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.

We consider continuous-time average consensus dynamics in which the agents' states are communicated through uniform quantizers. Solutions to the resulting system are defined in the Krasowskii sense and are proven to converge to conditions of "practical consensus". To cope with undesired chattering phenomena we introduce a hysteretic quantizer, and we study the convergence properties of the resulting dynamics by a hybrid system approach. (C) 2011 Elsevier Ltd. All rights reserved

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.

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