Smooth Morse-Lyapunov Functions Of Strong Attractors For Differential Inclusions With Applications

In this paper, we develop a smooth converse Lyapunov theorem for Morse decompositions of strong attractors of diferential inclusion:x′(t)∈F (x(t)), x(t)∈Rm,where F is an upper semicontinuous multivalued mapping on Rmwith compact convexvalues. Roughly speaking, let there be given a strong attractor A of the system withattraction basin and Morse decomposition M={M1,, Ml}. We will construct aradially unbounded function V∈C∞() such that(1) V is constant on each Morse set Mk;(2) V is strictly decreasing along any solution of the system in outside the Morsesets.Our result here is rather new even if we come back to the situation of smoothdynamical systems. The construction method of Lyapunov functions of attractors hereis quite diferent from that in [23]. It also difers signifcantly from those in otherliteratures. Our method seems to be more direct and simpler, and can be easily handled.Smooth Lyapunov functions are of crucial importance in the stability analysis, thedesign of feedback controls and the study of robustness property of asymptotic behavior.To illustrate the application of smooth Morse Lyapunov functions of attractors, we frstdiscuss the robust property of asymptotic stability with respect to the perturbations.Such results seem to be of particular interest in the theory of nonlinear control. We willshow that for any ε>0and compact set K, any solution x(t) of the perturbedsystem:x′(t)∈F (x(t))+h(t)starting from K will ultimately enter and stay in the ε neighborhood of an Morse setMjprovided that the perturbation is sufciently small, i.e.,x(t)∈B(Mj, ε) for t large enough.As an example to the application of smooth Lyapunov functions of attractors, weare interested in the topological simplicity of attractors from the point of view of shapetheory. In general the geometric structure of an attractor may be very complicated.Yet despite all that, here we prove that an attractor A of (3.1) has the shape of anyopen admissible neighborhood O of itself. Particularly we see that global attractorshave the shape of a single point. These results extend some known ones on smoothdynamical systems in [48] etc. to nonsmooth ones, and they also show that, from thetopological point of view, the strong attractors of nonsmooth dynamical systems are nomore complicated than those of the smooth systems.