## $$H_{\infty}$$ analysis of nonlinear stochastic time-delay systems.(English)Zbl 1153.93355

Summary: The $$H_{\infty}$$ analysis problem is studied for a general class of nonlinear stochastic systems with time-delay. The stochastic systems are described in terms of stochastic functional differential equations. The Razumikhin-type lemma is employed to establish sufficient conditions for the time-delay stochastic systems to be internally stable, and the $$H_{\infty}$$ analysis problem is studied in order to quantify the disturbance rejection attenuation level of the nonlinear stochastic time-delay system. In particular, the paper obtains the general conditions under which the $$L_2$$ gain of the system is less than or equal to a given constant. Some easy-to-test criteria are also given so as to determine whether the nonlinear stochastic time-delay system under investigation is internally stable and whether it achieves certain $$H_{\infty}$$ performance index. Finally, illustrative examples are provided to show the usefulness of the proposed theory.

### MSC:

 93B36 $$H^\infty$$-control 93C23 Control/observation systems governed by functional-differential equations 93E15 Stochastic stability in control theory
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