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 proposed cost functional and the convergence results.

In this paper an optimal control problem is considered, where the control variable lies in a measure space and the state variable fulfills an elliptic equation. This formulation leads to a sparse structure of the optimal control. In this setting we prove a new regularity result for the optimal state and the optimal control. Moreover, a finite element discretization based on [E. Casas, C. Clason, and K. Kunisch, SIAM J. Control Optim., 50 (2012), pp. 1735-1752] is discussed and a priori error estimates are derived, which significantly improve the estimates from that paper. Numerical examples for problems in two and three space dimensions illustrate our results.

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