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Necessary and sufficient Lyapunov characterizations of a particular type of asymptotic stability in hybrid dynamical systems are given. This type involves uniform bounds on the amount of ordinary time, but not the number of jumps, it takes solutions to converge to the asymptotically stable set. Connections of this asymptotic stability concept to Zeno behavior are explored. Necessary and sufficient conditions for Zeno and uniform Zeno stability are shown.
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Goebel et al. (2008) studied this question.
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