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Trends in PDE Constrained Optimization pp 109–131Cite as

Birkhäuser

Optimal Control of Nonlinear Hyperbolic Conservation Laws with Switching

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

Abstract

We consider optimal control problems governed by nonlinear hyperbolic conservation laws at junctions and analyze in particular the Fréchet-differentiability of the reduced objective functional. This is done by showing that the control-to-state mapping of the considered problems satisfies a generalized notion of differentiability. We consider both, the case where the controls are the initial and the boundary data as well as the case where the system is controlled by the switching times of the node condition. We present differentiability results for the considered problems in a quite general setting including an adjoint-based gradient representation of the reduced objective function.

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Acknowledgements

The authors gratefully acknowledge the support of the German Research Foundation (DFG) within the Priority Program 1253 “Optimization with Partial Differential Equations” under grant UL158/8-1. Moreover, we gratefully acknowledge discussions with T. I. Seidman.

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Authors and Affiliations

  1. Department of Mathematics, Technische Universität Darmstadt, Dolivostr. 15, 64293, Darmstadt, Germany

    Sebastian Pfaff & Stefan Ulbrich

  2. Department of Mathematics, Universität Erlangen-Nürnberg, Cauerstr. 11, 91058, Erlangen, Germany

    Günter Leugering

Authors
  1. Sebastian Pfaff
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  2. Stefan Ulbrich
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  3. Günter Leugering
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Correspondence to Sebastian Pfaff .

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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Pfaff, S., Ulbrich, S., Leugering, G. (2014). Optimal Control of Nonlinear Hyperbolic Conservation Laws with Switching. 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_8

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