  - We discuss the approximation of distributed null controls for partial
differential equations. The main purpose is to determine an approximation of
controls that drives the solution from a prescribed initial state at the
initial time to the zero target at a prescribed final time. As a non trivial
example, we mainly focus on the Stokes system for which the existence of
square-integrable controls have been obtained in [Fursikov \& Imanuvilov,
Controllability of Evolution Equations, 1996]) via Carleman type estimates. We
introduce a least-squares formulation of the controllability problem, and we
show that it allows the construction of strong convergent sequences of
functions toward null controls for the Stokes system. The approach consists
first in introducing a class of functions satisfying a priori the boundary
conditions in space and time-in particular the null controllability condition
at time T-, and then finding among this class one element satisfying the
system. This second step is done by minimizing a quadratic functional, among
the admissible corrector functions of the Stokes system. We also discuss
briefly the direct problem for the steady Navier-Stokes system. The method does
not make use of any duality arguments and therefore avoid the ill-posedness of
dual methods, when parabolic type equation are considered.
