Control Problems

Abstract

Finally, the techniques discussed in the previous chapters are applied to control problems involving elliptic boundary value problems. That is, we consider boundary value problems which are affected on (part of) the boundary ∂Ω by some control. This control, in turn, is determined by minimizing some quadratic functional involving the natural norms of the state of the system and the control. Since these natural norms involve at least one ‘broken’ norm, they are usually avoided in finite element methods. The optimality conditions of such a minimization problem lead to a representation of the control in terms of the state. Together with the boundary value problem these conditions constitute a linear operator equation consisting of two weakly coupled saddle point problems. We will derive this system in Section 6.2 in the context of the general saddle point problems discussed in Section 4.1.1 and show that it is well-posed.