We consider a hybrid system consisting of two flexible beams connected by a point mass. The constant of rotational inertia is assumed to be nonzero. In a previous paper we have proved that, in the presence of the point mass, the system is well posed in asymmetric spaces in which solutions have one more degree of regularity to one side of the mass.We are interested in the problem of controllability when the control acts on the free extreme of one of the beams. We prove that when the control time is large enough the system is exactly controllable in an asymmetric space. This result is sharp. The proofs combine classical techniques from asymptotic analysis and the theory of nonharmonic Fourier series.