]> 8.8 Conﬁrmation of the results by the Wiener – Hopf method

### 8.8 Conﬁrmation of the results by the Wiener – Hopf method

Our objective is to justify and conﬁrm the results obtained via the formal spectral–factorization method. To do this, an approach similar to that of  will be used. First of all, we show that there is a unique $u\in {\text{L}}^{2}\left(0,\infty \right)$ minimizing (8.2). Next, using the Wiener–Hopf technique, the Laplace transform $û\in {\text{H}}^{2}\left({\Pi }^{+}\right)$ of $u$ will be found. Finally, the formula obtained for $û$ will be compared with an expression for $û$ arising from (8.22).

Notice that the output function $y$ of system (8.5) can be represented as

$y={y}_{h}+{y}_{n},$

where

 ${y}_{h}\left(t\right)={c}_{0}^{\ast }{A}^{k}\varphi \left(t-kr-r\right),\phantom{\rule{2em}{0ex}}t\in \left[kr,\left(k+1\right)r\right]$ (8.34)

denotes the homogeneous part of the output corresponding to the response to a nonzero initial condition $\varphi$ on $\left[-r,0\right]$ and $u=0$, and where

 ${y}_{n}\left(t\right)=\sum _{k=0}^{\infty }{c}_{0}^{\ast }{A}^{k}{b}_{0}\left(T\left[\left(k+1\right)r\right]u\right)\left(t\right)$ (8.35)

denotes the nonhomogeneous part of the output corresponding to response to $\varphi =0$ on $\left[-r,0\right]$ and a nonzero forcing term $u$. Here ${\left\{T\left(t\right)\right\}}_{t\ge 0}$ is the semigroup of right shifts on ${\text{L}}^{2}\left(0,\infty \right)$.

Now,

${∥{y}_{h}∥}_{{\text{L}}^{2}\left(0,\infty \right)}^{2}=\sum _{k=0}^{\infty }{\int }_{kr}^{kr+r}{\left|{c}_{0}^{\ast }{A}^{k}\varphi \left(t-kr-r\right)\right|}^{2}dt\le \sum _{k=0}^{\infty }{\left|{\left({A}^{\ast }\right)}^{k}{c}_{0}\right|}_{{\mathbb{ℝ}}^{2}}^{2}{∥\varphi ∥}_{{\text{L}}^{2}\left(-r,0;{\mathbb{ℝ}}^{2}\right)}^{2}.$

The spectrum of the matrix $A$ lies in the open unit disk, and thus ${y}_{h}\in {\text{L}}^{2}\left(0,\infty \right)$ for any initial condition $\varphi \in {\text{L}}^{2}\left(-r,0;{\mathbb{ℝ}}^{2}\right)$.

Similar arguments lead, through the estimate

${∥{y}_{n}∥}_{{\text{L}}^{2}\left(0,\infty \right)}\le \left(\sum _{k=0}^{\infty }\left|{c}_{0}^{\ast }{A}^{k}{b}_{0}\right|\right){∥u∥}_{{\text{L}}^{2}\left(0,\infty \right)},$

to the conclusion that the mapping $\mathcal{ℒ}:{\text{L}}^{2}\left(0,\infty \right)\ni u↦{y}_{n}\in {\text{L}}^{2}\left(0,\infty \right)$ is linear and bounded. Thus (8.2) can be rewritten as

 $J={∥{y}_{h}+\mathcal{ℒ}u∥}_{{\text{L}}^{2}\left(0,\infty \right)}^{2}+{∥u∥}_{{\text{L}}^{2}\left(0,\infty \right)}^{2}$ (8.36)

and, by Curtain and Pritchard [15, Theorem 12.1, p. 254], $J$ achieves its minimal value at $u\in {\text{L}}^{2}\left(0,\infty \right)$ deﬁned as the unique solution of the equation

 $\left({\mathcal{ℒ}}^{\ast }\mathcal{ℒ}+I\right)u=-{\mathcal{ℒ}}^{\ast }{y}_{h}$ (8.37)

A direct solution of (8.37) is complicated, to determine the optimal control function $u\in {\text{L}}^{2}\left(0,\infty \right)$, the Wiener–Hopf technique will be useful. By Parseval’s theorem, instead of minimizing (8.36) one can minimize

 $J=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }\left[{\left|ĝ\left(j\omega \right)û\left(j\omega \right)+{ŷ}_{h}\left(j\omega \right)\right|}^{2}+{\left|û\left(j\omega \right)\right|}^{2}\right]d\omega$ (8.38)

where $ĝ\in {\text{H}}^{\infty }\left({\Pi }^{+}\right)$ is given by

 $ĝ\left(s\right)={e}^{-sr}{c}_{0}^{\ast }{\left(I-{e}^{-sr}A\right)}^{-1}{b}_{0}=\frac{a{e}^{-sr}}{1+b{e}^{-2sr}}$ (8.39)

and where $û$, ${ŷ}_{h}\in {\text{H}}^{2}\left({\Pi }^{+}\right)$, with ${ŷ}_{h}$ given by

 ${ŷ}_{h}\left(s\right)=\frac{a{e}^{-sr}}{1+b{e}^{-2sr}}\left[-b{e}^{-sr}{\stackrel{̂}{\varphi }}_{1F}\left(s\right)+{\stackrel{̂}{\varphi }}_{2F}\left(s\right)\right]$ (8.40)

are the Laplace transform of $g$, $u$, and ${y}_{h}$ respectively, and ${\stackrel{̂}{\varphi }}_{1F}\left(s\right)$ and ${\stackrel{̂}{\varphi }}_{2F}\left(s\right)$ are the ﬁnite Laplace transforms of ${\varphi }_{1}$ and ${\varphi }_{2}$, the components of $\varphi$. Under the Laplace transform, the operator $\mathcal{ℒ}$ is similar to the multiplication operator ${\text{H}}^{2}\left({\Pi }^{+}\right)\ni û↦ĝû\in {\text{H}}^{2}\left({\Pi }^{+}\right)$ with $ĝ\in {\text{H}}^{\infty }\left({\Pi }^{+}\right)$ given by (8.39). Observe that (8.39) agrees with (8.11).

Applying once more the result from [15, Theorem 12.1, p. 254] we establish that $J$ achieves its minimum at $û\in {\text{L}}^{2}\left(j\mathbb{ℝ}\right)$ (on a larger space), given by

$û\left(j\omega \right)=\frac{-{ŷ}_{h}\left(j\omega \right)ĝ\left(-j\omega \right)}{1+{\left|ĝ\left(j\omega \right)\right|}^{2}},\phantom{\rule{2em}{0ex}}\omega \in \mathbb{ℝ}.$

Hence, by the orthogonal projection theorem,

${û}_{opt}\left(s\right)={P}_{+}\left[j\omega ↦\frac{-{ŷ}_{h}\left(j\omega \right)ĝ\left(-j\omega \right)}{1+{\left|ĝ\left(j\omega \right)\right|}^{2}}\right]\left(s\right),$

where ${P}_{+}$ denotes the orthogonal projector from ${\text{L}}^{2}\left(j\mathbb{ℝ}\right)$ onto ${\text{H}}^{2}\left({\Pi }^{+}\right)$, which – by the uniqueness theorem for harmonic functions – may be treated as a closed subspace of ${\text{L}}^{2}\left(j\mathbb{ℝ}\right)$. The spectral–factorization of Section 8.4 will be used to determine ${û}_{opt}$. Notice that $\stackrel{̂}{\vartheta }$, $1∕\stackrel{̂}{\vartheta }\in {\text{H}}^{\infty }\left({\Pi }^{+}\right)$, where $\stackrel{̂}{\vartheta }$ is the spectral factor deﬁned by (8.13) (with $p$ and $q$ given by (8.14)). Now

${û}_{opt}\left(s\right)=\frac{1}{\stackrel{̂}{\vartheta }\left(s\right)}{P}_{+}\left[j\omega ↦\frac{-{ŷ}_{h}\left(j\omega \right)ĝ\left(-j\omega \right)}{\stackrel{̂}{\vartheta }\left(-j\omega \right)}\right]\left(s\right).$

Taking into account the formulae (8.13), (8.39) and (8.40), we get

 $\begin{array}{cc}\hfill \frac{-{ŷ}_{h}\left(s\right)ĝ\left(-s\right)}{\stackrel{̂}{\vartheta }\left(-s\right)}=& \hfill \left(1-{p}^{2}\right)\phantom{\rule{0ex}{0ex}}\frac{-q{e}^{-sr}}{1+b{e}^{-2sr}}\left[{\stackrel{̂}{\varphi }}_{1F}\left(s\right)+{e}^{-sr}{\stackrel{̂}{\varphi }}_{2F}\left(s\right)\right]+\\ \hfill & \hfill +\left(1-{p}^{2}\right)\phantom{\rule{0ex}{0ex}}\frac{1}{p+q{e}^{2sr}}\left[{\stackrel{̂}{\varphi }}_{2F}\left(s\right)+{q}^{2}{e}^{sr}{\stackrel{̂}{\varphi }}_{1F}\left(s\right)\right]\end{array}$ (8.41)

By replacing in (8.40) the initial vector $\left[\begin{array}{c}{\varphi }_{1}\hfill \\ {\varphi }_{2}\hfill \end{array}\right]$ by

$\frac{\left(1-{p}^{2}\right)q}{a}\left[\begin{array}{c}\hfill \frac{1}{b}{\varphi }_{2}\hfill \\ \hfill -{\varphi }_{1}\hfill \end{array}\right],$

we establish that the ﬁrst term in the right–hand side of (8.41) is the Laplace transform of a function which belongs to ${\text{L}}^{2}\left(0,\infty \right)$, or equivalently that the expression represents a function in ${\text{H}}^{2}\left({\Pi }^{+}\right)$. The second term in the right–hand side of (8.41) has poles in ${\Pi }^{+}$ and therefore does not belong to ${\text{H}}^{2}\left({\Pi }^{+}\right)$. Hence

 $\begin{array}{c}\hfill {û}_{opt}\left(s\right)=\frac{1}{\stackrel{̂}{\vartheta }\left(s\right)}\left(1-{p}^{2}\right)\phantom{\rule{0ex}{0ex}}\frac{-q{e}^{-sr}}{1+b{e}^{-2sr}}\left[{\stackrel{̂}{\varphi }}_{1F}\left(s\right)+{e}^{-sr}{\stackrel{̂}{\varphi }}_{2F}\left(s\right)\right]=\hfill \\ \hfill =\left(1-{p}^{2}\right)\phantom{\rule{0ex}{0ex}}\frac{-q{e}^{-sr}}{p+q{e}^{-2sr}}\left[{\stackrel{̂}{\varphi }}_{1F}\left(s\right)+{e}^{-sr}{\stackrel{̂}{\varphi }}_{2F}\left(s\right)\right]\hfill \end{array}$ (8.42)

According to the results presented in Section 8.6, (8.42) should agree with

 $û\left(s\right)=-{g}^{#}{\left(sI-{\mathcal{𝒜}}_{c}\right)}^{-1}\varphi$ (8.43)

From (8.19) and (8.20), we ﬁnd

 $\begin{array}{ccc}\hfill -{g}^{#}x& \hfill =\hfill & \left(1-p\right){x}_{2}\left(0\right)+\left(b-q\right){x}_{1}\left(-r\right)=\left(1-p\right)\left[{x}_{2}\left(0\right)-q{x}_{1}\left(-r\right)\right]=\hfill \\ \hfill & \hfill =\hfill & \left(1-p\right)\left[{x}_{2}\left(0\right)+p{x}_{2}\left(0\right)\right]=\left(1-{p}^{2}\right){x}_{2}\left(0\right)\phantom{\rule{2em}{0ex}}\forall x\in D\left({\mathcal{𝒜}}_{c}\right)\hfill \end{array}$ (8.44)

Thus, to evaluate the right–hand side of (8.43), complete knowledge of ${\left(sI-{\mathcal{𝒜}}_{c}\right)}^{-1}\varphi$ is not required: it is enough to observe that

 $\left({\left(sI-{\mathcal{𝒜}}_{c}\right)}^{-1}\varphi \right)\left(0\right)=\frac{{e}^{-sr}}{p+q{e}^{-2sr}}\left[\begin{array}{cc}\hfill -q{e}^{-sr}\hfill & \hfill p\hfill \\ \hfill -q\hfill & \hfill -q{e}^{-sr}\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\stackrel{̂}{\varphi }}_{1F}\left(s\right)\hfill \\ \hfill {\stackrel{̂}{\varphi }}_{2F}\left(s\right)\hfill \end{array}\right]$ (8.45)

On substituting (8.45) and (8.44) in (8.43), we get (8.42), which conﬁrms the validity of the results obtained via the formal spectral–factorization method. Correctness of the expression (8.44) for the optimal feedback implies, via the closed–loop Lyapunov equation (8.28), the correctness of the optimal cost expression obtained in Section 8.7.

Remark 8.8.1. The operators:

$\begin{array}{c}{\text{H}}^{2}\left({\Pi }^{+}\right)\ni {ŷ}_{h}↦{P}_{+}\left[j\omega ↦\frac{-{ŷ}_{h}\left(j\omega \right)ĝ\left(-j\omega \right)}{1+{\left|ĝ\left(j\omega \right)\right|}^{2}}\right]\in {\text{H}}^{2}\left({\Pi }^{+}\right)\hfill \\ {\text{H}}^{2}\left({\Pi }^{+}\right)\ni {ŷ}_{h}↦{P}_{+}\left[j\omega ↦\frac{-{ŷ}_{h}\left(j\omega \right)ĝ\left(-j\omega \right)}{\stackrel{̂}{\vartheta }\left(-j\omega \right)}\right]\in {\text{H}}^{2}\left({\Pi }^{+}\right)\hfill \end{array}$

are called the Toeplitz operators with symbols $\left[j\omega ↦-\frac{ĝ\left(-j\omega \right)}{1+{\left|ĝ\left(j\omega \right)\right|}^{2}}\right]\in {\text{L}}^{\infty }\left(j\mathbb{ℝ}\right)$ and $\left[j\omega ↦-\frac{ĝ\left(-j\omega \right)}{\stackrel{̂}{\vartheta }\left(-j\omega \right)}\right]\in {\text{L}}^{\infty }\left(j\mathbb{ℝ}\right)$, respectively.