## 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.

### MSC:

 93D15 Stabilization of systems by feedback 93D30 Lyapunov and storage functions
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