Exponential Stability of Impulsive Stochastic Functional Differential Systems with Delayed Impulses

Abstract

A class of generalized impulsive stochastic functional differential systems with delayed impulses is considered. By employing piecewise continuous Lyapunov functions and the Razumikhin techniques, several criteria on the exponential stability and uniform stability in terms of two measures for the mentioned systems are obtained, which show that unstable stochastic functional differential systems may be stabilized by appropriate delayed impulses. Based on the stability results, delayed impulsive controllers which mean square exponentially stabilize linear stochastic delay systems are proposed. Finally, numerical examples are given to verify the effectiveness and advantages of our results.

Stability of Stochastic Differential Delay Systems with Delayed Impulses

“…Moreover, it is also shown that if a continuous dynamic system is stable, then, under some conditions, the delayed impulses do not destroy the stability of the systems. Our results can generalize some existing results in [20,21].…”

“…Moreover, when 3 −2 = 1/2, 3 −1 = 1/2, 3 = 4, for ∈ N, we have Π ∞ =1 < 5 and ∑ ∞ =1 ( − 1) = +∞. Then, Theorem 3 can be used, but the results in [20,21] cannot be applicable to this case.…”

On Input-to-State Stability of Impulsive Stochastic Systems with Time Delays

“…Impulsive stochastic delayed systems incorporate impulses effects, stochastic perturbations, and time delays in one system simultaneously. During the last decade, there has been extensive interest in the study of force-free delayed impulsive stochastic systems; we refer to [24][25][26][27][28] and references therein. However, the corresponding theory for impulsive stochastic systems with external inputs has been relatively less developed.…”

Stability Analysis of Impulsive Stochastic Functional Differential Equations with Delayed Impulses via Comparison Principle and Impulsive Delay Differential Inequality

“…In recent years, the stability analysis of impulsive stochastic functional equations which include delay equations is interesting to many investigators, and many results of stability criteria of these equations have been reported (see, e.g., [23][24][25][26][27][28][29]). Very recently, [30,31] took environment noise into account and generalized delayed impulses to stochastic equations. In particular, applying the Lyapunov functions couples with Razumikhin techniques, [30] investigates both moment and almost sure exponential stability of impulsive stochastic functional differential equations with delayed impulses (ISFDEs-DI), and several Razumikhin-type criteria on the exponential stability and uniform stability in terms of two measures for the equations were established in [31].…”