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Optimal Feedback Control Problem for the Bingham Model with Periodical Boundary Conditions on Spatial Variables

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The paper is devoted to an optimal feedback control problem for the Bingham model with periodic conditions on spatial variables. An interpretation of the problem is given in the form of operator inclusion with multivalued right-hand side. On the base of a topological approximation approach to studying hydrodynamics problems, and the degree theory of multivalued vector fields, the existence of solutions of this inclusion is deduced. Then it is proved that among the solutions of the problem in question, there is a solution minimizing a given cost functional.

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References

  1. 1.

    F. Abergel and R. Temam, “On Some Control Problems in Fluid Mechanics,” Theor. Comput. Fluid Dyn., 1, No. 6, 303–325 (1990).

  2. 2.

    H. Choi, R. Temam, P. Moin, and J. Kim, “Feedback control for unsteady flow and its application to Burgers equation,” J. Fluid Mech., 253, 509–543 (1993).

  3. 3.

    G. Duvant and J. L. Lions, Inequalities in Mechanics and Physics [in Russian], Nauka, Moscow (1980).

  4. 4.

    A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications [in Russian], Nauchn. Kniga, Novosibirsk (1999).

  5. 5.

    M. D. Gunzburger, L. Hou, and T. P. Svobodny, “Boundary velocity control of incompressible flow with an application to viscous drag reduction,” SIAM J. Control Optim., 30, No. 1, 167–181 (1992).

  6. 6.

    J. U. Kim, “On the initial-boundary value problem for a Bingham fluid in a three dimensional domain,” Trans. Amer. Math. Soc., 304, No. 2, 751–770 (1987).

  7. 7.

    M. A. Krasnosel’skii, P. P. Zabreiko, E. I. Pustyl’nik, and P. E. Sobolevskii, Integral Operators in Spaces of Summable Functions [in Russian], Nauka, Moscow (1966).

  8. 8.

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

  9. 9.

    J. L. Lions and E. Magenes, Problemés aux Limites non Homogénes et Applications, Dunod, Paris (1968).

  10. 10.

    V. V. Obukhovskii, P. Zecca, and V. G. Zvyagin, “Optimal feedback control in the problem of the motion of a viscoelastic fluid,” Topol. Methods Nonlinear Anal., 23, No. 2, 323–337 (2004).

  11. 11.

    G. A. Seregin, “On the dynamical system associated with two dimensional equations of the motion of Bingham fluid,” Zap. Nauchn. Semin. POMI, 188, 128–142 (1991).

  12. 12.

    V. V. Shelukhin, “Bingham viscoplastic as a limit of non-Newtonian fluids,” J. Math. Fluid Mechanic, 4, No. 2, 109–127 (2002).

  13. 13.

    R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, Philadelphia, Pennsylvania (1983).

  14. 14.

    R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis [in Russian], Mir, Moscow (1987).

  15. 15.

    A. V. Zvyagin, “An optimal control problem with feedback for a mathematical model of the motion of weakly concentrated water polymer solutions with objective derivative,” Siberian Math. J., 54, No. 4, 807–825 (2013).

  16. 16.

    A. V. Zvyagin, “Optimal feedback control for a thermoviscoelastic model of Voigt fluid motion,” Doklady Math., 468, No. 3, 251–253 (2016).

  17. 17.

    V. G. Zvyagin, “Topological approximation approach to study of mathematical problems of hydrodynamics,” J. Math. Sci., 201, No. 6, 830–858 (2014).

  18. 18.

    V. G. Zvyagin and M. Yu. Kuz’min, “On an optimal control problem in the Voight model of the motion of a viscoelastic fluid,” J. Math. Sci., 149, No. 5, 1618–1627 (2008).

  19. 19.

    V. G. Zvyagin and M. V. Turbin, “Optimal feedback control in the mathematical model of low concentrated agueous polymer solutions,” J. Optim. Theory Appl., 148, No. 1, 146–163 (2011).

  20. 20.

    V. G. Zvyagin and M. V. Turbin, Mathematical Problems of the Hydrodynamics of Viscoelastic Media [Russian], Moscow (2012).

  21. 21.

    V. Zvyagin, V. Obukhovskii, and A. Zvyagin, “On inclusions with multivalued operators and their applications to some optimization problems,” J. Fixed Point Theory Appl., 16, 27—82 (2014).

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Correspondence to V. G. Zvyagin.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 477, 2018, pp. 54–86.

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Zvyagin, V.G., Zvyagin, A.V. & Turbin, M.V. Optimal Feedback Control Problem for the Bingham Model with Periodical Boundary Conditions on Spatial Variables. J Math Sci (2020) doi:10.1007/s10958-020-04667-7

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