  This paper is devoted to the partial null controllability issue of parabolic
linear systems with n equations. Given a bounded domain in R N, we study the
effect of m localized controls in a nonempty open subset only controlling p
components of the solution (p, m < n). The first main result of this paper is a
necessary and sufficient condition when the coupling and control matrices are
constant. The second result provides, in a first step, a sufficient condition
of partial null controllability when the matrices only depend on time. In a
second step, through an example of partially controlled 2x2 parabolic system,
we will provide positive and negative results on partial null controllability
when the coefficients are space dependent.
