Abstract second-order boundary control systems

Abstract

Many physical systems, such as vibrating elastic bodies, acoustic media and electromagnetic fields, are governed by second-order evolution equations $\ddot{x}+D\dot{x}+Sx=0$ and interact with their environment through their boundary. The classical analysis via the operator matrix $\mathcal{A}=\left[\begin{array}{cc}0& I\\ -S& -D\end{array}\right]$ requires $S$ to be self-adjoint, which typically fixes a boundary condition a priori and thus excludes boundary control. We therefore use $S$ only to define the energy space and govern the dynamics by a maximal operator $S_{\max }\supset S$ whose domain carries no boundary condition. Based on a factorization $S=A^{\ast }A$, we construct a boundary triplet for the associated operator matrix and define boundary inputs and outputs of Robin type, parametrized by an operator $P$ on the boundary space. We show that the resulting system is impedance passive and internally well-posed if $P$ is a contraction, and that for $D=0$ this covers all boundary conditions under which the free dynamics generate a contraction semigroup. Moreover, we show that the position-momentum formulation (e.g., $\left[\begin{array}{cc}0& I\\ \Delta & 0\end{array}\right]$ for the wave equation) and the strain-momentum formulation (e.g., $\left[\begin{array}{cc}0& \nabla \\ \mathrm{div}& 0\end{array}\right]$) describe the same open system, including its boundary inputs and outputs. We illustrate our results by an $n$-dimensional wave equation and Maxwell's equations.