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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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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DOI: https://doi.org/10.1007/s00245-012-9178-9