SMOOTH MORSE-LYAPUNOV FUNCTIONS OF STRONG ATTRACTORS FOR DIFFERENTIAL INCLUSIONS

This paper is concerned with a smooth converse Lyapunov theorem for Morse decompositions of strong attractors of differential inclusion x'(t) is an element of F(x(t)), where F is an upper semicontinuous multivalued mapping on R-m with compact convex values. Roughly speaking, let there be given a strong attractor A of the system with attraction basin Omega and Morse decomposition M = {M-1, ... , M-l}. We will construct a radially unbounded function V is an element of C-infinity(Omega) such that (1) V is constant on each Morse set M-k and (2) V is strictly decreasing along any solution of the system in Omega outside the Morse sets.