2013/08/26 by Pierluigi Colli, Colli, Pierluigi, M. Hassan Farshbaf-Shaker +3
Computer Science · Materials Science · Mathematics · #35K61 #49K20 #74M15 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Optimization and Control (math.OC) #Solidification and crystal growth phenomena
paper · pdf · doi:10.48550/arxiv.1308.5617
openalex publication_date 2013/08/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.