An explicit construction of strong Lyapunov functions for systems satisfying conditions of the La Salle theorem

We present a construction of a (strong) Lyapunov function whose derivative is negative definite along the solutions of the system using another (weak) Lyapunov function whose derivative along the solutions of the system is negative semidefinite. The construction can be carried out if a Lie algebraic condition that involves the (weak) Lyapunov function and the system vector field is satisfied. Our main result extends to general nonlinear systems the strong Lyapunov function construction presented in Fauboug, L., et al. (June 2000), which was valid only for homogeneous systems.