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We establish that differential inclusions corresponding to upper semicontinuous multifunctions are strongly asymptotically stable if and only if there exists a smooth Lyapunov function. Since well-known concepts of generalized solutions of differential equations with discontinuous right-hand side can be described in terms of solutions of certain related differential inclusions involving upper semicontinuous multifunctions, this result gives a Lyapunov characterization of asymptotic stability of either Filippov or Krasovskii solutions for differential equations with discontinuous right-hand side. In the study of weak (as opposed to strong) asymptotic stability, the existence of a smooth Lyapunov function is rather exceptional. However, the methods employed in treating the strong case of asymptotic stability are applied to yield a necessary condition for the existence of a smooth Lyapunov function for weakly asymptotically stable differential inclusions; this is an extension to the context of Lyapunov functons of Brockett's celebrated``covering condition'' from continuous feedback stabilization theory. Academic Press

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