Stability analysis of second-order differential systems with Erlang distribution random impulses

Shuorui Zhang,Jitao Sun

doi:10.1186/1687-1847-2013-4

Abstract

Differential systems with random impulses are a new kind of mathematical models. In this paper, we put forward a model of second-order impulsive differential systems with Erlang distribution random impulses. Sufficient conditions are obtained for oscillation in mean and p-moment stability of this model respectively. An example is presented to illustrate the efficiency of the results obtained.

Highlights

1 Introduction It is recognized that the impulsive differential system is an effective model for many real world phenomena, it has been widely used in the study of physics, engineering, information and communications technology, etc. in the past years and a lot of valuable results have been obtained
Most researchers concern about two kinds of impulse times: fixed impulse times and varying impulse times, which mean that the impulse time is some functions of the ‘state x’ [ – ]
The impulse phenomena sometimes happen at random times, and any solution of systems driven by this kind of impulses is a stochastic process, which is very different from those of differential systems with impulses at fixed moments and varying impulse times [ ]

Summary

Introduction

It is recognized that the impulsive differential system is an effective model for many real world phenomena, it has been widely used in the study of physics, engineering, information and communications technology, etc. in the past years and a lot of valuable results have been obtained (see [ – ] and references therein).For impulsive differential systems, most researchers concern about two kinds of impulse times: fixed impulse times and varying impulse times, which mean that the impulse time is some functions of the ‘state x’ [ – ]. Wu and Meng first introduced random impulsive ordinary differential equations and investigated the boundedness of solutions to these models by Lyapunov’s direct method in [ ]. In [ ], Wu et al discussed the existence and uniqueness in mean square of solutions to certain impulsive differential systems by employing the Cauchy-Schwarz inequality, Lipschitz condition and techniques in stochastic analysis. In [ ], Anguraj et al presented the existence and exponential stability of mild solutions of semilinear differential equations with random impulses.

Results

Conclusion