Skip to main content
SpringerLink
Log in

On the Stabilization of the Bresse Beam with Kelvin–Voigt Damping

  • Published:
Applied Mathematics & Optimization Aims and scope Submit manuscript

Abstract

The aim of this paper is to study the theoretical and numerical stability of the Bresse system in one-dimensional bounded domain with viscoelastic Kelvin–Voigt damping. We first showed the well posedness of the system. Then stability is obtained by applying the multiplicative techniques. Later a numerical scheme is proposed and analyzed. Finally a priori error estimate is established.

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

Fig. 1

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Afilal, M., Guesmia, A., Soufyane, A.: New stability results for a linear thermoelastic bresse system with second sound. Appl. Math. Optim. (2019). https://doi.org/10.1007/s00245-019-09560-7

  2. Ahn, J., Stewart, D.E.: A viscoelastic Timoshenko beam with dynamic frictionless impact. Discret. Contin. Dyn. Syst. Ser. B 12(1), 122 (2009)

    MathSciNet  MATH  Google Scholar 

  3. Alabau Boussouira, F., Muñoz Rivera, J.E., Almeida Jr., D.S.: Stability to weak dissipative Bresse system. J. Math. Anal. Appl. 374(2), 481–498 (2011)

    Article  MathSciNet  Google Scholar 

  4. Bernardi, C., Copetti, M.: Discretization of a nonlinear dynamic thermoviscoelastic Timoshenko beam model. Z. Angew. Math. Mech. 97, 532549 (2017)

    Article  MathSciNet  Google Scholar 

  5. Bresse, J.A.C.: Cours de Mécanique Appliquée. Mallet-Bachelier, Paris (1859)

    MATH  Google Scholar 

  6. Chen, S., Liu, K., Liu, Z.: Spectrum and stability of elastic systems with global or local Kelvin-Voigt damping. SIAM J. Appl. Math. 59, 651–668 (1998)

    Article  MathSciNet  Google Scholar 

  7. Chen, Z.J., Liu, W.J., Chen, D.Q.: General decay rates for a laminated beam with memory. Taiwanese J. Math. (2019). https://doi.org/10.11650/tjm/181109

  8. Copetti, M., Fernàndez, J.R.: A dynamic contact problem involving a Timoshenko beam model Appl. Numer. Math. 63, 117–128 (2013)

    Article  MathSciNet  Google Scholar 

  9. Crouzeix, M., Rappaz, J.: On Numerical Approximation in Bifurcation Theory. Masson, Paris (1990)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Fatori, L.H., Monteiro, R.N.: The optimal decay rate for a weak dissipative bresse system. App. Math. Lett. 25, 600–604 (2012)

    Article  MathSciNet  Google Scholar 

  12. Guesmia, A., Kafini, M.: Bresse system with infinite memories. Math. Methods Appl. Sci. 38(11), 2389–2402 (2015)

    Article  MathSciNet  Google Scholar 

  13. Guesmia, A., Kirane, M.: Uniform and weak stability of Bresse system with two infinite memories. Z. Angew. Math. Phys. 67, 139 (2016)

    Article  MathSciNet  Google Scholar 

  14. Guesmia, A., Massaoudi, S.A.: On the control of a viscoelastic damped Timoshenko-type system. Appl. Math. Comput. 26(2), 589597 (2008)

    MathSciNet  Google Scholar 

  15. Guesmia, A., Massaoudi, S.A.: General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping. Math. Methods Appl. Sci. 32(16), 21022122 (2009)

    Article  MathSciNet  Google Scholar 

  16. Haraux, A.: Semi-groupes lineéaires et eéquations d’évolution lineéaires peériodiques. Publication du Laboratoire d’Analyse Numeérique No. 78011, Universiteé Pierre et Marie Curie, Paris (1978)

  17. Keddi, A., Apalara, T., Messaoudi, S.: Exponential and polynomial decay in a thermoelastic-bresse system with second sound. Appl. Math. Optim. 77(2), 315–341 (2018)

    Article  MathSciNet  Google Scholar 

  18. Komornik, V.: Exact Controllability and Stabilization: The Multiplier Method. RAM Res. Appl. Math. Wiley, Chicester (1994)

    MATH  Google Scholar 

  19. Lagnese, J.: Boundary Stabilization of Thin Plates. SIAM Studies in Appl. Math. SIAM, Philadelphia (1989)

    Book  Google Scholar 

  20. Lagnese, J.E., Leugering, G., Schmidt, E.J.P.G.: Modelling of dynamic networks of thin thermoelastic beams. Math. Methods Appl. Sci. 16, 327–358 (1993)

    Article  MathSciNet  Google Scholar 

  21. Lagnese, J.E., Leugering, G., Schmidt, J.P.G.: Modelling Analysis and Control of Dynamic Elastic Multi-Link Structures, Systems Control Found. Appl. Birkhauser Boston Inc., Boston (1994)

    MATH  Google Scholar 

  22. Li, D., Zhang, C., Hu, Q., Zhang, H.: Energy decay rate of Bresse system with nonlinear localized damping. Br. J. Math. Comput. Sci. 4(12), 1665–1677 (2014)

    Article  Google Scholar 

  23. Li, G., Luan, Y., Liu, W.: Well-posedness and exponential stability of a thermoelastic-Bresse system with second sound and delay. Hacet. J. Math. Stat. (2019). https://doi.org/10.15672/hujms.568332

  24. Lions, J.L.: Contrôlabilité exacte et stabilisation de systèmes distribués, vol. 1. Masson, Paris (1988)

    MATH  Google Scholar 

  25. Lions, J.L.: Contrôlabilité exacte et stabilisation de systèmes distribués, vol. 2. Masson, Paris (1988)

    MATH  Google Scholar 

  26. Liu, K., Liu, Z.: Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping. SIAM J. Cont. Optim 36(3), 1086–1098 (1998)

    Article  MathSciNet  Google Scholar 

  27. Liu, Z., Rao, B.: Energy decay of the thermoelastic Bresse system. Z. Angew. Math. Phys. 60(1), 54–69 (2009)

    Article  MathSciNet  Google Scholar 

  28. Liu, W.J., Zhao, W.F.: Stabilization of a thermoelastic laminated beam with past history. Appl. Math. Optim. 80(1), 103–133 (2019)

    Article  MathSciNet  Google Scholar 

  29. Massaoudi, S., Hashem Hassan, J.: Neu general decay results in a finite-memory Bresse system. Commun. Pure Appl. Anal. 18(4), 1637–1662 (2019)

    Article  MathSciNet  Google Scholar 

  30. Rincon, M., Copetti, M.: Numerical analysis for a locally damped wave equation. J. Appl. Anal. Comput. 3, 169–182 (2013)

    MathSciNet  MATH  Google Scholar 

  31. Tatar, N.: Stabilization of a viscoelastic Timoshenko beam. Appl. Anal. 92(1), 27–43 (2013)

    Article  MathSciNet  Google Scholar 

  32. Wehbe, A., Youssef, W.: Stabilization of the uniform Timoshenko beam by one locally distributed feedback. Appl. Anal. 7, 1067–1078 (2009)

    Article  MathSciNet  Google Scholar 

  33. Wehbe, A., Youssef, W.: Exponential and polynomial stability of an elastic bresse system with two locally distributed feedbacks. J. Math. Phys. 51, 1–17 (2010)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Faculty of Science, Beirut Arab University, Debbieh, Lebanon

    Toufic El Arwadi

  2. Faculty of Sciences, Lebanese University, Hadath, Beirut, Lebanon

    Wael Youssef

Authors
  1. Toufic El Arwadi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wael Youssef
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Toufic El Arwadi.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

El Arwadi, T., Youssef, W. On the Stabilization of the Bresse Beam with Kelvin–Voigt Damping. Appl Math Optim 83, 1831–1857 (2021). https://doi.org/10.1007/s00245-019-09611-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-019-09611-z

Keywords

Search

Navigation