Stability of linear systems of differential equations with random jump linear solutions in Hilbert space

Stability of linear systems of differential equations with random jump linear solutions in Hilbert space

The conditions of stability in the middle and in the mean square solutions of stochastic differential equations with random perturbations in Hilbert spaces are obtained.

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Бібліографічні деталі
Дата:2013
Автор: Nikitin, A.V.
Формат: Стаття
Мова:English
Опубліковано: Інститут кібернетики ім. В.М. Глушкова НАН України 2013
Назва видання:Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/86488
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Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:Stability of linear systems of differential equations with random jump linear solutions in Hilbert space / А.V. Nikitin // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2013. — Вип. 8. — С. 237-240. — Бібліогр.: 6 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
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spelling irk-123456789-864882016-04-15T13:14:13Z Stability of linear systems of differential equations with random jump linear solutions in Hilbert space Nikitin, A.V. The conditions of stability in the middle and in the mean square solutions of stochastic differential equations with random perturbations in Hilbert spaces are obtained. Отримані умови стійкості в середньому та в середньому квадратичному розв’язків стохастичних диференціальних рівнянь з випадковими збуреннями у гільбертових просторах. 2013 Article Stability of linear systems of differential equations with random jump linear solutions in Hilbert space / А.V. Nikitin // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2013. — Вип. 8. — С. 237-240. — Бібліогр.: 6 назв. — англ. 2308-5878 http://dspace.nbuv.gov.ua/handle/123456789/86488 519.21 en Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки Інститут кібернетики ім. В.М. Глушкова НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language English
description The conditions of stability in the middle and in the mean square solutions of stochastic differential equations with random perturbations in Hilbert spaces are obtained.
format Article
author Nikitin, A.V.
spellingShingle Nikitin, A.V.
Stability of linear systems of differential equations with random jump linear solutions in Hilbert space
Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки
author_facet Nikitin, A.V.
author_sort Nikitin, A.V.
title Stability of linear systems of differential equations with random jump linear solutions in Hilbert space
title_short Stability of linear systems of differential equations with random jump linear solutions in Hilbert space
title_full Stability of linear systems of differential equations with random jump linear solutions in Hilbert space
title_fullStr Stability of linear systems of differential equations with random jump linear solutions in Hilbert space
title_full_unstemmed Stability of linear systems of differential equations with random jump linear solutions in Hilbert space
title_sort stability of linear systems of differential equations with random jump linear solutions in hilbert space
publisher Інститут кібернетики ім. В.М. Глушкова НАН України
publishDate 2013
url http://dspace.nbuv.gov.ua/handle/123456789/86488
citation_txt Stability of linear systems of differential equations with random jump linear solutions in Hilbert space / А.V. Nikitin // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2013. — Вип. 8. — С. 237-240. — Бібліогр.: 6 назв. — англ.
series Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки
work_keys_str_mv AT nikitinav stabilityoflinearsystemsofdifferentialequationswithrandomjumplinearsolutionsinhilbertspace
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fulltext Серія: Фізико-математичні науки. Випуск 8 237 UDC 519.21 А. V. Nikitin, Ph. D. Bukovinian State Finance and Economics University, Chernivtsi STABILITY OF LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS WITH RANDOM JUMP LINEAR SOLUTIONS IN HILBERT SPACE The conditions of stability in the middle and in the mean square solutions of stochastic differential equations with random perturbations in Hilbert spaces are obtained. Key words: Hilbert space, stability, generators operator Mar- kov process. Introduction. The statement about the stability or instability of the first approach are formulated as follows: if the solution of linear equations, built on the original stable or unstable, then the solution of the original equation will also be under stable or unstable. This assertion is true under certain assumptions that are reasonable and at work. Analysis of the beha- vior of solutions of stochastic differential equations of Ito-Skorokhod de- voted a number of works, including [1–6]. Formulation of the problem. Consider a Markov process ( )t , which can be in states 1 2, ,..., ,...n   with probabilities ( ) { ( ) }k kp t t    ( 1,2,...),k  (1) which satisfy the system of differential equations 1 ( ) ( )k ks s k dp t p t dt      . (2) This constant coefficients ks ( 1,2,...)k  satisfy the relation 0ks  , 1 0ks s     , 0ks  ( )k s ( , 1,2,...)k s  . The Markov process ( )t abruptly changes its value at random times jt ( 0,1, 2,...)j  . Along with Markov processes consider the system of linear differen- tial equations ( ) ( ) dX t AX t dt  , jt t ( 0,1, 2,...)j  , (3) which time jump jt changes its value under the law ( 0) ( 0)j ks jX t C X t   , (4) © А. V. Nikitin, 2013 Математичне та комп’ютерне моделювання 238 the transition of a random process from a state s to a state k ( )k s , ksC — constant matrix. Solution of system (3) is a random process. The paper investigates the stability of the average and the stability of the mean square solutions of linear differential equations (3). Thus the derived equations to determine the necessary and sufficient conditions for asymptotic stability. Difference approximation of differential equations. Turn of differen- tial equations (2) and (3) the difference, putting nt nh ( 0,1,2,...)n  , 0h  . The system of differential equations (2) is approximated by a system of difference equations 1 1 ( ) ( ) ( )k n k n ks s n s p t p t h p t       ( 1, 2,...)k  . (5) Let the conditions 1 0kkh  ( 1,2,...)k  are true. The system of differential equations (5) describes Markov chain of infinite, can be in states 1 2, ,..., ,...n   with probabilities ( ) { ( ) }k kp t t    ( 1,2,...)k  . The system of differential equations (3) is approximated by a system of difference equations 1 1( , )n n n n nX X hA X    , (6) where ( , )k sA A   ( 1, 2,...)s  ; 1( , ) ( )k s ksA h C E    ( )k s . (7) Moment equation for a system of difference equations (6). Let ( , , )nf t X  — density distribution of discrete- continuous random process X , that can be represented as 1 ( , , ) ( , ) ( )n k n k k f t X f t X        , (8) where ( )  — the Dirac delta function. Partial density distribution ( , )k nf t X satisfying the conditions 1 1 1 1 1 1 1 1, ( , ) (1 ) ( ( ( , )) ) | det( ( , )) | ( , ( ( , )) ) | det( ( , )) , ( 1, 2,...) k n kk k n k s k s ks s n k s k s s s q f t X h f t E hA X E hA ha f t E hA Q Q X E hA Q Q k                         (10) Expand the terms of the equation (10) the degree of the parameter h , in anticipation of differentiation ( , )kf t X ( 1, 2,...)k  for all arguments. Thus we arrive at a system of differential equations with partial derivatives Серія: Фізико-математичні науки. Випуск 8 239 1 1 1 ( , ) ( , ) ( , ) ( , ) | det , ( 1, 2,...) k k k k p k ks s ks ks s f t X Df t X A X f t X S A t DX f t C X C k             (11) Denote the mathematical expectation vector X and matrix XX  symbols 1 * * * 1 ( ) ( , ) ( , ) ; ( ) ( , ) ( , ) m m m m k kE E k kE E M t O Xf t X dX Xf t X dX D t OO XX f t X dX XX f t X dX                  (12) Vector (t)M and matrix (t)D are defined by equations 1 ( ) ( )k k M t M t     , ( ) ( , ) m k k E M t Xf t X dX  , ( 1, 2,...)k  1 ( ) ( )k k D t D t     , *( ) ( , ) m k k E D t XX f t X dX  , ( 1, 2,...)k  . Vectors (t)kM ( 1, 2,...)k  satisfying the system of differential equ- ations 1 (t) q k k k ks ks k s dM A M C M dt     ( 1,2,...)k  (13) is derived from the system (11) by Matrix multiplication XX  and than by integration over the entire space mE . Further assume that the partial density distribution (0, )kf X , ( 1, 2,...)k  have finite moments of first and second order. From the sys- tem of equations (13) and (14) we obtain the following theorem. Theorem. To trivial solutions of linear differential equations (3) with random jumps solution of the form (4), caused by the Markov process ( )t (1) be asymptotically stable on average, it is necessary and sufficient that the trivial solution of system (13) was asymptotically stable. Conclusions. We built the moment equations, which follows from the stability of the initial stability of stochastic equations in a Hilbert space. References: 1. Гихман И. И. Стохастические дифференциальные уравнения и их приме- нение / И. И. Гихман, А. В. Скороход. — К. : Наука. думка, 1982. — 612 с. Математичне та комп’ютерне моделювання 240 2. Скороход А. В. Асимптотические методы теории стохастических дифферен- циальных уравнений / А. В. Скороход. — К. : Наука. думка, 1987. — 328 с. 3. Korolyuk V. S. Stochastic Models of Systems / V. S. Korolyuk, V. V. Koro- lyuk. — Kluwer : Dordrecht, 1999. — 185 p. 4. Koroliuk V. Stochastic Systems in Merging Phase Space / V. Koroliuk, N. Limnios // World Scientific Publishing. — 2005. — 330 p. 5. Хусаинов Д. Я. Метод функций Ляпунова в исследовании устойчивости дифференциальных функциональных систем / Д. Я. Хусаинов, А. В. Ша- тырко. — К. : Издательство ИНТИ, 1997. — 236 с. 6. Царьков Е. Ф. Случайные возмущения дифференциальных — функцио- нальных уравнений / Е. Ф. Царьков. – Рига : Издательство «Зинатне», 1989. — 429 с. Отримані умови стійкості в середньому та в середньому квадрати- чному розв’язків стохастичних диференціальних рівнянь з випадко- вими збуреннями у гільбертових просторах. Ключові слова: Гільбертовий простір, стійкість, твірний опе- ратор марківського процесу. 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/BleedOffset [ 0 0 0 0 ] /ConvertColors /ConvertToRGB /DestinationProfileName (sRGB IEC61966-2.1) /DestinationProfileSelector /UseName /Downsample16BitImages true /FlattenerPreset << /PresetSelector /MediumResolution >> /FormElements true /GenerateStructure false /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles true /MarksOffset 6 /MarksWeight 0.250000 /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /DocumentCMYK /PageMarksFile /RomanDefault /PreserveEditing true /UntaggedCMYKHandling /UseDocumentProfile /UntaggedRGBHandling /LeaveUntagged /UseDocumentBleed false >> ] >> setdistillerparams << /HWResolution [600 600] /PageSize [419.528 595.276] >> setpagedevice

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