The purpose of the paper is to show how asymptotic properties, first of all stochastic Lyapunov stability, of linear stochastic functional differential equations can be studied via the property of solvability of the equation in certain pairs of spaces of stochastic processes, the property which we call input-to-state stability with respect to these spaces. Input-to-state stability and hence the desired asymptotic properties can be effectively verified by means of a special regularization, also known as “theW-method” in the literature. How this framework provides verifiable conditions of different kinds of stochastic stability is shown.

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Àííîòàöèÿ. Èññëåäóþòñÿ âîïðîñû ãëîáàëüíîé ýêñïîíåíöèàëüíîé póñòîé÷èâîñòè (2 ≤ p < ∞) ñèñòåì íåëèíåéíûõ äèôôåðåíöèàëüíûõ óðàâíåíèé Èòî ñ çàïàçäûâàíèÿìè ñïåöèàëüíîãî âèäà íà îñíîâå òåîðèè ïîëîaeèòåëüíî îáðàòèìûõ ìàòðèö. Äëÿ ýòîãî ïðèìåíÿþòñÿ èäåè è ìåòîäû, ðàç-ðàáîòàííûå Í.Â.Àçáåëåâûì è åãî ó÷åíèêàìè äëÿ èññëåäîâàíèÿ âîïðîñîâ óñòîé÷èâîñòè äåòåðìèíè-ðîâàííûõ ôóíêöèîíàëüíîäèôôåðåíöèàëüíûõ óðàâíåíèé. Ïðèâîäÿòñÿ äîñòàòî÷íûå óñëîâèÿ ãëî-áàëüíîé ýêñïîíåíöèàëüíîé póñòîé÷èâîñòè (2 ≤ p < ∞) ñèñòåì íåëèíåéíûõ äèôôåðåíöèàëüíûõ óðàâíåíèé Èòî ñ çàïàçäûâàíèÿìè â òåðìèíàõ ïîëîaeèòåëüíîé îáðàòèìîñòè ìàòðèö, ïîñòðîåííûõ ïî èñõîäíîé ñèñòåìå. Ïðîâåðÿåòñÿ âûïîëíèìîñòü ýòèõ óñëîâèé äëÿ êîíêðåòíûõ óðàâíåíèé.1. Ââåäåíèå.Âîïðîñàì óñòîé÷èâîñòè ðåøåíèé ñèñòåì ñî ñëó÷àéíûìè ïàðàìåòðàìè ïîñâÿùåíî áîëüøîå êî-ëè÷åñòâî ðàáîò. Äîñòàòî÷íî ïîëíûé èõ ñïèñîê ïðèâåäåí â ìîíîãðàôèÿõ [1] [4].  îñíîâíîì, â ýòèõ ðàáîòàõ èññëåäîâàíèå ñòîõàñòè÷åñêîé óñòîé÷èâîñòè ïðîâîäèòñÿ òðàäèöèîííûìè ìåòîäàìè, îñíîâàííûìè íà ôóíêöèîíàëàõ ËÿïóíîâàÊðàñîâñêîãîÐàçóìèõèíà. Îäíàêî ïðèìåíåíèå ýòèõ ìåòîäîâ âî ìíîãèõ ñëó÷àÿõ âñòðå÷àåò ñåðü¼çíûå òðóäíîñòè. Ïîýòîìó ýôôåêòèâíûå ïðèçíàêè óñòîé÷èâîñòè îáû÷íî óäàåòñÿ äîêàçûâàòü ëèøü äëÿ ñðàâíèòåëüíî óçêèõ êëàññîâ ñòîõàñòè÷åñêèõ äèôôåðåíöèàëüíûõ óðàâíåíèé ñ ïîñëåäåéñòâèåì. Ñ äðóãîé ñòîðîíû, â òåîðèè óñòîé÷èâîñòè äå-òåðìèíèðîâàííûõ ôóíêöèîíàëüíîäèôôåðåíöèàëüíûõ óðàâíåíèé âûñîêóþ ýôôåêòèâíîñòü ïî-êàçàë Wìåòîä, ò.å. ìåòîä ïðåîáðàçîâàíèÿ èñõîäíîãî óðàâíåíèÿ ñ ïîìîùüþ âñïîìîãàòåëüíîãî óðàâíåíèÿ, ðàçðàáîòàííûé Í.Â.Àçáåëåâûì è åãî ó÷åíèêàìè. Öåëüþ òàêîãî ïðåîáðàçîâàíèÿ ÿâ-ëÿåòñÿ ïîëó÷åíèå èíòåãðàëüíîãî óðàâíåíèÿ, äëÿ êîòîðîãî ïðîùå èññëåäîâàòü íóaeíûå ñâîéñòâà ðåøåíèé.Äëÿ íåëèíåéíûõ äèôôåðåíöèàëüíûõ óðàâíåíèé Èòî ñ ïîñëåäåéñòâèåì âîïðîñû óñòîé÷èâî-ñòè èçó÷åíû íåäîñòàòî÷íî.  ðàáîòàõ [5], [6] èññëåäîâàëèñü âîïðîñû ëîêàëüíîé óñòîé÷èâîñòè ðåøåíèé íåëèíåéíûõ ñòîõàñòè÷åñêèõ äèôôåðåíöèàëüíûõ óðàâíåíèé ñ ïîñëåäåéñòâèåì ñ ïîìî-ùüþ Wìåòîäà.  ñëó÷àå ëèíåéíûõ óðàâíåíèé ëîêàëüíàÿ óñòîé÷èâîñòü ðåøåíèé è ãëîáàëüíàÿ óñòîé÷èâîñòü ðåøåíèé ýêâèâàëåíòíû, â íåëèíåéíîì aeå ñëó÷àå èç ãëîáàëüíîé óñòîé÷èâîñòè ðå-øåíèÿ ñëåäóåò ëîêàëüíàÿ óñòîé÷èâîñòü ýòîãî aeå ðåøåíèÿ, íî îáðàòíîå íåâåðíî. Êðîìå òîãî, â ñëó÷àå ëèíåéíûõ óðàâíåíèé èç ëîêàëüíîé óñòîé÷èâîñòè íåêîòîðîãî ðåøåíèÿ óðàâíåíèÿ ñëåäóåò ëîêàëüíàÿ óñòîé÷èâîñòü ëþáîãî ðåøåíèÿ ýòîãî aeå óðàâíåíèÿ, à â ñëó÷àå íåëèíåéíûõ óðàâíåíèé ýòîò ôàêò òàêaeå íåâåðåí. íàñòîÿùåé ðàáîòå èññëåäóþòñÿ âîïðîñû ãëîáàëüíîé ýêñïîíåíöèàëüíîé póñòîé÷èâîñòè (2 ≤ p < ∞) ñèñòåì íåëèíåéíûõ äèôôåðåíöèàëüíûõ óðàâíåíèé Èòî ñ çàïàçäûâàíèÿìè ñïåöèàëüíîãî âèäà. Ïðè ýòîì ïðèìåíÿþòñÿ ïðèíöèïû Wìåòîäà è òåîðèÿ ïîëîaeèòåëüíî îáðàòèìûõ ìàòðèö. Îòëè÷èå îò êëàññè÷åñêîãî Wìåòîäà ñîñòîèò â òîì, ÷òî êàaeäîå óðàâíåíèå ñèñòåìû ïðåîáðàçóåò-ñÿ íåçàâèñèìî îò îñòàëüíûõ, à êàaeäàÿ êîìïîíåíòà ðåøåíèÿ îöåíèâàåòñÿ îòäåëüíî. Òàêîé ïîäõîä, â ñî÷åòàíèè ñî ñïåöèàëüíûì âèäîì óðàâíåíèÿ, ïîçâîëÿåò ïîëó÷èòü íîâûå ðåçóëüòàòû íå òîëüêî äëÿ íåëèíåéíûõ, íî è äëÿ ëèíåéíûõ óðàâíåíèé, êàê âàaeíîì ÷àñòíîì ñëó÷àå óðàâíåíèé íåëè-íåéíûõ. Äëÿ ïîëó÷åíèÿ îöåíîê â ñòàòüå èñïîëüçîâàí ìåòîä ð...