Sparse Dirichlet optimal control problems

In this paper, we analyze optimal control problems governed by an elliptic partial differential equation, in which the control acts as the Dirichlet data. Box constraints for the controls are imposed and the cost functional involves the state and possibly a sparsity-promoting term, but not a Tikhonov regularization term. Two different discretizations are investigated: the variational approach and a full discrete approach. For the latter, we use continuous piecewise linear elements to discretize the control space and numerical integration of the sparsity-promoting term. It turns out that the best way to discretize the state equation is to use the Carstensen quasi-interpolant of the boundary data, and a new discrete normal derivative of the adjoint state must be introduced to deal with this. Error estimates, optimization procedures and examples are provided.