We consider a space-time variational formulation of a PDE-constrained optimal
control problem with box constraints on the control and a parabolic PDE with
Robin boundary conditions. In this setting, the optimal control problem reduces
to an optimization problem for which we derive necessary and sufficient
optimality conditions.
Next, we introduce a space-time (tensorproduct) discretization using finite
elements in space and piecewise linear functions in time. This setting is known
to be equivalent to a Crank-Nicolson time stepping scheme for parabolic
problems. The optimization problem is solved by a projected gradient method. We
show numerical comparisons for problems in 1d, 2d and 3d in space. It is shown
that the classical semi-discrete primal-dual setting is more efficient for
small problem sizes and moderate accuracy. However, the space-time
discretization shows good stability properties and even outperforms the
classical approach as the dimension in space and/or the desired accuracy
increases.Comment: 20 page