Article Contents
Article Contents

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

J. B. acknowledges partial support from the National Science Centre of Poland Grant 2017/25/B/ST1/00051 and the National Research Foundation of South Africa Grant 82770. The research was completed while A. B. was a doctoral candidate in the Interdisciplinary Doctoral School at L´od´z University of Technology, Poland

• 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 $\Gamma$.

Mathematics Subject Classification: Primary: 35R02, 05C50; Secondary: 35F46, 05C90.

 Citation:

• Figure 1.  Starlike network of channels

Figure 2.  The reconstructed multi digraph $\boldsymbol{\Gamma}$. It is seen that it cannot describe a flow on $\Gamma$ as $\varpi_5$ and $\varpi_6$ must flow in the same direction

Figure 3.  The reconstructed multi digraph $\boldsymbol{\Gamma}$ for (46), (47)

Figure 4.  A network $\Gamma$ realizing the flow (48), (49)

Figure 5.  Multi digraphs $G_1$ with 3 sources and two sinks and $G_2$ with all sources and all sinks grouped into a single source and a single sink

Figure 6.  The line digraph for both $G_1$ and $G_2$

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