Exact controllability for evolutionary imperfect transmission problems

In this paper we study the asymptotic behaviour of an exact controllability problem for a second order linear evolution equation defined in a two-component composite with ε-periodic disconnected inclusions of size ε. On the interface we prescribe a jump of the solution that varies according to a real parameter γ. In particular, we suppose that -1<=1. The case when γ=1 is the most interesting and delicate one, since the homogenized problem is represented by a coupled system of a P.D.E. and an O.D.E., giving rise to a memory effect. Our approach to exact controllability consists in applying the Hilbert Uniqueness Method, introduced by J.-L. Lions, which leads us to the construction of the exact control as the solution of a transposed problem. Our main result proves that the exact control and the corresponding solution of the ε-problem converge to the exact control of the homogenized problem and to the corresponding solution respectively.