Abstract
In this paper, the Razumikhin approach is applied to the study of both p-th moment and almost sure stability on a general decay for a class of impulsive stochastic functional differential systems with Markovian switching. Based on the Lyapunov-Razumikhin methods, some sufficient conditions are derived to check the stability of impulsive stochastic functional differential systems with Markovian switching. One numerical example is provided to demonstrate the effectiveness of the results.
Highlights
Impulsive stochastic systems with Markovian switching is a class of hybrid dynamical systems, which is composed of both the logical switching rule of continuous-time finite-state Markovian process and the state represented by a stochastic differential system [1]
Since the delay phenomenon and the Markovian switching exists among impulsive stochastic systems, the whole systems become more complex and may oscillate or be not stable, we introduce Razumikhin-type theorems and Lyapunov methods to give the conditions that make the systems stable
We shall establish some criteria on the p-th moment exponential stability and almost exponential stability for system (1) by using the Razumikhin technique and Lyapunov functions
Summary
Impulsive stochastic systems with Markovian switching is a class of hybrid dynamical systems, which is composed of both the logical switching rule of continuous-time finite-state Markovian process and the state represented by a stochastic differential system [1]. Recalled that Razumikhin developed this technique to study the stability of deterministic systems with delay in [6] [7], Mao extended this technique to stochastic functional differential systems [8] This technique has become very popular in recent years since it is extensively applied to investigate many phenomena in physics, biology, finance, etc. The main aim of the present paper is attempt to investigate the p-th moment and almost sure stability on a general decay of impulsive stochastic delay differential systems with Markovian switching.
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