Telegraph systems on networks and port-Hamiltonians. Ⅱ. Network realizability

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 \begin{document}$ 2\times 2 $\end{document} hyperbolic equations on a metric graph \begin{document}$ \Gamma $\end{document} . This is achieved by interpreting the matrix of the boundary conditions as a potential map of vertex connections of \begin{document}$ \Gamma $\end{document} and then showing that, under the derived assumptions, that matrix can be used to determine the adjacency matrix of \begin{document}$ \Gamma $\end{document} .