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Ïðè ýòîì ïðåäïîëàãàåòñÿ, ÷òî Y0 - âåùåñòâåííîå ãèëüáåðòîâî ïðîñòðàíñòâî (·, ·)0 è k · k0 - ñêàëÿðíîå ïðîèçâåäåíèå è íîðìà ñîîòâåòñòâåííî. Ïðåäïîëîãàåòñÿ òàê æå, ÷òî A : D(A) ⊂ Y0 → Y0 - çàìêíóòûéíåîãðàíè÷åííûé ïëîòíî îïðåäåëåííûé ëèíåéíûé îïåðàòîð. Ãèëüáåðòîâî ïðîñòðàíñòâî Y1 îïðåäåëÿåòñÿ êàê D(A) è ñíàáæàåòñÿ ñêàëÿðíûì ïðîèçâåäåíèåì(y, η)1 := ((βI − A)y, (βI − A)η)0 ,y, η ∈ D(A) ,(22)13ãäå β ∈ ρ(A) ∩ R - ïðîèçâîëüíîå ôèêñèðîâàííîå ÷èñëî, ñóùåñòâîâàíèå êîòîðîãî ïðåäïîëàãàåòñÿ (ρ(A) - ðåçîëüâåíòíîå ìíîæåñòâî îïåðàòîðà A).Ãèëüáåðòîâî ïðîñòðàíñòâî Y−1 îïðåäåëÿåòñÿ êàê çàìûêàíèå ïðîñòðàíñòâà Y0 îòíîñèòåëüíî íîðìû kyk−1 := k(βI − A)−1 yk0 . Ïðåäïîëàãàåòñÿ, ÷òîìû èìååì ïëîòíûå è íåïðåðûâíûå âëîæåíèÿ(23)Y1 ⊂ Y0 ⊂ Y−1 .Ñêîáêà äâîéñòâåííîñòè (·, ·)−1,1 íà Y−1 × Y1 îïðåäåëÿåòñÿ îäíîçíà÷íûì ïðîäîëæåíèåì ïî íåïðåðûâíîñòè ôóíêöèîíàëîâ (·, y)0 ñ y ∈ Y1 íà Y−1 .
Äëÿïðîèçâîëüíîãî ÷èñëà T > 0 îïðåäåëèì íîðìó äëÿ èçìåðèìûõ ïî Áîõíåðóôóíêöèé èç L2 (0, T ; Yj ), j = 1, 0, −1 ñ ïîìîùüþ ôîðìóëû ZTkyk2,j :=1/2ky(t)k2j dt.(24)0 äàëüíåéøåì òàêàÿ òðîéêà ïðîñòðàíñòâ Y1 ⊂ Y0 ⊂ Y−1 íàçûâàåòñÿ ãèëüáåðòîâîé.Ïóñòü WT - ïðîñòðàíñòâî ôóíêöèé y(·) ∈ L2 (0, T ; Y1 ), äëÿ êîòîðûõẏ(·) ∈ L2 (0, T ; Y−1 ), ñíàáæåííîå íîðìîéky(·)kWT := ky(·)k22,1 + kẏ(·)k22,−11/2(25).Ïðåäïîëàãàåòñÿ, ÷òî Ξ è Z - äâà äðóãèõ ãèëüáåðòîâûõ ïðîñòðàíñòâà ñîñêàëÿðíûìè ïðîèçâåäåíèÿìè (·, ·)Ξ , (·, ·)Z è íîðìàìè k · kΞ , k · kZ , ñîîòâåòñòâåííî.Âìåñòå ñ âûøå ââåäåííûì îïåðàòîðîì A : Y1 → Y−1 ðàññìàòðèâàþòñÿëèíåéíûå îãðàíè÷åííûå îïåðàòîðû B : Ξ → Y−1 , C : Y1 → Z, ìíîãîçíà÷íîåîòîáðàæåíèå φ : R+ × Z → 2Ξ è îòîáðàæåíèå ψcont : Y1 → R+ .
Îòíîñèòåëüíîφ ïðåäïîëàãàåì, ÷òî ýòî ïîëóíåïðåðûâíàÿ ñâåðõó ôóíêöèÿ, ψcont - âûïóêëàÿ,ïîëóíåïðåðûâíàÿ ñíèçó ôóíêöèÿ, ψcont 6≡ +∞. Ðàññìîòðèì ýâîëþöèîííîåâàðèàöèîííîå íåðàâåíñòâî ñ ìíîãîçíà÷íîé íåëèíåéíîñòüþ â âèäå(ẏ − Ay − Bξ, η − y)−1,1 + ψcont (η) − ψcont (y) ≥ 0 ,∀ η ∈ Y1 ,z(t) = Cy(t) , ξ(t) ∈ φ(t, z(t)) , y(0) = y0 ∈ Y0 .(26)(27)Ââåä¼ì äðóãîå ïîíÿòèå ïðîöåññîâ èç òåîðèè óïðàâëåíèÿ, êîòîðîå ïîÿâëÿåòñÿâ ðàáîòå Ëèõòàðíèêîâà À.Ë., ßêóáîâè÷à Â.À. ([3]).14Îïðåäåëåíèå 8.y(·) ∈ WT ∩ C(0, T ; Y0 ) íàçûâàåòñÿ ðåøåíèåìñèñòåìû (26), (27) íà ïðîìåæóòêå (0, T ), åñëè ñóùåñòâóåò ôóíêöèÿ ξ(·) ∈L2 (0, T ; Ξ) òàêàÿ, ÷òî äëÿ ïî÷òè âñåõ t ∈ (0, T ) ñîîòíîøåíèÿ (26), (27)RTâûïîëíåíû è 0 ψcont (y(t))dt < +∞.
Ïàðà {ξ(·), y(·)} íàçûâàåòñÿ ïðîöåññîì.Ôóíêöèÿ ξ(·) íàçûâàåòñÿ ñåëåêòîðîì îòíîñèòåëüíî ðåøåíèÿ y(·).Ôóíêöèÿ ðàçäåëå 4.2 äîêàçûâàþòñÿ äîñòàòî÷íûå ÷àñòîòíûå óñëîâèÿ óñòîé÷èâîñòè íà êîíå÷íîì èíòåðâàëå.Ââåä¼ì ïîíÿòèå óñòîé÷èâîñòè íà êîíå÷íîì èíòåðâàëå äëÿ âàðèàöèîííîãî íåðàâåíñòâà:Îïðåäåëåíèå 9. Íåðàâåíñòâî (26), (27) íàçûâàåòñÿ (α, β, t0 , T )-óñòîé÷èâûì,y(·) èç íåðàâåíñòâà ky(t0 )k0 < αt ∈ [t0 , t0 + T ).åñëè äëÿ êàæäîãî ðåøåíèÿky(t)k0 < βäëÿ âñåõâûòåêàåò, ÷òîÑôîðìóëèðóåì òåîðåìó î äîñòàòî÷íûõ óñëîâèÿõ óñòîé÷èâîñòè:Òåîðåìà 4.íåïðåðûâíûéRJ := [t0 , t0 + T ) - âðåìåííîé èíòåðâàë è ñóùåñòâóþòôóíêöèîíàë Φ : J × Y0 → R è èíòåãðèðóåìàÿ ôóíêöèÿ g : J →Ïóñòüòàêèå, ÷òî ñëåäóþùèå óñëîâèÿ âûïîëíåíû:ZΦ(t, y(t)) − Φ(s, y(s)) <t(28)g(τ )dτss, t ∈ J, s < t, è ïðîèçâîëüíûõ ôóíêöèé y(·) ∈ WT ∩ C(0, T ; Y0 )÷òî α ≤ ky(t)k0 ≤ β äëÿ âñåõ t ∈ J;Z tg(τ )dτ ≤min Φ(t, y) − max Φ(s, y)(29)äëÿ âñåõòàêèõ,säëÿ âñåõy∈Y0 :kyk0 =βy∈Y0 :kyk0 =αs, t ∈ J, s < t.Òîãäà íåðàâåíñòâî (26), (27)(α, β, t0 , T )-óñòîé÷èâî.Ðàçäåë 4.3 ïîñâÿùåí ïðîãíîçèðîâàíèþ ïîòåðè (α, β, t0 , T )-óñòîé÷èâîñòè. ðàçäåëå 4.4 èññëåäóþòñÿ ýâîëþöèîííûå âàðèàöèîííûå íåðàâåíñòâà ñíåëèíåéíîñòÿìè òèïà ãèñòåðåçèñà.Ðàññìîòðèì ýâîëþöèîííîå âàðèàöèîííîå íåðàâåíñòâî ñ îïåðàòîðîì ãèñòåðåçèñà â êà÷åñòâå íåëèíåéíîñòè â âèäå(ẏ − Ay − Bξ, η − y)−1,1 + ψcont (η) − ψcont (y) ≥ 0 ,∀ η ∈ Y1 ,z(t) = Cy(t) , ξ(t) = φ(z, ξ0 )(t) , y(0) = y0 ∈ Y0 , ξ0 ∈ E(z(0))(30)(31)15ãäå A, B, C - òàêèå æå, êàê â ñèñòåìå (26)-(27), ïðîñòðàíñòâà Ξ, Z, Y−1 , Y0 , Y1- òàêèå æå, êàê â ïðåäûäóùåì ðàçäåëå.Çäåñü φ - ñèëüíî íåïðåðûâíûé îïåðàòîð ãèñòåðåçèñà:φ : D(φ) ⊂ W 1,2 (0, T ; Z) × Ξ → W 1,2 (0, T ; Ξ),(32)ãäå W 1,2 (0, T ; Z) è W 1,2 (0, T ; Ξ) - ïðîñòðàíñòâà Ñîáîëåâà ôóíêöèé íà ïðîìåæóòêå âðåìåíè (0, T ) ñî çíà÷åíèÿìè â Z èëè Ξ ñîîòâåòñòâåííî.
 êîíöåðàçäåëà 4.3 â êà÷åñòâå èëëþñòðàöèè ÷àñòîòíûõ ìåòîäîâ ïðèâîäèòñÿ çàäà÷àíàãðåâà ñòåðæíÿ, äëÿ êîòîðîé äîêàçûâàåòñÿ âûïîëíåíèå ÷àñòîòíîãî óñëîâèÿ. ðàçäåëå 4.5 èññëåäóåòñÿ óñòîé÷èâîñòü íà êîíå÷íîì èíòåðâàëå âðåìåíèýâîëþöèîííûõ âàðèàöèîííûõ íåðàâåíñòâ ñ íåëèíåéíîñòÿìè òèïà ãèñòåðåçèñàè îïåðàòîðàìè âûõîäà âèäà(ẏ − Ay − Bξ, η − y)−1,1 + ψcont (η) − ψcont (y) ≥ 0 ,∀ η ∈ Y1 ,(33)z(t) = Cy(t) , ξ(t) ∈ φ(z, ξ0 )(t) , y(0) = y0 ∈ Y0 , ξ(0) = ξ0 ∈ φ(t, z(t)) .
(34)r(t) = Dy(t) + Eξ(t),(35)ãäå îïåðàòîðû A, B, C , ïðîñòðàíñòâà Ξ, Z, Y−1 , Y0 , Y1 - òàêèå æå, êàê â ïðåäûäóùåì ðàçäåëå, R - äðóãîå ãèëüáåðòîâî ïðîñòðàíñòâî ñ íîðìîé k · kR .  ÷àñòíîñòè, èìååì A ∈ L(Y0 , Y−1 ), B ∈ L(Ξ, Y−1 ), C ∈ L(Y−1 , Z), D ∈ L(Y1 , R)è E ∈ L(Ξ, R). Âåëè÷èíà r(t) íàçûâàåòñÿ âûõîäîì ñèñòåìû. Îòíîñèòåëüíîòàêîãî âûõîäà r(t) äîêàçûâàåòñÿ òåîðåìà îá óñòîé÷èâîñòè íà êîíå÷íîì ïðîìåæóòêå âðåìåíè. çàêëþ÷åíèè ïåðå÷èñëåíû îñíîâíûå ðåçóëüòàòû äèññåðòàöèè.
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