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Attainability Issue for the Optimal Control Problem in Coefficients for a Degenerate Parabolic Variational Inequality

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We study the optimal control problem in coefficients for the degenerate parabolic variational inequality. Since problems of this type may have the Lavrent’ev effect, we consider the optimal control problem in a class of so-called H-admissible solutions. We substantiate the attainability of H-optimal pairs via the optimal solutions of some nondegenerate perturbed optimal control problems under the condition of solvability of the original degenerate problem.

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References

  1. S. E. Pastuhova, “Degenerate equations of monotone type: Lavrent’ev phenomenon and attainability problems,” Mathematics, 198, Issue 10, 1465–1494 (2007).

    MATH  Google Scholar 

  2. V. V. Zhikov, “Weighted Sobolev spaces,” Mathematics, 189, Issue 8, 27–58 (1998).

    MathSciNet  MATH  Google Scholar 

  3. V. V. Zhikov and S. E. Pastukhova, “Homogenization of degenerate elliptic equations,” Sib. Math. J., 25, Issue 1, 80–101 (2006).

    Article  MathSciNet  Google Scholar 

  4. P. I. Kogut and G. Leugering, “Optimal L1-control in coefficients for Dirichlet elliptic problems: H-optimal solutions,” Z. Anal. Anwend., 31, Issue 1, 31–53 (2012).

    Article  MathSciNet  Google Scholar 

  5. P. I. Kogut and G. Leugering, “Optimal L1-control in coefficients for Dirichlet elliptic problems: W-optimal solutions,” J. Optim. Theory Appl., 150, 205–232 (2011).

    Article  MathSciNet  Google Scholar 

  6. F. Murat, “Un contre-exemple pour le probléme de de contrôle dans les coefficients,” C.R.A.S. Paris, Sér. A, 273, 708–711 (1971).

    MATH  Google Scholar 

  7. G. Butazzo, G. Dal Maso, A. Garroni, and A. Malusa, “On the relaxed formulation of some shape optimization problems,” Adv. Math. Sci. Appl., 1, Issue 7, 1–24 (1997).

    MathSciNet  MATH  Google Scholar 

  8. J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer Verlag, New York (1971).

    Book  Google Scholar 

  9. F. Murat, “Compacité par compensation,” Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser., Issue 5, 489–507 (1978).

  10. A. N. Stanzhitskii and T. V. Dobrodzii, “Study of optimal control problems on the half-line by the averaging method,” Different. Equat., 47, 264–277 (2011).

    Article  Google Scholar 

  11. A. V. Sukretna and O. A. Kapustyan, “Approximate averaged synthesis of the problem of optimal control for a parabolic equation,” Ukr. Mat. Zh., 56, 1384–1394 (2004); English translation: Ukr. Math. J., 56, No. 10, 1653–1664 (2004).

  12. E. A. Kapustyan and A. G. Nakonechnyi, “Optimal bounded control synthesis for a parabolic boundary-value problem with fast oscillatory coefficients,” J. Automat. Inform. Sci., 31, 33–44 (1999).

    Article  Google Scholar 

  13. O. P. Kupenko, “Optimal control problems in coefficients for degenerate variational inequalities of monotone type. II. Attainability problem,” J. Comput. Appl. Math., 1, No. 107, 15–34 (2012).

    Google Scholar 

  14. A. A. Kovalevsky and Yu. S. Gorban, “Degenerate anisotropic variational inequalities with L1-data,” C. R. Math. Acad. Sci. Paris, 345, No. 8, 441–444 (2007).

    Article  MathSciNet  Google Scholar 

  15. V. Chiadó Piat and F. Serra Cassano, “Cassano some remarks about the density of smooth functions in weighted Sobolev spaces,” J. Convex Anal., 1, Issue 2, 135–142 (1994).

    MathSciNet  MATH  Google Scholar 

  16. V. V. Zhikov, “On Lavrentiev phenomenon,” Russ. J. Math. Phys., 3, Issue 2, 249–269 (1994).

    MathSciNet  Google Scholar 

  17. F. Nicolosi, P. Drabek, and A. Kufner, Nonlinear Elliptic Equations, Singular and Degenerate Cases, Univ. West Bohemia, Pilsen (1996).

    MATH  Google Scholar 

  18. E. Zeider, Nonlinear Analysis and Its Applications, IIA, IIB, Springer-Verlag, New York–Heidelberg (1990).

  19. V. Barbu, Optimal Control of Variational Inequalities, Pitman Adv. Publ. Program, London (1984).

  20. O. P. Kupenko, “Optimal control problems in coefficients for degenerate variational inequalities of monotone type. I. Existence of optimal solutions,” J. Comput. Appl. Math., 3, No. 106, 88–104 (2011).

    Google Scholar 

  21. I. G. Balanenko and P. I. Kogut, “H-optimal control in coefficients for Dirichlet parabolic problems,” Bull. Dnipropetrovsk Univ.: Ser. Comm. Math. Model. Differ. Equat. Theory, 18, Issue 8, 45–63 (2010).

    Google Scholar 

  22. N. V. Kasimova, Solvability Issue for Optimal Control Problem in Coefficients for Degenerate Parabolic Variational Inequality, Underst. Complex Syst. (2020) (to appear).

  23. C. D’Apice, U. De Maio, and P. I. Kogut, “Suboptimal boundary control for elliptic equations in critically perforated domains,” Ann. Inst. H. Poincaré Anal. Non Linéaire, 25, 1073–1101 (2008).

    Article  MathSciNet  Google Scholar 

  24. Ch. Gaevskii, K. Greger, and K. Zacharias, Nonlinear Operator Equations and Operator Differential Equations [in Russian], Mir, Moscow (1978).

    Google Scholar 

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Authors and Affiliations

  1. T. Shevchenko Kyiv National University, Acad. Hlushkov Avenue, 4, Kyiv, 03127, Ukraine

    N. V. Kasimova

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  1. N. V. Kasimova
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Correspondence to N. V. Kasimova.

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Published in Neliniini Kolyvannya, Vol. 23, No. 2, pp. 200–216, April–June, 2020.

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Kasimova, N.V. Attainability Issue for the Optimal Control Problem in Coefficients for a Degenerate Parabolic Variational Inequality. J Math Sci 258, 636–654 (2021). https://doi.org/10.1007/s10958-021-05571-4

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  • DOI: https://doi.org/10.1007/s10958-021-05571-4

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