Practical stability for systems depending on a smallparameter 1

Systems _ x(t) = F(t; x(t); ") depending on a small parameter " are considered. We introduce a concept of convergence of such a system to a system _ x(t) = G(x(t)). Assuming this convergence, we prove that global asymptotic stability for _ x(t) = G(x(t)) implies some notion of \practical stability" for _ x(t) = F(t; x(t); ") if F(; x; ") satisses a periodicity assumption. We apply this theory to a \practical stability" analysis of \fast time-varying" systems studied in periodic averaging, and of \highly oscillatory" systems studied by Sussmann and Liu 1]. We use this theory for the \practical stabilization" of a class of driftless control-aane systems.