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A trajectory based approach for stability robustness of systems with inputs

We show, for a definition of semiglobal practical external stability, that the stability property holds on semi-infinite time intervals if and only if it holds on arbitrarily long but finite time intervals. These results have immediate applications in analysis of stability properties of highly oscillatory systems with inputs using averaging or for systems with inputs that are slowly varying. Results are stated for general flows and the stability is given with respect to arbitrary (not necessarily compact) sets.