Set-valued Lyapunov functions for difference inclusions

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The paper relates set-valued Lyapunov functions to pointwise asymptotic stability in systems described by a difference inclusion. Pointwise asymptotic stability of a set is a property which requires that each point of the set be Lyapunov stable and that every solution to the inclusion, from a neighborhood of the set, be convergent and have the limit in the set. Weak set-valued Lyapunov functions are shown, via an argument resembling an invariance principle, to imply this property. Strict set-valued Lyapunov functions are shown, in the spirit of converse Lyapunov results, to always exist for closed sets that are pointwise asymptotically stable.