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Robustness of exponential stability of stochastic differential delay equations. (English) Zbl 0851.93074
Consider a linear stochastic delay differential equation driven by Brownian motion such that \(x(t) \equiv 0\) is a solution. The author shows that if the trivial solution of a corresponding deterministic non-delay equation is exponentially stable, then the given equation will be exponentially stable in mean square provided both the delay and the noise are small. In fact he shows that if an explicit inequality involving the delay, upper bounds on the noise intensity, and two matrices in the Lyapunov equation of the associated linear ODE are satisfied, then exponential stability in mean square follows. A generalization to nonlinear equations is indicated. Finally, the author provides two examples (a linear and a nonlinear one).

MSC:
93E15 Stochastic stability in control theory
93D09 Robust stability
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H20 Stochastic integral equations
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