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In this paper, we analyze space-time finite element methods for the numerical solution of distributed parabolic optimal control problems with energy regularization in the Bochner space L² (0, T;H^-1 () ). By duality, the related norm can be evaluated by means of the solution of an elliptic quasi-stationary boundary value problem. When eliminating the control, we end up with the reduced optimality system that is nothing but the variational formulation of the coupled forward-backward primal and adjoint equations. Using Babuška's theorem, we prove unique solvability in the continuous case. Furthermore, we establish the discrete inf-sup condition for any conforming space-time finite element discretization yielding quasi-optimal discretization error estimates. Various numerical examples confirm the theoretical findings. We emphasize that the energy regularization results in a more localized control with sharper contours for discontinuous target functions, which is demonstrated by a comparison with an L²-regularization and with a sparse optimal control approach.
Langer et al. (Fri,) studied this question.