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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

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  • 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.


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  • 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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