Optimal boundary control of a nonstandard viscous Cahn-Hilliard system with dynamic boundary condition

TL;DR

This paper addresses an optimal boundary control problem for a complex, nonlinear viscous Cahn-Hilliard system with dynamic boundary conditions, establishing differentiability, existence of controls, and optimality conditions.

Contribution

It introduces a novel control framework for a coupled PDE system with dynamic boundary conditions, including differentiability and optimality analysis.

Findings

Proved Fréchet differentiability of the control-to-state operator.

Established existence of optimal controls.

Derived first-order necessary optimality conditions.

Abstract

In this paper, we study an optimal boundary control problem for a model for phase separation taking place in a spatial domain that was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105-118. The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing control literature about this PDE system, we consider here a dynamic boundary condition involving the Laplace-Beltrami operator for the order parameter of the system, which models an additional nonconserving phase transition occurring on the surface of the domain. We show the Fr\'echet differentiability of the associated control-to-state operator in appropriate Banach spaces and derive results on the existence of optimal controls and on first-order necessary optimality conditions in terms of a variational inequality and the adjoint state system.