  In this work we recast parametrized time dependent optimal control problems
governed by partial differential equations in a saddle point formulation and we
propose reduced order methods as an effective strategy to solve them. Indeed,
on one hand parametrized time dependent optimal control is a very powerful
mathematical model which is able to describe several physical phenomena; on the
other hand, it requires a huge computational effort. Reduced order methods are
a suitable approach to have rapid and accurate simulations. We rely on
POD-Galerkin reduction over the physical and geometrical parameters of the
optimality system in a space-time formulation. Our theoretical results and our
methodology are tested on two examples: a boundary time dependent optimal
control for a Graetz flow and a distributed optimal control governed by time
dependent Stokes equations. With these two experiments the convenience of the
reduced order modelling is further extended to the field of time dependent
optimal control.
