On the homogenization of an optimal control problem in a domain perforated by holes of critical size and arbitrary shape

The paper studies the asymptotic behavior of the optimal control for the Poisson type boundary value problem in a domain perforated by holes of an arbitrary shape with Robin-type boundary conditions on the internal boundaries. The cost functional is assumed to be dependent on the gradient of the state and on the usual norm of the control. We consider the so-called “critical” relation between the problem parameters and the period of the structure $\varepsilon\to0$. Two “strange” terms arise in the limit. The paper extends, by first time in the literature, previous papers devoted to the homogenization of the control problem which always assumed the symmetry of the periodic holes.