In this paper we develop generalized Lyapunov and invariant set theorems fo
r nonlinear dynamical systems wherein all regularity assumptions on the Lya
punov function and the system dynamics are removed. In particular, local an
d global stability theorems are given using lower semicontinuous Lyapunov f
unctions. Furthermore, generalized invariant set theorems are derived where
in system trajectories converge to a union of largest invariant sets contai
ned in intersections over finite intervals of the closure of generalized Ly
apunov level surfaces. The proposed results provide transparent generalizat
ions to standard Lyapunov and invariant set theorems. (C) 1999 Elsevier Sci
ence B.V. All rights reserved.