A New Stability Criterion for Neutral Stochastic Delay Differential Equations with Markovian Switching

Wei Hu

doi:10.1155/2018/7814974

Abstract

In this short paper, a new stability theorem for neutral stochastic delay differential equations with Markovian switching is investigated by applying stochastic analysis technique and Razumikhin stability approach. A novel criterion of thepth moment exponential stability is derived for the related systems. The feature of the criterion shows that the estimated upper bound for the diffusion operator of Lyapunov function is allowed to be indefinite, even if to be unbounded, which can loosen the constraints of the existing results. Last, an example is provided to illustrate the usefulness and significance of the theoretical results.

Highlights

As is well-known, the stability is very hot topic in deterministic or stochastic dynamic systems
In [38], the author explained that the Markovian switching systems had been emerging as a convenient mathematical framework for the formulation of various design problems in different fields such as target tracking, fault tolerant control, and manufacturing processes, which can be seen as the motivation of wide practical use of the theoretic results for hybrid systems
Owing to their widely real applications, stochastic systems with Markovian switching have received a great deal of attention and many interesting results have been reported in the literature

Summary

Introduction

As is well-known, the stability is very hot topic in deterministic or stochastic dynamic systems (for instance, see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37] and references therein). Mao in [46, 47] applied Razumikhin approach to derive the exponential stability criteria for neutral stochastic delay differential systems with Markovian switching. The former results all need that the upper bound λ(t) for the diffusion operator of the Lyapunov function is negative definite for all t This restriction leads to the criteria having strong conservativeness in practice due to the fact that there are a large number of time-varying systems not satisfying the above conditions. Throughout this short paper, let (Ω, F, {Ft}t≥0, P) be a complete probability space with a natural filtration {Ft}t≥0 satisfying the usual condition (i.e., it is right continuous and F0 norm of contains all P-null sets).

Preliminaries

Main Results

An Example

Conclusion