]> 7.2.1 Admissible factor control vectors

#### 7.2.1 Admissible factor control vectors

Assume additionally that the semigroup ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ is EXS.

Deﬁnition 7.2.1. A vector $d\in \text{H}$ is called an admissible factor control vector if

 $\left({\int }_{0}^{\infty }S\left(t\right)du\left(t\right)dt\right)\in D\left(\mathcal{𝒜}\right)\phantom{\rule{2em}{0ex}}\forall u\in {\text{L}}^{2}\left(0,\infty \right)$ (7.1)

By EXS the operator $W$, $Wu:={\int }_{0}^{\infty }S\left(t\right)du\left(t\right)dt$ belongs to $L\left({\text{L}}^{2}\left(0,\infty \right),\text{H}\right)$. Since $\mathcal{𝒜}$ is closed and (7.1) is equivalent to the inclusion $R\left(W\right)\subset D\left(\mathcal{𝒜}\right)$ then applying the closed graph theorem we get $Q=\mathcal{𝒜}W\in L\left({\text{L}}^{2}\left(0,\infty \right),\text{H}\right)$. The operator $Q$ is called the reachability map. Moreover, the following is known [44, Theorem 4.2] with the reﬂection operator ${R}_{t}$ given by

$\left({R}_{t}u\right)\left(\tau \right):=\left\{\begin{array}{cc}\hfill u\left(t-\tau \right),\hfill & \phantom{\rule{1em}{0ex}}\tau \in \left[0,t\right]\hfill \\ \hfill 0,\hfill & \phantom{\rule{1em}{0ex}}\tau \ge t\hfill \end{array}\right\},\phantom{\rule{2em}{0ex}}u\in {\text{L}}^{2}\left(0,\infty \right).$

Theorem 7.2.1. Let $d\in \text{H}$ be an admissible factor control vector and let $u\in {\text{L}}^{2}\left(0,\infty \right)$. Then the function

$x\left(t\right):=Q{R}_{t}u=\mathcal{𝒜}{\int }_{0}^{t}S\left(t-\tau \right)du\left(\tau \right)d\tau$

is a weak solution of

 $\left\{\begin{array}{c}ẋ\left(t\right)=\mathcal{𝒜}\left[x\left(t\right)+du\left(t\right)\right]\hfill \\ x\left(0\right)=0\hfill \end{array}\right\}$ (7.2)

i.e. it is a continuous $\text{H}$–valued function of $t$ such that for $y\in D\left({\mathcal{𝒜}}^{\ast }\right)$, $t↦{〈x\left(t\right),y〉}_{\text{H}}$ is absolutely continuous and for almost all $t$ and all $y\in D\left({\mathcal{𝒜}}^{\ast }\right)$ we have

$\frac{d}{dt}{〈x\left(t\right),y〉}_{\text{H}}={〈x\left(t\right)+du\left(t\right),{\mathcal{𝒜}}^{\ast }y〉}_{\text{H}}.$

From [69, p. 258 - 259] we know that for almost all $t\ge 0$

 $\frac{d}{dt}{〈S\left(t\right){x}_{0},w〉}_{\text{H}}={〈S\left(t\right){x}_{0},{\mathcal{𝒜}}^{\ast }w〉}_{\text{H}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\forall {x}_{0}\in \text{H}\phantom{\rule{1em}{0ex}}\forall w\in D\left({\mathcal{𝒜}}^{\ast }\right)$ (7.3)

Now by Theorem 7.2.1 a unique weak solution of

$\left\{\begin{array}{c}ẋ\left(t\right)=\mathcal{𝒜}\left[x\left(t\right)+du\left(t\right)\right]\hfill \\ x\left(0\right)={x}_{0}\hfill \end{array}\right\}$

takes the form

$x\left(t\right)=S\left(t\right){x}_{0}+Q{R}_{t}u,\phantom{\rule{2em}{0ex}}{x}_{0}\in \text{H},\phantom{\rule{1em}{0ex}}u\in {\text{L}}^{2}\left(0,\infty \right).$