Abstract
In this paper, we investigate optimal control problems for Allen-Cahn variational inequalities with a dynamic boundary condition involving double obstacle potentials and the Laplace-Beltrami operator. The approach covers both the cases of distributed controls and of boundary controls. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. We prove existence of optimal controls and derive first-order necessary conditions of optimality. The general strategy is the following: we use the results that were recently established by two of the authors for the case of (differentiable) logarithmic potentials and perform a so-called “deep quench limit”. Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired first-order necessary optimality conditions for the case of (non-differentiable) double obstacle potentials.
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Acknowledgments
This paper was initiated during a visit of PC to the WIAS in Berlin. The kind hospitality and stimulating atmosphere of the WIAS are gratefully acknowledged. Some financial support comes from the MIUR-PRIN Grant 2010A2TFX2 “Calculus of Variations”.
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Colli, P., Farshbaf-Shaker, M.H. & Sprekels, J. A Deep Quench Approach to the Optimal Control of an Allen–Cahn Equation with Dynamic Boundary Conditions and Double Obstacles. Appl Math Optim 71, 1–24 (2015). https://doi.org/10.1007/s00245-014-9250-8
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DOI: https://doi.org/10.1007/s00245-014-9250-8