Optimal control for evolutionary imperfect transmission problems

We study the optimal control problem of a second order linear evolution equation defined in two-component composites with epsilon-periodic disconnected inclusions of size epsilon in presence of a jump of the solution on the interface that varies according to a parameter gamma. In particular here the case gamma<1 is analyzed. The optimal control theory, introduced by Lions, leads us to characterize the control as the solution of a set of equations, called optimality conditions. The main result of this paper proves that the optimal control of the epsilon-problem, which is the unique minimum point of a quadratic cost functional at epsilon level, converges to the optimal control of the homogenized problem with respect to a suitable limit cost functional. The main difficulties are to find the appropriate limit functional for the control of the homogenized system and to identify the limit of the controls.