  We study the controllability of a coupled system of linear parabolic
equations, with non-negativity constraint on the state. We establish two
results of controllability to trajectories in large time: one for diagonal
diffusion matrices with an "approximate" nonnegativity constraint, and a
another stronger one, with "exact" nonnegativity constraint, when all the
diffusion coefficients are equal and the eigenvalues of the coupling matrix
have nonnegative real part. The proofs are based on a "staircase" method.
Finally, we show that state-constrained controllability admits a positive
minimal time, even with weaker unilateral constraint on the state.
