Telegraph systems on networks and port-Hamiltonians. I. Boundary conditions and well-posedness

Jacek Banasiak; Adam Błoch

doi:10.3934/eect.2021046

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2022, Volume 11, Issue 4: 1331-1355. Doi: 10.3934/eect.2021046

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Telegraph systems on networks and port-Hamiltonians. I. Boundary conditions and well-posedness

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, Łódź University of Technology, Łódź, Poland
3.
International Scientific Laboratory of Applied Semigroup Research, South Ural State University, Chelyabinsk, Russia

Received: January 31, 2021

Revised: June 30, 2021

Early access: August 2021

Published: August 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 Łódź University of Technology, Poland

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The paper is concerned with a system of linear hyperbolic differential equations on a network coupled through general transmission conditions of Kirchhoff's-type at the nodes. We discuss the reduction of such a problem to a system of 1-dimensional hyperbolic problems for the associated Riemann invariants and provide a semigroup-theoretic proof of its well-posedness. A number of examples showing the relation of our results with recent research is also provided.

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Mathematics Subject Classification: Primary: 35R02; Secondary: 47D03, 35L40.

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