Well-posedness of linear first order Port-Hamiltonian Systems on multidimensional spatial domains

TL;DR

This paper establishes a mathematical framework for analyzing well-posedness and boundary conditions of linear first-order port-Hamiltonian systems on multidimensional domains, with applications to wave, Maxwell, and plate models.

Contribution

It introduces a boundary triple approach for port-Hamiltonian systems on multidimensional domains and develops the theory of quasi Gelfand triples for boundary control systems.

Findings

Characterization of boundary conditions for unique, energy-non-increasing solutions.

Development of a boundary triple framework for multidimensional port-Hamiltonian systems.

Application to wave, Maxwell, and Mindlin plate models.

Abstract

We consider a port-Hamiltonian system on a spatial domain $\Omega \subseteq \mathbb{R}^n$ that is bounded with Lipschitz boundary. We show that there is a boundary triple associated to this system. Hence, we can characterize all boundary conditions that provide unique solutions that are non-increasing in the Hamiltonian. As a by-product we develop the theory of quasi Gelfand triples. Adding ``natural'' boundary controls and boundary observations yields scattering/impedance passive boundary control systems. This framework can be applied to the wave equation, Maxwell equations and Mindlin plate model, and probably many more.