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**Feedback stabilization and Lyapunov functions.**
*(English)*
Zbl 0961.93047

The authors consider the problem of finding a feedback stabilizing law \(x \to k(x)\) associated with a control system \(x'(t) = f(x(t),u(t))\). Using the concept of positional strategies introduced in the framework of differential games by Krasovskii and Subbotin in 1988, assuming the existence of a Lyapunov Lipschitz function \(V\) defined on the complement of a sufficiently small ball around the origin, they prove the existence of piecewise constant feedback on a partition \(\pi:=\{ t_i \}_{i>0}\) of \([0,+\infty)\) with small enough diameter such that any \(\pi\)-trajectory starting from some neighborhood of a level set of \(V\), remains in this neighborhood until it reaches any given small ball around the origin in finite time.

In a second part of the paper the authors prove the existence of a Lyapunov function having the required property to get the previous existence result of “practical” stabilization. For this purpose they assume that \(f(\cdot,u)\) is Lipschitz and bounded on any bounded set and that the dynamical system satisfies a controllability property which expresses that a compact set \(A\) exists such that from any initial position in the complement of \(A\) starts a trajectory reaching \(A\) in a finite time. Then they show that there exists a lower semicontinuous Lyapunov function defined as a value function. This function is a solution to the Hamilton-Jacobi equation associated with some appropriate dynamical system. The paper ends with a robustness property of the feedback described in the first statement. Robustness is considered with respect to perturbations due to error measurements on the state position and on its velocity.

In a second part of the paper the authors prove the existence of a Lyapunov function having the required property to get the previous existence result of “practical” stabilization. For this purpose they assume that \(f(\cdot,u)\) is Lipschitz and bounded on any bounded set and that the dynamical system satisfies a controllability property which expresses that a compact set \(A\) exists such that from any initial position in the complement of \(A\) starts a trajectory reaching \(A\) in a finite time. Then they show that there exists a lower semicontinuous Lyapunov function defined as a value function. This function is a solution to the Hamilton-Jacobi equation associated with some appropriate dynamical system. The paper ends with a robustness property of the feedback described in the first statement. Robustness is considered with respect to perturbations due to error measurements on the state position and on its velocity.

Reviewer: Patrick Saint-Pierre (Paris)