  The goal of this article is to show a local exact controllability to smooth
(C2) trajectories for the 2-d density dependent incompressible Navier-Stokes
equations. Our controllability result requires some geometric condition on the
ow of the target trajectory, which is remanent from the transport equation
satisfied by the density. The proof of this result uses a fixed point argument
in suitable spaces adapted to a Carleman weight function that follows the ow of
the target trajectory. Our result requires the proof of new Carleman estimates
for heat and Stokes equations.
