The system is governed by the equations

$$\left\{\begin{array}{ccccc}\hfill 0& \hfill =\hfill & -\frac{\partial V\left(\theta ,\tau \right)}{\partial \theta}-RI\left(\theta ,\tau \right),\hfill & \hfill 0\le \theta \le 1,& \tau \ge 0\hfill \\ \hfill C\frac{\partial V\left(\theta ,\tau \right)}{\partial \tau}& \hfill =\hfill & -\frac{\partial I\left(\theta ,\tau \right)}{\partial \theta},\hfill & \hfill 0\le \theta \le 1,& \tau \ge 0\hfill \\ \hfill I\left(1,\tau \right)& \hfill =\hfill & 0,\hfill & \hfill & \tau \ge 0\hfill \\ \hfill V\left(0,\tau \right)& \hfill =\hfill & KV\left(1,\tau \right)+E,\hfill & \hfill & \tau \ge 0\hfill \\ \hfill V\left(\theta ,0\right)& \hfill =\hfill & 0,\hfill & \hfill 0\le \theta \le 1& \hfill \end{array}\right\}.$$

Problem 2.2.1. Determine the value $K\ne 1$ for which the integral performance index

$$J=\frac{1}{RC}{\int}_{0}^{\infty}{\left(V\left(1,\tau \right)-\frac{E}{1-K}\right)}^{2}d\tau $$

achieves its minimal value.

The substitution $x\left(\theta ,t\right)=V\left(\theta ,RCt\right)-\frac{E}{1-K}$ reduces the system equations to the form

and the performance index to the form

$$J={\int}_{0}^{1}{x}^{2}\left(1,t\right)dt$$ | (2.2) |

Let $\text{H}={\text{L}}^{2}\left(0,1\right)$, with standard inner product

$${\langle {x}_{1},{x}_{2}\rangle}_{\text{H}}={\int}_{0}^{1}{x}_{1}\left(\theta \right)\overline{{x}_{2}\left(\theta \right)}d\theta $$

be the state space. Taking $x\left(t\right)\left(\theta \right):=x\left(\theta ,t\right)$ we may write (2.1) as an abstract initial–value problem

with the unbounded linear operator

and with the initial condition

$${x}_{0}\left(\theta \right)=\frac{E}{K-1}$$ | (2.5) |

(note that ${x}_{0}\notin D\left(\mathcal{\mathcal{A}}\right)$). Simultaneously, the performance index (2.2) can be written formally as

$$J={\int}_{0}^{\infty}{\left|\mathcal{\mathcal{C}}x\left(t\right)\right|}^{2}dt$$ | (2.6) |

where $\mathcal{\mathcal{C}}$ is an unbounded linear functional deﬁned by the formula

$$\mathcal{\mathcal{C}}x=x\left(1\right)$$ | (2.7) |

Let us notice that there exists a unique $d\in \text{H}$, namely

$$d\left(\theta \right)=\frac{\theta}{K-1}$$ | (2.8) |

such that $\mathcal{\mathcal{C}}$ may be represented as

This is a simple but nontrivial example of the determination of the proportional controller setting, optimal with respect to a quadratic integral functional, steering the distributed – parameter plant of parabolic type through the boundary, with feedback from the boundary observation.

To solve our problem, we ﬁrst have to accomplish two fundamental tasks:

- (i)
- establish the well–posedness of the closed–loop system, i.e. verify that the linear closed–loop–system operator generates a semigroup,
- (ii)
- obtain an explicit expression for the performance index in terms of the system parameters – in the case of a positive solution of (i), this reduces to solving the Lyapunov operator equation.

Contrary to the classical problems met in mathematical physics, the presence of a feedback gives rise to non–selfadjoint operators, which is the source of several complications.

Our aim is to show that elementary spectral methods employing Riesz bases are an effective tool, allowing us to establish semigroup generation as well as express the solution of the Lyapunov operator equation in the series form. The bases are constructed from a system of eigenfunctions of the linear operator describing the closed–loop system. If, as in the given example, we can explicitly determine the spectrum, then a signiﬁcant further simpliﬁcation of the formulae expressing the performance index is possible.

Note that an example of parametric optimization of the distributed–parameter system, using the eigenfunctions series expansion, is given also in [73]. The method used there is based on a somewhat differently understood Lyapunov operator equation, and leads to another computational algorithm. Moreover, the performance index analysed there corresponds to a bounded observation operator $\mathcal{\mathcal{C}}$, and most of the reasoning has only intuitive justiﬁcation.