Asymptotic behaviour of some Optimal Control Problems

Abstract

The theory of homogenization of partial differential equations is a concept that deals with the study of the macroscopic behaviour of a composite medium through its microscopic properties. The aim of this thesis is to analyse the asymptotic behaviour of various classes of optimal control problems. Some preliminary concepts necessary for this study have been introduced in Chapter 1. The notion of control problems is introduced and a brief review of the existing literature on the homogenization of optimal control problems is given in Chapter 2. Survey of some existing results in the literature are presented and the difficulties presented by some of these problems are highlighted and discussed in the later chapters. In chapter 3 the asymptotic behaviour of a class of optimal control problems with varying control sets and the corresponding low cost control problems is studied. It also attempts to answer an open problem posed in [KP02]. The aymptotic behaviour of low cost control problems on perforated domains is studied in Chapter 4. Also the extended results for some problems discussed at the second chapter are given with an improved description. In Chapter 5, an optimal control problem with constraints on state is studied. The admissible control set involved in the problem is defined through the state variable. The problem is settled for both the fixed cost of the control and low cost control cases in both perforated and non-perforated settings. A state constraint problem with a different control set is given as an open problem. Many open problems are addressed and discussed in this chapter.