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A general stability result in a memory-type Timoshenko system
1. | King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics, Dhahran 31261, Saudi Arabia, Saudi Arabia |
References:
[1] |
F. Alabau-Boussouira and P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Acad. Sci. Paris, Ser. I, 347 (2009), 867-872. |
[2] |
F. Alabau-Bousosuira, Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control, Nonl. Differ. Eqns. Appl., 14 (2007), 643-669.
doi: 10.1007/s00030-007-5033-0. |
[3] |
F. Ammar-Khodja, A. Benabdallah, J. E. Muñoz Rivera and R. Racke, Energy decay for Timoshenko systems of memory type, J. Differ. Eqns, 194 (2003), 82-115.
doi: 10.1016/S0022-0396(03)00185-2. |
[4] |
V. I. Arnold, "Mathematical Methods of Classical Mechanics," Springer-Verlag, New York, 1989. |
[5] |
H. D. Fernández Sare and J. E. Muñoz Rivera, Stability of Timoshenko systems with past history, J. Math. Anal. Appl., 339 (2008), 482-502.
doi: 10.1016/j.jmaa.2007.07.012. |
[6] |
H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier's law, Arch. Rational Mech. Anal., 194 (2009), 221-251. |
[7] |
A. Guesmia and S. A. Messaoudi, On the control of solutions of a viscoelastic equation, Appl. Math. Comput., 206 (2008), 589-597.
doi: 10.1016/j.amc.2008.05.122. |
[8] |
A. Guesmia and S. A. Messaoudi, General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping, Math. Meth. Appl. Sci., 32 (2009), 2102-2122.
doi: 10.1002/mma.1125. |
[9] |
K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system, Math. models Meth. Appl. Sci., 18 (2008), 647-667.
doi: 10.1142/S0218202508002930. |
[10] |
J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam, SIAM J. Control Optim., 25 (1987), 1417-1429.
doi: 10.1137/0325078. |
[11] |
Y. Liu and S. Kawashima, Decay property for the Timoshenko system with memory-type dissipation, to be published in Math. models and Meth. Appl. Sci., 22 (2012), DOI No: 10.1142/S0218202511500126.
doi: 10.1142/S0218202511500126. |
[12] |
S. A. Messaoudi and M. I. Mustafa, On the internal and boundary stabilization of Timoshenko beams, Nonlinear Differential Eqns. Appl., 15 (2008), 655-671.
doi: 10.1007/s00030-008-7075-3. |
[13] |
S. A. Messaoudi and B. Said-Houari, Energy decay in a Timoshenko-type system of thermoelasticity of type III, J. Math. Anal. Appl., 348 (2008), 298-307
doi: 10.1016/j.jmaa.2008.07.036. |
[14] |
S. A. Messaoudi and B. Said -Houari, Energy decay in a Timoshenko-type system with history in thermoelasticity of type III, Adv. Differ. Eqns, 14 (2009), 375-400. |
[15] |
S. A. Messaoudi and M. I. Mustafa, On the stabilization of the Timoshenko system by a weak nonlinear dissipation, Math. Meth. Appl. Sci., 32 (2009), 454-469.
doi: 10.1002/mma.1047. |
[16] |
S. A. Messaoudi, M. Pokojovy and B. Said-Houari, Nonlinear Damped Timoshenko systems with second sound: Global existence and exponential stability, Math. Meth. Appl. Sci., 32 (2009), 505-534.
doi: 10.1002/mma.1049. |
[17] |
S. A. Messaoudi and B. Said -Houari, Uniform decay in a Timoshenko -type system with past history, J. Math. Anal. Appl., 360 (2009), 459-475.
doi: 10.1016/j.jmaa.2009.06.064. |
[18] |
S. A. Messaoudi and M. I. Mustafa, A stability result in a memory-type Timoshenko system, Dyn. Syst. Appl., 18 (2009), 457-468. |
[19] |
J. E. Muñoz Rivera and R. Racke, Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability, J. Math. Anal. Appl., 276 (2002), 248-276.
doi: 10.1016/S0022-247X(02)00436-5. |
[20] |
J. E. Muñoz Rivera and R. Racke, Global stability for damped Timoshenko systems, Discrete Contin. Dyn. syst., 9 (2003), 1625-1639. |
[21] |
J. E. Muñoz Rivera and R. Racke, Timoshenko systems with indefinite damping, J. Math. Anal. Appl., 341 (2008), 1068-1083.
doi: 10.1016/j.jmaa.2007.11.012. |
[22] |
M. I. Mustafa and S. A. Messaoudi, General energy decay rates for a weakly damped Timoshenko system, J. Dyn. Control Syst., 16 (2010), 211-226
doi: 10.1007/s10883-010-9090-z. |
[23] |
C. A. Raposo, J. Ferreira, M. L. Santos and N. N. O. Castro, Exponential stability for the Timoshenko system with two weak dampings, Appl. Math. Lett., 18 (2005), 535-542.
doi: 10.1016/j.aml.2004.03.017. |
[24] |
A. Soufyane and A. Wehbe, Uniform stabilization for the Timoshenko beam by a locally distributed damping, Electron. J. Differ. Eqns, 29 (2003), 1-14. |
[25] |
S. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philosophical Magazine, 41 (1921), 744-746.
doi: 10.1080/14786442108636264. |
show all references
References:
[1] |
F. Alabau-Boussouira and P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Acad. Sci. Paris, Ser. I, 347 (2009), 867-872. |
[2] |
F. Alabau-Bousosuira, Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control, Nonl. Differ. Eqns. Appl., 14 (2007), 643-669.
doi: 10.1007/s00030-007-5033-0. |
[3] |
F. Ammar-Khodja, A. Benabdallah, J. E. Muñoz Rivera and R. Racke, Energy decay for Timoshenko systems of memory type, J. Differ. Eqns, 194 (2003), 82-115.
doi: 10.1016/S0022-0396(03)00185-2. |
[4] |
V. I. Arnold, "Mathematical Methods of Classical Mechanics," Springer-Verlag, New York, 1989. |
[5] |
H. D. Fernández Sare and J. E. Muñoz Rivera, Stability of Timoshenko systems with past history, J. Math. Anal. Appl., 339 (2008), 482-502.
doi: 10.1016/j.jmaa.2007.07.012. |
[6] |
H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier's law, Arch. Rational Mech. Anal., 194 (2009), 221-251. |
[7] |
A. Guesmia and S. A. Messaoudi, On the control of solutions of a viscoelastic equation, Appl. Math. Comput., 206 (2008), 589-597.
doi: 10.1016/j.amc.2008.05.122. |
[8] |
A. Guesmia and S. A. Messaoudi, General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping, Math. Meth. Appl. Sci., 32 (2009), 2102-2122.
doi: 10.1002/mma.1125. |
[9] |
K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system, Math. models Meth. Appl. Sci., 18 (2008), 647-667.
doi: 10.1142/S0218202508002930. |
[10] |
J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam, SIAM J. Control Optim., 25 (1987), 1417-1429.
doi: 10.1137/0325078. |
[11] |
Y. Liu and S. Kawashima, Decay property for the Timoshenko system with memory-type dissipation, to be published in Math. models and Meth. Appl. Sci., 22 (2012), DOI No: 10.1142/S0218202511500126.
doi: 10.1142/S0218202511500126. |
[12] |
S. A. Messaoudi and M. I. Mustafa, On the internal and boundary stabilization of Timoshenko beams, Nonlinear Differential Eqns. Appl., 15 (2008), 655-671.
doi: 10.1007/s00030-008-7075-3. |
[13] |
S. A. Messaoudi and B. Said-Houari, Energy decay in a Timoshenko-type system of thermoelasticity of type III, J. Math. Anal. Appl., 348 (2008), 298-307
doi: 10.1016/j.jmaa.2008.07.036. |
[14] |
S. A. Messaoudi and B. Said -Houari, Energy decay in a Timoshenko-type system with history in thermoelasticity of type III, Adv. Differ. Eqns, 14 (2009), 375-400. |
[15] |
S. A. Messaoudi and M. I. Mustafa, On the stabilization of the Timoshenko system by a weak nonlinear dissipation, Math. Meth. Appl. Sci., 32 (2009), 454-469.
doi: 10.1002/mma.1047. |
[16] |
S. A. Messaoudi, M. Pokojovy and B. Said-Houari, Nonlinear Damped Timoshenko systems with second sound: Global existence and exponential stability, Math. Meth. Appl. Sci., 32 (2009), 505-534.
doi: 10.1002/mma.1049. |
[17] |
S. A. Messaoudi and B. Said -Houari, Uniform decay in a Timoshenko -type system with past history, J. Math. Anal. Appl., 360 (2009), 459-475.
doi: 10.1016/j.jmaa.2009.06.064. |
[18] |
S. A. Messaoudi and M. I. Mustafa, A stability result in a memory-type Timoshenko system, Dyn. Syst. Appl., 18 (2009), 457-468. |
[19] |
J. E. Muñoz Rivera and R. Racke, Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability, J. Math. Anal. Appl., 276 (2002), 248-276.
doi: 10.1016/S0022-247X(02)00436-5. |
[20] |
J. E. Muñoz Rivera and R. Racke, Global stability for damped Timoshenko systems, Discrete Contin. Dyn. syst., 9 (2003), 1625-1639. |
[21] |
J. E. Muñoz Rivera and R. Racke, Timoshenko systems with indefinite damping, J. Math. Anal. Appl., 341 (2008), 1068-1083.
doi: 10.1016/j.jmaa.2007.11.012. |
[22] |
M. I. Mustafa and S. A. Messaoudi, General energy decay rates for a weakly damped Timoshenko system, J. Dyn. Control Syst., 16 (2010), 211-226
doi: 10.1007/s10883-010-9090-z. |
[23] |
C. A. Raposo, J. Ferreira, M. L. Santos and N. N. O. Castro, Exponential stability for the Timoshenko system with two weak dampings, Appl. Math. Lett., 18 (2005), 535-542.
doi: 10.1016/j.aml.2004.03.017. |
[24] |
A. Soufyane and A. Wehbe, Uniform stabilization for the Timoshenko beam by a locally distributed damping, Electron. J. Differ. Eqns, 29 (2003), 1-14. |
[25] |
S. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philosophical Magazine, 41 (1921), 744-746.
doi: 10.1080/14786442108636264. |
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