Approximate controllability for equations of fluid mechanics with a few body controls

The approximate controllability by means of a few controls is proven for 2D Navier-Stokes system in an infinite channel under generalized Navier boundary conditions in the finite direction and the periodicity assumption in the infinite direction. The set of controls is given explicitly, and do not depend on the viscosity of the fluid. The case of 1D Burgers system in a bounded interval under Dirichlet boundary conditions, where the controls are supported in an arbitrary small nonempty open subset, is also discussed.