Feedback Control Problem for Modified Kelvin-Voigt Model


This paper deals with the optimal feedback control problem for the modified Kelvin-Voigt model. The considered model describes the motion of weakly concentrated aqueous polymer solutions. In our case, the control function (the external force) depends on the velocity of the fluid. In such a way, the control is not selected from a finite set of available controls, but belongs to the image of some multi-valued map. The solution for the control problem of fluid motion is a pair: the velocity of the fluid and the control (the density of external forces). Since there can be many such pairs, the concept of optimal solution naturally arises, which gives a minimum to specified cost functional. For the considered optimal feedback control problem the existence theorem on weak solution is proved.

This is a preview of subscription content, access via your institution.


  1. 1.

    Filippov AF. On certain questions in the theory of optimal control. J Soc Ind Appl Math Ser A Control 1962;1(1):76–84.

    MathSciNet  Article  Google Scholar 

  2. 2.

    Aubin JP, Cellina A. Differential Inclusion. Berlin: Springer-Verlag; 1984.

    Google Scholar 

  3. 3.

    Kamenskii M, Obukhovskii V, Zecca P. Condensing multivalued maps and semilinear differential inclusions in banach spaces. Berlin-New York: Walter de Gruyter; 2001.

    Google Scholar 

  4. 4.

    Obukhovskii VV, Zecca P, Zvyagin VG. Optimal feedback control in the problem of the motion of a viscoelastic fluid. Topological Methods Nonlinear Anal 2004;23:323–37.

    MathSciNet  Article  Google Scholar 

  5. 5.

    Zvyagin VG, Zvyagin AV, Turbin MV. Optimal feedback control problem for the Bingham model with periodical boundary conditions on spatial variables. J Math Sci 2020;244:959–80.

    MathSciNet  Article  Google Scholar 

  6. 6.

    Zvyagin VG, Turbin MV. Optimal feedback control in the mathematical model of low concentrated agueous polymer solutions. J Optim Theory Appl 2011; 148(1):146–63.

    MathSciNet  Article  Google Scholar 

  7. 7.

    Zvyagin VG, Turbin MV. The study of initial-boundary value problems for mathematical models of the motion of Kelvin–Voigt fluids. J Math Sci 2010; 168(2):157–308.

    MathSciNet  Article  Google Scholar 

  8. 8.

    Plotnikov PI, Turbin MV, Ustiuzhaninova AS. Existence theorem for a weak solution of the optimal feedback control problem for the modified Kelvin-Voigt model of weakly concentrated aqueous polymer solutions. Dokl Math 2019; 100(2):433–5.

    Article  Google Scholar 

  9. 9.

    Pavlovsky VA. On theoretical description of weak aqueous solutions of polymers. Dokl Akad Nauk SSSR 1971;200(4):809–12. (in Russian).

    Google Scholar 

  10. 10.

    Oskolkov AP. Some quasilinear systems that arise in the study of the motion of viscous fluids. Zapiski Naucnyh Seminarov Leningradskogo Otdelenija Matematiceskogo Instituta im. V A Steklova Akad Nauk SSSR (LOMI) 1975;52:128–57. (in Russian).

    MathSciNet  Google Scholar 

  11. 11.

    Turbin MV, Ustiuzhaninova AS. The existence theorem for a weak solution to initial-boundary value problem for system of equations describing the motion of weak aqueous polymer solutions. Russian Mathematics 2019;63(8):54–69.

    Article  MATH  Google Scholar 

  12. 12.

    Kaya M, Celebi AO. Existence of weak solutions of the g-Kelvin-Voight equation. Math Comput Model 2009;49:497–504.

    MathSciNet  Article  Google Scholar 

  13. 13.

    Antontsev SN, Khompysh K h. Kelvin-Voight equation with p-Laplacian and damping term: Existence, uniqueness and blow-up. J Math Anal Appl 2017; 446(2):1255–73.

    MathSciNet  Article  Google Scholar 

  14. 14.

    Kalantarov VK. Attractors for some nonlinear problems of mathematical physics. Zapiski Naucnyh Seminarov Leningradskogo Otdelenija Matematiceskogo Instituta im. V A Steklova Akad Nauk SSSR (LOMI) 1986;152:50–54. (in Russian).

    Google Scholar 

  15. 15.

    Kalantarov VK, Titi ES. 2007. Global attractors and estimates of the number of degrees of determining modes for the 3D Navier-Stokes-Voight equations.arXiv:0705.3972v1.

  16. 16.

    Kalantarov VK, Levant B, Titi ES. Gevrey regularity for the attractor of the 3D Navier-Stokes-Voight equations. J Nonlinear Sci 2009;19:133–52.

    MathSciNet  Article  Google Scholar 

  17. 17.

    Temam R. Navier-stokes equations, theory and numerical analysis. Rhode Island: AMS Chelsea Providence; 2000.

    Google Scholar 

  18. 18.

    Ladyzhenskaya OA. The mathematical theory of viscous incompressible flow. New York: Gordon and Breach Science Publishers; 1969.

    Google Scholar 

  19. 19.

    Solonnikov VA. Estimates of Green’s tensors for some boundary-value problems. DAN SSSR 1960;130(5):988–91.

    MathSciNet  Google Scholar 

  20. 20.

    Vorovich II, Yudovich VI. Steady flow of a viscous incompressible fluid. Mat Sb 1961;53(4):393–428.

    MathSciNet  Google Scholar 

  21. 21.

    Fursikov AV. Optimal control of distributed systems. Theory and applications. Trans. of math. monographs, AMS Providence, vol 187; 2000.

  22. 22.

    Simon J. Compact sets in the space Lp, (0,T; B). Annali di Matematica 1986;146:65–96.

    Article  Google Scholar 

Download references


The research of the first author (Theorem 1 on the existence of a weak solution for the feedback control problem) was supported by the Russian Foundation for Basic Research, project no. 20-01-00051. The research of the second author (Theorem 2 on the existence of a solution minimizing the given cost functional) was supported by the Russian Science Foundation, project no. 19-11-00146.

Author information



Corresponding author

Correspondence to Mikhail Turbin.

Ethics declarations

Conflict of Interest

The authors declare that they have no conflict of interest.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ustiuzhaninova, A., Turbin, M. Feedback Control Problem for Modified Kelvin-Voigt Model. J Dyn Control Syst (2021).

Download citation


  • Feedback control
  • Control problem
  • Modified Kelvin-Voigt model
  • Weak solution
  • Multi-valued map
  • Existence theorem

Mathematics Subject Classification (2010)

  • 76D55
  • 49J20
  • 76A05
  • 49J53