We describe the asymptotic behaviour of the exact controls and the corresponding states of the wave equation in a ε- periodic two component composite with a jump of the solution on the interface depending on a parameter γ. The approach to the exact controllability process is developed by applying the Hilbert Uniqueness Method, introduced by J. -L. Lions. We prove that, according to the values of γ, the exact controls and the corresponding solutions of the ε-problems converge to the exact controls of different homogenized problems and to the corresponding solutions, respectively.
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