Practical stability for systems depending on a small parameter

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

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