For robust discretizations of the Navier-Stokes equations with small viscosity, standard
Galerkin schemes have to be augmented by stabilization terms due to the indefinite
convective terms and due to a possible lost of a discrete inf-sup condition. For optimal
control problems for fluids such stabilization have in general an undesired effect in the
sense that optimization and discretization do not commute. This is the case for the
combination of streamline upwind Petrov-Galerkin (SUPG) and pressure stabilized
Petrov-Galerkin (PSPG). In this work we study the effect of different stabilized finite
element methods to distributed control problems governed by singular perturbed Oseen
equations. In particular, we address the question whether a possible commutation error in
optimal control problems lead to a decline of convergence order. Therefore, we give
a priori estimates for SUPG/PSPG. In a numerical study for a flow with
boundary layers, we illustrate to which extend the commutation error affects the
accuracy.