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Approximate Optimal Regulator for Distributed Control Problem with Superposition Functional and Rapidly Oscillating Coefficients

  • Olena A. Kapustian
Chapter
Part of the Understanding Complex Systems book series (UCS)

Abstract

In this paper, we consider the optimal stabilization problem on infinite time interval for a parabolic equation with rapidly oscillating coefficients and non-decomposable quadratic cost functional with superposition type operator. In general, to find the exact formula of optimal regulator is not possible for such a problem, because the Fourier method cannot be directly applied. But the transition to the homogenized parameters greatly simplifies the structure of the problem. Assuming that the problem with the homogenized coefficients already admits optimal regulator, we ground approximate optimal control in the feedback form for the initial problem. We give an example of superposition operator for specific conditions in this paper.

References

  1. 1.
    Appell, J., Zabreiko, P.: Nonlinear Superposition Operators. Cambridge University Press, Cambridge (1990)CrossRefGoogle Scholar
  2. 2.
    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, 365–391 (1993)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Fursikov, A.V.: Optimal Control of Distributed Systems. Theory and Applications. AMS, Providence (1999)Google Scholar
  4. 4.
    Jikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994)CrossRefGoogle Scholar
  5. 5.
    Kapustian, O.A., Sobchuk, V.V.: Approximate homogenized synthesis for distributed optimal control problem with superposition type cost functional. Stat. Optim. Inf. Comput. 6 233–239 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kapustian, O.A., Sukretna, A.V.: Approximate averaged synthesis of the problem of optimal control for a parabolic equation. Ukr. Math. J. 56(10), 1653–1664 (2004)CrossRefGoogle Scholar
  7. 7.
    Kapustyan, O.A.: Approximate synthesis of optimal bounded control for a parabolic boundary-value problem. Ukr. Math. J. 54(12), 2067–2074 (2002)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kapustyan, E.A., Nakonechnyi, A.G.: Optimal bounded control synthesis for a parabolic boundary-value problem with fast oscillatory coefficients. J. Autom. Inf. Sci. 31(12), 33–44 (1999)CrossRefGoogle Scholar
  9. 9.
    Kapustyan, O.V., Kapustyan, O.A., Sukretna, A.V.: Approximate bounded synthesis for one weakly nonlinear boundary-value problem. Nonlinear Oscil. 12(3), 297–304 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kapustyan, O.V., Kapustian, O.A., Sukretna, A.V.: Approximate stabilization for a nonlinear parabolic boundary-value problem. Ukr. Math. J. 63(5), 759–767 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lions, J.-L.: Optimal Control of Systems, Governed by Partial Differential Equations. Springer, Berlin (1971)Google Scholar
  12. 12.
    Mashchenko, S.O.: A mathematical programming problem with the fuzzy set of indices of constraints. Cybern. Syst. Anal. 49(1), 62–68 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Mel’nik, V.S.: Minimization of superposition functionals. Cybern. Syst. Anal. 30(1), 41–46 (1994)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Pichkur, V.V., Sasonkina, M.S.: Practical stabilization of discrete control systems. Int. J. Pure Appl. Math. 81(6), 877–884 (2012)Google Scholar
  15. 15.
    Sell, G.R., You, Y.: Dynamics of Evolutionary Equations. Springer, New York (2002)CrossRefGoogle Scholar
  16. 16.
    Zhikov, V.V., Kozlov, S.M., Oleinik, O.A.: G-convergence of parabolic operators. Russ. Math. Surv. 36, 9–60 (1983)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Olena A. Kapustian
    • 1
  1. 1.Taras Shevchenko National University of KyivKyivUkraine

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