Periodic solutions and optimization problems for a class of semilinear parabolic control systems

Abstract

This paper is devoted to the study of the existence of periodic solutions for a class of control problems described by a semilinear parabolic equation. Related optimization problems are also considered. Periodic control problems and optimal periodic control problems for evolution equations described both by ordinary differential equations and by parabolic equations arise in many different situations. Examples are found in the theory of chemical processes [7], biomedical models [13], competing species ([6], and the extensive references therein), thermostat problems [3], [5], [18] and [20] (and the references therein). Several authors have also treated the existence of periodic solutions of differential inclusions in Banach spaces. These differential inclusions can model periodic control problems. In fact, under suitable assumptions, a large class of control problems can be reduced to the problem of finding solutions of a differential inclusion. We mention here the following papers: [8], [9], [10], [15], [16] and the references therein. The variety of techniques used in the quoted papers is quite large. Referring only to those papers dealing with parabolic equations, we recall that in [6] the properties of the Poincare map and the theory of ordered spaces