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 ε-periodic disconnected inclusions of size ε in presence of a jump of the solution on the interface that varies according to a parameter γ. In particular here the case γ<1 is analyzed. The optimal control theory, introduced by Lions (Optimal Control of System Governed by Partial Differential Equations, 1971), 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 ε-problem, which is the unique minimum point of a quadratic cost functional Jε, converges to the optimal control of the homogenized problem with respect to a suitable limit cost functional J∞. 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.