A Space-Time Variational Method for Optimal Control Problems

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. We propose to utilize a well-posed inf-sup stable framework of the PDE in appropriate Lebesgue-Bochner spaces. Next, we introduce a conforming simultaneous space-time (tensorproduct) discretization in these Lebesgue-Bochner spaces. 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 simultaneous space-time discretization shows good stability properties and even outperforms the classical approach as the dimension in space and/or the desired accuracy increases.