  We propose, analyze, and test new robust iterative solvers for systems of
linear algebraic equations arising from the space-time finite element
discretization of reduced optimality systems defining the approximate solution
of hyperbolic distributed, tracking-type optimal control problems with both the
standard $L^2$ and the more general energy regularizations. In contrast to the
usual time-stepping approach, we discretize the optimality system by space-time
continuous piecewise-linear finite element basis functions which are defined on
fully unstructured simplicial meshes. If we aim at the asymptotically best
approximation of the given desired state $y_d$ by the computed finite element
state $y_{\varrho h}$, then the optimal choice of the regularization parameter
$\varrho$ is linked to the space-time finite element mesh-size $h$ by the
relations $\varrho=h^4$ and $\varrho=h^2$ for the $L^2$ and the energy
regularization, respectively. For this setting, we can construct robust
(parallel) iterative solvers for the reduced finite element optimality systems.
These results can be generalized to variable regularization parameters adapted
to the local behavior of the mesh-size that can heavily change in the case of
adaptive mesh refinements. The numerical results illustrate the theoretical
findings firmly.
