Measure valued directional sparsity for parabolic optimal control problems

A directional sparsity framework allowing for measure valued controls in the spatial direction is proposed for parabolic optimal control problems. It allows for controls which are localized in space, where the spatial support is independent of time. Well-posedness of the optimal control problems is established and the optimality system is derived. It is used to establish structural properties of the minimizer. An a priori error analysis for finite element discretization is obtained, and numerical results illustrate the effects of the proposed cost functional and the convergence results.