# Dissipative extensions and port-Hamiltonian operators on networks

## TL;DR

This paper classifies boundary conditions for port-Hamiltonian PDEs on networks, including infinite ones with complex edge properties, using boundary systems to analyze dissipative extensions and operator behavior.

## Contribution

It introduces a new description for maximal dissipative extensions of skew-symmetric operators using boundary systems, applicable to complex network structures.

## Key findings

- Classified boundary conditions leading to contraction semigroups for port-Hamiltonian PDEs.
- Extended the theory to infinite networks with zero-length edge accumulations.
- Developed a framework for unbounded, non-negative weights on Hilbert spaces.

## 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.