A duality-based approach for linear parabolic optimal control problems

This paper is concerned with the optimal control problem governed by linear parabolic equation with box constraints on control variables. We employ the Fenchel duality scheme to derive an unconstrained dual problem. Compared with the primal problem, the objective functional of the dual problem includes a projection onto the box constraints. We prove the existence and uniqueness of solutions to the dual problem and derive the first-order optimality conditions. Furthermore, we investigate the saddle point property between the solutions of the primal problem and the solutions of the dual problem. To solve the dual problem, we design two implementable methods: the conjugate gradient method and the semi-smooth Newton method. The solutions of the primal problem can be easily obtained through the solutions of the dual problem. We demonstrate the effectiveness and accuracy of the proposed methods by solving three example problems.