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
${\text{C}}_{0}$–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.