]> 2.1 Introduction

2.1 Introduction

This chapter is devoted to the discussion of the Riesz–bases application in the analysis of distributed–parameter systems controlled by a finite–dimensional or conventional controller. Both boundary control and boundary observation are allowed. The Riesz bases are constructed from a system of eigenfunctions of the closed–loop system operator. They are an effective tool in proving that the closed–loop system is well–posed, i.e. it give rise to a C0–semigroup. These bases also enable us to construct Lyapunov functionals in the form of series expansions. The analysis is illustrated by a completely worked–out example where the proportional controller setting is optimized with respect to the ISE criterion. The controller is applied to a parabolic plant modeling a resistive–capacitive (noninductive) direct–current transmission line.