Skip to main content
SpringerLink
Log in

Asymptotic Analysis of an Optimal Control Problem Involving a Thick Two-Level Junction with Alternate Type of Controls

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We study the asymptotic behavior (as ε→0) of an optimal control problem in a plane thick two-level junction, which is the union of some domain and a large number 2N of thin rods with variable thickness of order \(\varepsilon =\mathcal{O}(N^{-1}).\) The thin rods are divided into two levels depending on the geometrical characteristics and on the controls given on their bases. In addition, the thin rods from each level are ε-periodically alternated and the inhomogeneous perturbed Fourier boundary conditions are given on the lateral sides of the rods. Using the direct method of the calculus of variations and the Buttazzo-Dal Maso abstract scheme for variational convergence of constrained minimization problems, the asymptotic analysis of this problem for different kinds of controls is made as ε→0.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Fleury, F., Sanchez-Palencia, E.: Asymptotic and spectral properties of the acoustic vibrations of body, perforated by narrow channels. Bull. Sci. Math. 2, 149–176 (1986)

    MathSciNet  Google Scholar 

  2. Sanchez-Hubert, J., Sanchez-Palencia, E.: Vibration and Coupling of Continuous Systems. Springer, Berlin/Heidelberg (1989)

    MATH  Google Scholar 

  3. Mel’nyk, T.A., Nazarov, S.A.: Asymptotic structure of spectrum of the Neumann problem in thin comb-like domain. C. R. Acad. Sci. Paris Ser. 1 319, 1343–1348 (1994)

    MATH  MathSciNet  Google Scholar 

  4. Mel’nyk, T.A.: Homogenization of the Poisson equation in a thick periodic junction. Z. Anal. Ihre Anwendungen 18(4), 953–975 (1999)

    MATH  MathSciNet  Google Scholar 

  5. Mel’nyk, T.A.: Homogenization of a perturbed parabolic problem in a thick periodic junction of type 3:2:1. Ukrainian Math. J. 52(11), 1737–1749 (2000)

    Article  MathSciNet  Google Scholar 

  6. De Maio, U., Mel’nyk, T.A., Perugia, C.: Homogenization of the Robin problem in a thick multilevel junction. Nonlinear Oscill. 7(3), 336–356 (2004)

    Article  MATH  Google Scholar 

  7. De Maio, U., Durante, T., Mel’nyk, T.A.: Asymptotic approximation for the solution to the Robin problem in a thick multilevel junction. Math. Models Methods Appl. Sci. 15(12), 1897–1921 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Mel’nyk, T.A., Vaschuk, P.S.: Homogenization of the Neumann-Fourier problem in a thick two-level junction of type 3:2:1. J. Math. Phys. Anal. Geom. 2(3), 318–337 (2006)

    MATH  Google Scholar 

  9. Zhikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin/Heidelberg (1994)

    MATH  Google Scholar 

  10. Braides, A., Defranceschi, A.: Homogenization of Multiple Integrals. Clarendon, Oxford (1998)

    MATH  Google Scholar 

  11. Buttazzo, G., Dal Maso, G.: Γ-convergence and optimal control problems. J. Optim. Theory Appl. 38, 385–407 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  12. Buttazzo, G.: Γ-convergence and its applications to some problem in the calculus of variations. In: School on Homogenization, ICTP, Trieste, 6–17 September 1993, pp. 38–61 (1994)

  13. Donato, P., Nabil, A.: Approximate controllability of linear parabolic equations in perforated domains. ESAIM: Control Optim. Calc. Var. 6, 21–38 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kesavan, S., Saint Jean Paulin, J.: Optimal control on perforated domains. J. Math. Anal. Appl. 229, 563–586 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  15. Saint Jean Paulin, J., Zoubairi, H.: Optimal control and “strange term” for the Stokes problem in perforated domains. Portugaliac Math. 59(2), 161–178 (2002)

    MATH  MathSciNet  Google Scholar 

  16. Cioranescu, D., Saint Jean Paulin, J.: Homogenization in open sets with holes. J. Math. Anal. Appl. 71, 590–607 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)

    MATH  Google Scholar 

  18. Denkowski, Z., 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(2), 365–391 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  19. De Maio, U., Gaudiello, A., Lefter, C.: Optimal control for a parabolic problem in a domain with highly oscillating boundary. Appl. Anal. 83(12), 1245–1264 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. Durante, T., Faella, L., Perugia, C.: Homogenization and behavior of optimal controls for the wave equation in domains with oscillating boundary. Nonlinear Differ. Equ. Appl. 14(5–6), 455–489 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kogut, P.I., Mel’nyk, T.A.: Limit analysis of one class of optimal control problems in thick singular junctions. J. Autom. Inf. Sci. 37, 8–24 (2005)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria dell’ Informazione e Matematica Applicata, Universita di Salerno, via Ponte don Melillo, 84084, Fisciano (SA), Italy

    T. Durante

  2. Department of Mathematical Physics, Faculty of Mathematics and Mechanics, National Taras Shevchenko University of Kiev, Volodymyrska str. 64, 01033, Kiev, Ukraine

    T. A. Mel’nyk

Authors
  1. T. Durante
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. A. Mel’nyk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. A. Mel’nyk.

Additional information

Communicated by F. Giannessi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Durante, T., 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, 205–225 (2010). https://doi.org/10.1007/s10957-009-9604-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-009-9604-6

Keywords

Search

Navigation