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${H}_{\infty }$ 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 Systems governed by functional-differential equations 93E15 Stochastic stability