Uniqueness results for Stokes equations and their consequences in linear and nonlinear control problems

The goal of this article is the study of the approximate controllability for two approximations of Navier Stokes equations with distributed controls. The method of proof combines a suitable linearization of the system with a fixed point argument. We then are led to study the approximate controllability of linear Stokes systems with potentials. We study both the case where there is no constraint on the control and the case where we search a control having one null component. In both cases, the problems is reduced to prove unique continuation results. This is done by means of Carleman estimates.