Dissipative extensions and port-Hamiltonian operators on networks
Authors
Marcus Waurick, Sven-Ake Wegner
Abstract
In this article we study port-Hamiltonian partial differential equations on certain one-dimensional manifolds. We classify those boundary conditions that give rise to contraction semigroups. As an application we study port-Hamiltonian operators on networks whose edges can have finite or infinite length. In particular, we discuss possibly infinite networks in which the edge lengths can accumulate zero and port-Hamiltonian operators with Hamiltonians that neither are bounded nor bounded away from zero. We achieve this, by first providing a new description for maximal dissipative extensions of skew-symmetric operators. The main technical tool used for this is the notion of boundary systems. The latter generalizes the classical notion of boundary triple(t)s and allows to treat skew-symmetric operators with unequal deficiency indices. In order to deal with fairly general variable coefficients, we develop a theory of possibly unbounded, non-negative, injective weights on an abstract Hilbert space.
Keywords
C 0 -semigroup; Port-Hamiltonian PDEs with singular weights; Maximal dissipative operators; Differential equations on infinite networks; Boundary triple(t); Quantum Graphs with vanishing edge lengths
Citation
- Journal: Journal of Differential Equations
- Year: 2020
- Volume: 269
- Issue: 9
- Pages: 6830–6874
- Publisher: Elsevier BV
- DOI: 10.1016/j.jde.2020.05.014
BibTeX
@article{Waurick_2020,
title={{Dissipative extensions and port-Hamiltonian operators on networks}},
volume={269},
ISSN={0022-0396},
DOI={10.1016/j.jde.2020.05.014},
number={9},
journal={Journal of Differential Equations},
publisher={Elsevier BV},
author={Waurick, Marcus and Wegner, Sven-Ake},
year={2020},
pages={6830--6874}
}
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