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Optimal Control of a Parabolic Equation with Dynamic Boundary Condition

Applied Mathematics & Optimization volume 67, pages 3–31 (2013)Cite this article

Abstract

We investigate a control problem for the heat equation. The goal is to find an optimal heat transfer coefficient in the dynamic boundary condition such that a desired temperature distribution at the boundary is adhered. To this end we consider a function space setting in which the heat flux across the boundary is forced to be an L p function with respect to the surface measure, which in turn implies higher regularity for the time derivative of temperature. We show that the corresponding elliptic operator generates a strongly continuous semigroup of contractions and apply the concept of maximal parabolic regularity. This allows to show the existence of an optimal control and the derivation of necessary and sufficient optimality conditions.

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Acknowledgements

We wish to thank our colleagues K. Gröger (Berlin), H. Amann (Zürich) and H. Vogt (Clausthal) for valuable discussions on the subject of the paper. Moreover, we thank the referees for the carefully reading of the paper.

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Authors and Affiliations

  1. Nonlinear Optimization and Inverse Problems, Weierstrass Institute for Applied Mathematics and Stochastics, Mohrenstrasse 39, 10117, Berlin, Germany

    D. Hömberg & K. Krumbiegel

  2. Partial Differential Equations, Weierstrass Institute for Applied Mathematics and Stochastics, Mohrenstrasse 39, 10117, Berlin, Germany

    J. Rehberg

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  1. D. Hömberg
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  2. K. Krumbiegel
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  3. J. Rehberg
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Correspondence to J. Rehberg.

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Hömberg, D., Krumbiegel, K. & Rehberg, J. Optimal Control of a Parabolic Equation with Dynamic Boundary Condition. Appl Math Optim 67, 3–31 (2013). https://doi.org/10.1007/s00245-012-9178-9

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Keywords

  • Parabolic equation
  • Mixed boundary condition
  • Maximal parabolic L p-regularity
  • Optimal control
  • Sufficient optimality conditions