The aim of this section is to present an exact method of evaluating the quadratic performance
index for inﬁnite–dimensional systems. Our method relies on solving the Lyapunov operator
equation with unbounded operators. For delay–differential systems of the neutral type an explicit
construction of the solution to the Lyapunov operator equation is given. The results are
illustrated by two examples of the determination of a classical controller setting. An
approximation method of evaluating the quadratic performance index is also provided
and compared with the exact method. Let us consider some examples motivating our
investigations.