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Ergodic geometry for non-elementary rank one manifolds
Exponential stabilization of a structure with interfacial slip
1. | University of Wollongong in Dubai, Dubai, United Arab Emirates |
2. | Department of Mathematics and Statistics, College of Sciences, King Fahd University of Petroleum and Minerals, P.O.Box. 5005, Dhahran 31261 |
References:
[1] |
Ammar-Khodja, A. Benabdallah and J. E. M. Rivera, Energy decay for Timoshenko system of memory type, J. Diff. Eqs., 194 (2003), 82-11.
doi: 10.1016/S0022-0396(03)00185-2. |
[2] |
C. F. Beards and I. M. A. Imam, The damping of plate vibration by interfacial slip between layers, Int. J. Mach. Tool. Des. Res., 18 (1978), 131-137.
doi: 10.1016/0020-7357(78)90004-5. |
[3] |
X.-G. Cao, D.-Y. Liu and G.-Q. Xu, Easy test for stability of laminated beams with structural damping and boundary feedback controls, J. Dynamical Control Syst., 13 (2007), 313-336.
doi: 10.1007/s10883-007-9022-8. |
[4] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. Ferreira, Existence and uniform decay for nonlinear viscoelastic equation with strong damping, Math. Meth. Appl. Sci., 24 (2001), 1043-1053.
doi: 10.1002/mma.250. |
[5] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. S. Prates Filho and J. A. Soriano, Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping, Diff. Integral Eqs., 14 (2001), 85-116. |
[6] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti and P. Martinez, General decay rate estimates for viscoelastic dissipative systems, Nonl. Anal.: T. M. A., 68 (2008), 177-193.
doi: 10.1016/j.na.2006.10.040. |
[7] |
M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42 (2003), 1310-1324.
doi: 10.1137/S0363012902408010. |
[8] |
M. De Lima Santos, Decay rates for solutions of a Timoshenko system with memory conditions at the boundary, Abstr. Appl. Anal., 7 (2002), 53-546.
doi: 10.1155/S1085337502204133. |
[9] |
X. S. Han and M. X. Wang, Global existence and uniform decay for a nonlinear viscoelastic equation with damping, Nonl. Anal.: T. M. A., 70 (2009), 3090-3098.
doi: 10.1016/j.na.2008.04.011. |
[10] |
S. W. Hansen and R. Spies, Structural damping in a laminated beam due to interfacial slip, J. Sound Vibration, 204 (1997), 183-202.
doi: 10.1006/jsvi.1996.0913. |
[11] |
Z. Liu and C. Pang, Exponential stability of a viscoelastic Timoshenko beam, Adv. Math. Sci. Appl., 8 (1998), 343-351. |
[12] |
A. Lo and N.-e. Tatar, Stabilization of a laminated beam with interfacial slip, Electron. J. Diff. Eqs., 129 (2015), 1-14. |
[13] |
M. Medjden and N.-e. Tatar, On the wave equation with a temporal nonlocal term, Dyn. Syst. Appl., 16 (2007), 665-672. |
[14] |
M. Medjden and N.-e. Tatar, Asymptotic behavior for a viscoelastic problem with not necessarily decreasing kernel, Appl. Math. Comput., 167 (2005), 1221-1235.
doi: 10.1016/j.amc.2004.08.035. |
[15] |
S. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl., 341 (2008), 1457-1467.
doi: 10.1016/j.jmaa.2007.11.048. |
[16] |
S. Messaoudi and M. I. Mustafa, A general result in a memory-type Timoshenko system, Comm. Pure Appl. Anal., (2013), 957-972.
doi: 10.3934/cpaa.2013.12.957. |
[17] |
S. 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] |
J. E. Munoz Rivera and F. P. Quispe Gomez, Existence and decay in non linear viscoelasticity, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 6 (2003), 1-37. |
[19] |
V. Pata, Exponential stability in linear viscoelasticity, Quart. Appl. Math., LXIV (2006), 499-513.
doi: 10.1090/S0033-569X-06-01010-4. |
[20] |
C. A. Rapaso, J. Ferreira, M. L. Santos and N. N. Castro, Exponential stabilization for the Timoshenko system with two weak dampings, Appl. Math. Lett., 18 (2005), 535-541.
doi: 10.1016/j.aml.2004.03.017. |
[21] |
J. E. M. Rivera and R. Racke, Mildly dissipative nonlinear Timoshenko systems: Global existence and exponential stability, J. Math. Anal. Appl., 276 (2002), 248-278.
doi: 10.1016/S0022-247X(02)00436-5. |
[22] |
J. E. M. Rivera and R. Racke, Global stability for damped Timoshenko systems, Discrete Contin. Dyn. Syst., 9 (2003), 1625-1639.
doi: 10.3934/dcds.2003.9.1625. |
[23] |
D. H. Shi and D. X. Feng, Exponential decay of Timoshenko beam with locally distributed feedback, IMA J. Math. Control Inform., 18 (2001), 395-403.
doi: 10.1093/imamci/18.3.395. |
[24] |
D. H. Shi, S. H. Hou and D. X. Feng, Feedback stabilization of a Timoshenko beam with an end mass, Int. J. Control, 69 (1998), 285-300.
doi: 10.1080/002071798222848. |
[25] |
A. Soufyane and Wehbe, Uniform stabilization for the Timoshenko beam by a locally distributed damping, Electron. J. Diff. Eqs., 29 (2003), 1-14. |
[26] |
N.-e. Tatar, Long time behavior for a viscoelastic problem with a positive definite kernel, Australian J. Math. Anal. Appl., 1 (2004), Article 5, 1-11. |
[27] |
N.-e. Tatar, Exponential decay for a viscoelastic problem with a singular problem, Zeit. Angew. Math. Phys., 60 (2009), 640-650.
doi: 10.1007/s00033-008-8030-1. |
[28] |
N.-e. Tatar, On a large class of kernels yielding exponential stability in viscoelasticity, Appl. Math. Comp., 215 (2009), 2298-2306.
doi: 10.1016/j.amc.2009.08.034. |
[29] |
N.-e. Tatar, Viscoelastic Timoshenko beams with occasionally constant relaxation functions, Appl. Math. Optim., 66 (2012), 123-145.
doi: 10.1007/s00245-012-9167-z. |
[30] |
N.-e. Tatar, Exponential decay for a viscoelastically damped Timoshenko beam, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 505-524.
doi: 10.1016/S0252-9602(13)60015-6. |
[31] |
N.-e. Tatar, Stabilization of a viscoelastic Timoshenko beam, Appl. Anal.: An International Journal, 92 (2013), 27-43.
doi: 10.1080/00036811.2011.587810. |
[32] |
J.-M. Wang, G.-Q. Xu and S.-P. Yung, Exponential stabilization of laminated beams with structural damping and boundary feedback controls, SIAM J. Control Optim., 44 (2005), 1575-1597.
doi: 10.1137/040610003. |
[33] |
G. Q. Xu, Feedback exponential stabilization of a Timoshenko beam with both ends free, Int. J. Control, 72 (2005), 286-297.
doi: 10.1080/00207170500095148. |
[34] |
Q. Yan et al., Boundary stabilization of nonuniform Timoshenko beam with a tipload, Chin. Ann. Math., 22 (2001), 485-494.
doi: 10.1142/S0252959901000450. |
show all references
References:
[1] |
Ammar-Khodja, A. Benabdallah and J. E. M. Rivera, Energy decay for Timoshenko system of memory type, J. Diff. Eqs., 194 (2003), 82-11.
doi: 10.1016/S0022-0396(03)00185-2. |
[2] |
C. F. Beards and I. M. A. Imam, The damping of plate vibration by interfacial slip between layers, Int. J. Mach. Tool. Des. Res., 18 (1978), 131-137.
doi: 10.1016/0020-7357(78)90004-5. |
[3] |
X.-G. Cao, D.-Y. Liu and G.-Q. Xu, Easy test for stability of laminated beams with structural damping and boundary feedback controls, J. Dynamical Control Syst., 13 (2007), 313-336.
doi: 10.1007/s10883-007-9022-8. |
[4] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. Ferreira, Existence and uniform decay for nonlinear viscoelastic equation with strong damping, Math. Meth. Appl. Sci., 24 (2001), 1043-1053.
doi: 10.1002/mma.250. |
[5] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. S. Prates Filho and J. A. Soriano, Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping, Diff. Integral Eqs., 14 (2001), 85-116. |
[6] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti and P. Martinez, General decay rate estimates for viscoelastic dissipative systems, Nonl. Anal.: T. M. A., 68 (2008), 177-193.
doi: 10.1016/j.na.2006.10.040. |
[7] |
M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42 (2003), 1310-1324.
doi: 10.1137/S0363012902408010. |
[8] |
M. De Lima Santos, Decay rates for solutions of a Timoshenko system with memory conditions at the boundary, Abstr. Appl. Anal., 7 (2002), 53-546.
doi: 10.1155/S1085337502204133. |
[9] |
X. S. Han and M. X. Wang, Global existence and uniform decay for a nonlinear viscoelastic equation with damping, Nonl. Anal.: T. M. A., 70 (2009), 3090-3098.
doi: 10.1016/j.na.2008.04.011. |
[10] |
S. W. Hansen and R. Spies, Structural damping in a laminated beam due to interfacial slip, J. Sound Vibration, 204 (1997), 183-202.
doi: 10.1006/jsvi.1996.0913. |
[11] |
Z. Liu and C. Pang, Exponential stability of a viscoelastic Timoshenko beam, Adv. Math. Sci. Appl., 8 (1998), 343-351. |
[12] |
A. Lo and N.-e. Tatar, Stabilization of a laminated beam with interfacial slip, Electron. J. Diff. Eqs., 129 (2015), 1-14. |
[13] |
M. Medjden and N.-e. Tatar, On the wave equation with a temporal nonlocal term, Dyn. Syst. Appl., 16 (2007), 665-672. |
[14] |
M. Medjden and N.-e. Tatar, Asymptotic behavior for a viscoelastic problem with not necessarily decreasing kernel, Appl. Math. Comput., 167 (2005), 1221-1235.
doi: 10.1016/j.amc.2004.08.035. |
[15] |
S. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl., 341 (2008), 1457-1467.
doi: 10.1016/j.jmaa.2007.11.048. |
[16] |
S. Messaoudi and M. I. Mustafa, A general result in a memory-type Timoshenko system, Comm. Pure Appl. Anal., (2013), 957-972.
doi: 10.3934/cpaa.2013.12.957. |
[17] |
S. 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] |
J. E. Munoz Rivera and F. P. Quispe Gomez, Existence and decay in non linear viscoelasticity, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 6 (2003), 1-37. |
[19] |
V. Pata, Exponential stability in linear viscoelasticity, Quart. Appl. Math., LXIV (2006), 499-513.
doi: 10.1090/S0033-569X-06-01010-4. |
[20] |
C. A. Rapaso, J. Ferreira, M. L. Santos and N. N. Castro, Exponential stabilization for the Timoshenko system with two weak dampings, Appl. Math. Lett., 18 (2005), 535-541.
doi: 10.1016/j.aml.2004.03.017. |
[21] |
J. E. M. Rivera and R. Racke, Mildly dissipative nonlinear Timoshenko systems: Global existence and exponential stability, J. Math. Anal. Appl., 276 (2002), 248-278.
doi: 10.1016/S0022-247X(02)00436-5. |
[22] |
J. E. M. Rivera and R. Racke, Global stability for damped Timoshenko systems, Discrete Contin. Dyn. Syst., 9 (2003), 1625-1639.
doi: 10.3934/dcds.2003.9.1625. |
[23] |
D. H. Shi and D. X. Feng, Exponential decay of Timoshenko beam with locally distributed feedback, IMA J. Math. Control Inform., 18 (2001), 395-403.
doi: 10.1093/imamci/18.3.395. |
[24] |
D. H. Shi, S. H. Hou and D. X. Feng, Feedback stabilization of a Timoshenko beam with an end mass, Int. J. Control, 69 (1998), 285-300.
doi: 10.1080/002071798222848. |
[25] |
A. Soufyane and Wehbe, Uniform stabilization for the Timoshenko beam by a locally distributed damping, Electron. J. Diff. Eqs., 29 (2003), 1-14. |
[26] |
N.-e. Tatar, Long time behavior for a viscoelastic problem with a positive definite kernel, Australian J. Math. Anal. Appl., 1 (2004), Article 5, 1-11. |
[27] |
N.-e. Tatar, Exponential decay for a viscoelastic problem with a singular problem, Zeit. Angew. Math. Phys., 60 (2009), 640-650.
doi: 10.1007/s00033-008-8030-1. |
[28] |
N.-e. Tatar, On a large class of kernels yielding exponential stability in viscoelasticity, Appl. Math. Comp., 215 (2009), 2298-2306.
doi: 10.1016/j.amc.2009.08.034. |
[29] |
N.-e. Tatar, Viscoelastic Timoshenko beams with occasionally constant relaxation functions, Appl. Math. Optim., 66 (2012), 123-145.
doi: 10.1007/s00245-012-9167-z. |
[30] |
N.-e. Tatar, Exponential decay for a viscoelastically damped Timoshenko beam, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 505-524.
doi: 10.1016/S0252-9602(13)60015-6. |
[31] |
N.-e. Tatar, Stabilization of a viscoelastic Timoshenko beam, Appl. Anal.: An International Journal, 92 (2013), 27-43.
doi: 10.1080/00036811.2011.587810. |
[32] |
J.-M. Wang, G.-Q. Xu and S.-P. Yung, Exponential stabilization of laminated beams with structural damping and boundary feedback controls, SIAM J. Control Optim., 44 (2005), 1575-1597.
doi: 10.1137/040610003. |
[33] |
G. Q. Xu, Feedback exponential stabilization of a Timoshenko beam with both ends free, Int. J. Control, 72 (2005), 286-297.
doi: 10.1080/00207170500095148. |
[34] |
Q. Yan et al., Boundary stabilization of nonuniform Timoshenko beam with a tipload, Chin. Ann. Math., 22 (2001), 485-494.
doi: 10.1142/S0252959901000450. |
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