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On the Unique Solvability of the Optimal Starting Control Problem for the Linearized Equations of Motion of a Viscoelastic Medium

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Abstract

We study an optimization problem for the linearized evolution equations of the Oldroyd model of motion of a viscoelastic medium. The equations are given in a bounded three-dimensional domain. The velocity distribution at the initial time is used as a control function. The objective functional is terminal. The existence of a unique optimal control is proved for a given set of admissible controls. A variational inequality characterizing the optimal control is derived.

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REFERENCES

  1. Saut, J.-C., Lectures on the mathematical theory of viscoelastic fluids, in Lectures on the Analysis of Nonlinear Partial Differential Equations. Part 3 , Somerville, 2013, pp. 325–393.

  2. Fang, D. and Zi, R., Global solutions to the Oldroyd-B model with a class of large initial data, SIAM J. Math. Anal., 2016, vol. 48, pp. 1054–1084.

    Article  MathSciNet  Google Scholar 

  3. Baranovskii, E.S., Steady flows of an Oldroyd fluid with threshold slip, Commun. Pure Appl. Anal., 2019, vol. 18, pp. 735–750.

    Article  MathSciNet  Google Scholar 

  4. Wan, R., Some new global results to the incompressible Oldroyd-B model, Z. Angew. Math. Phys., 2019, vol. 70, article ID 28.

  5. Fursikov, A.V., Optimal’noe upravlenie raspredelennymi sistemami. Teoriya i prilozheniya (Optimal Control of Distributed Systems. Theory and Applications), Novosibirsk: Nauchn. Kniga, 1999.

    Book  Google Scholar 

  6. Kuznetsov, A.V., Boundary optimal control in initial–boundary value problem for the model of viscoelastic medium with total derivative, Vestn. Voronezh. Gos. Univ. Ser. Fiz. Mat., 2008, no. 1, pp. 232–248.

  7. Doubova, A. and Fernandez-Cara, E., On the control of viscoelastic Jeffreys fluids, Syst. Control Lett., 2012, vol. 61, pp. 573–579.

    Article  MathSciNet  Google Scholar 

  8. Baranovskii, E.S., Optimal boundary control of nonlinear-viscous fluid flows, Sb. Math., 2020, vol. 211, no. 4, pp. 505–520.

    Article  MathSciNet  Google Scholar 

  9. Baranovskii, E.S., Domnich, A.A., and Artemov, M.A., Optimal boundary control of non-isothermal viscous fluid flow, Fluids, 2019, vol. 4, no. 3, article ID 133.

  10. Temam, R., Navier–Stokes Equations. Theory and Numerical Analysis, Amsterdam–New York–Oxford: North-Holland, 1979. Translated under the title: Uravneniya Nav’e–Stoksa. Teoriya i chislennyi analiz, Moscow: Mir, 1981.

  11. Baranovskii, E.S. and Artemov, M.A., Global existence results for Oldroyd fluids with wall slip, Acta Appl. Math., 2017, vol. 147, pp. 197–210.

    Article  MathSciNet  Google Scholar 

  12. Kinderlehrer, D. and Stampacchia, G., An Introduction to Variational Inequalities and Their Applications, New York–London–Toronto–Sydney–San Francisco: Academic Press, 1980. Translated under the title: Vvedenie v variatsionnye neravenstva i ikh prilozheniya, Moscow: Mir, 1983.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Voronezh State University, Voronezh, 394018, Russia

    M. A. Artemov

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  1. M. A. Artemov
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Correspondence to M. A. Artemov.

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Translated by V. Potapchouck

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Artemov, M.A. On the Unique Solvability of the Optimal Starting Control Problem for the Linearized Equations of Motion of a Viscoelastic Medium. Diff Equat 57, 1070–1075 (2021). https://doi.org/10.1134/S0012266121080115

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  • DOI: https://doi.org/10.1134/S0012266121080115

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