Volume 28, 2022
|Number of page(s)
|24 November 2022
Optimal control of anisotropic Allen-Cahn equations
Department of Mathematics, University of Regensburg, 93040 Regensburg, Germany
* Corresponding author: email@example.com
Accepted: 28 September 2022
This paper aims at solving an optimal control problem governed by an anisotropic Allen-Cahn equation numerically. Therefor we first prove the Frechet differentiability of an in time discretized parabolic control problem under certain assumptions on the involved quasilinearity and formulate the first order necessary conditions. As a next step, since the anisotropies are in general not smooth enough, the convergence behavior of the optimal controls is studied for a sequence of (smooth) approximations of the former quasilinear term. In addition the simultaneous limit in the approximation and the time step size is considered. For a class covering a large variety of anisotropies we introduce a certain regularization and show the previously formulated requirements. Finally, a trust region Newton solver is applied to various anisotropies and configurations, and numerical evidence for mesh independent behavior and convergence with respect to regularization is presented.
Mathematics Subject Classification: 35K59 / 49K20 / 49M41 / 65M60
Key words: Allen-Cahn equation / anisotropy / quasilinear parabolic equation / optimal control / regularization / discretization / optimality conditions
© The authors. Published by EDP Sciences, SMAI 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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