Partial null controllability of parabolic linear systems

This paper is devoted to the partial null controllability issue of parabolic linear systems with n equations. Given a bounded domain Ω in RN (N ∈ N∗), we study the effect of m localized controls in a nonempty open subset ω only controlling p components of the solution (p,m 6 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 2 × 2 parabolic system, we will provide positive and negative results on partial null controllability when the coefficients are space dependent.