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Exponential stability and exponential stabilization of singularly perturbed stochastic systems with time-varying delay. (English) Zbl 1207.93112
Summary: 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 $$\varepsilon$$-dependent Linear Matrix Inequalities (LMIs). It is shown that if this set of LMIs for $$\varepsilon =0$$ are feasible then the system is exponentially stable in mean square for sufficiently small $$\varepsilon \geqslant 0$$. 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 $$\varepsilon =0$$, then the system is $$\varepsilon$$-uniformly exponentially stable for all sufficiently small $$\varepsilon \geqslant 0$$. Based on the stability criteria, an $$\varepsilon$$-independent state-feedback controller that stabilizes the system for sufficiently small $$\varepsilon \geqslant 0$$ is derived. Finally, numerical examples are presented, which show our results are effective and useful.

##### MSC:
 93E15 Stochastic stability in control theory 93E03 Stochastic systems in control theory (general) 93D30 Lyapunov and storage functions 93C70 Time-scale analysis and singular perturbations in control/observation systems
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