Optimal Boundary Control of a Viscous Cahn--Hilliard System with Dynamic Boundary Condition and 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 Appl. Math. Optim., 2014) 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.

Optimal distributed control of a diffuse interface model of tumor growth

“…While there exist many contributions concerning the well-posedness of various types of Cahn-Hilliard systems, only a few deal with their optimal control. In this connection, we mention the papers [12,13,28,40], which deal with zero Neumann boundary conditions like (1.6), while in the recent papers [7,8,13,14] dynamic boundary conditions have been studied. A num-ber of papers also investigates optimal control problems for convective Cahn-Hilliard systems (cf.…”

A boundary control problem for the pure Cahn–Hilliard equation with dynamic boundary conditions

“…[16,22]. However, about the optimal control of viscous or non-viscous Cahn-Hilliard systems with dynamic boundary conditions of the form (1.4), we only know of the papers [10] and [6] dealing with the viscous case; to the best of our knowledge, the present contribution is the first paper treating the optimal control of the pure Cahn-Hilliard system with dynamic boundary conditions. The technique used in our approach essentially consists in starting from the known results for τ > 0 and then letting the parameter τ tend to zero.…”

“…for some positive constants η and C and for every r ∈ D. This condition, earlier introduced in [4] in relation with the Allen-Cahn equation with dynamic boundary conditions (see also [11]), is then used in [9] (as well as in [6] and [10]) to deal with the Cahn-Hilliard system. This complements [14], where some kind of an opposite inequality is assumed.…”

“…Namely, the problem we want to address consists in minimizing a proper cost functional depending on both the control u Γ and the associate state (y, y Γ ). Among several possibilities, we choose the cost functional 6) where the functions z Q , z Σ and the nonnegative constants b Q , b Σ , b 0 are given. The control problem then consists in minimizing (1.6) subject to the state system and to the constraint u Γ ∈ U ad , where the control box U ad is given by…”

“…The technique used in our approach essentially consists in starting from the known results for τ > 0 and then letting the parameter τ tend to zero. In doing that, we use some of the ideas of [7] and [6], which deal with the Allen-Cahn and the viscous Cahn-Hilliard equations, respectively, and address similar control problems related to the nondifferentiable double obstacle potential by seeing it as a limit of logarithmic doublewell potentials.…”