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

Published online by Cambridge University Press:  26 August 2011

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

Keywords

Homogenizationquasilinear optimal control problemthick multilevel junctionasymptotic behaviorsingular perturbation
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 2 , April 2012 , pp. 583 - 610
Copyright
© EDP Sciences, SMAI, 2011

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References

Blanchard, D. and Gaudiello, A., Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem. ESAIM : COCV 9 (2003) 449–460. Google Scholar
Blanchard, D., Gaudiello, A. and Griso, G., Junction of a periodic family of elastic rods with 3d plate. Part I. J. Math. Pures Appl. 88 (2007) 1–33 (Part I); 88 (2007) 149–190 (Part II). Google Scholar
Blanchard, D., Gaudiello, A. and Mel’nyk, T.A., Boundary homogenization and reduction of dimention in a Kirchhoff-Love plate. SIAM J. Math. Anal. 39 (2008) 1764–1787. Google Scholar
G. Buttazzo, Γ-convergence and its applications to some problem in the calculus of variations, in School on Homogenization, ICTP, Trieste, 1993 (1994) 38–61.
Buttazzo, G. and Dal Maso, G., Γ-convergence and optimal control problems. J. Optim. Theory Appl. 38 (1982) 385–407. Google Scholar
Chechkin, G.A., Chechkina, T.P., D’Apice, C., De-Maio, U. and Mel’nyk, T.A., Asymptotic analysis of a boundary value problem in a cascade thick junction with a random transmission zone. Appl. Anal. 88 (2009) 1543–1562. Google Scholar
De Maio, U., Gaudiello, A. and Lefter, C., optimal control for a parabolic problem in a domain with highly oscillating boundary. Appl. Anal. 83 (2004) 1245–1264. Google Scholar
De Maio, U., Durante, T. and Mel’nyk, T.A., Asymptotic approximation for the solution to the Robin problem in a thick multi-level junction. Math. Models Methods Appl. Sci. 15 (2005) 1897–1921. Google Scholar
Denkowski, Z. and Mortola, S., Asymptotic behavior of optimal solutions to control problems for systems described by differential inclusions corresponding to partial differential equations. J. Optim. Theory Appl. 78 (1993) 365–391. Google Scholar
Durante, T. and Mel’nyk, T.A., Asymptotic analysis of an optimal control problem involving a thick two-level junction with alternate type of controls. J. Optim. Theory Appl. 144 (2010) 205–225. Google Scholar
Durante, T., Faella, L. and Perugia, C., Homogenization and behaviour of optimal controls for the wave equation in domains with oscillating boundary. Nonlinear Differ. Equ. Appl. 14 (2007) 455–489. Google Scholar
Kesavan, S. and Paulin, J. Saint Jean, Optimal control on perforated domains. J. Math. Anal. Appl. 229 (1999) 563–586. Google Scholar
Y.I. Lavrentovich, T.V. Knyzkova and V.V. Pidlisnyuk, The potential of application of new nanostructural materials for degradation of pesticides in water, in Proceedings of the 7th Int. HCH and Pesticides Forum Towards the establishment of an obsolete POPS/pecticides stockpile fund for Central and Eastern European countries and new independent states, Kyiv, Ukraine (2003) 167–169.
Lenczner, M., Multiscale model for atomic force microscope array mechanical behavior. Appl. Phys. Lett. 90 (2007) 091908; doi : 10.1063/1.2710001. Google Scholar
J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, Berlin (1971).
S.E. Lyshevshi, Mems and Nems : Systems, Devices, and Structures. CRC Press, Boca Raton, FL (2002).
Mel’nyk, T.A., Homogenization of the Poisson equation in a thick periodic junction. Z. f. Anal. Anwendungen 18 (1999) 953–975. Google Scholar
Mel’nyk, T.A., Homogenization of a perturbed parabolic problem in a thick periodic junction of type 3  : 2  : 1. Ukr. Math. J. 52 (2000) 1737–1749. Google Scholar
Mel’nyk, T.A., Homogenization of a boundary-value problem with a nonlinear boundary condition in a thick junction of type 3  : 2  : 1. Math. Models Meth. Appl. Sci. 31 (2008) 1005–1027. Google Scholar
Mel’nyk, T.A. and Chechkin, G.A., Asymptotic analysis of boundary value problems in thick three-dimensional multi-level junctions. Math. Sb. 200 3 (2009) 49–74 (in Russian); English transl. : Sb. Math. 200 (2009) 357–383. Google Scholar
Mel’nyk, T.A. and Nazarov, S.A., Asymptotic structure of the spectrum in the problem of harmonic oscillations of a hub with heavy spokes. Dokl. Akad. Nauk Russia 333 (1993) 13–15 (in Russian); English transl. : Russian Acad. Sci. Dokl. Math. 48 (1994) 28–32. Google Scholar
Mel’nyk, T.A. and Nazarov, S.A., Asymptotic structure of the spectrum of the Neumann problem in a thin comb-like domain. C.R. Acad Sci. Paris, Ser. 1 319 (1994) 1343–1348. Google Scholar
Mel’nyk, T.A. and Nazarov, S.A., Asymptotics of the Neumann spectral problem solution in a domain of thick comb type. Trudy Seminara imeni I.G. Petrovskogo 19 (1996) 138–173 (in Russian); English transl. : J. Math. Sci. 85 (1997) 2326–2346. Google Scholar
Mel’nyk, T.A. and Sadovyj, D. Yu., Homogenization of elliptic problems with alternating boundary conditions in a thick two-level junction of type 3 :2 :2. J. Math. Sci. 165 (2010) 67–90. Google Scholar
Mel’nyk, T.A., Nakvasiuk, Iu.A. and Wendland, W.L., Homogenization of the Signorini boundary-value problem in a thick junction and boundary integral equations for the homogenized problem. Math. Meth. Appl. Sci. 34 (2011) 758–775. Google Scholar