Periodic solutions for a class of evolution inclusions

TL;DR

This paper establishes existence and approximation results for periodic solutions of evolution inclusions involving subdifferentials, with applications to nonlinear parabolic control systems.

Contribution

It provides new existence theorems for both convex and nonconvex evolution inclusions and demonstrates strong relaxation for convex solutions.

Findings

Existence of periodic solutions for the considered evolution inclusion.

Approximation of convex solutions by extremal trajectories.

Application to a nonlinear parabolic control system.

Abstract

We consider a periodic evolution inclusion defined on an evolution triple of spaces. The inclusion involves also a subdifferential term. We prove existence theorems for both the convex and the nonconvex problem, and we also produce extremal trajectories. Moreover, we show that every solution of the convex problem can be approximated uniformly by certain extremal trajectories (strong relaxation). We illustrate our results by examining a nonlinear parabolic control system.