Skip to main content
SpringerLink
Log in

Uniform Exponential Decay for Viscous Damped Systems*

  • Chapter
  • First Online:
Book cover Advances in Phase Space Analysis of Partial Differential Equations

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 78))

Summary

We consider a class of viscous damped vibrating systems. We prove that, under the assumption that the damping term ensures the exponential decay for the corresponding inviscid system, then the exponential decay rate is uniform for the viscous one, regardless what the value of the viscosity parameter is. Our method is mainly based on a decoupling argument of low and high frequencies. Low frequencies can be dealt with because of the effectiveness of the damping term in the inviscid case while the dissipativity of the viscous term guarantees the decay of the high-frequency components. This method is inspired in previous work by the authors on time-discretization schemes for damped systems in which a numerical viscosity term needs to be added to ensure the uniform exponential decay with respect to the time-step parameter.

2000 AMS Subject Classification: Primary: 35B35, 93D15, Secondary: 74D05, 35L90

* This work has been partially supported by grant MTM 2008-03541 of the MICINN (Spain).

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

Access this chapter

Log in via an institution
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 PDF
  • Read on any device
  • Instant download
  • Own it forever
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. B. Allibert. Contrôle analytique de l’équation des ondes et de l’équation de Schrödinger sur des surfaces de revolution. Commun. Partial Differential Equations, 23(9–10):1493–1556, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  2. C. Bardos, G. Lebeau, and J. Rauch. Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim., 30(5):1024–1065, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  3. P. Bechouche and A. Jüngel. Inviscid limits of the complex Ginzburg-Landau equation. Commun. Math. Phys., 214(1):201–226, 2000.

    Article  MATH  Google Scholar 

  4. N. Burq. Contrôle de l’équation des plaques en présence d’obstacles strictement convexes. Mém. Soc. Math. Fr. (N.S.), (55):126, 1993.

    Google Scholar 

  5. N. Burq and P. Gérard. Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes. C. R. Acad. Sci. Paris Sér. I Math., 325(7):749–752, 1997.

    MATH  Google Scholar 

  6. S. Cox and E. Zuazua. The rate at which energy decays in a damped string. Commun. Partial Differential Equations, 19(1-2):213–243, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  7. S. Cox and E. Zuazua. The rate at which energy decays in a string damped at one end. Indiana Univ. Math. J., 44(2):545–573, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  8. S. Ervedoza, C. Zheng, and E. Zuazua. On the observability of time-discrete conservative linear systems. J. Funct. Anal., 254(12):3037–3078, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  9. S. Ervedoza and E. Zuazua. Perfectly matched layers in 1-d: Energy decay for continuous and semi-discrete waves. Numer. Math., 109(4):597–634, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  10. S. Ervedoza and E. Zuazua. Uniformly exponentially stable approximations for a class of damped systems. J. Math. Pures Appl., 91:20–48, 2009.

    Google Scholar 

  11. A. Haraux. Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Port. Math., 46(3):245–258, 1989.

    MATH  MathSciNet  Google Scholar 

  12. L. I. Ignat and E. Zuazua. Dispersive properties of a viscous numerical scheme for the Schrödinger equation. C. R. Math. Acad. Sci. Paris, 340(7):529–534, 2005.

    MATH  MathSciNet  Google Scholar 

  13. S. Jaffard. Contrôle interne exact des vibrations d’une plaque rectangulaire. Port. Math., 47(4):423–429, 1990.

    MATH  MathSciNet  Google Scholar 

  14. G. Lebeau. Contrôle de l’équation de Schrödinger. J. Math. Pures Appl. (9), 71(3):267–291, 1992.

    MATH  MathSciNet  Google Scholar 

  15. G. Lebeau. Équations des ondes amorties. Séminaire sur les Équations aux Dérivées Partielles, 1993–1994, École Polytech., 1994.

    Google Scholar 

  16. J.-L. Lions. Contrôlabilité exacte, Stabilisation et Perturbations de Systèmes Distribués. Tome 1. Contrôlabilité exacte, volume RMA 8. Masson, 1988.

    Google Scholar 

  17. S. Machihara and Y. Nakamura. The inviscid limit for the complex Ginzburg-Landau equation. J. Math. Anal. Appl., 281(2):552–564, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  18. A. Münch and A. F. Pazoto. Uniform stabilization of a viscous numerical approximation for a locally damped wave equation. ESAIM Control Optim. Calc. Var., 13(2):265–293 (electronic), 2007.

    Article  MATH  MathSciNet  Google Scholar 

  19. K. Ramdani, T. Takahashi, and M. Tucsnak. Uniformly exponentially stable approximations for a class of second order evolution equations—application to LQR problems. ESAIM Control Optim. Calc. Var., 13(3):503–527, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  20. L. R. Tcheugoué Tebou and E. Zuazua. Uniform boundary stabilization of the finite difference space discretization of the 1-d wave equation. Adv. Comput. Math., 26(1-3):337–365, 2007.

    Article  MathSciNet  Google Scholar 

  21. L.R. Tcheugoué Tébou and E. Zuazua. Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity. Numer. Math., 95(3):563–598, 2003.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques de Versailles, Université de Versailles Saint-Quentin-en-Yvelines, avenue des États Unis, 78035, Versailles, France

    Sylvain Ervedoza

  2. Enrique Zuazua BASQUE CENTER for APPLIED MATHEMATICS, Bizkaia Technology Park, Building 500, E-48160, Derio-Basque, Country-Spain

    Enrique Zuazua

Authors
  1. Sylvain Ervedoza
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Enrique Zuazua
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sylvain Ervedoza .

Editor information

Editors and Affiliations

  1. Dipto. Matematica, Università di Bologna, Piazza di Porta San Donato 5, Bologna, 40126, Italy

    Antonio Bove

  2. Dipto. Scienze Matematiche, Università di Trieste, via A. Valerio 12/1, Trieste, 34127, Italy

    Daniele Del Santo

  3. Dipto. Matematica, Università Pisa, Largo Bruno Pontecorvo 5, Pisa, 56127, Italy

    M.K. Venkatesha Murthy

Additional information

Dedicated to Ferruccio Colombini with friendship

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Birkhäuser Boston

About this chapter

Cite this chapter

Ervedoza, S., Zuazua, E. (2009). Uniform Exponential Decay for Viscous Damped Systems*. In: Bove, A., Del Santo, D., Murthy, M. (eds) Advances in Phase Space Analysis of Partial Differential Equations. Progress in Nonlinear Differential Equations and Their Applications, vol 78. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4861-9_6

Download citation

Publish with us

Policies and ethics

Search

Navigation