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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 |
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. |
show all references
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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