LaSalle-Type Theorems for General Nonlinear Stochastic Functional Differential Equations by Multiple Lyapunov Functions

Xueyan Zhao,Feiqi Deng,Xiaojing Zhong

doi:10.1155/2014/486191

Abstract

We investigate LaSalle-type theorems for general nonlinear stochastic functional differential equations. With some preliminaries on lemmas and the derivation techniques, we establish three LaSalle-type theorems for the general nonlinear stochastic functional differential equations via multiple Lyapunov functions. For the typical special case with estimations involving|xt|pfor the derivatives of the Lyapunov functions, a theorem is established as the corollary of the main theorem. At the end of the paper, an example is given to illustrate the usage of the method proposed and show the advantage of the results obtained.

Highlights

As it is well known, the Lyapunov function method is the most widely used tool to establish criteria for stability or other asymptotic properties of dynamic systems governed by differential equations or difference equations
We investigate LaSalle-type theorems for general nonlinear stochastic functional differential equations
LaSalle established a very important theorem named LaSalle invariance principle or LaSalle’s theorem [1], which weakened the condition of the Lyapunov function method on the negative definiteness of the derivatives of the Lyapunov functions along the solutions of the equations, and it has been widely used in the theory of ordinary differential equations

Summary

Introduction

As it is well known, the Lyapunov function method is the most widely used tool to establish criteria for stability or other asymptotic properties of dynamic systems governed by differential equations or difference equations. With this method, the derivatives of the Lyapunov functions or their upper bounds are often desired to be negative definite. With some preliminaries on lemmas and the derivation techniques, we establish three LaSalle-type theorems for the general nonlinear stochastic functional differential equations via multiple Lyapunov functions. At the end of the paper, an example is given to illustrate the usage of the method proposed in the paper

Preliminaries

Main Results

Example

Conclusion