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March  2011, 10(2): 667-686. doi: 10.3934/cpaa.2011.10.667

## Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks

 1 Mathématiques, Image et Applications Pôle Sciences et Technologies, Université de la Rochelle, France 2 Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany 3 United Arab Emirates University, P.O. Box 17551 Al Ain, United Arab Emirates

Received  April 2010 Revised  October 2010 Published  December 2010

We study the exponential stability of the Timoshenko beam system by interior time-dependent delay term feedbacks. The beam is clamped at the two hand points subject to two internal feedbacks: one with a time-varying delay and the other without delay. Using the variable norm technique of Kato, it is proved that the system is well-posed whenever an hypothesis between the weight of the delay term in the feedback, the weight of the term without delay and the wave speeds. By introducing an appropriate Lyapunov functional the exponential stability of the system is proved. Under the imposed constrain on the weights of the feedbacks and the wave speeds, the exponential decay of the energy is established via a suitable Lyapunov functional.
Citation: Mokhtar Kirane, Belkacem Said-Houari, Mohamed Naim Anwar. Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks. Communications on Pure and Applied Analysis, 2011, 10 (2) : 667-686. doi: 10.3934/cpaa.2011.10.667
##### References:
 [1] C. Abdallah, P. Dorato, J. Benitez-Read and R. Byrne, Delayed positive feedback can stabilize oscillatory system, ACC. San Francisco, (1993), 3106-3107. [2] 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 (1986), 152-156. doi: doi:10.1137/0324007. [3] T. Kato, Linear and quasilinear equations of evolution of hyperbolic type, C.I.M.E., II (1976), 125-191. [4] J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam, SIAM J. Control. Optim., 25 (1987), 1417-1429. doi: doi:10.1137/0325078. [5] 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 (2006), 1561-1585. doi: doi:10.1137/060648891. [6] S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Diff. Int. Equs., 21 (2008), 935-958. [7] S. Nicaise, C. Pignotti and J. Valein, Exponential stability of the wave equation with boundary time-varying delay,, submitted., (). [8] S. Nicaise and J. Valein, Stabilization of second order evolution equations with unbounded feedback with delay, ESAIM Control. Optim. Calc. Var., 16 (2010), 420-456. doi: doi:10.1051/cocv/2009007. [9] S. Nicaise, J. Valein and E. Fridman, Stabilization of the heat and the wave equations with boundary time-varying delays, DCDS-S., 2 (2009), 559-581. doi: doi:10.3934/dcdss.2009.2.559. [10] C. A. Raposo, J. Ferreira, M. L. Santos and N. N. O. Castro, Expoenetial stability for the Timoshenko system with two weak dampings, Appl. Math. Letters, 18 (2005), 535-541. doi: doi:10.1016/j.aml.2004.03.017. [11] J. E. Munoz Rivera and R. Racke, Timoshenko systems with indefinite damping, J. Math. Anal. Appl., 341 (2008), 1068-1083. doi: doi:10.1016/j.jmaa.2007.11.012. [12] B. Said-Houari and Y. Laskri, A stability result of a timoshenko system with a delay term in the internal feedback,, Appl. Math. Comput., (). [13] D-H. Shi and D-X. Feng, Exponential decay of Timoshenko beam with locally distributed feedback, IMA J. Math. Cont. Inf., 18 (2001), 395-403. doi: doi:10.1093/imamci/18.3.395. [14] A. Soufyane and A. Wehbe, Exponential stability for the Timoshenko beam by a locally distributed damping, Electron. J. Differential Equations, 29 (2003), 1-14. [15] I. H. Suh and Z. Bien, Use of time delay action in the controller design, IEEE Trans. Automat. Control, 25 (1980), 600-603. doi: doi:10.1109/TAC.1980.1102347. [16] S. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismaticbars, Philisophical. Magazine, 41 (1921), 744-746. [17] C. Q. Xu, S. P. Yung and L. K. Li, Stabilization of the wave system with input delay in the boundary control, ESAIM: Control Optim. Calc. Var., 12 (2006), 770-785. doi: doi:10.1051/cocv:2006021.

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##### References:
 [1] C. Abdallah, P. Dorato, J. Benitez-Read and R. Byrne, Delayed positive feedback can stabilize oscillatory system, ACC. San Francisco, (1993), 3106-3107. [2] 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 (1986), 152-156. doi: doi:10.1137/0324007. [3] T. Kato, Linear and quasilinear equations of evolution of hyperbolic type, C.I.M.E., II (1976), 125-191. [4] J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam, SIAM J. Control. Optim., 25 (1987), 1417-1429. doi: doi:10.1137/0325078. [5] 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 (2006), 1561-1585. doi: doi:10.1137/060648891. [6] S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Diff. Int. Equs., 21 (2008), 935-958. [7] S. Nicaise, C. Pignotti and J. Valein, Exponential stability of the wave equation with boundary time-varying delay,, submitted., (). [8] S. Nicaise and J. Valein, Stabilization of second order evolution equations with unbounded feedback with delay, ESAIM Control. Optim. Calc. Var., 16 (2010), 420-456. doi: doi:10.1051/cocv/2009007. [9] S. Nicaise, J. Valein and E. Fridman, Stabilization of the heat and the wave equations with boundary time-varying delays, DCDS-S., 2 (2009), 559-581. doi: doi:10.3934/dcdss.2009.2.559. [10] C. A. Raposo, J. Ferreira, M. L. Santos and N. N. O. Castro, Expoenetial stability for the Timoshenko system with two weak dampings, Appl. Math. Letters, 18 (2005), 535-541. doi: doi:10.1016/j.aml.2004.03.017. [11] J. E. Munoz Rivera and R. Racke, Timoshenko systems with indefinite damping, J. Math. Anal. Appl., 341 (2008), 1068-1083. doi: doi:10.1016/j.jmaa.2007.11.012. [12] B. Said-Houari and Y. Laskri, A stability result of a timoshenko system with a delay term in the internal feedback,, Appl. Math. Comput., (). [13] D-H. Shi and D-X. Feng, Exponential decay of Timoshenko beam with locally distributed feedback, IMA J. Math. Cont. Inf., 18 (2001), 395-403. doi: doi:10.1093/imamci/18.3.395. [14] A. Soufyane and A. Wehbe, Exponential stability for the Timoshenko beam by a locally distributed damping, Electron. J. Differential Equations, 29 (2003), 1-14. [15] I. H. Suh and Z. Bien, Use of time delay action in the controller design, IEEE Trans. Automat. Control, 25 (1980), 600-603. doi: doi:10.1109/TAC.1980.1102347. [16] S. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismaticbars, Philisophical. Magazine, 41 (1921), 744-746. [17] C. Q. Xu, S. P. Yung and L. K. Li, Stabilization of the wave system with input delay in the boundary control, ESAIM: Control Optim. Calc. Var., 12 (2006), 770-785. doi: doi:10.1051/cocv:2006021.
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