On the practical global uniform asymptotic stability of stochastic differential equations

TL;DR

This paper analyzes the stability of stochastic differential equations using Lyapunov functions, focusing on global uniform boundedness and practical exponential stability without requiring the origin to be an equilibrium point.

Contribution

It extends Lyapunov stability analysis to stochastic systems where the equilibrium is not necessarily at the origin, providing new criteria for practical stability.

Findings

Established conditions for global uniform boundedness of solutions.

Derived criteria for practical uniform exponential stability.

Provided an example demonstrating the applicability of the results.

Abstract

The method of Lyapunov functions is one of the most effective ones for the investigation of stability of dynamical systems, in particular, of stochastic differential systems. The main purpose of the paper is the analysis of the stability of stochastic differential equations by using Lyapunov functions when the origin is not necessarily an equilibrium point. The global uniform boundedness and the global practical uniform exponential stability of so- lutions of stochastic differential equations based on Lyapunov techniques are investigated. Furthermore, an example is given to illustrate the applicability of the main result.