  The goal of the present article is to study controllability properties of
mixed systems of linear parabolic-transport equations, with possibly
non-diagonalizable diffusion matrix, on the one-dimensional torus. The
equations are coupled by zero or first order coupling terms, with constant
coupling matrices, without any structure assumptions on them. The distributed
control acts through a constant matrix operator on the system, so that there
might be notably less controls than equations, encompassing the case of
indirect and simultaneous controllability. More precisely, we prove that in
small time, such kind of systems are never controllable in appropriate Sobolev
spaces, whereas in large time, null-controllability holds, for sufficiently
regular initial data, if and and only if a spectral Kalman rank condition is
verified. We also prove that initial data that are not regular enough are not
controllable. Positive results are obtained by using the so-called fictitious
control method together with an algebraic solvability argument, whereas the
negative results are obtained by using an appropriate WKB construction of
approximate solutions for the adjoint system associated to the control problem.
As an application to our general results, we also investigate into details the
case of $2\times2$ systems (i.e., one pure transport equation and one parabolic
equation).
