## A smooth converse Lyapunov theorem for robust stability.(English)Zbl 0856.93070

The main result of the paper is a converse Lyapunov theorem for differential inclusions $$\dot x=F(x)$$; specifically, when $$F(x)$$ is Lipschitz, then asymptotic stability of the origin implies (and is clearly implied by) the existence of an infinitely differentiable Lyapunov function. This is a prime achievement, with a variety of applications to robust control design and perturbation analysis of differential equations. (Indeed, the differential inclusions formulation reflects the reviewer’s taste; the paper mentions this formulation, but addresses the equation $$\dot x=f(x,d(t))$$ with $$d(t)$$ a parameter in a compact set, and $$f(x,d)$$ Lipschitz.) The construction of the Lyapunov function is given in two related versions. The technical difficulties that arise, and the way they are solved, are of interest in their own. In addition, the paper provides an overview of possible applications, and gives concrete examples that demonstrate the technique. The paper is a big step forward in the area.

### MSC:

 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 93D30 Lyapunov and storage functions 34A60 Ordinary differential inclusions 93D20 Asymptotic stability in control theory 93D09 Robust stability
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