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Stabilization of Networked Hyperbolic Systems with Boundary Feedback

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Book cover Trends in PDE Constrained Optimization

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 165))

Abstract

We summarize recent theoretical results as well as numerical results on the feedback stabilization of first order quasilinear hyperbolic systems (on networks). For the stabilization linear feedback controls are applied at the nodes of the network. This yields the existence and uniqueness of a C 1-solution of the hyperbolic system with small C 1-norm. For this solution an appropriate L 2-Lyapunov function decays exponentially in time. This implies the exponential stability of the system. A numerical discretization of the Lyapunov function is presented and a numerical analysis shows the expected exponential decay for a class of first-order discretization schemes. As an application for the theoretical results the stabilization of the gas flow in fan-shaped pipe networks with compressors is considered.

This work has been supported by DFG SPP 1253, DAAD 508846 and DAAD D/0811409 (Procope 2009/10).

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Acknowledgements

This work has been supported by DFG GU376/7-1 and HE5386/8-1.

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Authors and Affiliations

  1. Department of Mathematics, University of Erlangen-Nuremberg, Cauerstr. 11, 91058, Erlangen, Germany

    Markus Dick, Martin Gugat & Günter Leugering

  2. IGPM, Department of Mathematics, RWTH Aachen University, Templergraben 55, 52056, Aachen, Germany

    Michael Herty & Sonja Steffensen

  3. School of Mathematical Sciences, Fudan University, Handan Rd. 220, 200433, Shanghai, China

    Ke Wang

Authors
  1. Markus Dick
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  2. Martin Gugat
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  3. Michael Herty
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  4. Günter Leugering
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  5. Sonja Steffensen
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  6. Ke Wang
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Corresponding author

Correspondence to Markus Dick .

Editor information

Editors and Affiliations

  1. Department Mathematik, Universität Erlangen-Nürnberg, Erlangen, Germany

    Günter Leugering

  2. Institut für Dynamik komplexer technisch, Max-Planck-Institut, Magdeburg, Germany

    Peter Benner

  3. Fakultät Bio- und Chemieingenieurwesen, Technische Universität Dortmund, Dortmund, Germany

    Sebastian Engell

  4. Institut für Mathematik, Humboldt-Universität zu Berlin, Berlin, Germany

    Andreas Griewank

  5. Mathematisches Institut, Universität Basel, Basel, Switzerland

    Helmut Harbrecht

  6. Fachbereich Mathematik Optimierung und Approximation, Universität Hamburg, Hamburg, Germany

    Michael Hinze

  7. Institut für Angewandte Mathematik, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany

    Rolf Rannacher

  8. Fachbereich Mathematik, Technische Universität Darmstadt, Darmstadt, Germany

    Stefan Ulbrich

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Dick, M., Gugat, M., Herty, M., Leugering, G., Steffensen, S., Wang, K. (2014). Stabilization of Networked Hyperbolic Systems with Boundary Feedback. In: Leugering, G., et al. Trends in PDE Constrained Optimization. International Series of Numerical Mathematics, vol 165. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-05083-6_31

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