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Trends in PDE Constrained Optimization pp 173–188Cite as

Birkhäuser

Optimal Control-Based Feedback Stabilization of Multi-field Flow Problems

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

Abstract

We discuss the numerical solution of the feedback stabilization problem for multi-field flow problems. Our approach is based on an analytical Riccati feedback concept derived by Raymond which allows to steer a perturbed flow back to its desired state, assumed to be a stationary, possibly unstable, flow profile. This concept, originally derived for incompressible flow fields described by the Navier-Stokes equations, uses a linear-quadratic regulator (LQR) approach for the linearized Navier-Stokes equations formulated on the space of divergence-free velocity fields. We extend this approach to a setting where the Navier-Stokes equations are coupled to a diffusion-convection equation describing the transport of a reactive species in a fluid. The stabilizing feedback control resulting from the LQR problem is obtained via solving the associated operator Riccati equation. We describe a numerical procedure to solve this Riccati equation numerically. This involves several technical difficulties on the algebraic level that we address in this report. We illustrate the performance of our method by a numerical example.

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Acknowledgements

We would like to thank Stephan Weller for his helpful advices regarding the handling of the FEM software NAVIER and René Schneider for many useful discussion throughout the whole project time. In addition, we would like to thank Martin Stoll, Andy Wathen, Matthias Heinkenschloss, and Jan Heiland for various discussions that extended our knowledge about optimal control for fluid mechanics.

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Authors and Affiliations

  1. Lehrstuhl für Angewandte Mathematik 3, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 11, 91058, Erlangen, Germany

    Eberhard Bänsch

  2. Research Group Computational Methods in Systems and Control Theory (CSC), Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg, Sandtorstr. 1, 39106, Magdeburg, Germany

    Peter Benner & Jens Saak

  3. Faculty of Mathematics, Research Group Mathematics in Industry and Technology (MiIT), Technische Universität Chemnitz, Reichenhainer Str. 39/41, 09126, Chemnitz, Germany

    Peter Benner, Jens Saak & Heiko K. Weichelt

Authors
  1. Eberhard Bänsch
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  2. Peter Benner
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  3. Jens Saak
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  4. Heiko K. Weichelt
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Corresponding author

Correspondence to Heiko K. Weichelt .

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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Bänsch, E., Benner, P., Saak, J., Weichelt, H.K. (2014). Optimal Control-Based Feedback Stabilization of Multi-field Flow Problems. 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_11

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