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The problems of almost sure (a.s.) stability and a.s. stabilization are investigated for hybrid stochastic systems (HSSs) with time-varying delays. The different time-varying delays in the drift part and in the diffusion part are considered. Based on nonnegative semimartingale convergence theorem, Hölder’s inequality, Doob’s martingale inequality, and Chebyshev’s inequality, some sufficient conditions are proposed to guarantee that the underlying nonlinear hybrid stochastic delay systems (HSDSs) are almost surely (a.s.) stable. With these conditions, a.s. stabilization problem for a class of nonlinear HSDSs is addressed through designing linear state feedback controllers, which are obtained in terms of the solutions to a set of linear matrix inequalities (LMIs). Two numerical simulation examples are given to show the usefulness of the results derived

Almost Sure Stability and Stabilization for Hybrid Stochastic Systems with Time-Varying Delays