  We consider space-time tracking type distributed optimal control problems for
the wave equation in the space-time domain $Q:= \Omega \times (0,T) \subset
{\mathbb{R}}^{n+1}$, where the control is assumed to be in the energy space
$[H_{0;,0}^{1,1}(Q)]^*$, rather than in $L^2(Q)$ which is more common. While
the latter ensures a unique state in the Sobolev space $H^{1,1}_{0;0,}(Q)$,
this does not define a solution isomorphism. Hence we use an appropriate state
space $X$ such that the wave operator becomes an isomorphism from $X$ onto
$[H_{0;,0}^{1,1}(Q)]^*$. Using space-time finite element spaces of piecewise
linear continuous basis functions on completely unstructured but shape regular
simplicial meshes, we derive a priori estimates for the error
$\|\widetilde{u}_{\varrho h}-\overline{u}\|_{L^2(Q)}$ between the computed
space-time finite element solution $\widetilde{u}_{\varrho h}$ and the target
function $\overline{u}$ with respect to the regularization parameter $\varrho$,
and the space-time finite element mesh-size $h$, depending on the regularity of
the desired state $\overline{u}$. These estimates lead to the optimal choice
$\varrho=h^2$ in order to define the regularization parameter $\varrho$ for a
given space-time finite element mesh size $h$, or to determine the required
mesh size $h$ when $\varrho$ is a given constant representing the costs of the
control. The theoretical results will be supported by numerical examples with
targets of different regularities, including discontinuous targets.
Furthermore, an adaptive space-time finite element scheme is proposed and
numerically analyzed.
