  We consider a distributed optimal control problem subject to a parabolic
evolution equation as constraint. The control will be considered in the energy
norm of the anisotropic Sobolev space $[H_{0;,0}^{1,1/2}(Q)]^\ast$, such that
the state equation of the partial differential equation defines an isomorphism
onto $H^{1,1/2}_{0;0,}(Q)$. Thus, we can eliminate the control from the
tracking type functional to be minimized, to derive the optimality system in
order to determine the state. Since the appearing operator induces an
equivalent norm in $H_{0;0,}^{1,1/2}(Q)$, we will replace it by a computable
realization of the anisotropic Sobolev norm, using a modified Hilbert
transformation. We are then able to link the cost or regularization parameter
$\varrho>0$ to the distance of the state and the desired target, solely
depending on the regularity of the target. For a conforming space-time finite
element discretization, this behavior carries over to the discrete setting,
leading to an optimal choice $\varrho = h_x^2$ of the regularization parameter
$\varrho$ to the spatial finite element mesh size $h_x$. Using a space-time
tensor product mesh, error estimates for the distance of the computable state
to the desired target are derived. The main advantage of this new approach is,
that applying sparse factorization techniques, a solver of optimal, i.e.,
almost linear, complexity is proposed and analyzed. The theoretical results are
complemented by numerical examples, including discontinuous and less regular
targets. Moreover, this approach can be applied also to optimal control
problems subject to non-linear state equations.
