  In this paper we investigate a priori error estimates for the space-time
Galerkin finite element discretization of an optimal control problem governed
by a simplified linear gradient enhanced damage model. The model equations are
of a special structure as the state equation consists of an elliptic PDE which
has to be fulfilled at almost all times coupled with an ODE that has to hold
true in almost all points in space. The state equation is discretized by a
piecewise constant discontinuous Galerkin method in time and usual conforming
linear finite elements in space. For the discretization of the control we
employ the same discretization technique which turns out to be equivalent to a
variational discretization approach. We provide error estimates of optimal
order both for the discretization of the state equation as well as for the
optimal control. Numerical experiments are added to illustrate the proven rates
of convergence.
