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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 & 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.   Google Scholar

[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.  doi: doi:10.1137/0324007.  Google Scholar

[3]

T. Kato, Linear and quasilinear equations of evolution of hyperbolic type,, C.I.M.E., (1976), 125.   Google Scholar

[4]

J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam,, SIAM J. Control. Optim., 25 (1987), 1417.  doi: doi:10.1137/0325078.  Google Scholar

[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.  doi: doi:10.1137/060648891.  Google Scholar

[6]

S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay,, Diff. Int. Equs., 21 (2008), 935.   Google Scholar

[7]

S. Nicaise, C. Pignotti and J. Valein, Exponential stability of the wave equation with boundary time-varying delay,, submitted., ().   Google Scholar

[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.  doi: doi:10.1051/cocv/2009007.  Google Scholar

[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.  doi: doi:10.3934/dcdss.2009.2.559.  Google Scholar

[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.  doi: doi:10.1016/j.aml.2004.03.017.  Google Scholar

[11]

J. E. Munoz Rivera and R. Racke, Timoshenko systems with indefinite damping,, J. Math. Anal. Appl., 341 (2008), 1068.  doi: doi:10.1016/j.jmaa.2007.11.012.  Google Scholar

[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., ().   Google Scholar

[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.  doi: doi:10.1093/imamci/18.3.395.  Google Scholar

[14]

A. Soufyane and A. Wehbe, Exponential stability for the Timoshenko beam by a locally distributed damping,, Electron. J. Differential Equations, 29 (2003), 1.   Google Scholar

[15]

I. H. Suh and Z. Bien, Use of time delay action in the controller design,, IEEE Trans. Automat. Control, 25 (1980), 600.  doi: doi:10.1109/TAC.1980.1102347.  Google Scholar

[16]

S. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismaticbars,, Philisophical. Magazine, 41 (1921), 744.   Google Scholar

[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.  doi: doi:10.1051/cocv:2006021.  Google Scholar

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.   Google Scholar

[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.  doi: doi:10.1137/0324007.  Google Scholar

[3]

T. Kato, Linear and quasilinear equations of evolution of hyperbolic type,, C.I.M.E., (1976), 125.   Google Scholar

[4]

J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam,, SIAM J. Control. Optim., 25 (1987), 1417.  doi: doi:10.1137/0325078.  Google Scholar

[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.  doi: doi:10.1137/060648891.  Google Scholar

[6]

S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay,, Diff. Int. Equs., 21 (2008), 935.   Google Scholar

[7]

S. Nicaise, C. Pignotti and J. Valein, Exponential stability of the wave equation with boundary time-varying delay,, submitted., ().   Google Scholar

[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.  doi: doi:10.1051/cocv/2009007.  Google Scholar

[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.  doi: doi:10.3934/dcdss.2009.2.559.  Google Scholar

[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.  doi: doi:10.1016/j.aml.2004.03.017.  Google Scholar

[11]

J. E. Munoz Rivera and R. Racke, Timoshenko systems with indefinite damping,, J. Math. Anal. Appl., 341 (2008), 1068.  doi: doi:10.1016/j.jmaa.2007.11.012.  Google Scholar

[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., ().   Google Scholar

[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.  doi: doi:10.1093/imamci/18.3.395.  Google Scholar

[14]

A. Soufyane and A. Wehbe, Exponential stability for the Timoshenko beam by a locally distributed damping,, Electron. J. Differential Equations, 29 (2003), 1.   Google Scholar

[15]

I. H. Suh and Z. Bien, Use of time delay action in the controller design,, IEEE Trans. Automat. Control, 25 (1980), 600.  doi: doi:10.1109/TAC.1980.1102347.  Google Scholar

[16]

S. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismaticbars,, Philisophical. Magazine, 41 (1921), 744.   Google Scholar

[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.  doi: doi:10.1051/cocv:2006021.  Google Scholar

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