Practical stability for systems depending on a small parameter

Systems (x) over dot (t) = F(t, x(t),epsilon) depending on a small parameter epsilon are considered. We introduce a concept of convergence of such a system to a system (x) over dot(t) = G(x(t)). Assuming this convergence, we prove that global asymptotic stability for (x)over dot(t) = G(x(t)) implies some notion of "practical stability" for (x) over dot (t) = F(t,x(t), epsilon) if F(., x, epsilon) satisfies 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-affine systems