  We consider space-time tracking optimal control problems for linear
para\-bo\-lic initial boundary value problems that are given in the space-time
cylinder $Q = \Omega \times (0,T)$, and that are controlled by the right-hand
side $z_\varrho$ from the Bochner space $L^2(0,T;H^{-1}(\Omega))$. So it is
natural to replace the usual $L^2(Q)$ norm regularization by the energy
regularization in the $L^2(0,T;H^{-1}(\Omega))$ norm. We derive a priori
estimates 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-time
finite element mesh-size $h$, and depending on the regularity of the desired
state $\bar{u}$. These estimates lead to the optimal choice $\varrho = h^2$.
The approximate state $\widetilde{u}_{\varrho h}$ is computed by means of a
space-time finite element method using piecewise linear and continuous basis
functions on completely unstructured simplicial meshes for $Q$. The theoretical
results are quantitatively illustrated by a series of numerical examples in two
and three space dimensions.
