Almost surely exponential stability of impulsive stochastic delay dynamical system with semi-Markov jump structure

Abstract

In this article, the problems of p-moment exponential stability and almost surely exponential stability of impulsive stochastic delay differential equations with semi-Markov jump structure are examined. Based on Lyapunov-Razumikhin method, Razumikhin-type theorems on criteria ensuring pth-moment exponential stability and almost surely exponential stability of trivial solution of a class of impulsive stochastic differential delay equations with semi-Markov jump structure are developed. The distinguishing feature of this present paper is that the system remains in a state for a random amount of time with a generalized distribution before the jump, unlike in the case of Markov jump in which the duration of stay in a state before jump is exponentially distributed.