Stability in the continuous case

We consider a time-invariant, finite-dimensional system of ordinary differential equations, whose right-hand side is continuous, but not Lipschitz continuous in general. For such a system, stability cannot be characterized in general by means of smooth Liapunov functions. We prove a new version of the converse of first Liapunov theorem. We give also some new conditions which allow us to verify, in different circumstances, whether a nonsmooth function is monotone along the solutions of the system.  2002 Elsevier Science (USA). All rights reserved.