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We consider space-time tracking optimal control problems for linearpara\-bo\-lic initial boundary value problems that are given in the space-timecylinder $Q = \Omega \times (0,T)$, and that are controlled by the right-handside $z_\varrho$ from the Bochner space $L^2(0,T;H^{-1}(\Omega))$. So it isnatural to replace the usual $L^2(Q)$ norm regularization by the energyregularization in the $L^2(0,T;H^{-1}(\Omega))$ norm. We derive a prioriestimates for the error $\|\widetilde{u}_{\varrho h} - \bar{u}\|_{L^2(Q)}$between the computed state $\widetilde{u}_{\varrho h}$ and the desired state$\bar{u}$ in terms of the regularization parameter $\varrho$ and the space-timefinite element mesh-size $h$, and depending on the regularity of the desiredstate $\bar{u}$. These estimates lead to the optimal choice $\varrho = h^2$.The approximate state $\widetilde{u}_{\varrho h}$ is computed by means of aspace-time finite element method using piecewise linear and continuous basisfunctions on completely unstructured simplicial meshes for $Q$. The theoreticalresults are quantitatively illustrated by a series of numerical examples in twoand three space dimensions.

Robust space-time finite element error estimates for parabolic distributed optimal control problems with energy regularization