Stabilisation des systèmes globalement asymptotiquement commandables

Given a globally asymptotically controllable system, we derive from an existence theorem of a semiconcave control-Lyapunov function, the construction of a globally asymptotically stable feedback in the Euler sense. The feedback is rather regular in the case of control systems affine in the control. Abridged English Version In this paper, we study systems of the general form ẋ(t) = f(x(t), u(t)) (1) where the state x(t) takes values in a Euclidian space R, the control u(t) takes values in a given compact set U , and f is locally Lipschitz in (x, u). A special element “0” is distinguished in U , and the state x = 0 of IR is an equilibrium point, i.e., f(0, 0) = 0. A trajectory will be an absolutely continuous function x(·) : [a, b] → IR associated to a control u(·) : [a, b] → IR for which (1) is verified almost everywhere in [a, b]. We now introduce our definitions and state the main results. Definition 1 The system (1) is Globally Asymptotically Controllable (abbreviated GAC) if the two following conditions are satisfied. Attractivity: For each ξ ∈ IR there is a control u : [0,∞[→ U that drives ξ asymptotically to 0. Lyapunov Stability: For each " > 0, there is a δ > 0 such that for each ξ ∈ IR with ‖ξ‖ ≤ δ there is a control u : [0,∞[→ U that drives ξ asymptotically to 0 and such that the corresponding trajectory x(·) satisfies ‖x(t‖ ≤ " for all t ≥ 0.