Abstract second-order boundary control systems

Abstract

We consider abstract second order systems of the form $\ddot{x}(t) + D \dot{x}(t) + Sx(t)=0$, which are typically analyzed via the operator matrix $\mathcal{A}=\left[\begin{smallmatrix} 0 & I \\ -S & -D \end{smallmatrix}\right]$ governing the free dynamics of the corresponding first-order in time formulation. While previous work (e.g. on spectral properties of) $\mathcal{A}$ has focused on self-adjoint uniformly positive $S$, we consider the more general case which comprises the situation where $S^*$ is symmetric, i.e., $S^*\subset S$. As we will show, this relaxation allows for a large freedom in view of boundary conditions. Our main contribution is the construction of a boundary triplet for the operator $\mathcal{A}$ and the definition of an associated boundary control system. We fully characterize the cases in which the latter is impedance resp. scattering passive in terms of the associated trace operators. Furthermore, based on a non-standard factorization of $S$ we introduce an equivalence transform of $\mathcal{A}$ that maps the abstract second-order system (e.g., $\mathcal{A} = \left[\begin{smallmatrix} 0 & I \\ Δ& -D \end{smallmatrix}\right]$ for the wave equation in position-momentum formulation) into widely-used alternative representation involving lower-order spatial derivatives on the jet space (i.e., $\left[\begin{smallmatrix} 0 & \nabla \\ \operatorname{div} & -D \end{smallmatrix}\right]$ corresponding to the wave equation in strain-momentum formulation). We illustrate the suggested approach on the example of a $n$-dimensional wave equation and a Maxwell equation.