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 & Continuous Dynamical Systems - A, 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.  doi: 10.1016/S0022-0396(03)00185-2.  Google Scholar

[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.  doi: 10.1016/0020-7357(78)90004-5.  Google Scholar

[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.  doi: 10.1007/s10883-007-9022-8.  Google Scholar

[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.  doi: 10.1002/mma.250.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.na.2006.10.040.  Google Scholar

[7]

M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation,, SIAM J. Control Optim., 42 (2003), 1310.  doi: 10.1137/S0363012902408010.  Google Scholar

[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.  doi: 10.1155/S1085337502204133.  Google Scholar

[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.  doi: 10.1016/j.na.2008.04.011.  Google Scholar

[10]

S. W. Hansen and R. Spies, Structural damping in a laminated beam due to interfacial slip,, J. Sound Vibration, 204 (1997), 183.  doi: 10.1006/jsvi.1996.0913.  Google Scholar

[11]

Z. Liu and C. Pang, Exponential stability of a viscoelastic Timoshenko beam,, Adv. Math. Sci. Appl., 8 (1998), 343.   Google Scholar

[12]

A. Lo and N.-e. Tatar, Stabilization of a laminated beam with interfacial slip,, Electron. J. Diff. Eqs., 129 (2015), 1.   Google Scholar

[13]

M. Medjden and N.-e. Tatar, On the wave equation with a temporal nonlocal term,, Dyn. Syst. Appl., 16 (2007), 665.   Google Scholar

[14]

M. Medjden and N.-e. Tatar, Asymptotic behavior for a viscoelastic problem with not necessarily decreasing kernel,, Appl. Math. Comput., 167 (2005), 1221.  doi: 10.1016/j.amc.2004.08.035.  Google Scholar

[15]

S. Messaoudi, General decay of solutions of a viscoelastic equation,, J. Math. Anal. Appl., 341 (2008), 1457.  doi: 10.1016/j.jmaa.2007.11.048.  Google Scholar

[16]

S. Messaoudi and M. I. Mustafa, A general result in a memory-type Timoshenko system,, Comm. Pure Appl. Anal., (2013), 957.  doi: 10.3934/cpaa.2013.12.957.  Google Scholar

[17]

S. Messaoudi and B. Said-Houari, Uniform decay in a Timoshenko-type system with past history,, J. Math. Anal. Appl., 360 (2009), 459.  doi: 10.1016/j.jmaa.2009.06.064.  Google Scholar

[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.   Google Scholar

[19]

V. Pata, Exponential stability in linear viscoelasticity,, Quart. Appl. Math., LXIV (2006), 499.  doi: 10.1090/S0033-569X-06-01010-4.  Google Scholar

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

[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.  doi: 10.1016/S0022-247X(02)00436-5.  Google Scholar

[22]

J. E. M. Rivera and R. Racke, Global stability for damped Timoshenko systems,, Discrete Contin. Dyn. Syst., 9 (2003), 1625.  doi: 10.3934/dcds.2003.9.1625.  Google Scholar

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

[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.  doi: 10.1080/002071798222848.  Google Scholar

[25]

A. Soufyane and Wehbe, Uniform stabilization for the Timoshenko beam by a locally distributed damping,, Electron. J. Diff. Eqs., 29 (2003), 1.   Google Scholar

[26]

N.-e. Tatar, Long time behavior for a viscoelastic problem with a positive definite kernel,, Australian J. Math. Anal. Appl., 1 (2004), 1.   Google Scholar

[27]

N.-e. Tatar, Exponential decay for a viscoelastic problem with a singular problem,, Zeit. Angew. Math. Phys., 60 (2009), 640.  doi: 10.1007/s00033-008-8030-1.  Google Scholar

[28]

N.-e. Tatar, On a large class of kernels yielding exponential stability in viscoelasticity,, Appl. Math. Comp., 215 (2009), 2298.  doi: 10.1016/j.amc.2009.08.034.  Google Scholar

[29]

N.-e. Tatar, Viscoelastic Timoshenko beams with occasionally constant relaxation functions,, Appl. Math. Optim., 66 (2012), 123.  doi: 10.1007/s00245-012-9167-z.  Google Scholar

[30]

N.-e. Tatar, Exponential decay for a viscoelastically damped Timoshenko beam,, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 505.  doi: 10.1016/S0252-9602(13)60015-6.  Google Scholar

[31]

N.-e. Tatar, Stabilization of a viscoelastic Timoshenko beam,, Appl. Anal.: An International Journal, 92 (2013), 27.  doi: 10.1080/00036811.2011.587810.  Google Scholar

[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.  doi: 10.1137/040610003.  Google Scholar

[33]

G. Q. Xu, Feedback exponential stabilization of a Timoshenko beam with both ends free,, Int. J. Control, 72 (2005), 286.  doi: 10.1080/00207170500095148.  Google Scholar

[34]

Q. Yan et al., Boundary stabilization of nonuniform Timoshenko beam with a tipload,, Chin. Ann. Math., 22 (2001), 485.  doi: 10.1142/S0252959901000450.  Google Scholar

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.  doi: 10.1016/S0022-0396(03)00185-2.  Google Scholar

[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.  doi: 10.1016/0020-7357(78)90004-5.  Google Scholar

[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.  doi: 10.1007/s10883-007-9022-8.  Google Scholar

[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.  doi: 10.1002/mma.250.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.na.2006.10.040.  Google Scholar

[7]

M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation,, SIAM J. Control Optim., 42 (2003), 1310.  doi: 10.1137/S0363012902408010.  Google Scholar

[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.  doi: 10.1155/S1085337502204133.  Google Scholar

[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.  doi: 10.1016/j.na.2008.04.011.  Google Scholar

[10]

S. W. Hansen and R. Spies, Structural damping in a laminated beam due to interfacial slip,, J. Sound Vibration, 204 (1997), 183.  doi: 10.1006/jsvi.1996.0913.  Google Scholar

[11]

Z. Liu and C. Pang, Exponential stability of a viscoelastic Timoshenko beam,, Adv. Math. Sci. Appl., 8 (1998), 343.   Google Scholar

[12]

A. Lo and N.-e. Tatar, Stabilization of a laminated beam with interfacial slip,, Electron. J. Diff. Eqs., 129 (2015), 1.   Google Scholar

[13]

M. Medjden and N.-e. Tatar, On the wave equation with a temporal nonlocal term,, Dyn. Syst. Appl., 16 (2007), 665.   Google Scholar

[14]

M. Medjden and N.-e. Tatar, Asymptotic behavior for a viscoelastic problem with not necessarily decreasing kernel,, Appl. Math. Comput., 167 (2005), 1221.  doi: 10.1016/j.amc.2004.08.035.  Google Scholar

[15]

S. Messaoudi, General decay of solutions of a viscoelastic equation,, J. Math. Anal. Appl., 341 (2008), 1457.  doi: 10.1016/j.jmaa.2007.11.048.  Google Scholar

[16]

S. Messaoudi and M. I. Mustafa, A general result in a memory-type Timoshenko system,, Comm. Pure Appl. Anal., (2013), 957.  doi: 10.3934/cpaa.2013.12.957.  Google Scholar

[17]

S. Messaoudi and B. Said-Houari, Uniform decay in a Timoshenko-type system with past history,, J. Math. Anal. Appl., 360 (2009), 459.  doi: 10.1016/j.jmaa.2009.06.064.  Google Scholar

[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.   Google Scholar

[19]

V. Pata, Exponential stability in linear viscoelasticity,, Quart. Appl. Math., LXIV (2006), 499.  doi: 10.1090/S0033-569X-06-01010-4.  Google Scholar

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

[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.  doi: 10.1016/S0022-247X(02)00436-5.  Google Scholar

[22]

J. E. M. Rivera and R. Racke, Global stability for damped Timoshenko systems,, Discrete Contin. Dyn. Syst., 9 (2003), 1625.  doi: 10.3934/dcds.2003.9.1625.  Google Scholar

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

[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.  doi: 10.1080/002071798222848.  Google Scholar

[25]

A. Soufyane and Wehbe, Uniform stabilization for the Timoshenko beam by a locally distributed damping,, Electron. J. Diff. Eqs., 29 (2003), 1.   Google Scholar

[26]

N.-e. Tatar, Long time behavior for a viscoelastic problem with a positive definite kernel,, Australian J. Math. Anal. Appl., 1 (2004), 1.   Google Scholar

[27]

N.-e. Tatar, Exponential decay for a viscoelastic problem with a singular problem,, Zeit. Angew. Math. Phys., 60 (2009), 640.  doi: 10.1007/s00033-008-8030-1.  Google Scholar

[28]

N.-e. Tatar, On a large class of kernels yielding exponential stability in viscoelasticity,, Appl. Math. Comp., 215 (2009), 2298.  doi: 10.1016/j.amc.2009.08.034.  Google Scholar

[29]

N.-e. Tatar, Viscoelastic Timoshenko beams with occasionally constant relaxation functions,, Appl. Math. Optim., 66 (2012), 123.  doi: 10.1007/s00245-012-9167-z.  Google Scholar

[30]

N.-e. Tatar, Exponential decay for a viscoelastically damped Timoshenko beam,, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 505.  doi: 10.1016/S0252-9602(13)60015-6.  Google Scholar

[31]

N.-e. Tatar, Stabilization of a viscoelastic Timoshenko beam,, Appl. Anal.: An International Journal, 92 (2013), 27.  doi: 10.1080/00036811.2011.587810.  Google Scholar

[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.  doi: 10.1137/040610003.  Google Scholar

[33]

G. Q. Xu, Feedback exponential stabilization of a Timoshenko beam with both ends free,, Int. J. Control, 72 (2005), 286.  doi: 10.1080/00207170500095148.  Google Scholar

[34]

Q. Yan et al., Boundary stabilization of nonuniform Timoshenko beam with a tipload,, Chin. Ann. Math., 22 (2001), 485.  doi: 10.1142/S0252959901000450.  Google Scholar

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