Practical stability for systems depending on a small parameter

Systems x˙(t)=F(t,x(t),ε) depending on a small parameter ε 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),ε) if F(·,x,ε) 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 (1991). We use this theory for the “practical stabilization” of a class of driftless control-affine systems