About stability of delay differential equations with square integrable level of stochastic perturbations. (English) Zbl 1426.60082
Summary: The long term behavior of solutions of stochastic delay differential equations with a fading stochastic perturbations is investigated. It is shown that if the level of stochastic perturbations fades on the infinity, for instance, if it is given by square integrable function, then an asymptotically stable deterministic system remains to be an asymptotically stable (in mean square).
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 34K50 | Stochastic functional-differential equations |
| 34K20 | Stability theory of functional-differential equations |
Keywords:
delay differential equations; fading stochastic perturbations; asymptotic mean square stabilityReferences:
| [1] | Gikhman, I. I.; Skorokhod, A. V., Stochastic Differential Equations (1972), Springer · Zbl 0169.48702 |
| [2] | Kolmanovskii, V. B.; Myshkis, A. D., Introduction to the Theory and Applications of Functional Differential Equations (1999), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0917.34001 |
| [3] | Shaikhet, L., Lyapunov Functionals and Stability of Stochastic Functional Differential Equations (2013), Springer Science & Business Media · Zbl 1277.34003 |
| [4] | Haynsworth, E. V., On the Schur Complement, (Basel Mathematical Notes. Basel Mathematical Notes, BMN, vol. 20 (1968)), 17 pages · Zbl 0155.06304 |
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