A Physically Motivated Class of Scattering Passive Linear Systems

We introduce a class of scattering passive linear systems motivated by examples from mathematical physics. The state space of the system is $X=H\oplus E$, where $H$ and $E$ are Hilbert spaces. We also have a Hilbert space $E_0$ which is dense in $E$, with continuous embedding, and $E_0'$ is the dual of $E_0$ with respect to the pivot space $E$. The input space is the same as the output space, and it is denoted by $U$. The semigroup generator has the structure $A= {\big[\begin{smallmatrix} 0 & -L \\ L^* & G-\frac{1}{2}K^* K\end{smallmatrix}\big]}$, where $L\in{\cal L}(E_0,H)$ and $K\in {\cal L}(E_0,U)$ are such that ${\big[\begin{smallmatrix}L \\ K\end{smallmatrix}\big]}$, with domain $E_0$, is closed as an unbounded operator from $E$ to $H\oplus U$. The operator $G\in{\cal L}(E_0,E_0')$ is such that ${\rm Re\,}\langle Gw_{_0},w_{_0}\rangle\leq 0$ for all $w_{_0} \in E_0$. The observation operator is $C=\left[\begin{matrix}0 & -K\end{matrix}\right]$, the control operator is $B=-C^*$, and the output equatio...