Skip to main content

Stabilization of Abstract Parabolic Systems

  • Chapter
Stabilization of Navier–Stokes Flows

Part of the book series: Communications and Control Engineering ((CCE))

  • 1241 Accesses

Abstract

We discuss here a few stabilization techniques for nonlinear parabolic-like equations in Hilbert spaces. The abstract theory of stabilization presented below captures most of the techniques to be developed for the specific problems which are treated in the next chapters. As a matter of fact, most of the stabilization results for Navier–Stokes equations can be formulated and proven for control systems in Hilbert spaces governed by so-called abstract parabolic systems to be defined below.

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 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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

References

  1. Barbu V (2003) Feedback stabilization of Navier–Stokes equations. ESAIM COCV 9:197–206

    Article  MathSciNet  MATH  Google Scholar 

  2. Barbu V (2009) The internal stabilization by noise of the linearized Navier–Stokes equation. ESAIM COCV (online)

    Google Scholar 

  3. Barbu V, Triggiani R (2004) Internal stabilization of Navier–Stokes equations with finite dimensional controllers. Indiana Univ. Math. J. 53:1443–1494

    Article  MathSciNet  MATH  Google Scholar 

  4. Barbu V, Triggiani R, Lasiecka I (2006) Abstract settings for tangential boundary stabilization of Navier–Stokes equations by high and low-gain feedback controllers. Nonlinear Anal. 64:2704–2746

    Article  MathSciNet  MATH  Google Scholar 

  5. Barbu V, Wang G (2003) Internal stabilization of semilinear parabolic systems. J. Math. Anal. Appl. 285:387–407

    Article  MathSciNet  MATH  Google Scholar 

  6. Barbu V, Wang G (2005) Feedback stabilization of periodic solutions to nonlinear parabolic-like evolution systems. Indiana Univ. Math. J. 54:1521–1546

    Article  MathSciNet  MATH  Google Scholar 

  7. Bensoussan A, Da Prato G, Delfour M (1992) Representation and Control of Infinite Dimensional Systems. Birkhäuser, Boston, Basel, Berlin

    MATH  Google Scholar 

  8. Henry D (1981) Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics 840. Springer, Berlin, Heidelberg, New York

    MATH  Google Scholar 

  9. Kato T (1966) Perturbation Theory of Linear Operators. Springer, Berlin, Heidelberg, New York

    Book  Google Scholar 

  10. Lasiecka I, Triggiani R (2000) Control Theory for Partial Differential Equations: Continuous and Approximation Theory. Cambridge University Press, Cambridge

    Book  Google Scholar 

  11. Pazy A (1983) Semigroups of Linear Operators. Springer, Berlin, Heidelberg, New York

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Viorel Barbu .

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag London Limited

About this chapter

Cite this chapter

Barbu, V. (2011). Stabilization of Abstract Parabolic Systems. In: Stabilization of Navier–Stokes Flows. Communications and Control Engineering. Springer, London. https://doi.org/10.1007/978-0-85729-043-4_2

Download citation

  • DOI: https://doi.org/10.1007/978-0-85729-043-4_2

  • Publisher Name: Springer, London

  • Print ISBN: 978-0-85729-042-7

  • Online ISBN: 978-0-85729-043-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics