Publication Cover
Applicable Analysis
An International Journal
Volume 93, 2014 - Issue 6
278
Views
19
CrossRef citations to date
0
Altmetric
Articles

Energy decay rates for a Timoshenko-type system of thermoelasticity of type III with constant delay

, &
Pages 1201-1216 | Received 03 Feb 2013, Accepted 04 Jul 2013, Published online: 26 Jul 2013

References

  • Timoshenko S. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos. Magn. 1921;41:744–746.
  • Kim JU, Renardy Y. Boundary control of the Timoshenko beam. SIAM J. Control Optim. 1987;25:1417–1429.
  • Messaoudi SA, Mustafa MI. On the internal and boundary stabilization of Timoshenko beams. Nonlinear Differen. Eqn. Appl. 2008;15:655–671.
  • Messaoudi SA, Mustafa MI. On the stabilization of the Timoshenko system by a weak nonlinear dissipation. Math. Meth. Appl. Sci. 2009;32:454–469.
  • Mustafa MI, Messaoudi SA. General energy decay rates for a weakly damped Timoshenko system. J. Dynam. Contr. Syst. 2010;16:211–226.
  • Raposo CA, Ferreira J, Santos ML, Castro NNO. Exponential stability for the Timoshenko system with two weak dampings. Appl. Math. Lett. 2005;18:535–541.
  • Muñoz Rivera JE, Racke R. Global stability for damped Timoshenko systems. Discrete Contin. Dyn. Syst. 2003;9:1625–1639.
  • Muñoz Rivera JE, Racke R. Timoshenko systems with indefinite damping. J. Math. Anal. Appl. 2008;341:1068–1083.
  • Soufyane A, Wehbe A. Uniform stabilization for the Timoshenko beam by a locally distributed damping. Electron. J. Differen. Eqn. 2003;2003:1–14.
  • Ammar-Khodja F, Benabdallah A, Muñoz Rivera JE, Racke R. Energy decay for Timoshenko systems of memory type. J. Differen. Eqn. 2003;194:82–115.
  • Fernández Sare HD, Muñoz Rivera JE. Stability of Timoshenko systems with past history. J. Math. Anal. Appl. 2008;339:482–502.
  • Guesmia A, Messaoudi SA. General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping. Math. Meth. Appl. Sci. 2009;32:2102–2122.
  • Guesmia A, Messaoudi SA. On the control of solutions of a viscoelastic equation. Appl. Math. Computat. 2008;206:589–597.
  • Messaoudi SA, Mustafa MI. A stability result in a memory-type Timoshenko system. Dyn. Syst. Appl. 2009;18:457–468.
  • Messaoudi SA, Said-Houari B. Uniform decay in a Timoshenko-type system with past history. J. Math. Anal. Appl. 2009;360:459–475.
  • Muñoz Rivera JE, Racke R. Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability. J. Math. Anal. Appl. 2002;276:248–276.
  • Green AE, Naghdi PM. A re-examination of the basic postulates of thermomechanics. Proc. Royal Soc. London A. 1991;432:171–194.
  • Green AE, Naghdi PM. On undamped heat waves in an elastic solid. J. Thermal Stress. 1992;15:253–264.
  • Green AE, Naghdi PM. Thermoelasticity without energy dissipation. J. Elasticity. 1993;31: 189–208.
  • Chandrasekharaiah DS. Hyperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev. 1998;51:705–729.
  • Messaoudi SA, Said-Houari B. Energy decay in a Timoshenko-type system of thermoelasticity of type III. JMAA. 2008;348:298–307.
  • Messaoudi SA, Said-Houari B. Energy decay in a Timoshenko-type system with history in thermoelasticity of type III. Adv. Differen. Eqn. 2009;4:375–400.
  • Kafini M. General energy decay in a Timoshenko-type system of thermoelasticity of type III with a viscoelastic damping. J. Math. Anal. Appl. 2011;375:523–537.
  • Suh IH, Bien Z. Use of time delay action in the controller design. IEEE Trans. Automat. Contr. 1980;25:600–603.
  • Datko R. Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM J. Contr. Optim. 1988;26:697–713.
  • Datko R, Lagnese J, Polis MP. An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J, Contr. Optim. 1986;24:152–156.
  • Racke R. Instability of coupled systems with delay. Commun. Pure Appl. Anal. 2012;11: 1753–1773.
  • Liu K. Locally distributed control and damping for the conservative systems. SIAM J. Contr. Optim. 1997;35:1574–1590.
  • Zuazua E. Exponential decay for the semi-linear wave equation with locally distributed damping. Commun. Partial Differen. Eqn. 1990;15:205–235.
  • Nicaise S, Pignotti C. Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Contr. Optim. 2006;45:1561–1585.
  • Ait Benhassi EM, Ammari K, Boulite S, Maniar L. Feedback stabilization of a class of evolution equations with delay. J. Evol. Eqn. 2009;9:103–121.
  • Mustafa MI. Uniform stability for thermoelastic systems with boundary time-varying delay. J. Math. Anal. Appl. 2011;383:490–498.
  • Nicaise S, Pignotti C, Valein J. Exponential stability of the wave equation with boundary time-varying delay. Discrete Contin. Dyn. Syst. Ser. S. 2011;4:693–722.
  • Nicaise S, Valein J, Fridman E. Stability of the heat and the wave equations with boundary time-varying delays. Discrete Contin. Dyn. Syst. 2009;2:559–581.
  • Said-Houari B, Laskri Y. A stability result of a Timoshenko system with a delay term in the internal feedback. Appl. Math. Computat. 2010;217:2857–2869.
  • Kirane M, Said-Houari B, Anwar MN. Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks. Commun. Pure Appl. Anal. 2011;10:667–686.
  • Nicaise S, Pignotti C. Stabilization of the wave equation with boundary or internal distributed delay. Differen. Int. Eqn. 2008;21:935–958.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.