Optimal control and “strange” term arising from homogenization of the Poisson equation in the perforated domain with the Robin-type boundary condition in the critical case

The present paper is devoted to the study of the asymptotic behavior of the optimal control for the boundary value problem in an $\varepsilon$-periodically perforated domain with linear Robin-type boundary condition, when the period of the structure tends to zero, and the problem parameters, diameter of perforations and adsorption coefficient, take critical values.