Optimal control of an Allen-Cahn equation with singular potentials and dynamic boundary condition

TL;DR

This paper develops a framework for optimal control of Allen-Cahn equations with singular nonlinearities and dynamic boundary conditions, establishing existence, differentiability, and optimality conditions for controls.

Contribution

It introduces the first analysis of optimal control problems for Allen-Cahn equations with singular nonlinearities and dynamic boundary conditions, including existence and optimality conditions.

Findings

Proved well-posedness and regularity of the state equation.

Established existence of optimal controls and differentiability of the control-to-state map.

Derived first-order necessary and second-order sufficient optimality conditions.

Abstract

In this paper, we investigate optimal control problems for Allen-Cahn equations with singular nonlinearities and a dynamic boundary condition involving singular nonlinearities 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. Parabolic problems with nonlinear dynamic boundary conditions involving the Laplace-Beltrami operation have recently drawn increasing attention due to their importance in applications, while their optimal control was apparently never studied before. In this paper, we first extend known well-posedness and regularity results for the state equation and then show the existence of optimal controls and that the control-to-state mapping is twice continuously Fr\'echet differentiable between appropriate function spaces. Based on these results, we establish the first-order necessary optimality conditions in terms of a variational inequality and the adjoint state equation, and we prove second-order sufficient optimality conditions.