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.

[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.

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[37]

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

[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.

[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.

[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.

[41]

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

[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.

[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.

[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.

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.

[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.

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[37]

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

[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.

[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.

[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.

[41]

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

[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.

[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.

[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.

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