Skip to main content

Part of the book series: International Mathematical Series ((IMAT,volume 6))

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. A. Agrachev and R. V. Gamkrelidze, Exponential representation of flows and chronological calculus, Math. USSR Sb. 35 (1979), 727–785.

    Article  MATH  Google Scholar 

  2. A. A. Agrachev and Yu. L. Sachkov, Lectures on Geometric Control Theory, Springer-Verlag, 2004.

    Google Scholar 

  3. A. A. Agrachev and A. V. Sarychev, On reduction of smooth system linear in control, Math. USSR Sb. 58 (1987), 15–30.

    Article  MATH  Google Scholar 

  4. A. A. Agrachev and A. V. Sarychev, Navier–Stokes equation controlled by degenerate forcing: controllability of finite-dimensional approximations, In: Proc. Intern. Conf. “Physics and Control 2003, St.-Petersburg, Russia, August 20-22, 2003,” CD ROM, 1346-1351.

    Google Scholar 

  5. A. A. Agrachev and A. V. Sarychev, Controllability of the Navier–Stokes equation by few low modes forcing, Dokl. Math. Sci. 69 (2004), no. 1/2, 112–115.

    MATH  Google Scholar 

  6. A. A. Agrachev and A. V. Sarychev, Navier–Stokes equations: Controllability by means of low modes forcing, J. Math. Fluid Mech. 7 (2005), 108–152.

    Article  MATH  MathSciNet  Google Scholar 

  7. A. A. Agrachev and A. V. Sarychev, Controllability of 2D Euler and Navier–Stokes equations by degenerate forcing, Commun. Math. Phys. 265 (2006), 673–697.

    Article  MATH  MathSciNet  Google Scholar 

  8. V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1997.

    Google Scholar 

  9. V. I. Arnold, Lectures on Partial Differential Equations, Springer-Verlag, 2004.

    Google Scholar 

  10. V. I. Arnold and B. M. Khesin, Topological Methods in Hydrodynamics, Springer-Verlag, New-York, 1998.

    MATH  Google Scholar 

  11. J.-M. Coron, Return method: some applications to flow control, In: Mathematical Control Theory, ICTP Lecture Notes Series Vol. 8. Parts 1–2, 2002.

    Google Scholar 

  12. B. Dubrovin, S. Novikov, and A. Fomenko, Modern Geometry – Methods and Applications. Part I, Springer-Verlag, New York, 1984.

    MATH  Google Scholar 

  13. W. E and J. C. Mattingly, Ergodicity for the Navier–Stokes equation with degenerate random forcing: finite-dimensional approximation, Comm. Pure Appl. Math. 54 (2001), no. 11, 1386–1402.

    Article  MathSciNet  Google Scholar 

  14. A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, Am. Math. Soc., Providence, 2000.

    Google Scholar 

  15. A. V. Fursikov and O. Yu. Imanuilov, Exact controllability of the Navier–Stokes and Boussinesq equations, Russian Math. Surv. 54 (1999), no. 3, 565–618.

    Article  MATH  Google Scholar 

  16. R. V. Gamkrelidze, On some extremal problems in the theory of differential equations with applications to the theory of optimal control, J. Soc. Ind. Appl. Math. Ser. A: Control 3 (1965), 106–128.

    MathSciNet  Google Scholar 

  17. R. V. Gamkrelidze, Principles of Optimal Control Theory, Plenum Press, New York, 1978.

    MATH  Google Scholar 

  18. V. Jurdjevic, Geometric Control Theory, Cambridge University Press, Cambridge, 1997.

    MATH  Google Scholar 

  19. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1966.

    Google Scholar 

  20. E. J. McShane, Generalized curves, Duke Math. J. 6 (1940), 513–536.

    Article  MATH  MathSciNet  Google Scholar 

  21. S. Rodrigues, Navier–Stokes equation on the rectangle: controllability by means of low mode forcing, J. Dyn. Control Syst. 12 (2006), 517–562.

    Article  MATH  MathSciNet  Google Scholar 

  22. S. Rodrigues, Navier–Stokes Equation on a Plane Bounded Domain: Continuity Properties for Controllability, ISTE, Hermes Publ., 2007. [To appear]

    Google Scholar 

  23. S. Rodrigues, Controlled PDE on Compact Riemannian Manifolds: Controllability Issues, In: Proc. Workshop Mathematical Control Theory and Finance, Lisbon, Portugal, 2007. [Submitted]

    Google Scholar 

  24. S. Rodrigues, 2D Navier-Stokes equation: A saturating set for the half-sphere. [Private communication]

    Google Scholar 

  25. M. Romito, Ergodicity of finite-dimensional approximations of the 3D Navier–Stokes equations forced by a degenerate noise, J. Stat. Phys. 114 (2004), no. 1-2, 155-177.

    Article  MATH  MathSciNet  Google Scholar 

  26. A. Shirikyan, Approximate controllability of three-dimensional Navier–Stokes equations, Commun. Math. Phys. 266 (2006), no. 1, 123–151.

    Article  MATH  MathSciNet  Google Scholar 

  27. A. Shirikyan, Exact controllability in projections for three-dimensional Navier–Stokes equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 24 (2007), 521-537.

    Google Scholar 

  28. L. C. Young, Generalized curves and the existence of an attained absolute minimum in the calculus of variations, Compt. Rend. Soc. Sci. Lett. Varsovie, cl. III 30 (1937), 212–234.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Agrachev, A., Sarychev, A. (2008). Solid Controllability in Fluid Dynamics. In: Bardos, C., Fursikov, A. (eds) Instability in Models Connected with Fluid Flows I. International Mathematical Series, vol 6. Springer, New York, NY. https://doi.org/10.1007/978-0-387-75217-4_1

Download citation

Publish with us

Policies and ethics