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Robustness of exponential stability of stochastic differential delay equations

Mao, X. (1996) Robustness of exponential stability of stochastic differential delay equations. IEEE Transactions on Automatic Control, 41 (3). pp. 442-447. ISSN 0018-9286

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Regard the stochastic differential delay equation dx(t) = [(A + Ā(t))x(t) + (B + B̄(t - τ))x(t - τ)] dt + g(t, x(t), x(t - τ)) dw(t) as the result of the effects of uncertainly, stochastic perturbation, and time lag to a linear ordinary differential equation ẋ(t) = (A + B)x(t). Assume the linear system is exponentially stable. In this paper we shall characterize how much the uncertainty, stochastic perturbation, and time lag the linear system can bear such that the stochastic delay system remains exponentially stable. The result will also be extended to nonlinear systems.