Mean square exponential and non-exponential asymptotic stability of impulsive stochastic Volterra equations

Dianli Zhao,Dong Han

doi:10.1186/1029-242x-2011-9

Dianli Zhao, Dong Han

Open Access

https://doi.org/10.1186/1029-242x-2011-9

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Abstract

In this article, some inequalities on convolution equations are presented firstly. The mean square stability of the zero solution of the impulsive stochastic Volterra equation is studied by using obtained inequalities on Liapunov function, including mean square exponential and non-exponential asymptotic stability. Several sufficient conditions for the mean square stability are presented. Results in this article indicate that not only the impulse intensity but also the time of impulse can influence the stability of the systems. At last, an example is given to show application of some obtained results. Mathematics classification Primary(2000): 60H10, 60F15, 60J70, 34F05.

Highlights

Study on the stability of stochastic differential equations has gained lots of attention over the last years
Taniguchi [1] studied the exponential stability for stochastic delay partial differential equations by use of the energy method which overcomes the difficulty of constructing the Liapunov functional on delay differential equations
Wan and Duan [2] extended the result of Taniguchi [1] to be applied to more general stochastic partial differential equations with memory

Summary

Introduction

Study on the stability of stochastic differential equations has gained lots of attention over the last years. If there exist positive numbers c1, c2 and V Î C1,2(R+ × Rn, R+) satisfying (i) c1 ||x||p ≤ V (t, x) ≤ c2 ||x||p; (ii) there exist two continuous and integrable functions k, h : R+ ® R+ and constant a. Satisfying (i) c1 ||x||p ≤ V (t, x) ≤ c2 ||x||p; (ii) there exist continuous and integrable function k : R+ ® R+ and positive constant a such that for any i = 1, 2,. LV(t, x(t)) ≤ −aV(t, x(t)) + k(t − s)V(s, x(s))ds, t ∈ (τi−1, τi) τi−1 holds when E||x (τi-1)||2 0; (iii) there exist positive constants ωi such that for any i = 1, 2,..., we have (iv) there exists 0

Mean square non-exponential asymptotic stability

Example