2012/12/11 by Pierluigi Colli, Colli, Pierluigi, Jürgen Sprekels +1
Computer Science · Materials Science · Mathematics · #49K15 #49K20 #74M15 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Solidification and crystal growth phenomena
paper · pdf · doi:10.48550/arxiv.1212.2359
openalex publication_date 2012/12/11 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
In this paper, we investigate optimal control problems for Allen-Cahn\nequations with singular nonlinearities and a dynamic boundary condition\ninvolving singular nonlinearities and the Laplace-Beltrami operator. The\napproach covers both the cases of distributed controls and of boundary\ncontrols. The cost functional is of standard tracking type, and box constraints\nfor the controls are prescribed. Parabolic problems with nonlinear dynamic\nboundary conditions involving the Laplace-Beltrami operation have recently\ndrawn increasing attention due to their importance in applications, while their\noptimal control was apparently never studied before. In this paper, we first\nextend known well-posedness and regularity results for the state equation and\nthen show the existence of optimal controls and that the control-to-state\nmapping is twice continuously Fr 'echet differentiable between appropriate\nfunction spaces. Based on these results, we establish the first-order necessary\noptimality conditions in terms of a variational inequality and the adjoint\nstate equation, and we prove second-order sufficient optimality conditions.\n