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

Jacek Banasiak; Adam Błoch; Jacek Banasiak; Adam Błoch

doi:10.3934/nhm.2021024

Networks and Heterogeneous Media

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

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

Jacek Banasiak ^{1,2,3
,},
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: 01 March 2021 Revised: 01 October 2021
Primary: 35R02, 05C50; Secondary: 35F46, 05C90

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 $.
- Hyperbolic systems on networks,
- 1-D hyperbolic systems,
- graph realizability,
- line graphs,
- adjacency matrix
Citation: Jacek Banasiak, Adam Błoch. Telegraph systems on networks and port-Hamiltonians. Ⅱ. Network realizability[J]. Networks and Heterogeneous Media, 2022, 17(1): 73-99. doi: 10.3934/nhm.2021024

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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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