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June  2019, 8(2): 423-445. doi: 10.3934/eect.2019021

Decay rate of the Timoshenko system with one boundary damping

Laboratoire de Mathématiques et ses Applications de Valenciennes, FR CNRS 2956, Institut des Sciences et Techniques de Valenciennes, Université Polytechnique Hauts-de-France, Le Mont Houy, 59313 VALENCIENNES Cedex 9, FRANCE

* Corresponding author: Virginie Régnier

Received  October 2017 Revised  August 2018 Published  March 2019

In this paper, we study the indirect boundary stabilization of the Timoshenko system with only one dissipation law. This system, which models the dynamics of a beam, is a hyperbolic system with two wave speeds. Assuming that the wave speeds are equal, we prove exponential stability. Otherwise, we show that the decay rate is of exponential or polynomial type. Note that the results hold without the technical assumptions on the coefficients coming from the multiplier method: a sharp analysis of the behaviour of the resolvent operator along the imaginary axis is performed to avoid those artificial restrictions.

Citation: Denis Mercier, Virginie Régnier. Decay rate of the Timoshenko system with one boundary damping. Evolution Equations & Control Theory, 2019, 8 (2) : 423-445. doi: 10.3934/eect.2019021
References:
[1]

F. Abdallah, Stabilisation et Approximation de Certains Systèmes Distribués par Amortisement Dissipatif et de Signe Indéfini, Ph.D thesis, Lebanese University and Université de Valenciennes et du Hainaut Cambrésis, 2013. Google Scholar

[2]

F. AbdallahD. Mercier and S. Nicaise, Spectral analysis and exponential or polynomial stability of some indefinite sign damped problems, Evolution Equations and Control Theory, 2 (2013), 1-33.  doi: 10.3934/eect.2013.2.1.  Google Scholar

[3]

F. Alabau, Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control, NoDEA Nonlinear Diff. Eqns Appl., 14 (2007), 643-669.  doi: 10.1007/s00030-007-5033-0.  Google Scholar

[4]

F. Ammar-KodjaA. BenabdallahJ. E. Munoz-Rivera and R. Racke, Energy decay for Timoshenko systems of memory type, J. Diff. Equations, 194 (2003), 82-115.  doi: 10.1016/S0022-0396(03)00185-2.  Google Scholar

[5]

F. Ammar-KodjaS. Kerbal and A. Soufyane, Stabilization of the nonuniform Timoshenko beam, J. Math. Anal. Appl., 327 (2007), 525-538.  doi: 10.1016/j.jmaa.2006.04.016.  Google Scholar

[6]

W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 305 (1988), 837-852.  doi: 10.1090/S0002-9947-1988-0933321-3.  Google Scholar

[7]

M. BassamD. MercierS. Nicaise and A. Wehbe, Stabilisation frontière indirecte du système de Timoshenko, C. R. Acad. Sc. Paris, Sér. I, 349 (2011), 379-384.  doi: 10.1016/j.crma.2011.03.011.  Google Scholar

[8]

M. BassamD. MercierS. Nicaise and A. Wehbe, Polynomial stability of the Timoshenko system by one boundary damping, J. Math. Anal and Appl., 425 (2015), 1177-1203.  doi: 10.1016/j.jmaa.2014.12.055.  Google Scholar

[9]

C. D. Benchimol, A note on weak stabilizability of contraction semi-groups, SIAM J. Control Optim., 16 (1978), 373-379.  doi: 10.1137/0316023.  Google Scholar

[10]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478.  doi: 10.1007/s00208-009-0439-0.  Google Scholar

[11]

D. Feng and W. Zhang, Nonlinear feedback control of Timoshenko beam, Science in China (Series A), 38 (1995), 918-927.   Google Scholar

[12]

I. Gohberg and M. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Spaces, Translation of Mathematical Monographs, Vol. 18, American Mathematical Society, 1969.  Google Scholar

[13]

M. Grobbelaar-Van Dalsen, Uniform stability for the Timoshenko beam with tip load, J. Math. Anal. Appl., 361 (2010), 392-400.  doi: 10.1016/j.jmaa.2009.06.059.  Google Scholar

[14]

B-Z. Guo, Riesz basis approch to the stabilization of a flexible beam with a tip mass, SIAM J. Control Optim., 39 (2001), 1736-1747.  doi: 10.1137/S0363012999354880.  Google Scholar

[15]

W. HeS. Zhang and S. Ge, Boundary output-feedback stabilization of a timoshenko beam using disturbance observer, IEEE Transactions on Industrial Electronics, 60 (2013), 5186-5194.  doi: 10.1109/TIE.2012.2219835.  Google Scholar

[16]

F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. of Diff. Eqs, 1 (1985), 43-56.   Google Scholar

[17]

J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam, SIAM J. Control Optim., 25 (1987), 1417-1429.  doi: 10.1137/0325078.  Google Scholar

[18]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644.  doi: 10.1007/s00033-004-3073-4.  Google Scholar

[19]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, 398 Research Notes in Mathematics, Champman & Hall/CRC, 1999.  Google Scholar

[20]

S. A. Messaoudi and M. I. Mustafa, A stability result in a memory-type Timoshenko system, Dynamic Systems and Applications, 18 (2009), 457-468.   Google Scholar

[21]

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.  Google Scholar

[22]

S. A. Messaoudi and M. I. Mustafa, On the internal and boundary stabilization of Timoshenko beams, NoDEA Nonlinear Diff. Eqns Appl., 15 (2008), 655-671.  doi: 10.1007/s00030-008-7075-3.  Google Scholar

[23]

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.  Google Scholar

[24]

J. E. Muñoz 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.  Google Scholar

[25]

J. E. Muñoz Rivera and H. D. Fernández Sare, Stability of Timoshenko systems with past history, J. Math. Anal. Appl., 339 (2008), 482-502.  doi: 10.1016/j.jmaa.2007.07.012.  Google Scholar

[26]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[27]

J. Prüss, On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.  Google Scholar

[28]

C. A. RaposoJ. FerreiraM. L. Santos and N. N. O. Castro, Exponential stability for the Timoshenko system with two week dampings, Applied Mathematics Letters, 18 (2005), 535-541.  doi: 10.1016/j.aml.2004.03.017.  Google Scholar

[29]

D. Shi and D. Feng, Exponential decay rate of the energy of a Timoshenko beam with locally distributed feedback, ANZIAM J., 44 (2002), 205-220.  doi: 10.1017/S1446181100013900.  Google Scholar

[30]

A. Soufyane, Stabilisation de la poutre de Timoshenko, C. R. Acad. Sci. Paris, Sér. I Math., 328 (1999), 731-734.  doi: 10.1016/S0764-4442(99)80244-4.  Google Scholar

[31]

A. Soufyane and A. Wehbe, Uniform stabilization for the Timoshenko beam by a locally distributed damping, Electronic Journal of Differential Equation, 29 (2003), 1-14.   Google Scholar

[32]

S. W. Taylor, Boundary Control of the Timoshenko Beam with Variable Physical Characteristics, Research Report 356, Dept. Math., Univ. Auckland, 1998. Google Scholar

[33]

Q. P. VuJ. M. WangG. Q. Xu and S. P. Yung, Spectral analysis and system of fundamental solutions for Timoshenko beams, Applied Mathematics Letters, 18 (2005), 127-134.  doi: 10.1016/j.aml.2004.09.001.  Google Scholar

[34]

A. Wehbe and W. Youssef, Stabilization of the uniform Timoshenko beam by one locally distributed feedback, Applicable Analysis, 88 (2009), 1067-1078.  doi: 10.1080/00036810903156149.  Google Scholar

[35]

L. ZietsmanN. F. J. van Rensburg and A. J. van der Merwe, A Timoshenko beam with tip body and boundary damping, Wave Motion, 39 (2004), 199-211.  doi: 10.1016/j.wavemoti.2003.08.003.  Google Scholar

show all references

References:
[1]

F. Abdallah, Stabilisation et Approximation de Certains Systèmes Distribués par Amortisement Dissipatif et de Signe Indéfini, Ph.D thesis, Lebanese University and Université de Valenciennes et du Hainaut Cambrésis, 2013. Google Scholar

[2]

F. AbdallahD. Mercier and S. Nicaise, Spectral analysis and exponential or polynomial stability of some indefinite sign damped problems, Evolution Equations and Control Theory, 2 (2013), 1-33.  doi: 10.3934/eect.2013.2.1.  Google Scholar

[3]

F. Alabau, Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control, NoDEA Nonlinear Diff. Eqns Appl., 14 (2007), 643-669.  doi: 10.1007/s00030-007-5033-0.  Google Scholar

[4]

F. Ammar-KodjaA. BenabdallahJ. E. Munoz-Rivera and R. Racke, Energy decay for Timoshenko systems of memory type, J. Diff. Equations, 194 (2003), 82-115.  doi: 10.1016/S0022-0396(03)00185-2.  Google Scholar

[5]

F. Ammar-KodjaS. Kerbal and A. Soufyane, Stabilization of the nonuniform Timoshenko beam, J. Math. Anal. Appl., 327 (2007), 525-538.  doi: 10.1016/j.jmaa.2006.04.016.  Google Scholar

[6]

W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 305 (1988), 837-852.  doi: 10.1090/S0002-9947-1988-0933321-3.  Google Scholar

[7]

M. BassamD. MercierS. Nicaise and A. Wehbe, Stabilisation frontière indirecte du système de Timoshenko, C. R. Acad. Sc. Paris, Sér. I, 349 (2011), 379-384.  doi: 10.1016/j.crma.2011.03.011.  Google Scholar

[8]

M. BassamD. MercierS. Nicaise and A. Wehbe, Polynomial stability of the Timoshenko system by one boundary damping, J. Math. Anal and Appl., 425 (2015), 1177-1203.  doi: 10.1016/j.jmaa.2014.12.055.  Google Scholar

[9]

C. D. Benchimol, A note on weak stabilizability of contraction semi-groups, SIAM J. Control Optim., 16 (1978), 373-379.  doi: 10.1137/0316023.  Google Scholar

[10]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478.  doi: 10.1007/s00208-009-0439-0.  Google Scholar

[11]

D. Feng and W. Zhang, Nonlinear feedback control of Timoshenko beam, Science in China (Series A), 38 (1995), 918-927.   Google Scholar

[12]

I. Gohberg and M. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Spaces, Translation of Mathematical Monographs, Vol. 18, American Mathematical Society, 1969.  Google Scholar

[13]

M. Grobbelaar-Van Dalsen, Uniform stability for the Timoshenko beam with tip load, J. Math. Anal. Appl., 361 (2010), 392-400.  doi: 10.1016/j.jmaa.2009.06.059.  Google Scholar

[14]

B-Z. Guo, Riesz basis approch to the stabilization of a flexible beam with a tip mass, SIAM J. Control Optim., 39 (2001), 1736-1747.  doi: 10.1137/S0363012999354880.  Google Scholar

[15]

W. HeS. Zhang and S. Ge, Boundary output-feedback stabilization of a timoshenko beam using disturbance observer, IEEE Transactions on Industrial Electronics, 60 (2013), 5186-5194.  doi: 10.1109/TIE.2012.2219835.  Google Scholar

[16]

F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. of Diff. Eqs, 1 (1985), 43-56.   Google Scholar

[17]

J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam, SIAM J. Control Optim., 25 (1987), 1417-1429.  doi: 10.1137/0325078.  Google Scholar

[18]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644.  doi: 10.1007/s00033-004-3073-4.  Google Scholar

[19]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, 398 Research Notes in Mathematics, Champman & Hall/CRC, 1999.  Google Scholar

[20]

S. A. Messaoudi and M. I. Mustafa, A stability result in a memory-type Timoshenko system, Dynamic Systems and Applications, 18 (2009), 457-468.   Google Scholar

[21]

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.  Google Scholar

[22]

S. A. Messaoudi and M. I. Mustafa, On the internal and boundary stabilization of Timoshenko beams, NoDEA Nonlinear Diff. Eqns Appl., 15 (2008), 655-671.  doi: 10.1007/s00030-008-7075-3.  Google Scholar

[23]

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.  Google Scholar

[24]

J. E. Muñoz 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.  Google Scholar

[25]

J. E. Muñoz Rivera and H. D. Fernández Sare, Stability of Timoshenko systems with past history, J. Math. Anal. Appl., 339 (2008), 482-502.  doi: 10.1016/j.jmaa.2007.07.012.  Google Scholar

[26]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[27]

J. Prüss, On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.  Google Scholar

[28]

C. A. RaposoJ. FerreiraM. L. Santos and N. N. O. Castro, Exponential stability for the Timoshenko system with two week dampings, Applied Mathematics Letters, 18 (2005), 535-541.  doi: 10.1016/j.aml.2004.03.017.  Google Scholar

[29]

D. Shi and D. Feng, Exponential decay rate of the energy of a Timoshenko beam with locally distributed feedback, ANZIAM J., 44 (2002), 205-220.  doi: 10.1017/S1446181100013900.  Google Scholar

[30]

A. Soufyane, Stabilisation de la poutre de Timoshenko, C. R. Acad. Sci. Paris, Sér. I Math., 328 (1999), 731-734.  doi: 10.1016/S0764-4442(99)80244-4.  Google Scholar

[31]

A. Soufyane and A. Wehbe, Uniform stabilization for the Timoshenko beam by a locally distributed damping, Electronic Journal of Differential Equation, 29 (2003), 1-14.   Google Scholar

[32]

S. W. Taylor, Boundary Control of the Timoshenko Beam with Variable Physical Characteristics, Research Report 356, Dept. Math., Univ. Auckland, 1998. Google Scholar

[33]

Q. P. VuJ. M. WangG. Q. Xu and S. P. Yung, Spectral analysis and system of fundamental solutions for Timoshenko beams, Applied Mathematics Letters, 18 (2005), 127-134.  doi: 10.1016/j.aml.2004.09.001.  Google Scholar

[34]

A. Wehbe and W. Youssef, Stabilization of the uniform Timoshenko beam by one locally distributed feedback, Applicable Analysis, 88 (2009), 1067-1078.  doi: 10.1080/00036810903156149.  Google Scholar

[35]

L. ZietsmanN. F. J. van Rensburg and A. J. van der Merwe, A Timoshenko beam with tip body and boundary damping, Wave Motion, 39 (2004), 199-211.  doi: 10.1016/j.wavemoti.2003.08.003.  Google Scholar

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