We prove a generalized Liapunov theorem which guarantees practical asymptotic stability. Based on this theorem, we show that if the averaged system x/spl dot/=f/sub av/(x) corresponding to x/spl dot/=f(x,t) is globally asymptotically stable then, starting from an arbitrarily large set of initial conditions, the trajectories of x/spl dot/=f(x, t//spl epsiv/) converge uniformly to an arbitrarily small residual set around the origin when /spl epsiv/<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/spl dot/=f(x) in terms of the Poincare map.
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