Skip to main content
SpringerLink
Log in

Asymptotic analysis and error estimates for an optimal control problem with oscillating boundaries

  • Published:
ANNALI DELL'UNIVERSITA' DI FERRARA Aims and scope Submit manuscript

Abstract

In this article, we consider a distributed optimal control problem associated with the Laplacian in a domain with rapidly oscillating boundary. For simplicity, we consider a rectangular region in 2d with oscillations on one part of the boundary. We consider two types of functionals, namely a functional involving the L 2-norm of the state variable and another one involving its H 1-norm. The homogenization of the optimality system is obtained and then we derive appropriate error estimates in both cases.

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. Achdou Y., Pironneau O., Valentin F.: Effective boundary conditions for laminar flows over periodic rough boundaries. J. Comput. Phys. 147(1), 187–218 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Allaire, G., Amar, M.: Boundary layer tails in periodic homogenization. ESAIM Control Optim. Calc. Var. 4, 209–243 (1999, electronic)

  3. Amirat Y., Bodart O.: Boundary layer correctors for the solution of Laplace equation in a domain with oscillating boundary. Z. Anal. Anwendungen 20(4), 929–940 (2001)

    MathSciNet  MATH  Google Scholar 

  4. Amirat, Y., Bodart, O., De Maio, U., Gaudiello, A.: Asymptotic approximation of the solution of the Laplace equation in a domain with highly oscillating boundary. SIAM J. Math. Anal. 35 no. 6, 1598–1616 (2004, electronic)

    Google Scholar 

  5. Amirat, Y., Simon, J.: Riblets and drag minimization. Optimization methods in partial differential equations (South Hadley, MA, 1996), Contemp. Math., vol. 209, pp. 9–17. American Mathematical Society, Providence (1997)

  6. Arrieta J.M., Bruschi S.M.: Rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a Lipschitz deformation. Math. Models Methods Appl. Sci. 17(10), 1555–1585 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Barbu, V.: Mathematical methods in optimization of differential systems. Mathematics and its Applications, vol. 310, Kluwer, Dordrecht, Translated and revised from the 1989 Romanian original (1994)

  8. Bensoussan A., Lions J.-L., Papanicolaou G.: Asynptotic Analysis for Periodic Structures. Ams Chelsea Publishing, New York (1978)

    Google Scholar 

  9. Birnir B., Hou S., Wellander N.: Derivation of the viscous Moore–Greitzer equation for aeroengine flow. J. Math. Phys. 48(6), 06520931 (2007)

    Article  MathSciNet  Google Scholar 

  10. Bonder J.F., Orive R., Rossi J.D.: The best Sobolev trace constant in a domain with oscillating boundary. Nonlinear Anal. 67(4), 1173–1180 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Brizzi R., Chalot J.-P.: Boundary homogenization and Neumann boundary value problem. Ricerche Mat. 46(2), 341–387 (1997)

    MathSciNet  MATH  Google Scholar 

  12. Bucur D., Feireisl E., Nečasová Š., Wolf J.: On the asymptotic limit of the Navier–Stokes system on domains with rough boundaries. J. Differ. Equ. 244(11), 2890–2908 (2008)

    Article  MATH  Google Scholar 

  13. Cioranescu D., Donato P.: An Introduction to Homogenization. Oxford University Press, Oxford (1999)

    MATH  Google Scholar 

  14. Fursikov, A.V.: Optimal control of distributed systems. Theory and applications, Translations of Mathematical Monographs, vol. 187. American Mathematical Society, Providence (2000)

  15. Gaudiello A.: Asymptotic behaviour of non-homogeneous Neumann problems in domains with oscillating boundary. Ricerche Mat. 43(2), 239–292 (1994)

    MathSciNet  MATH  Google Scholar 

  16. Gaudiello A., Hadiji R., Picard C.: Homogenization of the Ginzburg–Landau equation in a domain with oscillating boundary. Commun. Appl. Anal. 7(2–3), 209–223 (2003)

    MathSciNet  MATH  Google Scholar 

  17. Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of the Second Order. Springer, Berlin (1977)

    Google Scholar 

  18. Grisvard, P.: Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics, vol. 24. Pitman (Advanced Publishing Program), Boston (1985)

  19. Jikov V.V., Kozlov S.M., Oleĭnik O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994)

    Book  Google Scholar 

  20. Kesavan S., SaintJean Paulin J.: Homogenization of an optimal control problem. SIAM J. Control Optim. 35(5), 1557–1573 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kesavan S., SaintJean Paulin J.: Optimal control on perforated domains. J. Math. Anal. Appl. 229(2), 563–586 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Landis E.M., Panasenko G.P.: A theorem on the asymptotic behavior of the solutions of elliptic equations with coefficients that are periodic in all variables, except one. Dokl. Akad. Nauk SSSR 235(6), 1253–1255 (1977)

    MathSciNet  Google Scholar 

  23. Lions, J.-L.: Optimal control of systems governed by partial differential equations. Translated from the French by S. K. Mitter. Die Grundlehren der mathematischen Wissenschaften, Band 170, Springer, New York (1971)

  24. Lions J.-L.: Some Methods in the Mathematical Analysis of Systems and their Control. Kexue Chubanshe (Science Press), Beijing (1981)

    MATH  Google Scholar 

  25. Lions, J.-L.: Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Recherches en Mathématiques Appliquées (Research in Applied Mathematics), vol. 8, Masson, Paris, 1988, Contrôlabilité exacte. (Exact controllability), With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch

  26. Lions, J.-L.: Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 2, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 9, Masson, Paris, 1988, Perturbations

  27. Moore F.K., Greitzer E.M.: A theory of post-stall transients in axial compression systems: Part 1 development of equations. Trans. ASME J. Eng. Gas Turbines Power 108, 68–76 (1986)

    Article  Google Scholar 

  28. Moore F.K., Greitzer E.M.: A theory of post-stall transients in axial compression systems: Part 2 application. Trans. ASME J. Eng. Gas Turbines Power 108, 231–239 (1986)

    Article  Google Scholar 

  29. Muthukumar T., Nandakumaran A.K.: Darcy-type law associated to an optimal control problem. Electron. J. Differ. Equ. no 16, 12 (2008)

    Google Scholar 

  30. Muthukumar T., Nandakumaran A.K.: Homogenization of low-cost control problems on perforated domains. J. Math. Anal. Appl. 351(1), 29–42 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Neuss N., Neuss-Radu M., Mikelić A.: Effective laws for the Poisson equation on domains with curved oscillating boundaries. Appl. Anal. 85(5), 479–502 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Raymond, J.-P.: Optimal control of partial differential equations. http://www.math.univ-toulouse.fr/~raymond/book-ficus.pdf, Institut de Mathématiques de Toulouse, Université Paul Sabatier, 31062 Toulouse Cedex, France

  33. Tartar, L.: The general theory of homogenization. Lecture Notes of the Unione Matematica Italiana, vol. 7, Springer, Berlin (A personalized introduction) (2009)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Science, Bangalore, India

    A. K. Nandakumaran & Ravi Prakash

  2. Institut de Mathématiques de Toulouse, Université Paul Sabatier and CNRS, 31062, Toulouse Cedex, France

    J.-P. Raymond

Authors
  1. A. K. Nandakumaran
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ravi Prakash
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J.-P. Raymond
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J.-P. Raymond.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nandakumaran, A.K., Prakash, R. & Raymond, JP. Asymptotic analysis and error estimates for an optimal control problem with oscillating boundaries. Ann Univ Ferrara 58, 143–166 (2012). https://doi.org/10.1007/s11565-011-0135-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11565-011-0135-3

Keywords

Mathematics Subject Classification (2000)

Search

Navigation