Optimal boundary control of a viscous Cahn-Hilliard system with dynamic boundary condition and double obstacle potentials

TL;DR

This paper develops an optimal boundary control framework for a viscous Cahn-Hilliard system with dynamic boundary conditions and double obstacle potentials, establishing existence and first-order optimality conditions through a deep quench limit approach.

Contribution

It introduces a novel method to derive optimality conditions for Cahn-Hilliard systems with non-differentiable potentials using a deep quench limit technique.

Findings

Proved existence of optimal controls.

Derived first-order necessary optimality conditions.

Extended the approach to non-differentiable double obstacle potentials.

Abstract

In this paper, we investigate optimal boundary control problems for Cahn-Hilliard variational inequalities with a dynamic boundary condition involving double obstacle potentials and the Laplace-Beltrami operator. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. We prove existence of optimal controls and derive first-order necessary conditions of optimality. The general strategy, which follows the lines of the recent approach by Colli, Farshbaf-Shaker, Sprekels (see the preprint arXiv:1308.5617) to the (simpler) Allen-Cahn case, is the following: we use the results that were recently established by Colli, Gilardi, Sprekels in the preprint arXiv:1407.3916 [math.AP] for the case of (differentiable) logarithmic potentials and perform a so-called "deep quench limit". Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired first-order necessary optimality conditions for the case of (non-differentiable) double obstacle potentials.