  The scope of this paper is the analysis and approximation of an optimal
control problem related to the Allen-Cahn equation. A tracking functional is
minimized subject to the Allen-Cahn equation using distributed controls that
satisfy point-wise control constraints. First and second order necessary and
sufficient conditions are proved. The lowest order discontinuous Galerkin - in
time - scheme is considered for the approximation of the control to state and
adjoint state mappings. Under a suitable restriction on maximum size of the
temporal and spatial discretization parameters $k$, $h$ respectively in terms
of the parameter $\epsilon$ that describes the thickness of the interface
layer, a-priori estimates are proved with constants depending polynomially upon
$1/ \epsilon$. Unlike to previous works for the uncontrolled Allen-Cahn problem
our approach does not rely on a construction of an approximation of the
spectral estimate, and as a consequence our estimates are valid under low
regularity assumptions imposed by the optimal control setting. These estimates
are also valid in cases where the solution and its discrete approximation do
not satisfy uniform space-time bounds independent of $\epsilon$. These
estimates and a suitable localization technique, via the second order condition
(see
\cite{Arada-Casas-Troltzsch_2002,Casas-Mateos-Troltzsch_2005,Casas-Raymond_2006,Casas-Mateos-Raymond_2007}),
allows to prove error estimates for the difference between local optimal
controls and their discrete approximation as well as between the associated
state and adjoint state variables and their discrete approximations
