Maxwell's Equations as a Scattering Passive Linear System

Abstract

We consider Maxwell's equations on a bounded domain $\Omega\subset{\mathbb R}^3$ with Lipschitz boundary $\Gamma$, with boundary control and boundary observation. Relying on an abstract framework developed by us in an earlier paper, we define a scattering passive linear system that corresponds to Maxwell's equations and investigate its properties. The state of the system is $\left[\begin{smallmatrix} {\bf B}\\ {\bf D} \end{smallmatrix}\right]$, where ${\bf B}$ and ${\bf D}$ are the magnetic and electric flux densities, and the state space of the system is $X=E\oplus E$, where $E=L^2(\Omega;{\mathbb R}^3)$. We assume that $\Gamma_0$ and $\Gamma_1$ are disjoint, relatively open subsets of $\Gamma$ such that $\overline\Gamma_0\cup\overline\Gamma_1= \Gamma$. We consider $\Gamma_0$ to be a superconductor, which means that on $\Gamma_0$ the tangential component of the electric field is forced to be zero. The input and output space $U$ consists of tangential vector fields of class $L^2$ on $\Gamma_1$. The input and output at any moment are suitable linear combinations of the tangential components of the electric and magnetic fields. The semigroup generator has the structure $A=\big[\begin{smallmatrix}0 & -L \\ L^* & G- \gamma^* R \gamma \end{smallmatrix}\big]P$, where $L={\rm rot}$ (with a suitable domain), $\gamma$ is the tangential component trace operator restricted to $\Gamma_1$, $R$ is a strictly positive pointwise multiplication operator on $U$ (that can be chosen arbitrarily), and $P^{-1}=\left[\begin{smallmatrix}\mu & 0\\ 0 & \varepsilon\end{smallmatrix}\right]$ is another strictly positive pointwise multiplication operator (acting on $X$). The operator $-G$ is pointwise multiplication with the conductivity $g\geq 0$ of the material in $\Omega$. The system is scattering conservative iff $g=0$.