Skip to main content
SpringerLink
Log in

Boundary Feedback Stabilization of Kirchhoff-Type Timoshenko System

  • Published:
Journal of Dynamical and Control Systems Aims and scope Submit manuscript

Abstract

In this work, the stabilization problem of the nonlinear vibrating Timoshenko systems of Kirchhoff-type with boundary control conditions is considered. By virtue of the multiplier method, the explicit energy decay rates for solutions of the system are established, depending on boundary control feedback. In the view of control, the result of this work implies that, by choosing suited feedback boundary controls, the Kirchhoff-type Timoshenko system can be achieved by various decay rates, not only exponential and polynomial.

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. Kirchhoff G. Vorlesungen über Mechanik. Teubner, Stuttgart 1883.

  2. Timoshenko S. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Phil Mag. 1921;41:744–746.

    Article  Google Scholar 

  3. Lagnese JE, Leugering G. Uniform stabilization of a nonlinear beam by nonlinear boundary feedback. J Differ Equ. 1991;91:355–388.

    Article  MATH  MathSciNet  Google Scholar 

  4. Araruna FD, Zuazua E. Controllability of the Kirchhoff system for beams as a limit of the Mindlin-Timoshenko system. SIAM J Control Optim. 2008;47(4):1909–1938.

    Article  MATH  MathSciNet  Google Scholar 

  5. Said-Houari B, Messaoudi SA, Guesmia A. General decay of solutions of a nonlinear system of viscoelastic wave equations. Nonlinear Differ Equ Appl. 2011;18:659–684.

    Article  MATH  MathSciNet  Google Scholar 

  6. Raposo CA, Ferreira J, Santos ML, Castro NNO. Exponential stability for the Timoshenko system with two weak dampings. Appl Math Let. 2005;18:535–541.

    Article  MATH  MathSciNet  Google Scholar 

  7. Soufyane A, Wehbe A. Uniform stabilization for the Timoshenko beam by a locally distributed damping. Electronic J Differ Equ. 2003;29:1–14.

    MathSciNet  Google Scholar 

  8. Muñoz Rivera JE, Racke R. Timoshenko systems with indefinite damping. J Math Anal Appl. 2008;341:1068–1083.

    Article  MATH  MathSciNet  Google Scholar 

  9. Ammar-Khodja F, Benabdallah A, Muñoz Rivera JE, Racke R. Energy decay for Timoshenko systems of memory type. J Differ Equ. 2003;194(1):82–115.

    Article  MATH  Google Scholar 

  10. Messaoudi SA, Mustafa MI. A stability result in a memory-type Timoshenko system. Dyn Syst Appl. 2009;18:457–468.

    MATH  MathSciNet  Google Scholar 

  11. Tatar N. Viscoelastic Timoshenko beams with occasionally constant relaxation functions. Appl Math Optim. 2012;66:123–145.

    Article  MATH  MathSciNet  Google Scholar 

  12. Wu YH, Xue XP. Decay rate estimates for the quasi-linear Timoshenko system with nonlinear control and damping terms. J Math Phys. 2011;52:093502.

    Article  MathSciNet  Google Scholar 

  13. Aassila M. Stabilization of a nonlinear Timoshenko beam. Z Angew Math Phys. 2002;53:747–768.

    Article  MATH  MathSciNet  Google Scholar 

  14. Muñoz Rivera JE, Racke R. Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability. J Math Anal Appl. 2002;276:248–276.

    Article  MATH  MathSciNet  Google Scholar 

  15. Messaoudi SA, Pokojovy M, Said-Houari B. Nonlinear damped Timoshenko systems with second sound global existence and exponential stability. Math Methods Appl Sci. 2009;32:505–534.

    Article  MATH  MathSciNet  Google Scholar 

  16. Messaoudi SA, Said-Houari B. Energy decay in a Timoshenko-type system of thermoelasticity of type III. J Math Anal Appl. 2008;348:298–307.

    Article  MATH  MathSciNet  Google Scholar 

  17. Sare HDF, Racke R. On the stability of damped Timoshenko systems Cattaneo versus Fourier law. Arch Rational Mech Anal. 2009;194:221–251.

    Article  MATH  Google Scholar 

  18. Djebabla A, Tatar N. Exponential stabilization of the Timoshenko system by a thermo-viscoelastic damping. J Dyn Control Syst. 2010;16(2):189–210.

    Article  MATH  MathSciNet  Google Scholar 

  19. Morgul O. Dynamic boundary control of a Euler-Bernoulli beam. IEEE Trans Autom Control. 1992;37(5):639–642.

    Article  MathSciNet  Google Scholar 

  20. Wang JJ, Li QH. Active vibration control methods of axially moving materials—a review. J Vib Control. 2004;10:475–491.

    Article  MATH  Google Scholar 

  21. Kim JU, Renardy Y. Boundary control of the Timoshenko beam. SIAM J Control Optim. 1987;25(6):1417–1429.

    Article  MATH  MathSciNet  Google Scholar 

  22. Yan QX. Boundary stabilization of Timoshenko beam. Syst Sci Math Sci. 2000;13(3):376–384.

    MATH  Google Scholar 

  23. Ammar-Khodja F, Kerbal S, Soufyane AE. Stabilization of the nonuniform Timoshenko beam. J Math Anal Appl. 2007;327(1):525–538.

    Article  MATH  MathSciNet  Google Scholar 

  24. Aouadi M, Soufyane A. Decay of the timoshenko beam with thermal effect and memory boundary conditions. J Dyn Control Syst. 2013;19(1):33–46.

    Article  MATH  MathSciNet  Google Scholar 

  25. Santos ML. Decay rates for solutions of a Timoshenko system with a memory condition at the boundary. Abstr Appl Anal. 2002;7:531–546.

    Article  MATH  MathSciNet  Google Scholar 

  26. Messaoudi SA, Soufyane A. Boundary stabilization of memory type in thermoelasticity of type III. Appl Anal. 2008;87:13–28.

    Article  MATH  MathSciNet  Google Scholar 

  27. Messaoudi SA, Mustafa M. On the internal and boundary stabilization of Timoshenko beams. Nonlinear Differ Equ Appl. 2008;15:655–671.

    Article  MATH  MathSciNet  Google Scholar 

  28. Kobayashi T, Sakamoto T. Adaptive stabilization of Kirchhoffs non-linear strings by boundary displacement feedback. Math Meth Appl Sci. 2007;30:1209–1221.

    Article  MATH  MathSciNet  Google Scholar 

  29. Shahruz SM. Boundary control of a non-linear axially moving string. Int J Robust Nonlinear Control. 2000;10:17–25.

    Article  MATH  MathSciNet  Google Scholar 

  30. Renardy M, Rogers R. An introduction to partial differential equations. New York: Springer Verlag; 1993.

    MATH  Google Scholar 

  31. Martinez P. A new method to obtain decay rate estimates for dissipative systems. ESAIM Control Optim Calc Var. 1999;4:419–444.

    Article  MATH  MathSciNet  Google Scholar 

  32. Washzu K. Variational methods in elasticity and plasticity. London-New York: Pergamon; 1968.

    Google Scholar 

Download references

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant 11271099.

Author information

Authors and Affiliations

  1. Department of Mathematics, Harbin University of Science and Technology, Harbin, 150080, People’s Republic of China

    Yuhu Wu

  2. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China

    Xiaoping Xue

Authors
  1. Yuhu Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaoping Xue
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuhu Wu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, Y., Xue, X. Boundary Feedback Stabilization of Kirchhoff-Type Timoshenko System. J Dyn Control Syst 20, 523–538 (2014). https://doi.org/10.1007/s10883-014-9229-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10883-014-9229-4

Keywords

Mathematics Subject Classification (2010)

Search

Navigation