Homogenization of quasilinear optimal control problems involving a thick multilevel junction of type 3 : 2 : 1^∗

¹ Dipartimento di Ingegneria dell’ Informazione e Matematica Applicata, Universita di Salerno, via Ponte don Melillo, 84084 Fisciano ( SA), Italy
durante@diima.unisa.it
² Department of Mathematical Physics, Faculty of Mechanics & Mathematics, National Taras Shevchenko University, Volodymyrska str. 64, 01033 Kyiv, Ukraine
melnyk@imath.kiev.ua

Received: 1 July 2010
Revised: 6 December 2010

Abstract

We consider quasilinear optimal control problems involving a thick two-level junction Ω_ε which consists of the junction body Ω₀ and a large number of thin cylinders with the cross-section of order 𝒪(ε²). The thin cylinders are divided into two levels depending on the geometrical characteristics, the quasilinear boundary conditions and controls given on their lateral surfaces and bases respectively. In addition, the quasilinear boundary conditions depend on parameters ε, α, β and the thin cylinders from each level are ε-periodically alternated. Using the Buttazzo–Dal Maso abstract scheme for variational convergence of constrained minimization problems, the asymptotic analysis (as ε → 0) of these problems are made for different values of α and β and different kinds of controls. We have showed that there are three qualitatively different cases. Application for an optimal control problem involving a thick one-level junction with cascade controls is presented as well.