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Homogenization of quasilinear optimal control problems involving a thick multilevel junction of type 3 : 2 : 1. (English) Zbl 1246.49005

Summary: We consider quasilinear optimal control problems involving a thick two-level junction \(\Omega _{\varepsilon }\) which consists of the junction body \(\Omega _{0}\) and a large number of thin cylinders with the cross-section of order \(\mathcal O(\varepsilon ^{2})\). 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 \(\varepsilon , \alpha , \beta \) and the thin cylinders from each level are \(\varepsilon \)-periodically alternated. Using the Buttazzo-Dal Maso abstract scheme for variational convergence of constrained minimization problems, the asymptotic analysis (as \(\varepsilon \rightarrow 0\)) of these problems are made for different values of \(\alpha \) and \(\beta \) and different kinds of controls. We show 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.

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35B40 Asymptotic behavior of solutions to PDEs
74K30 Junctions
93C70 Time-scale analysis and singular perturbations in control/observation systems
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References:

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