Lyapunov Stability of the Generalized Stochastic Pantograph Equation
The purpose of the paper is to study stability properties of the generalized stochastic pantograph equation, the main feature of which is the presence of unbounded delay functions. This makes the stability analysis rather different from the classical one. Our approach consists in linking different k...
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| Format: | Article |
| Language: | English |
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Wiley
2018-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2018/7490936 |
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| author | Ramazan Kadiev Arcady Ponosov |
| author_facet | Ramazan Kadiev Arcady Ponosov |
| author_sort | Ramazan Kadiev |
| collection | DOAJ |
| description | The purpose of the paper is to study stability properties of the generalized stochastic pantograph equation, the main feature of which is the presence of unbounded delay functions. This makes the stability analysis rather different from the classical one. Our approach consists in linking different kinds of stochastic Lyapunov stability to specially chosen functional spaces. To prove stability, we check that the solutions of the equation belong to a suitable space of stochastic processes, instead of searching for an appropriate Lyapunov functional. This gives us possibilities to study moment stability, stability with probability 1, and many other stability properties in an efficient way. We show by examples how this approach works in practice, putting emphasis on delay-independent stability conditions for the generalized stochastic pantograph equation. The framework can be applied to any stochastic functional differential equation with finite dimensional initial conditions. |
| format | Article |
| id | doaj-art-bc8350205ebd47a497651eb7bb014cf0 |
| institution | Kabale University |
| issn | 2314-4629 2314-4785 |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-bc8350205ebd47a497651eb7bb014cf02025-02-03T06:11:15ZengWileyJournal of Mathematics2314-46292314-47852018-01-01201810.1155/2018/74909367490936Lyapunov Stability of the Generalized Stochastic Pantograph EquationRamazan Kadiev0Arcady Ponosov1Dagestan Research Center of the Russian Academy of Sciences and Department of Mathematics, Dagestan State University, Makhachkala 367005, RussiaNorwegian University of Life Sciences, Faculty of Sciences and Technology, P.O. Box 5003, N-1432 Ås, NorwayThe purpose of the paper is to study stability properties of the generalized stochastic pantograph equation, the main feature of which is the presence of unbounded delay functions. This makes the stability analysis rather different from the classical one. Our approach consists in linking different kinds of stochastic Lyapunov stability to specially chosen functional spaces. To prove stability, we check that the solutions of the equation belong to a suitable space of stochastic processes, instead of searching for an appropriate Lyapunov functional. This gives us possibilities to study moment stability, stability with probability 1, and many other stability properties in an efficient way. We show by examples how this approach works in practice, putting emphasis on delay-independent stability conditions for the generalized stochastic pantograph equation. The framework can be applied to any stochastic functional differential equation with finite dimensional initial conditions.http://dx.doi.org/10.1155/2018/7490936 |
| spellingShingle | Ramazan Kadiev Arcady Ponosov Lyapunov Stability of the Generalized Stochastic Pantograph Equation Journal of Mathematics |
| title | Lyapunov Stability of the Generalized Stochastic Pantograph Equation |
| title_full | Lyapunov Stability of the Generalized Stochastic Pantograph Equation |
| title_fullStr | Lyapunov Stability of the Generalized Stochastic Pantograph Equation |
| title_full_unstemmed | Lyapunov Stability of the Generalized Stochastic Pantograph Equation |
| title_short | Lyapunov Stability of the Generalized Stochastic Pantograph Equation |
| title_sort | lyapunov stability of the generalized stochastic pantograph equation |
| url | http://dx.doi.org/10.1155/2018/7490936 |
| work_keys_str_mv | AT ramazankadiev lyapunovstabilityofthegeneralizedstochasticpantographequation AT arcadyponosov lyapunovstabilityofthegeneralizedstochasticpantographequation |