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

Jacek Banasiak; Adam Błoch

doi:10.3934/nhm.2021024

Article Contents

2022, Volume 17, Issue 1: 73-99. Doi: 10.3934/nhm.2021024

This issue Previous Article Next Article Well-posedness theory for nonlinear scalar conservation laws on networks

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

Jacek Banasiak^1,2,3, and
Adam Błoch^2,

1.
Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa
2.
Institute of Mathematics, Lódź University of Technology Lódź, Poland
3.
International Scientific Laboratory of Applied Semigroup Research South Ural State University, Chelyabinsk, Russia

Received: February 28, 2021

Revised: September 30, 2021

Published: February 2022

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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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 $.

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

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Figure 1. Starlike network of channels

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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

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

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

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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

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

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