  We consider an optimal control problem constrained by a parabolic partial
differential equation (PDE) with Robin boundary conditions. We use a well-posed
space-time variational formulation in Lebesgue--Bochner spaces with minimal
regularity. The abstract formulation of the optimal control problem yields the
Lagrange function and Karush--Kuhn--Tucker (KKT) conditions in a natural
manner. This results in space-time variational formulations of the adjoint and
gradient equation in Lebesgue--Bochner spaces with minimal regularity.
Necessary and sufficient optimality conditions are formulated and the
optimality system is shown to be well-posed.
  Next, we introduce a conforming uniformly stable simultaneous space-time
(tensorproduct) discretization of the optimality system in these
Lebesgue--Boch\-ner spaces. Using finite elements of appropriate orders in
space and time for trial and test spaces, this setting is known to be
equivalent to a Crank--Nicolson time-stepping scheme for parabolic problems.
Differences to existing methods are detailed.
  We show numerical comparisons with time-stepping methods. The space-time
method shows good stability properties and requires fewer degrees of freedom in
time to reach the same accuracy.
