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

  • Eberhard Bänsch
  • Peter Benner
  • Jens Saak
  • Heiko K. WeicheltEmail author
Chapter
Part of the International Series of Numerical Mathematics book series (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.

Keywords

Coupled flow control Navier-Stokes equations Diffusion-convection equation Riccati-based feedback 

Mathematics Subject Classification (2010)

76D55 93D15 93C20 15A24 

Notes

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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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Eberhard Bänsch
    • 1
  • Peter Benner
    • 2
    • 3
  • Jens Saak
    • 2
    • 3
  • Heiko K. Weichelt
    • 3
    Email author
  1. 1.Lehrstuhl für Angewandte Mathematik 3Friedrich-Alexander-Universität Erlangen-NürnbergErlangenGermany
  2. 2.Research Group Computational Methods in Systems and Control Theory (CSC)Max Planck Institute for Dynamics of Complex Technical Systems MagdeburgMagdeburgGermany
  3. 3.Faculty of Mathematics, Research Group Mathematics in Industry and Technology (MiIT)Technische Universität ChemnitzChemnitzGermany

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