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

Tiziana Durante; Taras A. Mel’nyk

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 2, page 583-610
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Durante, Tiziana, and Mel’nyk, Taras A.. "Homogenization of quasilinear optimal control problems involving a thick multilevel junction of type 3 : 2 : 1∗." ESAIM: Control, Optimisation and Calculus of Variations 18.2 (2012): 583-610. <http://eudml.org/doc/277820>.

@article{Durante2012,
abstract = {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. },
author = {Durante, Tiziana, Mel’nyk, Taras A.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Homogenization; quasilinear optimal control problem; thick multilevel junction; asymptotic behavior; singular perturbation; homogenization},
language = {eng},
month = {7},
number = {2},
pages = {583-610},
publisher = {EDP Sciences},
title = {Homogenization of quasilinear optimal control problems involving a thick multilevel junction of type 3 : 2 : 1∗},
url = {http://eudml.org/doc/277820},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Durante, Tiziana
AU - Mel’nyk, Taras A.
TI - Homogenization of quasilinear optimal control problems involving a thick multilevel junction of type 3 : 2 : 1∗
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2012/7//
PB - EDP Sciences
VL - 18
IS - 2
SP - 583
EP - 610
AB - 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.
LA - eng
KW - Homogenization; quasilinear optimal control problem; thick multilevel junction; asymptotic behavior; singular perturbation; homogenization
UR - http://eudml.org/doc/277820
ER -

References

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