The results of this paper concern exact controllability to thetrajectories for a coupled system of semilinear heat equations. Wehave transmission conditions on the interface and Dirichlet boundaryconditions at the external part of the boundary so that the system can beviewed as a single equation with discontinuous coefficients in theprincipal part. Exact controllability to the trajectories is proved when weconsider distributed controls supported in the part of the domain where thediffusion coefficient is the smaller and if the nonlinear term f(y) growsslower than |y|log3/2(1+|y|) at infinity. In the proof we use nullcontrollability results for the associate linear system and globalCarleman estimates with explicit bounds or combinations of several ofthese estimates. In order to treat the terms appearing on theinterface, we have to construct specific weight functions depending ongeometry.
