Navier--Stokes equations: controllability by means of low modes forcing

We study controllability issues for 2D and 3D Navier-Stokes (NS) sys- tems with periodic boundary conditions. The systems are controlled by a degenerate (applied to few low modes) forcing. Methods of differential geometric/Lie algebraic control theory are used to establish global control- lability of finite-dimensional Galerkin approximations of 2D and 3D NS and Euler systems, global controllability in finite-dimensional projection of 2D NS system and L2-approximate controllability for 2D NS system. Beyond these main goals we obtain results on boundedness and contin- uous dependence of trajectories of 2D NS system on degenerate forcing, when the space of forcings is endowed with so called relaxation metric.