Liapunov and lagrange stability: Inverse theorems for discontinuous systems

The main result of this paper is a converse Liapunov theorem which applies to systems of ordinary differential equations with a discontinuous righthand side. We treat both the problem of local stability of an equilibrium position and the problem of boundedness of solutions. In particular, we show that in order to achieve a necessary and sufficient condition in terms of continuous Liapunov functions, the classical definitions need to be strengthened in a convenient way. This work was motivated by the recently renewed interest in stabilization by discontinuous feedback and analysis of the state evolution with respect to bounded inputs. To achieve a more general treatment, the exposition is developed in the framework of differential inclusions theory.