Exponential Stability for a Nonlinear Timoshenko System with Distributed Delay

Main Article Content

Lamine Bouzettouta
Fahima Hebhoub
Karima Ghennam
Sabrina Benferdi


This paper is concerned with a nonlinear Timoshenko system modeling clamped thin elastic beams with distributed delay time. The distributed delay is defined on feedback term associated to the equation for rotation angle. Under suitable assumptions on the data, we establish the exponential stability of the system under the usual equal wave speeds assumption.

Article Details


  1. T.A. Apalara, Well-posedness and exponential stability for a linear damped Timoshenko system with second sound and internal distributed delay. Electron. J. Differ. Equ. 2014 (2014), 254.
  2. T.A. Apalara, Uniform decay in weakly dissipative timoshenko system with internal distributed delay feedbacks, Acta Math. Sci. 36 (2016), 815-830.
  3. L. Bouzettouta, D. Abdelhak, Exponential stabilization of the full von K ´arm ´an beam by a thermal effect and a frictional damping and distributed delay, J. Math. Phys. 60 (2019), 041506.
  4. L. Bouzettouta, S. Zitouni, Kh. Zennir. and H. Sissaoui, Stability of Bresse system with internal distributed delay. J. Math. Comput. Sci. 7(1) (2017), 92-118.
  5. R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim. 24(1) (1986), 152-156.
  6. B.W. Feng and M. L. Pelicer, Global existence and exponential stability for a nonlinear Timoshenko system with delay. Bound. Value Probl. 2015 (2015), Article ID 206.
  7. H.E. Khochemane, A. Djebabla, S. Zitouni, L. Bouzettouta, Well-posedness and general decay of a nonlinear damping porous-elastic system with infinite memory, J. Math. Phys. 61 (2020), 021505.
  8. H.E. Khochemane, L. Bouzettouta, A. Guerouah, Exponential decay and well-posedness for a one-dimensional porouselastic system with distributed delay, Appl. Anal. (2019), 1-15. https://doi.org/10.1080/00036811.2019.1703958.
  9. H. E. Khochemane , S. Zitouni and L. Bouzettouta, Stability result for a nonlinear damping porous-elastic system with delay term, Nonlinear studies, 27(2) (2020), 1-17.
  10. S. A. Messaoudi and B. Said-Houari, Energy decay in a Timoshenko-type system for thermoelasticity of type III with distributed delay and past history, Electron. J. Differ. Equ. 2018 (2018), 75.
  11. M.I. Mustafa, A uniform stability result for thermoelasticity of type III with boundary distributed delay, J. Math. Anal. Appl. 415 (2014), 148-158.
  12. S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Control Optim. 45(5) (2006), 1561-1585.
  13. S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay. Electron. J. Differ. Equ. 2011 (2011), 41.
  14. C. Pignotti, A note on stabilization of locally damped wave equations with time delay, Syst. Control Lett. 61 (2012), 92-97.
  15. S. P. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Phil. Mag. Ser. 6(41) (1921), 744-746.
  16. S. Zitouni, L. Bouzettouta, Kh. Zennir and D. Ouchenane, Exponential decay of thermo-elastic Bresse system with distributed delay term, Hacettepe J. Math. Stat. 47(5) (2018), 1216-1230.