Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 183729, 21 pages
http://dx.doi.org/10.1155/2012/183729
Research Article

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

Hua Yang,1,2 Huisheng Shu,3 Xiu Kan,1 and Yan Che1

1School of Information Science and Technology, Donghua University, Shanghai 200051, China
2College of Information Science and Engineering, Shanxi Agricultural University, Taigu 030801, China
3Department of Applied Mathematics, Donghua University, Shanghai 200051, China

Received 21 June 2012; Accepted 1 August 2012

Academic Editor: Bo Shen

Copyright © 2012 Hua Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.