Efficient block preconditioners for integral constrained elliptic optimal control problems with finite element approximations

Abstract

In this paper, focusing on a distributed optimal control problem for the elliptic equations with integral control constraint, we propose efficient block diagonal preconditioners to solve the corresponding linearized algebraic system with finite element methods. We derive the first order necessary and sufficient optimality conditions for an integral constraint on the control, and divide the discretized optimal conditions into three matrix forms. Then block-diagonal preconditioners for the corresponding linear algebraic systems are constructed. With respect to both the mesh size and the regularization parameter, the robustness of our preconditioners is proved in detail. In fact, the condition numbers of the preconditioned matrices are a constant for different parameters. Meanwhile, based on the equivalent matrix forms, an algorithm is proposed for this kind of constrained optimal control problems. Numerical experiments are given to depict the efficiency of our proposed preconditioners.