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

Nathanael Skrepek

doi:10.3934/eect.2020098

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2021, Volume 10, Issue 4: 965-1006. Doi: 10.3934/eect.2020098 open access

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

Nathanael Skrepek^,

University of Wuppertal, School of Mathematics and Natural Science, Gaußstraße 20, D-42119 Wuppertal, Germany

^* Corresponding author: skrepek@uni-wuppertal.de
^* Corresponding author: skrepek@uni-wuppertal.de

Received: September 30, 2019

Revised: May 31, 2020

Early access: October 2020

Published: December 2021

This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 765579

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Abstract

We consider a port-Hamiltonian system on an open spatial domain $ \Omega \subseteq \mathbb{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.

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Mathematics Subject Classification: 47D06, 93C20, 35Q93, 47F05.

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Figure 1. Illustration of a quasi Gelfand triple, where $ D_{+} = \text{dom} \ \iota_{+} $ and $ D_{-} = \text{dom} \ \iota_{-} $

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