  The distributed null controllability for coupled parabolic systems with
non-diagonalizable diffusion matrices with a reduced number of controls has
been studied in the case of constant matrices. On the other hand, boundary
controllability issues and distributed controllability with non-constant
coefficients for this kind of systems is not completely understood. In this
paper, we analyze the boundary controllability properties of a class of coupled
parabolic systems with non-diagonalizable diffusion matrices in the constant
case and the distributed controllability of a 2 x 2 non-diagonalizable
parabolic system with space-dependent coefficients. For the boundary
controllability problem, our strategy relies on the moment method. For the
distributed controllability problem, our findings provide positive and negative
control results by using the Fattorini-Hautus test and a fictitious control
strategy.
