A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems

In this paper, a priori error estimates for space-time finite element discretizations of optimal control problems governed by semilinear parabolic PDEs and subject to pointwise control constraints are derived. We extend the approach from Meidner and Vexler (SIAM Control Optim 47(3):1150-1177, 2008; SIAM Control Optim 47(3):1301-1329, 2008) where linear-quadratic problems have been considered, discretizing the state equation by usual conforming finite elements in space and a discontinuous Galerkin method in time. Error estimates for controls discretized by piecewise constant functions in time and cellwise constant functions in space are derived in detail and we explain how error estimate for further discretization approaches, e. g., cellwise linear discretization in space, the postprocessing approach from Meyer and Rösch (SIAM J Control Optim 43:970-985, 2004), and the variationally discrete approach from Hinze (J Comput Optim Appl 30:45-63, 2005) can be obtained. In addition, we derive an estimate for a setting with finitely many time-dependent controls.