On homogenization of a mixed boundary optimal control problem

Abstract

We study the asymptotic behaviour of an optimal control problem for the Ukawa equation in a thick multi-structure with different types and classes of admissible boundary controls. This thick multi-structure consists of a domain (the junction's body) and a large number of $\varepsilon$-periodically situated thin cylinders. We consider two types of boundary controls, namely, the Dirichlet $H^{1/2}$-controls on the bases $\Gamma_{\varepsilon}$ of thin cylinders, and the Neumann $L^2$-controls on their 'vertical' sides. We present some ideas and results concerning of the asymptotic analysis for such problems as ${\varepsilon}\to 0$ and derive conditions under which the homogenized problem can be recovered in the explicit form. We show that the mathematical description of the homogenized optimal boundary control problem is different from the original one. These differences appear not only in the control constraints, limit cost functional, state equations, and boundary conditions, but also in the type of admissible controls for the limit problem - one of them is the Dirichlet $L^2$-control, whereas the second one is appeared as the distributed $L^2$-control.