10.1 Introduction

The aim of this chapter is to introduce the concept of controllability for control systems defined in infinite dimensional spaces. In particular, the controlled heat equation is studied in the framework of semigroup theory. Basic concepts from functional analysis are prerequisites for understanding this chapter.

In Section 10.2, example of an infinite dimensional linear control system is given. In Section 10.3, uniformly continuous and strongly continuous semigroups are defined with examples. The definition of infinitesimal generator of a semigroup and related examples are provided in Section 10.4. In Section 10.5, important properties of C0-semigroups and uniformly continuous semigroups and their generators are discussed. The mild solution of initial value problems (IVPs) in terms of semigroups is also introduced in this section. In Section 10.6, approximate controllability of infinite dimensional systems and ill-posed problems are defined, and examples are provided. In Section 10.7, computation of a control function for an approximately controllable system as a solution to an ill-posed problem is demonstrated using the linear heat equation. Section 10.8 contains a brief discussion on a semilinear control system and its controllability using the corresponding linear system.