American Institute of Mathematical Sciences

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November  2016, 36(11): 6285-6306. doi: 10.3934/dcds.2016073

Exponential stabilization of a structure with interfacial slip

Assane Lo 1, and  Nasser-eddine Tatar 2,

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

Received  May 2015 Revised  July 2016 Published  August 2016

Two exponential stabilization results are proved for a vibrating structure subject to an interfacial slip. More precisely, the structure consists of two identical beams of Timoshenko type and clamped together but allowing for a longitudinal movement between the layers. We will stabilize the system through a transverse friction and also through a viscoelastic damping.
Keywords: Timoshenko system, slip dynamic, vibration reduction, Exponential stabilization, multiplier technique..
Mathematics Subject Classification: 34B05, 34D05, 34H0.
Citation: Assane Lo, Nasser-eddine Tatar. Exponential stabilization of a structure with interfacial slip. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6285-6306. doi: 10.3934/dcds.2016073
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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