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doi: 10.3934/eect.2022004
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The influence of the physical coefficients of a Bresse system with one singular local viscous damping in the longitudinal displacement on its stabilization

Université Polytechnique Hauts-de-France, CÉRAMATHS/DEMAV, Valenciennes, France

* Corresponding author: Haidar.Badawi@uphf.fr

Received  September 2021 Early access January 2022

In this paper, we investigate the stabilization of a linear Bresse system with one singular local frictional damping acting in the longitudinal displacement, under fully Dirichlet boundary conditions. First, we prove the strong stability of our system. Next, using a frequency domain approach combined with the multiplier method, we establish the exponential stability of the solution if the three waves have the same speed of propagation. On the contrary, we prove that the energy of our system decays polynomially with rates $ t^{-1} $ or $ t^{-\frac{1}{2}} $.

Citation: Mohammad Akil, Haidar Badawi. The influence of the physical coefficients of a Bresse system with one singular local viscous damping in the longitudinal displacement on its stabilization. Evolution Equations and Control Theory, doi: 10.3934/eect.2022004
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, Math. Methods Appl. Sci., 41 (2018), 1876-1907.  doi: 10.1002/mma.4717.

[2]

M. AfilalA. GuesmiaA. Soufyane and M. Zahri, On the exponential and polynomial stability for a linear Bresse system, Math. Methods Appl. Sci., 43 (2020), 2626-2645.  doi: 10.1002/mma.6070.

[3]

M. Akil, H. Badawi, S. Nicaise and A. Wehbe, On the stability of Bresse system with one discontinuous local internal Kelvin–Voigt damping on the axial force, Z. Angew. Math. Phys., 72 (2021), Paper No. 126, 27 pp. doi: 10.1007/s00033-021-01558-y.

[4]

M. AkilH. BadawiS. Nicaise and A. Wehbe, Stability results of coupled wave models with locally memory in a past history framework via nonsmooth coefficients on the interface, Math. Methods Appl. Sci., 44 (2021), 6950-6981.  doi: 10.1002/mma.7235.

[5]

M. AkilY. ChitourM. Ghader and A. Wehbe, Stability and exact controllability of a Timoshenko system with only one fractional damping on the boundary, Asymptot. Anal., 119 (2020), 221-280.  doi: 10.3233/ASY-191574.

[6]

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.

[7]

F. Alabau BoussouiraJ. E. Muñoz Rivera and D. da 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.

[8]

M. AlvesL. FatoriM. Jorge Silva and R. Monteiro, Stability and optimality of decay rate for a weakly dissipative Bresse system, Math. Methods Appl. Sci., 38 (2015), 898-908.  doi: 10.1002/mma.3115.

[9]

W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852.  doi: 10.1090/S0002-9947-1988-0933321-3.

[10]

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.

[11]

M. BassamD. MercierS. Nicaise and A. Wehbe, Stability results of some distributed systems involving mindlin-Timoshenko plates in the plane, ZAMM Z. Angew. Math. Mech., 96 (2016), 916-938.  doi: 10.1002/zamm.201500172.

[12]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780.  doi: 10.1007/s00028-008-0424-1.

[13]

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.

[14]

J. A. C. Bresse, Cours de Mécanique Appliquée, Mallet Bachelier, 1859.

[15]

P. R. de Lima and H. D. Fernández Sare, Stability of thermoelastic Bresse systems, Z. Angew. Math. Phys., 70 (2019), Paper No. 3, 33 pp. doi: 10.1007/s00033-018-1057-z.

[16]

T. El Arwadi and W. Youssef, On the stabilization of the Bresse beam with Kelvin–Vigt damping, Appl. Math. Optim., 83 (2021), 1831-1857.  doi: 10.1007/s00245-019-09611-z.

[17]

L. H. FatoriM. de Oliveira Alves and H. D. F. Sare, Stability conditions to Bresse systems with indefinite memory dissipation, Appl. Anal., 99 (2020), 1066-1084.  doi: 10.1080/00036811.2018.1520982.

[18]

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.

[19]

L. H. Fatori and J. E. Muñoz Rivera, Rates of decay to weak thermoelastic Bresse system, IMA J. Appl. Math., 75 (2010), 881-904.  doi: 10.1093/imamat/hxq038.

[20]

T.-E. GhoulM. Khenissi and B. Said-Houari, On the stability of the Bresse system with frictional damping, J. Math. Anal. Appl, 455 (2017), 1870-1898.  doi: 10.1016/j.jmaa.2017.04.027.

[21]

A. Guesmia, Asymptotic stability of Bresse system with one infinite memory in the longitudinal displacements, Mediterr. J. Math., 14 (2017), Paper No. 49, 19 pp. doi: 10.1007/s00009-017-0877-y.

[22]

A. Guesmia and M. Kafini, Bresse system with infinite memories, Math. Methods Appl. Sci., 38 (2015), 2389-2402.  doi: 10.1002/mma.3228.

[23]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. 

[24]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-66282-9.

[25]

J. E. LagneseG. Leugering and E. J. P. G. Schmidt, Modelling of dynamic networks of thin thermoelastic beams, Math. Methods Appl. Sci., 16 (1993), 327-358.  doi: 10.1002/mma.1670160503.

[26]

J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures, Birkhäuser Boston, 1994. doi: 10.1007/978-1-4612-0273-8.

[27]

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.

[28]

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.

[29]

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.

[30]

J. E. Muñoz Rivera and M. G. Naso, Boundary stabilization of bresse systems, Z. Angew. Math. Phys., 70 (2019), Paper No. 56, 16 pp. doi: 10.1007/s00033-019-1102-6.

[31]

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

[32]

N. Noun and A. Wehbe, Stabilisation faible interne locale de système élastique de Bresse, C. R. Math. Acad. Sci. Paris, 350 (2012), 493-498.  doi: 10.1016/j.crma.2012.04.003.

[33]

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

[34]

J. Prüss, On the spectrum of $C_{0}$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.

[35]

J. SorianoW. Charles and R. Schulz, Asymptotic stability for Bresse systems, J. Math. Anal. Appl., 412 (2014), 369-380.  doi: 10.1016/j.jmaa.2013.10.019.

[36]

A. Soufyane and A. Wehbe, Uniform stabilization for the Timoshenko beam by a locally distributed damping, Electron. J. Differential Equations, (2003), 14 PP.

[37]

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.

[38]

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, 17 PP. 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, Math. Methods Appl. Sci., 41 (2018), 1876-1907.  doi: 10.1002/mma.4717.

[2]

M. AfilalA. GuesmiaA. Soufyane and M. Zahri, On the exponential and polynomial stability for a linear Bresse system, Math. Methods Appl. Sci., 43 (2020), 2626-2645.  doi: 10.1002/mma.6070.

[3]

M. Akil, H. Badawi, S. Nicaise and A. Wehbe, On the stability of Bresse system with one discontinuous local internal Kelvin–Voigt damping on the axial force, Z. Angew. Math. Phys., 72 (2021), Paper No. 126, 27 pp. doi: 10.1007/s00033-021-01558-y.

[4]

M. AkilH. BadawiS. Nicaise and A. Wehbe, Stability results of coupled wave models with locally memory in a past history framework via nonsmooth coefficients on the interface, Math. Methods Appl. Sci., 44 (2021), 6950-6981.  doi: 10.1002/mma.7235.

[5]

M. AkilY. ChitourM. Ghader and A. Wehbe, Stability and exact controllability of a Timoshenko system with only one fractional damping on the boundary, Asymptot. Anal., 119 (2020), 221-280.  doi: 10.3233/ASY-191574.

[6]

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.

[7]

F. Alabau BoussouiraJ. E. Muñoz Rivera and D. da 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.

[8]

M. AlvesL. FatoriM. Jorge Silva and R. Monteiro, Stability and optimality of decay rate for a weakly dissipative Bresse system, Math. Methods Appl. Sci., 38 (2015), 898-908.  doi: 10.1002/mma.3115.

[9]

W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852.  doi: 10.1090/S0002-9947-1988-0933321-3.

[10]

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.

[11]

M. BassamD. MercierS. Nicaise and A. Wehbe, Stability results of some distributed systems involving mindlin-Timoshenko plates in the plane, ZAMM Z. Angew. Math. Mech., 96 (2016), 916-938.  doi: 10.1002/zamm.201500172.

[12]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780.  doi: 10.1007/s00028-008-0424-1.

[13]

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.

[14]

J. A. C. Bresse, Cours de Mécanique Appliquée, Mallet Bachelier, 1859.

[15]

P. R. de Lima and H. D. Fernández Sare, Stability of thermoelastic Bresse systems, Z. Angew. Math. Phys., 70 (2019), Paper No. 3, 33 pp. doi: 10.1007/s00033-018-1057-z.

[16]

T. El Arwadi and W. Youssef, On the stabilization of the Bresse beam with Kelvin–Vigt damping, Appl. Math. Optim., 83 (2021), 1831-1857.  doi: 10.1007/s00245-019-09611-z.

[17]

L. H. FatoriM. de Oliveira Alves and H. D. F. Sare, Stability conditions to Bresse systems with indefinite memory dissipation, Appl. Anal., 99 (2020), 1066-1084.  doi: 10.1080/00036811.2018.1520982.

[18]

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.

[19]

L. H. Fatori and J. E. Muñoz Rivera, Rates of decay to weak thermoelastic Bresse system, IMA J. Appl. Math., 75 (2010), 881-904.  doi: 10.1093/imamat/hxq038.

[20]

T.-E. GhoulM. Khenissi and B. Said-Houari, On the stability of the Bresse system with frictional damping, J. Math. Anal. Appl, 455 (2017), 1870-1898.  doi: 10.1016/j.jmaa.2017.04.027.

[21]

A. Guesmia, Asymptotic stability of Bresse system with one infinite memory in the longitudinal displacements, Mediterr. J. Math., 14 (2017), Paper No. 49, 19 pp. doi: 10.1007/s00009-017-0877-y.

[22]

A. Guesmia and M. Kafini, Bresse system with infinite memories, Math. Methods Appl. Sci., 38 (2015), 2389-2402.  doi: 10.1002/mma.3228.

[23]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. 

[24]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-66282-9.

[25]

J. E. LagneseG. Leugering and E. J. P. G. Schmidt, Modelling of dynamic networks of thin thermoelastic beams, Math. Methods Appl. Sci., 16 (1993), 327-358.  doi: 10.1002/mma.1670160503.

[26]

J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures, Birkhäuser Boston, 1994. doi: 10.1007/978-1-4612-0273-8.

[27]

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.

[28]

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.

[29]

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.

[30]

J. E. Muñoz Rivera and M. G. Naso, Boundary stabilization of bresse systems, Z. Angew. Math. Phys., 70 (2019), Paper No. 56, 16 pp. doi: 10.1007/s00033-019-1102-6.

[31]

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

[32]

N. Noun and A. Wehbe, Stabilisation faible interne locale de système élastique de Bresse, C. R. Math. Acad. Sci. Paris, 350 (2012), 493-498.  doi: 10.1016/j.crma.2012.04.003.

[33]

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

[34]

J. Prüss, On the spectrum of $C_{0}$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.

[35]

J. SorianoW. Charles and R. Schulz, Asymptotic stability for Bresse systems, J. Math. Anal. Appl., 412 (2014), 369-380.  doi: 10.1016/j.jmaa.2013.10.019.

[36]

A. Soufyane and A. Wehbe, Uniform stabilization for the Timoshenko beam by a locally distributed damping, Electron. J. Differential Equations, (2003), 14 PP.

[37]

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.

[38]

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, 17 PP. doi: 10.1063/1.3486094.

Figure 1.  Geometric description of the function $ a(x) $
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