Optimal Control for Allen-Cahn Equations Enhanced by Model Predictive Control

Abstract

The Allen-Cahn equation is a simple model of a nonlinear reaction-diffusion process. It is often used to model interface motion in time, e.g. phase separation in alloys. It has applications in many areas including material sciences, biology, geology, as well as image processing. We will consider a simple scalar Allen-Cahn equation subject to distributed control. Here, the nonlinear reaction term is obtained from using the standard double-well potential, leading to a cubic nonlinearity. We will describe a nonlinear feedback control strategy based on the concept of Model Predictive Control (MPC). We also show how to obtain the open-loop trajectory and control using numerical techniques for PDE-constrained optimization. The feedback control scheme is then applied to the spatially semi-discretized nonlinear optimal control problem. For the prediction and control step within the MPC scheme, we apply a linear-quadratic regulator/Gaussian design problem. The arising computational challenge consisting in solving the associated large-scale algebraic Riccati equations has already been shown in the literature to be feasible using reasonably fine discretizations.

Optimal Control of a Quasilinear Parabolic Equation and its Time Discretization

“…To the best of our knowledge there does not exist any mathematical treatment on the optimal control of anisotropic phase-field models so far. Optimal control of isotropic Allen-Cahn variational equations are studied, e.g., in [8][9][10][11][12] and of Cahn-Hilliard variational (in-)equalities in [13][14][15] and references therein. Let us mention results given in the context of anisotropic Allen-Cahn equations.…”

Optimal control of a quasilinear parabolic equation and its time discretization

“…Optimal control of isotropic Allen-Cahn variational equations are studied e.g. in [3,5,14,20,34] and of Cahn-Hilliard variational equalities in [24,15,25] and references therein. Let us mention results given in the context of anisotropic Allen-Cahn equations.…”