Stability and Stabilization of Discontinuous Systems and Nonsmooth Lyapunov Functions

Abstract

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.

Nonsmooth Lyapunov functions and discontinuous Carathéodory systems

“…This notion is analogous to the notion of set-valued derivative of a map with respect to a differential inclusion introduced in Shevitz and Paden (1994) and improved in Bacciotti and Ceragioli (1999) (note that in Bacciotti and Ceragioli (1999), Clarke regular functions instead of nonpathological functions were used; the analogous set-valued derivative for nonpathological functions has been studied in Ceragioli (2000)). …”

“…We consider nonsmooth Lyapunov functions: nonsmoothness of Lyapunov functions is in fact unavoidable when studying stability properties of discontinous systems. In particular, Lipschitz continuous Lyapunov functions which are also nonpathological (Valadier, 1989) are considered and a notion of derivative introduced in Shevitz and Paden (1994), improved in Bacciotti and Ceragioli (1999) and then used in Ceragioli (1999, 2003), is used throughout the paper. The paper consists of two main section.…”

Uma extensão do princípio de invariância para sistemas chaveados afins

“…Uma das ferramentas mais importantes para estudar o comportamento assintótico das soluções de sistemas dinâmicosé o Princípio de Invariância de LaSalle. Este resultado foi primeiramente desenvolvido para equações diferenciais ordinárias 2 autônomas definidas em espaço de dimensão finita [6] e depois o resultado foi estendido para outras classes de sistemas dinâmicos incluindo equações diferenciais funcionais [4], sistemas descontínuos [2] e sistemas chaveados [3,5,9,12].…”

Stability properties of equilibrium sets of non-linear mechanical systems with dry friction and impact

“…one dry friction element) is discussed. Moreover, in [3,6,43] the Lyapunov stability of an equilibrium point in the equilibrium set is shown. Most papers are limited to either one-degreeof-freedom systems or to systems exhibiting only one switching boundary.…”