The Stabilization Problem: AGAS and SRS Feedbacks

The Stabilization Problem: AGAS and SRS Feedbacks Ludovic Rifford1 Institut Girard Desargues, Universite Claude Bernard, 43, Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex - France, 1 The Problem Throughout this paper, M denotes a smooth manifold of dimension n. We are given a control system on M of the form, x˙ = f(x, u) := m ∑ i=1 uifi(x), (1) where f1, · · · , fm are smooth vector fields on M and where the control u = (u1, · · · , um) belongs to Bm, the closed unit ball in IRm. Throughout the paper, “smooth” means always “of class C∞”. Such a control system is said to be Globally Asymptotically Controllable at the point O ? M (abbreviated GAC in the sequel) if the following two properties are satisfied: 1. Attractivity: For each x ? M there exists a control u(·) : IR≥0 ? Bm such that the corresponding trajectory x(·;x, u(·)) of (1) tends to O as t tends to infinity. 2. Lyapunov stability: For each neighborhood V of O, there exists some neigh- borhood U ofO such that if x ? U then the control u(·) above can be chosen such that x(t;x, u(·)) ? V , ?t ≥ 0.

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• semiconcave functions


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