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Abstract. We study stability and stabilizability properties of systems with discontinuous righthand side (with solutions intended in Filippov's sense) by means of locally Lipschitz continuous and regular Lyapunov functions. The stability result is obtained in the more general context of differential inclusions. Concerning stabilizability, we focus on systems affine with respect to the input: we give some sufficient conditions for a system to be stabilized by means of a feedback law of the Jurdjevic-Quinn type.Résumé. Onétudie les propriétés de stabilité et stabilisation des systèmes avec second membre discontinu (les solutionsétant prises dans le sens de Filippov) au moyen des fonctions de Lyapunov lipchitziennes et régulières. Le résultat de stabilité est obtenu dans le contexte plus général des inclusions différentielles. En ce qui concerne la stabilisation, onétudie des systèmes affines par rapport au contrôle : on donne des conditions suffisantes pour la stabilisation au moyen d'un retour d'état du type de Jurdjevic et Quinn.

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In this paper we address the problem of extending La Salle Invariance Principle to switched system. We prove an extension of the invariance principle relative to dwell-time switched solutions, and a second one relative to constrained switched systems.

Differential equations with discontinuous righthand side and solutions intended in Carathéodory sense are considered. For these equations sufficient conditions which guarantee both Lyapunov stability and asymptotic stability in terms of nonsmooth Lyapunov functions are given. An invariance principle is also proven.