Abstract. This paper is concerned with the structure of asymptotically stabilizing feedbacks for a nonlinear control system on R n . We first introduce a family of discontinuous, piecewise smooth vector fields and derive a number of properties enjoyed by solutions of the corresponding O.D.E's. We then define a class of "patchy feedbacks" which are obtained by patching together a locally finite family of smooth controls. Our main result shows that, if a system is asymptotically controllable at the origin, then it can be stabilized by a piecewise constant patchy feedback control.Résumé. Dans cet article, on considère la structure de lois de feedback qui stabilisent asymptotiquement un système de contrôle non linéaire. Nousétudions une famille de champs de vecteurs discontinus, réguliers par morceaux, et démontrons de nombreuses propriétés satisfaites par leséquations différen-tielles ordinaires correspondantes. En outre, nous définissons une classe de "feedbacks rapiécés" qui sont obtenus par la superposition d'une famille localement finie de contrôles réguliers. Notre résultat principal montre que, si le système est asymptotiquement contrôlableà l'origine, alors il peutêtre stabilisé par un "feedback rapiécé", constant par morceaux.
Abstract. The paper is concerned with patchy vector fields, a class of discontinuous, piecewise smooth vector fields that were introduced in [A-B] to study feedback stabilization problems. We prove the stability of the corresponding solution set w.r.t. a wide class of impulsive perturbations. These results yield the robusteness of patchy feedback controls in the presence of measurement errors and external disturbances.
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