A discontinuous Galerkin finite element method for an optimal
control problem related to semilinear parabolic PDE's is examined.
The schemes under consideration are discontinuous in time but
conforming in space. Convergence of discrete schemes of arbitrary
order is proven. In addition, the convergence of discontinuous
Galerkin approximations of the associated optimality system to the
solutions of the continuous optimality system is shown. The proof
is based on stability estimates at arbitrary time points under
minimal regularity assumptions, and a discrete compactness
argument for discontinuous Galerkin schemes (see Walkington
[SINUM (June 2008) (submitted), preprint available at http://www.math.cmu.edu/~noelw], Sects. 3, 4).