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.

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.

Full text

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document

Boundary control problem and

optimality conditions

for the Cahn–Hilliard equation

with dynamic boundary conditions

\begin

centerPierluigi Colli*(1)*

e-mail: pierluigi.colli@unipv.it

Andrea Signori*(2)*

e-mail: andrea.signori02@universitadipavia.it

(1) Dipartimento di Matematica “F. Casorati”, Università di Pavia

via Ferrata 5, 27100 Pavia, Italy

(2) Dipartimento di Matematica e Applicazioni, Università di Milano–Bicocca

via Cozzi 55, 20125 Milano, Italy