêëþ÷åíèåÑîðìóëèðóåì êðàòêî îñíîâíûå ðåçóëüòàòû ðàáîòû:1.