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Well-posedness of Networked Hyperbolic Systems of Balance Laws

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Constrained Optimization and Optimal Control for Partial Differential Equations

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 160))

Abstract

We present an overview on recent results concerning hyperbolic systems on networks. We present a summary of theoretical results on existence, uniqueness and stability. The established theory extends previously known results on the Cauchy problem for nonlinear, 2×2 hyperbolic balance laws. The proofs are based on Wave-Front Tracking and therefore we present detailed results on the Riemann problem first.

Mathematics Subject Classification (2000). 35L65, 49J20.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Erlangen–Nürnberg, D-91058, Erlangen, Germany

    Martin Gugat & Günther Leugering

  2. Department of Mathematics, RWTH Aachen University, D-52056, Aachen, Germany

    Michael Herty

  3. Department of Mathematics, TU Kaiserslautern, D-67663, Kaiserslautern, Germany

    Axel Klar

  4. Department of Mathematics, University of Stuttgart, D-70569, Stuttgart, Germany

    Veronika Schleper

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  1. Martin Gugat
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  2. Michael Herty
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  3. Axel Klar
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  4. Günther Leugering
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  5. Veronika Schleper
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Correspondence to Martin Gugat .

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Editors and Affiliations

  1. , Department Mathematik, Universität Erlangen-Nürnberg, Martensstr. 3, Erlangen, 91058, Germany

    Günter Leugering

  2. , Fakultät Bio- und Chemieingenieurwesen, Technische Universität Dortmund, Geschossbau 2, Dortmund, 44221, Germany

    Sebastian Engell

  3. , Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, Berlin, 12489, Germany

    Andreas Griewank

  4. , Fachbereich Mathematik, Universität Hamburg, Bundesstr. 55, Hamburg, 20146, Germany

    Michael Hinze

  5. , Institut für Angewandte Mathematik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 293, Heidelberg, 69120, Germany

    Rolf Rannacher

  6. , FB 4 - Department Mathematik, Universität Trier, Trier, 54286, Germany

    Volker Schulz

  7. Fakultät für Mathematik, M1, Lehrstuhl für Mathematische Optimierung, Technische Universität München, Boltzmannstr. 3, Garching bei München, 85748, Germany

    Michael Ulbrich

  8. , Fachbereich Mathematik, Technische Universität Darmstadt, Dolivostrasse 15, Darmstadt, 64293, Germany

    Stefan Ulbrich

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Gugat, M., Herty, M., Klar, A., Leugering, G., Schleper, V. (2012). Well-posedness of Networked Hyperbolic Systems of Balance Laws. In: Leugering, G., et al. Constrained Optimization and Optimal Control for Partial Differential Equations. International Series of Numerical Mathematics, vol 160. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0133-1_7

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