Skip to main content
SpringerLink
Log in

Finite-dimensional controllability for systems of Navier-Stokes type

  • Control Theory
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

We consider the problem of partial controllability for an evolution equation with a quadratic nonlinearity, in which one should provide, at a given time, a given projection of the solution onto some finite-dimensional subspace by using the action of external forces that belong to one and the same subspace at each time. On the basis of estimates for the solution of a subdifferential Cauchy problem for a system of Navier-Stokes type, we prove the controllability and the existence of a control with minimum norm. We consider applications to the Navier-Stokes equations of a viscous incompressible fluid and a heat convection model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Temam, R., Navier-Stokes Equations. Theory and Numerical Analysis, Amsterdam: North-Holland, 1977. Translated under the title Uravneniya Nav’e-Stoksa. Teoriya i chislennyi analiz, Moscow: Mir, 1981.

    MATH  Google Scholar 

  2. Fursikov, A.V., Optimal’noe upravlenie raspredelennymi sistemami. Teoriya i prilozheniya (Optimal Control for Distributed Systems. Theory and Applications), Novosibirsk, 1999.

  3. Fursikov, A.V. and Emanuilov, O.Yu., Exact Controllability of Navier-Stokes and Boussinesq Equations, Uspekhi Mat. Nauk, 1999, vol. 54, no. 3 (327), pp. 93–146.

    MathSciNet  Google Scholar 

  4. Barbu, V., Feedback Stabilization of Navier-Stokes Equations, ESAIM Control Optim. Calc. Var., 2003, vol. 9, pp. 197–205.

    MATH  MathSciNet  Google Scholar 

  5. Barbu, V. and Triggiani, R., Internal Stabilization of Navier-Stokes Equations with Finite-Dimensional Controllers, Indiana Univ. Math. J., 2004, vol. 53, no. 5, pp. 1443–1494.

    Article  MATH  MathSciNet  Google Scholar 

  6. Agrachev, A.A. and Sarychev, A.V., Navier-Stokes Equations: Controllability by Means of Low Modes Forcing, J. Math. Fluid Mech., 2005, vol. 7, pp. 108–152.

    Article  MATH  MathSciNet  Google Scholar 

  7. Shirikyan, A., Exact Controllability in Projections for Three-Dimensional Navier-Stokes Equations, Analyse Non Lineaire, 2007, vol. 24, pp. 521–537.

    Article  MATH  MathSciNet  Google Scholar 

  8. Barbu, V., Analysis and Control of Nonlinear Infinite Dimensional Systems, New York, 1993.

  9. Barbu, V., The Time Optimal Control of Navier-Stokes Equations, Systems Control Lett., 1997, vol. 30, pp. 93–100.

    Article  MATH  MathSciNet  Google Scholar 

  10. Chebotarev, A.Yu., Subdifferential Inverse Problems for Evolution Navier-Stokes Systems, J. Inverse Ill-Posed Probl., 2000, vol. 8, no. 3, pp. 275–287.

    MathSciNet  Google Scholar 

  11. Chebotarev, A.Yu., Subdifferential Boundary Value Problems of Magnetic Fluid Dynamics, Differ. Uravn., 2007, vol. 43, no. 12, pp. 1700–1709.

    MathSciNet  Google Scholar 

  12. Kato, T. and Fujita, H., On the Nonstationary Navier-Stokes System, in Rend. Sem. Mat. Univ. Padova 32, 1962, pp. 243–260.

Download references

Author information

Authors and Affiliations

  1. Institute for Applied Mathematics, Far-East Branch of the Russian Academy of Sciences, Vladivostok, Russia

    A. Yu. Chebotarev

Authors
  1. A. Yu. Chebotarev
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © A.Yu. Chebotarev, 2010, published in Differentsial’nye Uravneniya, 2010, Vol. 46, No. 10, pp. 1495–1503.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chebotarev, A.Y. Finite-dimensional controllability for systems of Navier-Stokes type. Diff Equat 46, 1498–1506 (2010). https://doi.org/10.1134/S0012266110100149

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266110100149

Keywords

Search

Navigation