  The paper is devoted to studying controllability properties for 3D
Navier-Stokes equations in a bounded domain. We establish a sufficient
condition under which the problem in question is exactly controllable in any
finite-dimensional projection. Our sufficient condition is verified for any
torus in $R^3$. The proofs are based on a development of a general approach
introduced by Agrachev and Sarychev in the 2D case. As a simple consequence of
the result on controllability, we show that the Cauchy problem for the 3D
Navier-Stokes system has a unique strong solution for any initial function and
a large class of external forces.
