  In this paper we will consider distributed Linear-Quadratic Optimal Control
Problems dealing with Advection-Diffusion PDEs for high values of the P\'eclet
number. In this situation, computational instabilities occur, both for steady
and unsteady cases. A Streamline Upwind Petrov-Galerkin technique is used in
the optimality system to overcome these unpleasant effects. We will apply a
finite element method discretization in an optimize-then-discretize approach.
Concerning the parabolic case, a stabilized space-time framework will be
considered and stabilization will also occur in both bilinear forms involving
time derivatives. Then we will build Reduced Order Models on this
discretization procedure and two possible settings can be analyzed: whether or
not stabilization is needed in the online phase, too. In order to build the
reduced bases for state, control, and adjoint variables we will consider a
Proper Orthogonal Decomposition algorithm in a partitioned approach. It is the
first time that Reduced Order Models are applied to stabilized parabolic
problems in this setting. The discussion is supported by computational
experiments, where relative errors between the FEM and ROM solutions are
studied together with the respective computational times.
