  This paper deals with the distributed and boundary controllability of the so
called Leray-$\alpha$ model. This is a regularized variant of the Navier-Stokes
system ($\alpha$ is a small positive parameter) that can also be viewed as a
model for turbulent flows. We prove that the Leray-$\alpha$ equations are
locally null controllable, with controls bounded independently of $\alpha$. We
also prove that, if the initial data are sufficiently small, the controls
converge as $\alpha \to 0^+$ to a null control of the Navier-Stokes equations.
We also discuss some other related questions, such as global null
controllability, local and global exact controllability to the trajectories,
etc.
