  Variational approaches have been used successfully as a strategy to take
advantage from real data measurements. In several applications, this approach
gives a means to increase the accuracy of numerical simulations. In the
particular case of fluid dynamics, it leads to optimal control problems with
non standard cost functionals which, when constraint to the Navier-Stokes
equations, require a non-standard theoretical frame to ensure the existence of
solution. In this work, we prove the existence of solution for a class of such
type of optimal control problems. Before doing that, we ensure the existence
and uniqueness of solution for the 3D stationary Navier-Stokes equations, with
mixed-boundary conditions, a particular type of boundary conditions very common
in applications to biomedical problems.
