Basic results on pointwise asymptotic stability and set-valued Lyapunov functions

Pointwise asymptotic stability of a set, for a difference inclusion, 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. It is equivalent to asymptotic stability for a single equilibrium, but is different in general, especially for noncompact sets of equilibria. Set-valued Lyapunov functions are set-valued mappings which characterize pointwise asymptotic stability in a way similar to how Lyapunov functions characterize asymptotic stability. It is shown here, via an argument resembling an invariance principle, that weak set-valued Lyapunov functions imply pointwise asymptotic stability. Strict set-valued Lyapunov functions are shown, in the spirit of converse Lyapunov results, to always exist for pointwise asymptotically stable closed sets.