Volume 18, Number 2, April-June 2012
|Page(s)||583 - 610|
|Published online||26 August 2011|
Homogenization of quasilinear optimal control problems involving a thick multilevel junction of type 3 : 2 : 1∗
Dipartimento di Ingegneria dell’ Informazione e Matematica
Applicata, Universita di Salerno, via Ponte don Melillo, 84084
Fisciano ( SA), Italy
2 Department of Mathematical Physics, Faculty of Mechanics & Mathematics, National Taras Shevchenko University, Volodymyrska str. 64, 01033 Kyiv, Ukraine
Revised: 6 December 2010
We consider quasilinear optimal control problems involving a thick two-level junction Ωε 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.
Mathematics Subject Classification: 49J20 / 35B27 / 35B40 / 74K30
Key words: Homogenization / quasilinear optimal control problem / thick multilevel junction / asymptotic behavior / singular perturbation
© EDP Sciences, SMAI, 2011
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