  In this paper we study the internal exact controllability for a second order
linear evolution equation defined in a two-component domain. On the interface
we prescribe a jump of the solution proportional to the conormal derivatives,
meanwhile a homogeneous Dirichlet condition is imposed on the exterior
boundary. Due to the geometry of the domain, we apply controls through two
regions which are neighborhoods of a part of the external boundary and of the
whole interface, respectively. Our approach to internal exact controllability
consists in proving an observability inequality by using the Lagrange
multipliers method. Eventually we apply the Hilbert Uniqueness Method,
introduced by J.-L. Lions, which leads to the construction of the exact control
through the solution of an adjoint problem. Finally we find a lower bound for
the control time depending not only on the geometry of our domain and on the
matrix of coefficients of our problem but also on the coefficient of
proportionality of the jump with respect to the conormal derivatives.
