С.В. Конягин - Вариационное исчисление и оптимальное управление, страница 3

Àíàëîãè÷íî äëÿ ïðàâîãî ñåìåéñòâà t0 (α) = τ + α; t1 (α) =(r)(r)t1 ; x0 (α) = x̂(τ ); x1 (α) = x1 ; x(r) (t, 1) = x̂(t), t ∈ [τ, t1 ] (ñì. ðèñ 1à). Ìû òðåáóåì(·)÷òîáû ó x (t, α) áûëè íåïðåðûâíûå ÷àñòíûå è ñìåøàííûå ïðîèçâîäíûå è, çíà÷èò,Ââåäåì îáîçíà÷åíèÿ:äëÿ íèõ âûïîëíÿåòñÿ òåîðåìà î âàðèàöèè èíòåãðàëüíîãî óíêöèîíàëà.12(a)x̂t0τ(b)x̂t1t0èñ. 1: Ñåìåéñòâà óíêöèé äëÿP (·)τ(a) èt1H(·)(b)Òåïåðü ðàññìîòðèì äâà èíòåãðàëà:(l)t1Z (α)L(t, x(l) (t, α), ẋ(l) (t, α))dt;J(l) (α) =(l)t0 (α)(r)t1Z (α)L(t, x(r) (t, α), ẋ(r) (t, α))dt;J(r) (α) =(r)t0 (α)Ò.ê.

ñîîòâåòñòâóþùèå àáñöèññû è îðäèíàòû ñîâïàäàþò, òî ñåìåéñòâà êðèâûõ "left" è"right" ñêëåèâàþòñÿ, ïîýòîìó ìû ìîæåì ðàññìîòðåòü íîâîå ñåìåéñòâî âèäà:(x(l) (t, α), åñëè t ≤ τ + α;x(t, α) =x(r) (t, α), åñëè t ≥ τ + α;Ïðè ýòîì′∀α x(t, α) ∈ P C [t0 , t1 ]J(l) (α) + J(r) (α) =Zt1kx(t, α) − x̂(t)kC → 0èïðèα → 0.ÒîãäàL(t, x(t, α), ẋ(t, α))dt = J(x(·, α)) = F (α)t0x̂(·) äîñòàâëÿåò ñèëüíûé ýêñòðåìóì, òî óíêöèÿ F (α) äîñòèãàåò ýêñòðåìóìà′′′′α = 0, ñëåäîâàòåëüíî ïî òåîðåìå Ôåðìà F (0) = 0 ⇒ F (0) = J(l) (0)+J(r) (0) = {ïîòåîðåìå î âàðèàöèè èíòåãðàëüíîãî óíêöèîíàëà} = −H(τ − 0) + H(τ + 0).

Èòàê,H(τ − 0) = H(τ + 0) è íåïðåðûâíîñòü H ïîêàçàíà. Ïðàâäà ïîêà íåÿñíî, ñóùåñòâóþòÅñëèïðèëè ðàññìîòðåííûå ñåìåéòñâà "left" è "right". Çàäàäèì èõ ÿâíûì îáðàçîì:x(l) (t, α) = x̂(t) −t − t0(x̂(τ + α) − x̂(τ ))τ + α − t0x(l) (τ + α, α) = x̂(τ + α) −τ + α − t0(x̂(τ + α) − x̂(τ )) = x̂(τ )τ + α − t0Ëåãêî ïðîâåðèòü, ÷òî ýòî ñåìåéñòâî óäîâëåòâîðÿåò âñåì óñëîâèÿì íà ãëàäêîñòü.x̂ ∈ C 2 .

Àíàëîãè÷íî ñòðîèòñÿ è ñåìåéñòâî "right".Çäåñü-òî è ïîíàäîáèòñÿ óñëîâèå13Ñëåäîâàòåëüíî ìû ïîêàçàëè, ÷òî äëÿH(τ + 0).x̂(·) ∈ C 2 [t0 , τ ], x̂(·) ∈ C 2 [τ, t1 ] H(τ − 0) =x̂(·) ∈ C 1 [t0 , τ ], x̂(·) ∈ C 1 [τ, t1 ] è äîêàæåì, ÷òî p(τ − 0) =(l)(l)p(τ + 0). Ñíîâà áóäåì ñòðîèòü 2 ñåìåéñòâà êðèâûõ (ðèñ. 1á): t0 (α) = t0 ; t1 (α) = τ ;(l)(l)(r)(r)x0 (α) = x0 ; x1 (α) = x̂(τ ) + α; x(l) (t, 0) = x̂(t), t ∈ [t0 , τ ] è t0 (α) = τ ; t1 (α) = t1 ;(r)(r)x0 (α) = x̂(τ ) + α; x1 (α) = x1 ; x(r) (t, 0) = x̂(t), t ∈ [τ, t1 ]. Òîãäà ïîëó÷àåìÏðåäïîëîæèì òåïåðü, ÷òî′0 = F (0) = p(τ − 0) − p(τ + 0)⇒ p(τ − 0) = p(τ + 0). ÿâíîì âèäå ñåìåéñòâî "left" çàäàåòñÿ ñëåäóþùèì îáðàçîì:x(l) = x̂(t) + αt − t0.τ − t0ñëó÷àé: ïóñòü τ - íå åäèíñòâåííàÿ òî÷êà ðàçðûâà ïðîèçâîäíîé. ⇒ Îáùèé∃ t̃0 , t̃1 ⊂ [t 0 , t1 ] : x̂(·) ∈ C 1 t̃0 , τ , x̂(·) ∈ C 1 τ, t̃1 . x̂ äîñòàâëÿåò ñèëüíûé ýêñòðåìóìíà îòðåçêå t̃0 , t̃1 ⇒ çàäà÷à ñâåäåíà ê óæå ðàçîáðàííîé âûøå.

Çàìå÷àíèå 6.2 Ýòà òåîðåìà ðàáîòàåò è äîêàçûâàåòñÿ àíàëîãè÷íî è äëÿâåêòîðíûõ óíêöèé.Íåîáõîäèìûå è äîñòàòî÷íûå óñëîâèÿâòîðîãî ïîðÿäêà äëÿ ñëàáîãî ýêñòðåìóìà âïðîñòåéøåé çàäà÷å êëàññè÷åñêîãîâàðèàöèîííîãî èñ÷èñëåíèÿ.7ÏóñòüL, Lx , Lẋ , Lxx , Lxẋ , Lẋẋ- íåïðåðûâíû âçàäà÷ó:J(x(·)) =Zt1U ⊂ R3èL = L(t, x, ẋ).àññìîòðèìL(t, x(t), ẋ(t))dt −→ min(max)t0x(t0 ) = x0 ; x(t1 ) = x1 ;Ïóñòüx̂- ñëàáûé ìèíèìóì.h(·) ∈ C01 [t0 , t1 ]h(·) ∈ C 1èh(t0 ) = h(t1 ) = 0.ðàçëîæèì L â ðÿä Òåéëîðà:˙ + αḣ(t) = L̂(t) + αL̂x (t)h(t) + αL̂ẋ (t)ḣ(t)+L t, x̂(t) + αh(t), x̂(t)àññìîòðèìJ(x̂(·) + αh(·)) = F (α),ò.å+α2α2L̂xx (t)h2 (t) + α2 L̂ẋx (t)h(t)ḣ(t) + L̂ẋẋ (t)ḣ2 (t) +22o(α2 )| {z }ðàâíîìåðíî ïîÑëåäîâàòåëüíîF (α) =Zt1t0Zt1 L̂x (t)h(t) + L̂ẋ (t)ḣ(t) dt+L̂(t)dt + αt014tα2+2Zt1 t0Êàê ìû çíàåì,L̂xx (t)h2 (t) + 2L̂ẋx (t)h(t)ḣ(t) + L̂ẋẋ (t)ḣ2 (t) dt + o(α2 )Rt1 L̂x (t)h(t) + L̂ẋ (t)ḣ(t) dt = δJ(x̂(·), h(·)) = 0t0Ââåäåì îáîçíà÷åíèÿ:A(t) := L̂xx (t); B(t) := L̂ẋx (t); C(t) := L̂ẋẋ ;Zt1 22Ah + 2Bhḣ + C ḣ dtK(h(·)) =t0⇒ F (α) = F (0) +x̂(·)α22K(h(·)) + o(α2 ) ⇒ limα→0- ñëàáûé ìèíèìóìF (α)−F (0)α2= 21 K(h(·))⇒ F (α) ≥ F (0) ⇒1F (α) − F (0)= K(h(·))2α→0α20 ≤ limÌû ïîëó÷èëè íåîáõîäèìîå óñëîâèå ñëàáîãî ìèíèìóìà (ìàêñèìóìà):∀h(·) ∈ C01 [t0 , t1 ] K(h(·)) ≥ 0 (ñîîòâåòñòâåííî K(h(·)) ≤ 0).

Íî ýòîóñëîâèå ñëîæíîÌû áóäåì èñïîëüçîâàòü ñëåäóþùóþ èäåþ: ðàññìîòðèì óíêöèþω(·) ∈ C 1 [t0 , t1 ]ïðîâåðÿòü, ïîçæå ïîëó÷èì áîëåå óäîáíûå óñëîâèÿ.Zt1t0⇒Zt1t0′ω(t)h2 (t) dt = ω(t1 )h2 (t1 ) − ω(t0 )h2 (t0 ) = 0′ω(t)h2 (t) dt =Zt1 ω̇(t)h2 (t) + 2ω(t)h(t)ḣ(t) dt = 0t0Ñëåäîâàòåëüíî, íåîáõîäèìîå óñëîâèå ìîæíî ïåðåïèñàòü ñëåäóþùèì îáðàçîì:Zt1 22(A(t) + ω̇(t)) h (t) + 2 (B(t) + ω(t)) h(t)ḣ(t) + C(t)ḣ (t) dt ≥ 0K(h(·)) =t01Ìû ìîæåì âûáèðàòü w(·) ∈ C [t0 , t1 ] òàê, êàê íàì íðàâèòñÿ ⇒ ïîäáåðåì w(·) òàê,2÷òîáû (B(t) + w(t)) = (A(t) + ẇ(t))C(t) (C(t) 6= 0). Ýòî âûïîëíÿåòñÿ òîãäà è òîëüêîòîãäà, êîãäàw(·)- ðåøåíèå óðàâíåíèÿ èêêàòèẇ(t) =(B(t) + w(t))2− A(t)C(t)Òîãäà, êàê íåñëîæíî ïðîâåðèòü,K(h(·)) =Zt1t0C(t)2B(t) + w(t)h(t) + ḣ(t) dtC(t)15Îäíàêî,C(t)ìîæåò îáðàùàòüñÿ â íóëü íà[t0 , t1 ];òî ðåøåíèå óðàâíåíèÿ èêêàòè íà âñåì îòðåçêåäàæå åñëè ýòîãî íå ïðîèñõîäèò,[t0 , t1 ]ìîæåò íå ñóùåñòâîâàòü.Ýåêòèâíûì îêàçûâàåòñÿ èñïîëüçîâàíèå ëîêàëüíîãî ðåøåíèÿ óðàâíåíèÿ èêêàòè.Òåîðåìà 7.1 x̂(·) - ñëàáûé ìèíèìóì (ñëàáûé ìàêñèìóì) ⇒ ∀t ∈ [t , t ] L̂(L̂ẋẋ(t) ≤ 0 ñîîòâåòñòâåííî) (óñëîâèå Ëåæàíäðà)01ẋẋ (t)≥0τ ∈ [t0 , t1 ] : L̂ẋẋ (t) < 0.V ∋ τ : ∀t ∈ V C(t) = L̂ẋẋ (t) < 0.

 ýòîéÄîêàçûâàåì òåîðåìó äëÿ ìèíèìóìà. Ïóñòü ñóùåñòâóåòÒîãäà ñóùåñòâóåò è íåêîòîðàÿ îêðåñòíîñòüîêðåñòíîñòè èìååò ñìûñë óðàâíåíèå èêêàòè. Áîëåå òîãî, ïî òåîðåìå ñóùåñòâîâàíèÿ∃(t̃0 , t̃1 ) : τ ∈ (t̃0 , t̃1 ): íà [t̃0 , t̃1 ] ñóùåñòâóåò ðåøåíèå óðàâíåíèÿ èêêàòè.1ðàññìîòðèì h(·) ∈ C0 [t0 , t1 ]: h(t) ≡ 0 ïðè t ≤ t̃0 , t ≥ t̃1 . Èìååì C(t) < 0 èèç äèóðîâÒîãäà0 ≤ K(h(·)) =Zt̃1C(t)t̃02B(t) + w(t)h(t) + ḣ(t) dt ≤ 0C(t)Ò.î. ýòîò èíòåãðàë ðàâåí 0 è, çíà÷èò, ïîäèíòåãðàëüíîå âûðàæåíèå ðàâíî 0.ẇ(t)− A(t)+h(t) = ḣ(t) íà [t̃0 , t̃1 ].

Íî ïðîèçâîëüíàÿ h íå îáÿçàíàC(t)óäîâëåòâîðÿòü òàêîìó äèåðåíöèàëüíîìó óðàâíåíèþ. Ïðîòèâîðå÷èå. Ñëåäîâàòåëüíî,Çàïèøåì óðàâíåíèå Ýéëåðà äëÿ óíêöèîíàëà−K(h(·)) =R t1t0(Ah2 + 2Bhḣ + C ḣ2 )dt:d 2B(t)h(t) + 2C(t)ḣ(t) + 2A(t)h(t) + 2B(t)ḣ(t) = 0dtB T (t)). Ýòî óðàâíåíèåðåøåíèå, h(t0 ) = 0, h(·) - íå(â âåêòîðíîì ñëó÷àå äëÿ ïîñëåäíåãî ñëàãàåìîãî èìååìíàçûâàåòñÿ óðàâíåíèåì ßêîáè.

Ïóñòüh(·)- åãîòîæäåñòâåííûé íîëü. ÒîãäàÎïðåäåëåíèå 7.1 òî÷êà τ > t - ñîïðÿæåííàÿ (ñ òî÷êîé t ), åñëè h(τ ) = 000Äîêàæåì ñëåäóþùåå íåîáõîäèìîå óñëîâèå ñëàáîãî ýêñòðåìóìà:Òåîðåìà 7.2 Ïóñòü ẋ(·) - ñëàáûé ìèíèìóì (ìàêñèìóì), ∀t ∈ [t , t ] L̂ẋẋ (t)(L̂ẋẋ(t) < 0 ñîîòâåòñòâåííî). Òîãäà (t0, t1) íå ñîäåðæèò ñîïðÿæåííûõ òî÷åê.01> 0Àíàëîãè÷íàÿ òåîðåìà âåðíà è äëÿ ìíîãîìåðíîãî ñëó÷àÿ. Ïðàâäà òîãäà íåîáõîäèìàh(·), òîãäà êàê â îäíîìåðíîì ñëó÷àå äîñòàòî÷íîïðîâåðêà äëÿ ïðîèçâîëüíîé óíêöèèðàññìîòðåòü êîíêðåòíóþ óíêöèþ â ñèëó îäíîìåðíîñòè ïðîñòðàíñòâà ðåøåíèéäèóðàâíåíèÿ ßêîáè.ĥ(·) - ðåøåíèåíîëü, τ ∈ (t0 , t1 ).Ñíîâà äîêàçûâàåì äëÿ ìèíèìóìà è ñíîâà îò ïðîòèâíîãî. Ïóñòüóðàâíåíèÿ ßêîáè,ĥ(t0 ) = ĥ(τ ) = 0, ĥ(·)- íå òîæäåñòâåííûéÏîëîæèìK̃(h(·)) =Zτ(Ah2 + 2Bhḣ + C ḣ2 )dtt0ÿâëÿåòñÿ ýêñòðåìàëüþ äëÿ K̃(·), ò.ê.1óðàâíåíèÿ ßêîáè.

Çíà÷èò, ∀h(·) ∈ C0 [t0 , τ ] δ K̃(αĥ(·), h(·)) =Çàìåòèì, ÷òîK̃(αĥ(·)).∀α αĥ(·)Òîãäà16αĥ(·) - òîæå ðåøåíèå0. Ïîëîæèì f (α) =K̃(αĥ(·) + εĥ(·)) − K̃(αĥ(·))= δ K̃(αĥ(·), ĥ(·)) = 0, f (0) = 0ε→0εf ′ (α) = limÇíà÷èò,f ≡ 0, K̃(ĥ(·)) = 0 Íàì ïîíàäîáèòñÿ ñëåäóþùàÿ ëåììà (î ñêðóãëåíèè óãëîâ), êîòîðóþ ìû ïðèâåäåìáåç äîêàçàòåëüñòâà:Ëåììà 7.1 ðàññìîòðèì çàäà÷óK(h(·)) =Zt1M (t, h(t), ḣ(t))dt → mint0h(t0 ) = h0 , h(t1 ) = h1 , h̃(·) ∈ C01 [t0 , t1 ]1) Åñëè M ∈ C([t0, t1] × R2), K(h̃(·)) = h(·)∈Cmin[t ,t ] K(h(·)), òî10 0 1K(h̃(·)) =minh(·)∈P C01 [t0 ,t1 ]K(h(·))2) Åñëè U îêðåñòíîñòü ãðàèêà {(t, h̃(t)) : t ∈ [t0, t1]},K(h(·)), òîK(h̃(·)) =minh(·)∈C [t ,t ]10 0 1kh(·)−h̃(·)kC[t0 ,t1 ] <εK(h̃(·)) =minh(·)∈P C01 [t0 ,t1 ]kh(·)−h̃(·)kC[t0 ,t1 ] <εK(h(·))Èç ýòîé ëåììû ïîëó÷àåì ñëåäóþùèé ïîëåçíûé âûâîä: ïóñòüäîñòàâëÿåò ñëàáûé ìèíèìóì, ò.åK(h̃(·)) =minh(·)∈P C01 [t0 ,t1 ]Ïóñòü òåïåðüÒîãäà èmin1h(·)∈C0 [t0 ,t1 ]K(h(·))ÒîãäàK(h(·)) = 0(ĥ(t), t ∈ [t0 , τ ]h̃(t) =0, t > τK(h̃(·)) = 0,ò.å.

è íà óíêöèèK(h(·)) =Zt1h̃(·)äîñòèãàåòñÿ ñèëüíûé ìèíèìóì.(A(t)h2 (t) + 2B(t)h(t)ḣ(t) + C(t)ḣ2 (t))dtt0Ìû èìååì:h̃(·) ≡ 0, h(·)˙ . t > τ ⇒ p(t) = 0.p(t) = 2B(t)h̃(t) + 2C(t)h̃(t)˙ − 0) =lim p(t) = 2B(τ )h̃(τ ) + 2C(τ )h̃(τt→τ,t<τ17= {h̃äîñòàâëÿåò ñèëüíûé ìèíèìóì, ïî òåîðåìå Âåéåðøòðàññà-Ýðäìàíà}=˙= 2C(τ )ĥ(τ ) = 0Çíà÷èò,˙ĥ(τ ) = 0 ïî óñëîâèÿì Ëåæàíäðà. Íî ýòî íåâîçìîæíî, ò.ê. òîãäà ĥ(·) ≡ 0 ïîòåîðåìå ñóùåñòâîâàíèÿ è åäèíñòâåííîñòè ðåøåíèÿ ëèíåéíîãî îäíîðîäíîãî óðàâíåíèÿâòîðîãî ïîðÿäêà.Ïåðåéäåì òåïåðü ê äîñòàòî÷íûì óñëîâèÿì.Òåîðåìà 7.3 Ïóñòü x̂(·) - äîïóñòèìàÿ ýêñòðåìàëü, ∀t ∈ [t0, t1] L̂ẋẋ(t) > 0. Ïóñòüíà (t0, t1] íåò ñîïðÿæåííûõ òî÷åê. Òîãäà x̂(·) äîñòàâëÿåò ñëàáûé ìèíèìóì.Çàìåòèì, ÷òî åäèíñòâåííîå îòëè÷èå îò íåîáõîäèìîãî óñëîâèÿ - ýòî òî, ÷òî òåïåðüòðåáóåòñÿ, ÷òîáû è òî÷êàt1íå áûëà ñîïðÿæåííîé.Äëÿ äîêàçàòåëüñòâà ïîòðåáóþòñÿ ñëåäóþùèå ëåììû:Ëåììà 7.2 Ñóùåñòâóåò òàêîå ðåøåíèå h(·) óðàâíåíèÿ ßêîáè, ÷òî ∀th(t) > 0∈ [t0 , t1 ] Êàê ìû çíàåì, ñóùåñòâóåò ðåøåíèå h0 óðàâíåíèÿ ßêîáè: ∀t ∈ (t0 , t1 ] h0 (t) >0, h0 (t0 ) = 0, ḣ0 (t0 ) = 1.

àññìîòðèì ðåøåíèå h1 (·) óðàâíåíèÿ ßêîáè: h1 (t0 ) =1, ḣ1 (t0 ) = 0. ∃τ ∈ (t0 , t1 ) : min h1 (t) > 0. Ïîëîæèì δ = min h0 (t) > 0,t∈[t0 ,τ ]M = max |h1 (t)|.t∈[τ,t1 ]Âûáåðåìαòàê, ÷òîáût∈[τ,t1 ]0<α<δ.MÒîãäà ïîëîæèì h(t) = h0 (t) + αh1 (t). Ïðè t ∈ [t0 , τ ]t ∈ [τ, t1 ] h(t) ≥ h0 (t) − α|h1 (t)| ≥ δ − αM > 0 èìååìh(t) > 0 + 0 = 0,à ïðèÍàéäåì ðåøåíèå óðàâíåíèÿ èêêàòè.

Ïî ïðåäûäóùåé ëåììå íà âñåì îòðåçêåḣ(t). Çíà÷èò, èìååò ñìûñë ñëåäóþùàÿ ëåììà:îïðåäåëåíî îòíîøåíèåh(t)Ëåììà 7.3 w(t) = −B(t) − C(t) - ðåøåíèå óðàâíåíèÿ èêêàòè.ḣ(t)h(t)Óìíîæèì ðàâåíñòâî èç óñëîâèÿ ñëåâà è ñïðàâà íàh(t):w(t)h(t) = −B(t)h(t) − C(t)ḣ(t)d(w(t)h(t)) = ẇ(t)h(t) + w(t)ḣ(t)dtÑ äðóãîé ñòîðîíû â ñèëó óðàâíåíèÿ ßêîáèdd(w(t)h(t)) = (−B(t)h(t) − C(t)ḣ(t)) = −A(t)h(t) − B(t)ḣ(t)dtdtÏðèðàâíèâàÿ ïîëó÷åííûå âûðàæåíèÿ äëÿd(w(t)h(t)), ïîëó÷àåìdt(ẇ(t) + A(t))h(t) = (−w(t) − B(t))ḣ(t)18ÑëåäîâàòåëüíîB(t) + w(t)ḣ(t)− A(t)− A(t) = −(B(t) + w(t)) −ẇ(t) = − (B(t) + w(t))h(t)C(t)è, çíà÷èò,w(t)óäîâëåòâîðÿåò óðàâíåíèþ èêêàòè.Ïðîäîëæèì äîêàçàòåëüñòâî äîñòàòî÷íîãî óñëîâèÿ. àçëîæèìḣ(t))˙L(t, x̂(t)+h(t), x̂(t)+ïî îðìóëå Òåéëîðà:˙L(t, x̂(t) + h(t), x̂(t)+ ḣ(t)) = L̂(t) + L̂x (t)h(t) + L̂ẋ (t)ḣ(t) + 21 L̂xx (t)h2 (t) +L̂ẋx (t)h(t)ḣ(t) + 21 L̂ẋẋ (t)ḣ2 (t) + o(h2 (t) + ḣ2 (t)), ïîñëåäíåå ñëàãàåìîå ðàâíîìåðíî ïî t22˙ïðè h (t)+ ḣ (t) → 0.

Îòñþäà L(t, x̂(t)+h(t), x̂(t)+ḣ(t)) ≥ L̂(t)+ L̂x (t)h(t)+ L̂ẋ (t)ḣ(t)+11222L̂(t)h(t)+L̂(t)h(t)ḣ(t)+L̂(t)ḣ(t)−δ(h(t)+ḣ2 (t)) êàê òîëüêî kh(·)kC 1 [t0 ,t1 ] < εẋx2 xx2 ẋẋÈòàê,1≥2Ïîêàæåì,R t1 t0Zt1t0J(x̂(·) + h(·)) − J(x̂(·)) ≥(A(t) − 2δ)h2 (t) + 2B(t)h(t)ḣ(t) + (C(t) − 2δ)ḣ2 (t)dt÷òîýòîíåîòðèöàòåëüíî. âûðàæåíèåω̇(t)h (t) + 2ω(t)h(t)ḣ(t) dt = 0, ïîëó÷àåì, ÷òî äîñòàòî÷íî2Zt1t0Âñïîìèíàÿ,÷òîïîêàçàòü, ÷òî(A(t) − 2δ + ẇ(t))h2 (t) + 2(B(t) + w(t))h(t)ḣ(t) + (C(t) − 2δ)ḣ2 (t))dt ≥ 0(B(t)+w(t))2− (A(t) − 2δ) ìîäèèêàöèþ óðàâíåíèÿC(t)−2δ- åãî ðåøåíèå.