Global asymptotic stability for the averaged implies semi-global practical asymptotic stability for the actual

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. As another application of the generalized Liapunov theorem, one may recover the classical asymptotic stability result for periodic solutions of time-invariant systems x˙=f(x) in terms of the Poincare map