$$\left\{\begin{array}{ccc}\hfill {x}_{k+1}& \hfill =\hfill & {Ax}_{k}+{Bu}_{k}\hfill \\ \hfill {y}_{k}& \hfill =\hfill & {Qx}_{k}\hfill \end{array}\right\},$$

where $A$, $B$ and $Q$ are the linear and bounded multiplication operators given by

$$\left(Ax\right)\left(\theta \right)=Ax\left(\theta \right),\phantom{\rule{1em}{0ex}}\left(Bu\right)\left(\theta \right)={b}_{0}u\left(\theta \right),\phantom{\rule{2em}{0ex}}\left(Qx\right)\left(\theta \right)={c}_{0}^{\ast}x\left(\theta \right),$$

for $-r\le \theta \le 0$. Simultaneously, the performance index (8.2) can be represented as

$$J=\sum _{k=0}^{\infty}\left[{\langle {y}_{k},{y}_{k}\rangle}_{\text{Y}}+{\langle {u}_{k},{u}_{k}\rangle}_{\text{U}}\right].$$

The validity of our results can be conﬁrmed using Helton’s solution of the abstract discrete lq problem – see [47].

Some characteristic features of the system under study which differ from those of standard LQ–theory have already been observed. However, the main difference is that the realization equation (8.15) deﬁning ${g}^{\#}$ plays here a much more intrinsic role. Equation (8.15) is a substitute of the regulator equation $\mathcal{\mathscr{H}}b=g$ well known in the standard lq problem. For our special system such an equation cannot be met, since $d\notin D\left(A\right)$, ${g}^{\#}$ is unbounded, and yet we have (8.33). This remark applies also to the Riccati operator equation. Equations (8.27) and (8.28) are substitutes for the Riccati operator equations for the closed–loop and open–loop systems, respectively. Theory of the Riccati operator equation has been developed in [80] and [88].

It has been observed elsewhere that the spectral–factorization method is a powerful analytic tool in stability analysis (for the construction of a Lyapunov functional for semilinear systems) even when formally applied. Our investigation shows that the same happens for the lq problem. Hence, we can conclude that the spectral factorization method seems to apply to problems with boundary control and/or observation, this should be expected, since it is an input–output frequency–domain procedure. We expect that there is a generalization of the Callier–Winkin spectral factorization approach to the lq problem for systems with boundary control and observations. This is probably the case also for the new proof of the fundamental LQ–theory result presented in Chapter 6.

The use of the d’Alembert solutions can be eliminated but this leads to more complicated calculations in comparison with these presented in our investigations.

Although the optimal controller was constructed by the formal spectral – factorization method, the results of Sections 8.7 and 8.8 conﬁrm that they are correct.

It is possible to give a fairly general proof of the existence of a unique optimal control which does not require an explicit construction of the observability and input–output maps. Indeed, we have

$$y=\overline{P}{x}_{0}+\overline{F}u\phantom{\rule{2em}{0ex}}\forall {x}_{0}\in \text{H}\phantom{\rule{1em}{0ex}}\forall u\in {\text{L}}^{2}\left(0,\infty \right)$$

where $\overline{P}$ denotes the (extended) observability map and $\overline{F}$ stands for the (extended) input–output map, or in the former notation ${y}_{h}=\overline{P}{x}_{0}$ and ${y}_{n}=\mathcal{\mathcal{L}}u=\overline{F}u$. It suffices to prove that $\overline{P}\in L\left(\text{H},{\text{L}}^{2}\left(0,\infty \right)\right)$ and $\overline{F}\in L\left({\text{L}}^{2}\left(0,\infty \right)\right)$. Recall that $\mathcal{\mathcal{A}}$ deﬁned by (8.8) generates an EXS ${\text{C}}_{0}$–semigroup on $\text{H}$. Notice that

$${c}^{\#}x={h}^{\ast}\mathcal{\mathcal{A}}x={\langle \mathcal{\mathcal{A}}x,h\rangle}_{\text{H}},\phantom{\rule{2em}{0ex}}h=\vartheta {h}_{0},\phantom{\rule{1em}{0ex}}\vartheta :=\frac{a}{1+b},\phantom{\rule{1em}{0ex}}{h}_{0}:=\left[\begin{array}{c}\hfill b1\\ \hfill -1\end{array}\right]\in \text{H}$$

for $x\in D\left(\mathcal{\mathcal{A}}\right)$, and $d\notin D\left(\mathcal{\mathcal{A}}\right)$ but (7.5) holds giving ${c}^{\#}d=-\vartheta $. Since $\left(\mathcal{\mathscr{H}}x\right)\left(\eta \right)=Hx\left(\eta \right)$, where $H\in L\left({\mathbb{\mathbb{R}}}^{2}\right)$, $H={H}^{T}>0$ solves the discrete matrix Lyapunov equation

$${A}^{T}HA-H=-{c}_{0}{c}_{0}^{T},$$

is a unique bounded, self–adjoint, coercive solution of the Lyapunov operator equation

$${\langle \mathcal{\mathcal{A}}x,\mathcal{\mathscr{H}}x\rangle}_{\text{H}}+{\langle x,\mathcal{\mathscr{H}}\mathcal{\mathcal{A}}x\rangle}_{\text{H}}=-{a}^{2}{x}_{2}^{2}\left(-r\right)\phantom{\rule{2em}{0ex}}\forall x\in D\left(\mathcal{\mathcal{A}}\right),$$

then by Theorems 2.4.1 and 2.4.2 the observation functional ${c}^{\#}$ is admissible. This proves that $\overline{P}\in L\left(\text{H},{\text{L}}^{2}\left(0,\infty \right)\right)$ holds. By Exercise 8.3.1 the condition (7.14) is satisﬁed which jointly with the admissibility of ${c}^{\#}$ and (7.5) implies, using Theorem 7.4.1, that $\overline{F}\in L\left({\text{L}}^{2}\left(0,\infty \right)\right)$.

Exercise 8.9.1. Show that $d$ is an admissible factor control vector. The dual observed system is

$$\left\{\begin{array}{ccc}\hfill \u1e57\left(t\right)& \hfill =\hfill & {\mathcal{\mathcal{A}}}^{\ast}p\left(t\right)\hfill \\ \hfill q\left(t\right)& \hfill =\hfill & {d}^{\ast}{\mathcal{\mathcal{A}}}^{\ast}p\left(t\right)\hfill \end{array}\right\}$$

where ${\mathcal{\mathcal{A}}}^{\ast}$ stands for the adjoint operator of $\mathcal{\mathcal{A}}$,

$${\mathcal{\mathcal{A}}}^{\ast}p=-{p}^{\prime},\phantom{\rule{2em}{0ex}}D\left({\mathcal{\mathcal{A}}}^{\ast}\right)=\left\{p\in {\text{W}}^{1,2}\left(-r,0\right)\oplus {\text{W}}^{1,2}\left(-r,0\right):\phantom{\rule{0ex}{0ex}}p\left(-r\right)={A}^{T}p\left(0\right)\right\},$$

and for $p\in D\left({\mathcal{\mathcal{A}}}^{\ast}\right)$ we have

$$q={d}^{\ast}{\mathcal{\mathcal{A}}}^{\ast}p=\frac{1}{1+b}{\u3008\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}\right],\left[\begin{array}{c}\hfill {p}_{1}^{\prime}\hfill \\ \hfill {p}_{2}^{\prime}\hfill \end{array}\right]\u3009}_{\text{H}}.$$

Since $\left(\mathcal{\mathscr{H}}p\right)\left(\eta \right)=Hp\left(\eta \right)$, where $H\in L\left({\mathbb{\mathbb{R}}}^{2}\right)$, $H={H}^{T}>0$ solves the discrete matrix Lyapunov equation

$$AH{A}^{T}-H=-{b}_{0}{b}_{0}^{T},$$

is a unique bounded, self–adjoint, coercive solution of the Lyapunov operator equation

$${\langle {\mathcal{\mathcal{A}}}^{\ast}p,\mathcal{\mathscr{H}}p\rangle}_{\text{H}}+{\langle p,\mathcal{\mathscr{H}}{\mathcal{\mathcal{A}}}^{\ast}p\rangle}_{\text{H}}=-{p}_{2}^{2}\left(0\right)\phantom{\rule{2em}{0ex}}\forall p\in D\left({\mathcal{\mathcal{A}}}^{\ast}\right)$$

then by Theorems 2.4.1 and 2.4.2 this observation functional is admissible. Our statement follows from the duality Theorem 7.2.2.

The system (8.7) can also be written in the form

where $b=\left[\begin{array}{c}0\hfill \\ \delta \hfill \end{array}\right]\in {\text{H}}_{-1}$, here $\delta $ denotes the Dirac pseudofunction, while ${\text{H}}_{-1}$ is the completion of $\text{H}={\text{L}}^{2}\left(-r,0;{\mathbb{\mathbb{R}}}^{2}\right)$ with respect to the norm ${\u2225u\u2225}_{{\text{H}}_{-1}}={\u2225{\mathcal{\mathcal{A}}}^{-1}x\u2225}_{\text{H}}$. To see this, observe that the sequence ${\left\{{d}_{n}\right\}}_{n\in \mathbb{\mathbb{N}}}$ in $D\left(\mathcal{\mathcal{A}}\right)$, deﬁned by

$${d}_{n}\left(\theta \right)=\frac{1}{1+b}\left[\begin{array}{c}\phantom{\rule{1em}{0ex}}\phantom{\rule{0ex}{0ex}}1,\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{if}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-r\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\le \theta \le 0\hfill \\ \left\{\begin{array}{cccc}1,\hfill & \text{if}\hfill & \hfill -r& \le \theta \le -r{2}^{-n}\hfill \\ 1-\left(\frac{\theta}{r}{2}^{n}+1\right)\left(1+\kappa {\rho}^{-2}\right),\hfill & \text{if}\hfill & \hfill -r{2}^{-n}& \le \theta \le 0\hfill \end{array}\right\}\hfill \end{array}\right],$$

has the following properties:

- (i)
- ${d}_{n}\to d\phantom{\rule{1em}{0ex}}\text{in}\phantom{\rule{1em}{0ex}}\text{H}\phantom{\rule{1em}{0ex}}\text{as}\phantom{\rule{1em}{0ex}}n\to \infty $,
- (ii)
- $\mathcal{\mathcal{A}}{d}_{n}\to -b\phantom{\rule{1em}{0ex}}\text{in}\phantom{\rule{1em}{0ex}}{\text{H}}_{-1}\phantom{\rule{1em}{0ex}}\text{as}\phantom{\rule{1em}{0ex}}n\to \infty .$

Property (i) is obvious. By (i), the sequence ${\left\{\mathcal{\mathcal{A}}{d}_{n}\right\}}_{n\in \mathbb{\mathbb{N}}}$ is a Cauchy sequence in ${\text{H}}_{-1}$. The limit can be evaluated explicitly by considering the graphs of each component of $\mathcal{\mathcal{A}}{d}_{n}$. It is shown in [38] that the system (8.46) does not belong to the Pritchard–Salamon class [70] and consequently the lq problem formulated in this section cannot be solved by the approach presented in [70]. In Lasiecka and Triggiani ([62]), the output operator is generally assumed to be bounded, except [62, Section 4.2] where some unbounded operators are admitted only for the ﬁnite horizon lq problem, and therefore (8.46) cannot be handled by the methods used by these authors.

As we already said the construction of the optimal controller presented in this section has been conﬁrmed by O. Staffans [80] and then by M. Weiss and G. Weiss [88]. The last paper uses (8.46) as an abstract model of the system.