# Boundary control problem and optimality conditions for the Cahn-Hilliard equation with dynamic boundary conditions

## TL;DR

This paper investigates a boundary control problem for the Cahn-Hilliard equation with dynamic boundary conditions, establishing existence, differentiability, and optimality conditions for controls under specific potential assumptions.

## Contribution

It introduces a framework for analyzing optimal boundary controls for the Cahn-Hilliard equation with dynamic boundary conditions, including existence and first-order optimality conditions.

## Key findings

- Existence of optimal control established
- Control-to-state operator shown to be Fréchet differentiable
- First-order necessary optimality conditions derived

## Abstract

This paper is concerned with a boundary control problem for the Cahn--Hilliard equation coupled with dynamic boundary conditions. In order to handle the control problem, we restrict our analysis to the case of regular potentials defined on the whole real line, assuming the boundary potential to be dominant. The existence of optimal control, the Fr\'echet differentiability of the control-to-state operator between appropriate Banach spaces, and the first-order necessary conditions for optimality are addressed. In particular, the necessary condition for optimality is characterized by a variational inequality involving the adjoint variables.