Telegraph type systems on networks and port-Hamiltonians. II. Graph realizability

TL;DR

This paper establishes conditions under which port-Hamiltonian systems with linear boundary conditions on networks can be represented as hyperbolic equations on metric graphs, linking boundary matrices to graph adjacency.

Contribution

It provides a novel characterization connecting port-Hamiltonian boundary conditions with the realizability of the system as a hyperbolic PDE on a graph.

Findings

Derived conditions for port-Hamiltonian systems to be represented on graphs

Connected boundary matrices to graph adjacency matrices

Enhanced understanding of hyperbolic systems on network structures

Abstract

Hyperbolic systems on networks often can be written as systems of first order equations on an interval, coupled by transmission conditions at the endpoints, also called port-Hamiltonians. However, general results for the latter have been difficult to interpret in the network language. The aim of this paper is to derive conditions under which a port-Hamiltonian with general linear Kirchhoff's boundary conditions can be written as a system of $2\times 2$ hyperbolic equations on a metric graph $\Gamma$. This is achieved by interpreting the matrix of the boundary conditions as a potential map of vertex connections of $\Gamma$ and then showing that, under the derived assumptions, that matrix can be used to determine the adjacency matrix of $\Gamma$.