The first aim of this paper is to study, by means of the periodic unfolding method, the homogenization of an elliptic problem with weakly con- verging data, in an ε-periodic two component composite with an imperfect transmission condition on the interface. Then we exploit this result to de- scribe the asymptotic behaviour of the exact controls and the corresponding states of hyperbolic problems set in composites with the same structure and presenting the same condition on the interface. The exact controllability is developed by applying the Hilbert Uniqueness Method, introduced by J. -L. Lions, which leads us to the construction of the exact controls as solutions of suitable transposed problem.
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