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arXiv · 1905.01117

Dissipative extensions and port-Hamiltonian operators on networks

Marcus Waurick·Sven-Ake Wegner
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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.

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BibTeXRIS

Marcus Waurick, Sven-Ake Wegner. 2019-05-03. Dissipative extensions and port-Hamiltonian operators on networks. https://doi.org/10.1016/j.jde.2020.05.014

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