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Patchy vector fields and asymptotic stabilization. (English) Zbl 0924.34058
Summary: This paper is concerned with the structure of asymptotically stabilizing feedbacks for a nonlinear control system on. The authors first introduce a family of discontinuous, piecewise smooth vector fields and derive a number of properties enjoyed by solutions to the corresponding ODEs. They define a class of “patchy feedbacks” which are obtained by patching together a locally finite family of smooth controls. The 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.

MSC:
34H05 Control problems involving ordinary differential equations
93D15 Stabilization of systems by feedback
34D30 Structural stability and analogous concepts of solutions to ordinary differential equations
49K15 Optimality conditions for problems involving ordinary differential equations
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