Exponential stability and exponential stabilization of singularly perturbed stochastic systems with time-varying delay

In this paper, the problems of exponential stability and exponential stabilization for linear singularly perturbed stochastic systems with time-varying delay are investigated. First, an appropriate Lyapunov functional is introduced to establish an improved delay-dependent stability criterion. By applying free-weighting matrix technique and by equivalently eliminating time-varying delay through the idea of convex combination, a less conservative sufficient condition for exponential stability in mean square is obtained in terms of epsilon-dependent linear matrix inequalities (LMIs). It is shown that if this set of LMIs for epsilon = 0 are feasible then the system is exponentially stable in mean square for sufficiently small epsilon >= 0. Furthermore, it is shown that if a certain matrix variable in this set of LMIs is chosen to be a special form and the resulting LMIs are feasible for epsilon = 0, then the system is epsilon-uniformly exponentially stable for all sufficiently small epsilon >= 0. Based on the stability criteria, an epsilon-independent state-feedback controller that stabilizes the system for sufficiently small epsilon >= 0 is derived. Finally, numerical examples are presented, which show our results are effective and useful. Copyright (C) 2010 John Wiley & Sons, Ltd.