  In this paper we investigate a priori error estimates for the space-time
Galerkin finite element discretization of a simplified semilinear 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 a non-smooth, semilinear ODE that has to hold true in
almost all points in space. The system is discretized by a constant
discontinuous Galerkin method in time and usual conforming linear finite
elements in space. For the uncontrolled equation, we prove linear convergence
in time and an order of $\mathcal{O}(h^{\frac{3}{2}-\varepsilon})$ for the
discretization error in space. Our main result regarding the optimal control
problem is the uniform convergence of dG(0)cG(1)-discrete controls to $l\in
H^1_{\{0\}}(0,T;L^2(\Omega))$. Error estimates for the controls are established
via a quadratic growth condition. Numerical experiments are added to illustrate
the proven rates of convergence.
