A Deep Quench Approach to the Optimal Control of an Allen–Cahn Equation with Dynamic Boundary Conditions and Double Obstacles

Abstract

In this paper, we investigate optimal control problems for Allen-Cahn variational inequalities with a dynamic boundary condition involving double obstacle potentials and the Laplace-Beltrami operator. The approach covers both the cases of distributed controls and of boundary controls. 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 is the following: we use the results that were recently established by two of the authors in the paper [5] 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.

A Boundary Control Problem for the Viscous Cahn–Hilliard Equation with Dynamic Boundary Conditions

“…We are aware, however, of the recent contributions [11] and [6] for the corresponding Allen-Cahn equation. In particular, [11] treats both the cases of distributed and boundary controls for logarithmic-type potentials as in (1.3).…”

“…More precisely, both the existence of optimal controls and twice continuous Fréchet differentiability for the well-defined control-to-state mapping were established, as well as first-order necessary and second-order sufficient optimality conditions. The related paper [6] deals with the existence of optimal controls and the derivation of first-order necessary conditions of optimality for the more difficult case of the double obstacle potential. The method used consists in performing a so-called "deep quench limit" of the problem studied in [11].…”

A boundary control problem for the pure Cahn–Hilliard equation 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. 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.…”

Recent Results on the Cahn - Hilliard Equation with Dynamic Boundary Conditions

“…The technique adopted in [23] essentially consists in starting from the known results for τ > 0 and then letting the parameter τ tend to zero. In doing that, some of the ideas of [20] and [24] are used: indeed, these papers [20,24] deal with the Allen Cahn and the viscous Cahn Hilliard equations, respectively, and address similar control problems related to the nondierentiable double-obstacle potential f dobs dened by (0.4). Now, we think it is important to recall some related contributions.…”

“…Hilliard equation with dynamic boundary conditions, in analogy with the corresponding contributions for the Allen Cahn equation (see [19] and [20]). In particular, we review the results proved in the three research papers…”