Institut Girard Desargues

pages

English

Documents

Écrit par
Girard Desargues

Publié par
pefav

Lire un extrait

Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus

Découvre YouScribe en t'inscrivant gratuitement

Je m'inscris

Découvre YouScribe en t'inscrivant gratuitement

Je m'inscris

pages

English

Ebook

Lire un extrait

Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus

Publié par

pefav

Nombre de lectures

Langue

English

Institut Girard Desargues , , EXISTENCE OF LIPSCHITZ AND SEMICONCAVE CONTROL-LYAPUNOV FUNCTIONS LUDOVIC RIFFORD Abstract. Given a locally Lipschitz control system which is globally asymptotically controllable to the origin, we construct a control-Lyapunov function for the system which is Lipschitz on bounded sets and we de- duce the existence of another one which is semiconcave (and so locally Lipschitz) outside the origin. The proof relies on value functions and nonsmooth calculus. 1. Introduction This paper is concerned with the stabilization problem for a standard control system of the form x˙(t) = f(x(t), u(t)). Lyapunov-like techniques have been successfully used in many problems in control theory, such as sta- bilizability, asymptotic controllability and stability. Stabilization by smooth feedback has been a subject of research by many authors. Among them, Art- stein provided an important contribution (see [3]), proving that the control system admits a smooth Lyapunov function if and only if there is a stabi- lizing relaxed feedback. Moreover, if the system is affine in the control, it is further the case that there exists an ordinary stabilizing feedback continu- ous outside the origin. In general however such a feedback fails to exist, as pointed out by Sontag and Sussmann [24] and by Brockett [8] among others ([22],[12]).