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In this paper we will consider distributed Linear-Quadratic Optimal ControlProblems dealing with Advection-Diffusion PDEs for high values of the P\'ecletnumber. In this situation, computational instabilities occur, both for steadyand unsteady cases. A Streamline Upwind Petrov-Galerkin technique is used inthe optimality system to overcome these unpleasant effects. We will apply afinite element method discretization in a optimize-then-discretize approach.For the parabolic case, a space-time framework will be considered andstabilization will also occur in the bilinear forms involving time derivatives.Then we will build Reduced Order Models on this discretization procedure andtwo possible settings can be analyzed: whether or not stabilization is neededin the online phase, too. In order to build the reduced bases for state,control, and adjoint variables we will consider a Proper OrthogonalDecomposition algorithm in a partitioned approach. The discussion is supportedby computational experiments, where relative errors between the FEM and ROMsolutions are studied together with the respective computational times.

A Streamline upwind Petrov-Galerkin Reduced Order Method for Advection-Dominated Partial Differential Equations under Optimal Control