Stochastic averaging for asymptotic stability

Abstract

Basic theorems of stochastic averaging are developed. They remove or weaken the restrictions in the existing averaging theory and provide tools for studying the stability of a general class of nonlinear systems with a stochastic perturbation. The chapter focuses on asymptotic stability, with the original system required to satisfy an equilibrium condition. If the perturbation process satisfies a uniform strong ergodic condition and the equilibrium of the average system is exponentially stable, the original system is exponentially practically stable in probability. Under the condition that the equilibrium of the average system is exponentially stable, if the perturbation process is ϕ-mixing with an exponential mixing rate and exponentially ergodic, and the original system satisfies an equilibrium condition, the equilibrium of the original system is asymptotically stable in probability. For the case where the average system is globally exponentially stable and all the other assumptions are valid globally, a global result is obtained for the original system.