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 right-hand 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.