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

Abstract

We consider a port-Hamiltonian system on an open spatial domain Ω ⊆ R n with bounded 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 will be applied to the wave equation, Maxwell's equations and Mindlin plate model. Probably, there are even more applications.

Stability of the multidimensional wave equation in port-Hamiltonian modelling

“…This is a new aspect that arises for spatial multidimensional port-Hamiltonian systems as in the onedimensional spatial setting the compact embedding is always given. It is likely that most of the techniques presented in this article will translate for general linear port-Hamiltonian systems on multidimensional spatial domains (see [24]) like Maxwell's equations and the Mindlin plate model. Probably the crucial tool will be a unique continuation principle.…”

A Compactness Result for the div-curl System with Inhomogeneous Mixed Boundary Conditions for Bounded Lipschitz Domains and Some Applications

“…with dom A 0 := (w, v) ∈ H 1 Γ0 (Ω) × H Γ1 (div, Ω) σ Γ1 w + kν Γ1 = 0 . As before, by [8,Theorem 4.4] or [22,Example 8.9], A 0 is a generator of C 0 -semigroup. Again, we want to separate the static solutions from the dynamic system.…”

“…Here, we did ignore the equations div E = 0, div H = 0 and ν Γ0 H = 0. However, A 0 is a generator of a C 0 -semigroup by [22,Example 8.10] or [25,Section 5], where the input function is u = 0. (In these sources they regard boundary control systems and system nodes, respectively.…”

Characterisation for Exponential Stability of port-Hamiltonian Systems

“…In order to confirm that W (or, equivalently, W := W −1 ) is unitary, we appeal to (18) and obtain 18) and ( 19) from (b). Then we compute for y = (y, ŷ…”

“…It has since then triggered manifold research and ideas. For this we refer to [9,21,2,18,23] and the references therein, see also [8] for a survey. The basic idea is to describe a physical -mostly energy conserving or at least energy dissipating -phenomenon in terms of a partial differential equation in the underlying physical domain together with suitable boundary conditions.…”