Abstract second-order boundary control systems
Till Preuster, Timo Reis, Manuel Schaller
TL;DR
This work develops a boundary-control framework for abstract second-order evolution equations of the form $\ddot{x}+D\dot{x}+Sx=0$ by constructing a boundary triplet for the generator $\mathcal{A}$ and defining a boundary node to realize boundary inputs/outputs. It shows that impedance and scattering passivity can be completely characterized through trace operators, and introduces an energy-based equivalence transform based on a factorization $S=A^*A$ that maps the system to a lower-order jet representation, connecting with port-Hamiltonian formulations. The theory is demonstrated on two canonical PDEs—the $n$-dimensional wave equation with damping and Maxwell’s equations—establishing well-posed boundary-control systems and energy balance under appropriate boundary conditions. Overall, the framework provides a systematic, operator-theoretic approach to boundary control of second-order systems, enabling robust passivity analysis and flexible reformulations. The results have potential impact on boundary-control design and energy-based modeling in infinite-dimensional settings.
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.