]> 7.5.2 The step response of the system: the case of distributed observation and control

#### 7.5.2 The step response of the system: the case of distributed    observation and control

The model with distributed observation and control has been already constructed in Example 1.1.1 (see (1.10)). It is a special case of the system (7.4), because we can take

$d={\mathcal{𝒜}}^{-1}b,\phantom{\rule{2em}{0ex}}h={\left({\mathcal{𝒜}}^{\ast }\right)}^{-1}c$

with $b$ and $c$ given by (1.9). Moreover, $d\in D\left(\mathcal{𝒜}\right)$ and thus (7.5) holds. Since ${c}^{#}x={h}^{\ast }\mathcal{𝒜}x={c}^{\ast }x$ for a dense subset $D\left(\mathcal{𝒜}\right)$ then the observation functional ${c}^{#}$ extends uniquely to the bounded everywhere deﬁned functional ${c}^{\ast }$, ${c}^{\ast }x={〈x,c〉}_{\text{H}}$. This yields $\left(\overline{P}x\right)\left(t\right)={c}^{\ast }{e}^{t\mathcal{𝒜}}x$ and by EXS we have $\overline{P}=P\in L\left(\text{H},{\text{L}}^{2}\left(0,\infty \right)\right)$. Actually, $P$ is a HS operator which follows from Theorem 2.4.4. All assumptions of Lemma 7.2.1 hold and from (7.8) we get

 $y\left(t\right)={y}_{0}{y}_{1}\left(t\right),\phantom{\rule{2em}{0ex}}{y}_{1}\left(t\right)=\sum _{n=1}^{\infty }\left(\frac{{e}^{{\lambda }_{n}t}-1}{{\lambda }_{n}}\right){〈b,{e}_{n}〉}_{\text{H}}{〈c,{e}_{n}〉}_{\text{H}}$ (7.26)

where

 ${〈b,{e}_{n}〉}_{\text{H}}={\int }_{0}^{{\theta }_{0}}{e}_{n}\left(\theta \right)d\theta =\frac{1}{{∥{h}_{n}∥}_{\text{H}}}\left[{R}_{0}\frac{1-cos\left({\mu }_{n}{\theta }_{0}\right)}{{\mu }_{n}^{2}}+\frac{sin\left({\mu }_{n}{\theta }_{0}\right)}{{\mu }_{n}}\right]$ (7.27)

and

 $\begin{array}{cc}\hfill {〈c,{e}_{n}〉}_{\text{H}}=& {\int }_{{\theta }_{1}}^{{\theta }_{2}}{e}_{n}\left(\theta \right)d\theta =\hfill \\ \hfill =& \frac{1}{{∥{h}_{n}∥}_{\text{H}}}\left[{R}_{0}\frac{cos\left({\mu }_{n}{\theta }_{1}\right)-cos\left({\mu }_{n}{\theta }_{2}\right)}{{\mu }_{n}^{2}}+\frac{sin\left({\mu }_{n}{\theta }_{2}\right)-sin\left({\mu }_{n}{\theta }_{1}\right)}{{\mu }_{n}}\right]\hfill \end{array}$ (7.28)