Applied Mathematics and Optimization, Vol.82, No.2, 721-754, 2020
Optimal Control Problem for the Cahn-Hilliard/Allen-Cahn Equation with State Constraint
In this paper, we consider a distributed optimal control problem for the Cahn-Hilliard/Allen-Cahn equation with state-constraint. The objective is to force the coverageyto have some specified properties or achieve a certain goal. Since the cost functional is discontinuous, together with state constraint, we employ a new penalty functional by the approximation of the cost functional, in this case, we derive the necessary optimality conditions for the approximating optimal control problem. Finally, by considering the limits of the necessary optimality conditions we have obtained, we solve the optimal control problem and derive the necessary optimality conditions.
Keywords:Cahn-Hilliard;Allen-Cahn equation;State constraint;Optimal control;Optimality conditions;Adjoint state system