Semi-global practical asymptotic stability and averaging

Abstract

We prove a generalized Liapunov theorem which guarantees practical asymptotic stability. Based on this theorem, we show that if the averaged system x ̇ =f av (x) corresponding to x ̇ =f(x,t) is globally asymptotically stable then, starting from an arbitrarily large set of initial conditions, the trajectories of x ̇ =f(x,t/ε) converge uniformly to an arbitrarily small residual set around the origin when ε>0 is taken sufficiently small. In other words, the origin is semi-globally practically asymptotically stable.