September  2019, 18(5): 2789-2818. doi: 10.3934/cpaa.2019125

Optimal indirect stability of a weakly damped elastic abstract system of second order equations coupled by velocities

1. 

Lebanese University, Faculty of Sciences 1 and EDST, KALMA, Hadath-Beirut-Lebanon

2. 

Paris-Saclay University, L2S, 3 Rue Joliot Curie, Gif-sur-Yvette, France

Received  September 2018 Revised  January 2019 Published  April 2019

In this paper, by means of the Riesz basis approach, we study the stability of a weakly damped system of two second order evolution equations coupled through the velocities (see (1.1)). If the fractional order damping becomes viscous and the waves propagate with equal speeds, we prove exponential stability of the system and, otherwise, we establish an optimal polynomial decay rate. Finally, we provide some illustrative examples.

Citation: Farah Abdallah, Mouhammad Ghader, Ali Wehbe, Yacine Chitour. Optimal indirect stability of a weakly damped elastic abstract system of second order equations coupled by velocities. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2789-2818. doi: 10.3934/cpaa.2019125
References:
[1]

F. AbdallahM. Ghader and A. Wehbe, Stability results of a distributed problem involving bresse system with history and/or cattaneo law under fully dirichlet or mixed boundary conditions, Mathematical Methods in the Applied Sciences, 41 (2018), 1876-1907. doi: 10.1002/mma.4717. Google Scholar

[2]

F. AbdallahS. NicaiseJ. Valein and A. Wehbe, Stability results for the approximation of weakly coupled wave equations, C. R. Math. Acad. Sci. Paris, 350 (2012), 29-34. doi: 10.1016/j.crma.2011.12.004. Google Scholar

[3]

M. Afilal and F. Ammar Khodja, Stability of coupled second order equations, Comput. Appl. Math., 19 (2000), 91-107. Google Scholar

[4]

F. Alabau, Stabilisation frontière indirecte de systèmes faiblement couplés, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1015-1020. doi: 10.1016/S0764-4442(99)80316-4. Google Scholar

[5]

F. AlabauP. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equ., 2 (2002), 127-150. doi: 10.1007/s00028-002-8083-0. Google Scholar

[6]

F. Alabau-Boussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM J. Control Optim., 41 (2002), 511-541. doi: 10.1137/S0363012901385368. Google Scholar

[7]

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

[8]

F. Alabau-Boussouira and P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations, Comptes Rendus Mathematique, 347 (2009), 867-872. doi: 10.1016/j.crma.2009.05.011. Google Scholar

[9]

F. Alabau-BoussouiraP. Cannarsa and D. Guglielmi, Indirect stabilization of weakly coupled systems with hybrid boundary conditions, Mathematical Control and Related Fields, 1 (2011), 413-436. doi: 10.3934/mcrf.2011.1.413. Google Scholar

[10]

F. Alabau-BoussouiraP. Cannarsa and D. Sforza, Decay estimates for second order evolution equations with memory, Journal of Functional Analysis, 254 (2008), 1342-1372. doi: 10.1016/j.jfa.2007.09.012. Google Scholar

[11]

F. Alabau-Boussouira and M. Léautaud, Indirect stabilization of locally coupled wave-type systems, ESAIM Control Optim. Calc. Var., 18 (2012), 548-582. doi: 10.1051/cocv/2011106. Google Scholar

[12]

F. Alabau BoussouiraJ. E. Muñoz Rivera and D. d. S. Almeida Júnior, Stability to weak dissipative Bresse system, J. Math. Anal. Appl., 374 (2011), 481-498. doi: 10.1016/j.jmaa.2010.07.046. Google Scholar

[13]

F. Alabau-BoussouiraZ. Wang and L. Yu, A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities, ESAIM Control Optim. Calc. Var., 23 (2017), 721-749. doi: 10.1051/cocv/2016011. Google Scholar

[14]

F. Amma Khodja and A. Bader, Stabilizability of systems of one-dimensional wave equations by one internal or boundary control force, SIAM J. Control Optim., 39 (2001), 1833-1851. doi: 10.1137/S0363012900366613. Google Scholar

[15]

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

[16]

K. Ammari and M. Mehrenberger, Stabilization of coupled systems, Acta Math. Hungar., 123 (2009), 1-10. doi: 10.1007/s10474-009-8011-7. Google Scholar

[17]

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

[18]

E. M. A. BenhassiK. AmmariS. Boulite and L. Maniar, Exponential energy decay of some coupled second order systems, Semigroup Forum, 86 (2013), 362-382. doi: 10.1007/s00233-012-9440-0. Google Scholar

[19]

G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math., 39 (1982/82), 433-454. Google Scholar

[20]

S. P. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15-55. Google Scholar

[21]

S. P. Chen and R. Triggiani, Gevrey class semigroups arising from elastic systems with gentle dissipation, the case $0 < \alpha < \frac 12$, Proc. Amer. Math. Soc., 110 (1990), 401-415. doi: 10.2307/2048084. Google Scholar

[22]

R. F. Curtain and H. Zwart, An introduction to Infinite-dimensional Linear Systems Theory, volume 21 of Texts in Applied Mathematics, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6. Google Scholar

[23]

L. H. Fatori and R. N. Monteiro, The optimal decay rate for a weak dissipative Bresse system, Appl. Math. Lett., 25 (2012), 600-604. doi: 10.1016/j.aml.2011.09.067. Google Scholar

[24]

X. Fu, Sharp decay rates for the weakly coupled hyperbolic system with one internal damping, SIAM J. Control Optim., 50 (2012), 1643-1660. doi: 10.1137/110833051. Google Scholar

[25]

A. Guesmia and M. Kafini, Bresse system with infinite memories, Math. Methods Appl. Sci., 38 (2015), 2389-2405. doi: 10.1002/mma.3228. Google Scholar

[26]

D. Henry, Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations, London Mathematical Society Lecture Note Series, Cambridge University Press, 2005. doi: 10.1017/CBO9780511546730. Google Scholar

[27]

B. V. Kapitonov, Uniform stabilization and exact controllability for a class of coupled hyperbolic systems, Mat. Apl. Comput., 15 (1996), 199-212. doi: 10.1007/BF02106615. Google Scholar

[28]

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

[29]

J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-link Structures, Systems & Control, Foundations & Applications. Birkh¨auser Boston, Inc., Boston, MA, 1994. doi: 10.1007/978-1-4612-0273-8. Google Scholar

[30]

Z. Liu and B. Rao, Frequency domain approach for the polynomial stability of a system of partially damped wave equations, J. Math. Anal. Appl., 335 (2007), 860-881. doi: 10.1016/j.jmaa.2007.02.021. Google Scholar

[31]

Z. Liu and B. Rao, Energy decay rate of the thermoelastic Bresse system, Z. Angew. Math. Phys., 60 (2009), 54-69. doi: 10.1007/s00033-008-6122-6. Google Scholar

[32]

Z. Liu and Q. Zhang, A note on the polynomial stability of a weakly damped elastic abstract system, Z. Angew. Math. Phys., 66 (2015), 1799-1804. doi: 10.1007/s00033-015-0517-y. Google Scholar

[33]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, volume 398 of Chapman & Hall/CRC Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 1999. Google Scholar

[34]

P. Loreti and B. Rao, Optimal energy decay rate for partially damped systems by spectral compensation, SIAM J. Control Optim., 45 (2006), 1612-1632. doi: 10.1137/S0363012903437319. Google Scholar

[35]

Z.-H. Luo, B.-Z. Guo and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems with Applications, Communications and Control Engineering Series. Springer-Verlag London, Ltd., London, 1999. doi: 10.1007/978-1-4471-0419-3. Google Scholar

[36]

S. A. Messaoudi and M. I. Mustafa, On the internal and boundary stabilization of Timoshenko beams, NoDEA Nonlinear Differential Equations Appl.s, 45 (2008), 655-671. doi: 10.1007/s00030-008-7075-3. Google Scholar

[37]

N. Najdi and A. Wehbe, Weakly locally thermal stabilization of Bresse systems, Electron. J. Differential Equations, pages No. 182, 19, 2014. Google Scholar

[38]

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

[39]

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

[40]

D. L. Russell, A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl., 173 (1993), 339-385. doi: 10.1006/jmaa.1993.1071. Google Scholar

[41]

A. Soufyane, Uniform stability of displacement coupled second-order equations, Electron. J. Differential Equations, pages No. 25, 10, 2001. Google Scholar

[42]

S. P. 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. Google Scholar

[43]

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

[44]

A. Wehbe and W. Youssef, Exponential and polynomial stability of an elastic Bresse system with two locally distributed feedbacks, J. Math. Phys., 51 (2010), 103523. doi: 10.1063/1.3486094. Google Scholar

show all references

References:
[1]

F. AbdallahM. Ghader and A. Wehbe, Stability results of a distributed problem involving bresse system with history and/or cattaneo law under fully dirichlet or mixed boundary conditions, Mathematical Methods in the Applied Sciences, 41 (2018), 1876-1907. doi: 10.1002/mma.4717. Google Scholar

[2]

F. AbdallahS. NicaiseJ. Valein and A. Wehbe, Stability results for the approximation of weakly coupled wave equations, C. R. Math. Acad. Sci. Paris, 350 (2012), 29-34. doi: 10.1016/j.crma.2011.12.004. Google Scholar

[3]

M. Afilal and F. Ammar Khodja, Stability of coupled second order equations, Comput. Appl. Math., 19 (2000), 91-107. Google Scholar

[4]

F. Alabau, Stabilisation frontière indirecte de systèmes faiblement couplés, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1015-1020. doi: 10.1016/S0764-4442(99)80316-4. Google Scholar

[5]

F. AlabauP. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equ., 2 (2002), 127-150. doi: 10.1007/s00028-002-8083-0. Google Scholar

[6]

F. Alabau-Boussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM J. Control Optim., 41 (2002), 511-541. doi: 10.1137/S0363012901385368. Google Scholar

[7]

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

[8]

F. Alabau-Boussouira and P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations, Comptes Rendus Mathematique, 347 (2009), 867-872. doi: 10.1016/j.crma.2009.05.011. Google Scholar

[9]

F. Alabau-BoussouiraP. Cannarsa and D. Guglielmi, Indirect stabilization of weakly coupled systems with hybrid boundary conditions, Mathematical Control and Related Fields, 1 (2011), 413-436. doi: 10.3934/mcrf.2011.1.413. Google Scholar

[10]

F. Alabau-BoussouiraP. Cannarsa and D. Sforza, Decay estimates for second order evolution equations with memory, Journal of Functional Analysis, 254 (2008), 1342-1372. doi: 10.1016/j.jfa.2007.09.012. Google Scholar

[11]

F. Alabau-Boussouira and M. Léautaud, Indirect stabilization of locally coupled wave-type systems, ESAIM Control Optim. Calc. Var., 18 (2012), 548-582. doi: 10.1051/cocv/2011106. Google Scholar

[12]

F. Alabau BoussouiraJ. E. Muñoz Rivera and D. d. S. Almeida Júnior, Stability to weak dissipative Bresse system, J. Math. Anal. Appl., 374 (2011), 481-498. doi: 10.1016/j.jmaa.2010.07.046. Google Scholar

[13]

F. Alabau-BoussouiraZ. Wang and L. Yu, A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities, ESAIM Control Optim. Calc. Var., 23 (2017), 721-749. doi: 10.1051/cocv/2016011. Google Scholar

[14]

F. Amma Khodja and A. Bader, Stabilizability of systems of one-dimensional wave equations by one internal or boundary control force, SIAM J. Control Optim., 39 (2001), 1833-1851. doi: 10.1137/S0363012900366613. Google Scholar

[15]

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

[16]

K. Ammari and M. Mehrenberger, Stabilization of coupled systems, Acta Math. Hungar., 123 (2009), 1-10. doi: 10.1007/s10474-009-8011-7. Google Scholar

[17]

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

[18]

E. M. A. BenhassiK. AmmariS. Boulite and L. Maniar, Exponential energy decay of some coupled second order systems, Semigroup Forum, 86 (2013), 362-382. doi: 10.1007/s00233-012-9440-0. Google Scholar

[19]

G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math., 39 (1982/82), 433-454. Google Scholar

[20]

S. P. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15-55. Google Scholar

[21]

S. P. Chen and R. Triggiani, Gevrey class semigroups arising from elastic systems with gentle dissipation, the case $0 < \alpha < \frac 12$, Proc. Amer. Math. Soc., 110 (1990), 401-415. doi: 10.2307/2048084. Google Scholar

[22]

R. F. Curtain and H. Zwart, An introduction to Infinite-dimensional Linear Systems Theory, volume 21 of Texts in Applied Mathematics, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6. Google Scholar

[23]

L. H. Fatori and R. N. Monteiro, The optimal decay rate for a weak dissipative Bresse system, Appl. Math. Lett., 25 (2012), 600-604. doi: 10.1016/j.aml.2011.09.067. Google Scholar

[24]

X. Fu, Sharp decay rates for the weakly coupled hyperbolic system with one internal damping, SIAM J. Control Optim., 50 (2012), 1643-1660. doi: 10.1137/110833051. Google Scholar

[25]

A. Guesmia and M. Kafini, Bresse system with infinite memories, Math. Methods Appl. Sci., 38 (2015), 2389-2405. doi: 10.1002/mma.3228. Google Scholar

[26]

D. Henry, Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations, London Mathematical Society Lecture Note Series, Cambridge University Press, 2005. doi: 10.1017/CBO9780511546730. Google Scholar

[27]

B. V. Kapitonov, Uniform stabilization and exact controllability for a class of coupled hyperbolic systems, Mat. Apl. Comput., 15 (1996), 199-212. doi: 10.1007/BF02106615. Google Scholar

[28]

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

[29]

J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-link Structures, Systems & Control, Foundations & Applications. Birkh¨auser Boston, Inc., Boston, MA, 1994. doi: 10.1007/978-1-4612-0273-8. Google Scholar

[30]

Z. Liu and B. Rao, Frequency domain approach for the polynomial stability of a system of partially damped wave equations, J. Math. Anal. Appl., 335 (2007), 860-881. doi: 10.1016/j.jmaa.2007.02.021. Google Scholar

[31]

Z. Liu and B. Rao, Energy decay rate of the thermoelastic Bresse system, Z. Angew. Math. Phys., 60 (2009), 54-69. doi: 10.1007/s00033-008-6122-6. Google Scholar

[32]

Z. Liu and Q. Zhang, A note on the polynomial stability of a weakly damped elastic abstract system, Z. Angew. Math. Phys., 66 (2015), 1799-1804. doi: 10.1007/s00033-015-0517-y. Google Scholar

[33]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, volume 398 of Chapman & Hall/CRC Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 1999. Google Scholar

[34]

P. Loreti and B. Rao, Optimal energy decay rate for partially damped systems by spectral compensation, SIAM J. Control Optim., 45 (2006), 1612-1632. doi: 10.1137/S0363012903437319. Google Scholar

[35]

Z.-H. Luo, B.-Z. Guo and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems with Applications, Communications and Control Engineering Series. Springer-Verlag London, Ltd., London, 1999. doi: 10.1007/978-1-4471-0419-3. Google Scholar

[36]

S. A. Messaoudi and M. I. Mustafa, On the internal and boundary stabilization of Timoshenko beams, NoDEA Nonlinear Differential Equations Appl.s, 45 (2008), 655-671. doi: 10.1007/s00030-008-7075-3. Google Scholar

[37]

N. Najdi and A. Wehbe, Weakly locally thermal stabilization of Bresse systems, Electron. J. Differential Equations, pages No. 182, 19, 2014. Google Scholar

[38]

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

[39]

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

[40]

D. L. Russell, A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl., 173 (1993), 339-385. doi: 10.1006/jmaa.1993.1071. Google Scholar

[41]

A. Soufyane, Uniform stability of displacement coupled second-order equations, Electron. J. Differential Equations, pages No. 25, 10, 2001. Google Scholar

[42]

S. P. 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. Google Scholar

[43]

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

[44]

A. Wehbe and W. Youssef, Exponential and polynomial stability of an elastic Bresse system with two locally distributed feedbacks, J. Math. Phys., 51 (2010), 103523. doi: 10.1063/1.3486094. Google Scholar

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