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Patchy Vector Fields and Asymptotic Stabilization

Published online by Cambridge University Press:  15 August 2002

Fabio Ancona  and
Alberto Bressan
Fabio Ancona
Affiliation:
Dipartimento di Matematica and CIRAM, Università di Bologna, piazza Porta S. Donato 5, Bologna 40127, Italy; ancona@ciram3.ing.unibo.it.
Alberto Bressan
Affiliation:
S.I.S.S.A., via Beirut 4, Trieste 34014, Italy; bressan@sissa.it.
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Abstract

This paper is concerned with the structure of asymptotically stabilizing feedbacks for a nonlinear control system on $\mathbb{R}^n$. 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.

Keywords

Asymptotic controllabilityfeedback asymptotic stabilizationdiscontinuous feedbackdiscontinuous vector fields.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 4 , 1999 , pp. 445 - 471
Copyright
© EDP Sciences, SMAI, 1999

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