Asymptotic Analysis of an Optimal Control Problem Involving a Thick Two-Level Junction with Alternate Type of Controls

We study the asymptotic behavior (as ε→0) of an optimal control problem in a plane thick two-level junction, which is the union of some domain and a large number 2N of thin rods with variable thickness of order e = O(N^-1).\varepsilon =\mathcal{O}(N^{-1}). The thin rods are divided into two levels depending on the geometrical characteristics and on the controls given on their bases. In addition, the thin rods from each level are ε-periodically alternated and the inhomogeneous perturbed Fourier boundary conditions are given on the lateral sides of the rods. Using the direct method of the calculus of variations and the Buttazzo-Dal Maso abstract scheme for variational convergence of constrained minimization problems, the asymptotic analysis of this problem for different kinds of controls is made as ε→0.