The results of this paper concern exact controllability to the
trajectories for a coupled system of semilinear heat equations. We
have transmission conditions on the interface and Dirichlet boundary
conditions at the external part of the boundary so that the system can be
viewed as a single equation with discontinuous coefficients in the
principal part. Exact controllability to the trajectories is proved when we
consider distributed controls supported in the part of the domain where the
diffusion coefficient is the smaller and if the nonlinear term f(y) grows
slower than |y|log3/2(1+|y|) at infinity. In the proof we use null
controllability results for the associate linear system and global
Carleman estimates with explicit bounds or combinations of several of
these estimates. In order to treat the terms appearing on the
interface, we have to construct specific weight functions depending on
geometry.