doi: 10.3934/dcdsb.2022090
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On the asymptotic stability of a Bresse system with two fractional damping terms: Theoretical and numerical analysis

Laboratory of Mathematical Techniques (LTM), Department of Mathematics, Faculty of Mathematics and Computer Sciences, University of Batna 2 -Mostefa Benboulaïd-, Fesdis, Batna 05078, Algeria

* Corresponding author: Toufik Bentrcia

Received  September 2021 Revised  April 2022 Early access May 2022

The aim of this work is to investigate the asymptotic stability of a viscoelastic Bresse system in one dimensional bounded domain. In this context, we introduce two internal damping terms expressed using the generalized Caputo fractional derivative. By adopting a diffusive representation, we show the well-posedness of the proposed system and we prove some decay results. In order to validate the theoretical findings, we implement a finite difference method and we conduct intensive numerical simulations. Moreover, we provide some insights on the convergence of the elaborated numerical scheme.

Citation: Toufik Bentrcia, Abdelaziz Mennouni. On the asymptotic stability of a Bresse system with two fractional damping terms: Theoretical and numerical analysis. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022090
References:
[1]

Z. AchouriN. E. Amroun and A. Benaissa, The Euler-Bernoulli beam equation with boundary dissipation of fractional derivative type, Math. Methods Appl. Sci., 40 (2017), 3837-3854.  doi: 10.1002/mma.4267.

[2]

M. AfilalA. GuesmiaA. Soufiyane 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]

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.

[4]

D. da S. Almeida Júnior and J. E. Muñoz Rivera, Stability criterion to explicit finite difference applied to the Bresse system, Afr. Mat., 26 (2015), 761-778.  doi: 10.1007/s13370-014-0244-0.

[5]

M. de O. AlvesL. H. FatoriM. A. Jorge Silva and R. N. 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.

[6]

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.

[7]

A. Benaissa and A. Kasmi, Well-posedeness and energy decay of solutions to a Bresse system with a boundary dissipation of fractional derivative type, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4361-4395.  doi: 10.3934/dcdsb.2018168.

[8]

A. Benaissa and S. Rafa, Well-posedness and energy decay of solutions to a wave equation with a general boundary control of diffusive type, Math. Nachr., 292 (2019), 1644-1673.  doi: 10.1002/mana.201800224.

[9]

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.

[10]

J. A. C. Bresse, Cours de Méchanique Appliquée, 1$^{st}$ edition, Mallet Bachelier, Paris, 1859.

[11]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, 1$^{st}$ edition, Springer, New York, 2011.

[12]

W. CharlesJ. A. SorianoF. A. Falcão Nascimento and J. H. Rodrigues, Decay rates for Bresse system with arbitrary nonlinear localized damping, J. Differential Equations, 255 (2013), 2267-2290.  doi: 10.1016/j.jde.2013.06.014.

[13]

U J. Choi and R. C. MacCamy, Fractional order Volterra equations with applications to elasticity, J. Math. Anal. Appl., 139 (1989), 448-464.  doi: 10.1016/0022-247X(89)90120-0.

[14]

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

[15]

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.

[16]

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.

[17]

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.

[18]

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.

[19]

A. Guesmia, Non-exponential and polynomial stability results of a Bresse system with one infinite memory in the vertical displacement, Nonauton. Dyn. Syst., 4 (2017), 78-97.  doi: 10.1515/msds-2017-0008.

[20]

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

[21]

A. Guesmia and M. Kirane, Uniform and weak stability of Bresse system with two infinite memories, Z. Angew. Math. Phys., 67 (2016), Art. 124, 39 pp. doi: 10.1007/s00033-016-0719-y.

[22]

H. HaddarJ. R. Li and D. Matignon, Efficient solution of a wave equation with fractional-order dissipative terms, J. Comput. Appl. Math., 234 (2010), 2003-2010.  doi: 10.1016/j.cam.2009.08.051.

[23]

L. S. Hahn and B. Epstein, Classical Complex Analysis, 1$^{st}$ edition, Jones & Bartlett Learning, Sudbury, 1996.

[24]

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

[25]

A. Khemmoudj and T. Hamadouche, Boundary stabilization of a Bresse-type system, Math. Methods Appl. Sci., 39 (2016), 3282-3293.  doi: 10.1002/mma.3773.

[26]

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

[27]

P. D. Lax and R. D. Richtmyer, Survey of the stability of linear finite difference equations Part I An equivalence theorem, Comm. Pure Appl. Math., 9 (1956), 267-293.  doi: 10.1002/cpa.3160090206.

[28]

C. Li and M. Cai, Theory and Numerical Approximations of Fractional Integrals and Derivatives, 1$^{st}$ edition, Siam, Philadelphia, 2020.

[29]

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.

[30]

D. Matignon and C. Prieur, Asymptotic stability of Webster-Lokshin equation, Math. Control Relat. Fields, 4 (2014), 481-500.  doi: 10.3934/mcrf.2014.4.481.

[31]

B. Mbodje, Wave energy decay under fractional derivative controls, IMA J. Math. Control Inform., 23 (2006), 237-257.  doi: 10.1093/imamci/dni056.

[32]

B. MbodjeG. MontsenyJ. Audounet and P. Benchimol, Optimal control for fractionally damped flexible systems, Proceedings of IEEE International Conference on Control and Applications, 2 (1994), 1329-1333.  doi: 10.1109/CCA.1994.381303.

[33]

S. A. Messaoudi and J. H. Hassan, New general decay results in a finite-memory Bresse system, Commun. Pure Appl. Anal., 18 (2019), 1637-1662.  doi: 10.3934/cpaa.2019078.

[34]

G. Montseny, Diffusive representation of pseudo-differential time-operators, ESAIM: Proceedings, 5 (1998), 159-175.  doi: 10.1051/proc:1998005.

[35]

G. E. B. Moraes and M. A. J. Silva, Arched beams of Bresse type: Observability and application in thermoelasticity, Nonlinear Dynamics, 103 (2021), 2365-2390.  doi: 10.1007/s11071-021-06243-3.

[36]

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.

[37]

J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.

[38]

S. RifoO. V. Villagran and J. E. Muñoz Rivera, The lack of exponential stability of the hybrid Bresse system, J. Math. Anal. Appl., 436 (2016), 1-15.  doi: 10.1016/j.jmaa.2015.11.041.

[39]

B. Said-Houari, Global existence and decay estimates for the solution of a nonlinear Bresse system, Nonlinear Anal., 172 (2018), 180-199.  doi: 10.1016/j.na.2018.03.007.

[40]

M. L. Santos and D. da S. Almeida Júnior, Numerical exponential decay to dissipative Bresse system, J. Appl. Math., 2010 (2010), Art. ID 848620, 17 pp. doi: 10.1155/2010/848620.

[41]

M. de L. SantosA. Soufyane and D. da S. Almeida Júnior, Asymptotic behavior to Bresse system with past history, Quart. Appl. Math., 73 (2015), 23-54.  doi: 10.1090/S0033-569X-2014-01382-4.

[42] G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3$^{rd}$ edition, Clarendon Press, Oxford, 1985. 
[43]

J. A. SorianoJ. E. Muñoz Rivera and L. H. Fatori, Bresse system with indefinite damping, J. Math. Anal. Appl., 387 (2012), 284-290.  doi: 10.1016/j.jmaa.2011.08.072.

[44]

A. Soufyane and B. Said-Houari, The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system, Evol. Equ. Control Theory, 3 (2014), 713-738.  doi: 10.3934/eect.2014.3.713.

[45]

N. E. Tatar, The decay rate for a fractional differential equation, J. Math. Anal. Appl., 295 (2004), 303-314.  doi: 10.1016/j.jmaa.2004.01.047.

[46]

N. E. Tatar, On a boundary controller of fractional type, Nonlinear Anal., 72 (2010), 3209-3215.  doi: 10.1016/j.na.2009.12.017.

[47]

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]

Z. AchouriN. E. Amroun and A. Benaissa, The Euler-Bernoulli beam equation with boundary dissipation of fractional derivative type, Math. Methods Appl. Sci., 40 (2017), 3837-3854.  doi: 10.1002/mma.4267.

[2]

M. AfilalA. GuesmiaA. Soufiyane 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]

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.

[4]

D. da S. Almeida Júnior and J. E. Muñoz Rivera, Stability criterion to explicit finite difference applied to the Bresse system, Afr. Mat., 26 (2015), 761-778.  doi: 10.1007/s13370-014-0244-0.

[5]

M. de O. AlvesL. H. FatoriM. A. Jorge Silva and R. N. 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.

[6]

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.

[7]

A. Benaissa and A. Kasmi, Well-posedeness and energy decay of solutions to a Bresse system with a boundary dissipation of fractional derivative type, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4361-4395.  doi: 10.3934/dcdsb.2018168.

[8]

A. Benaissa and S. Rafa, Well-posedness and energy decay of solutions to a wave equation with a general boundary control of diffusive type, Math. Nachr., 292 (2019), 1644-1673.  doi: 10.1002/mana.201800224.

[9]

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.

[10]

J. A. C. Bresse, Cours de Méchanique Appliquée, 1$^{st}$ edition, Mallet Bachelier, Paris, 1859.

[11]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, 1$^{st}$ edition, Springer, New York, 2011.

[12]

W. CharlesJ. A. SorianoF. A. Falcão Nascimento and J. H. Rodrigues, Decay rates for Bresse system with arbitrary nonlinear localized damping, J. Differential Equations, 255 (2013), 2267-2290.  doi: 10.1016/j.jde.2013.06.014.

[13]

U J. Choi and R. C. MacCamy, Fractional order Volterra equations with applications to elasticity, J. Math. Anal. Appl., 139 (1989), 448-464.  doi: 10.1016/0022-247X(89)90120-0.

[14]

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

[15]

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.

[16]

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.

[17]

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.

[18]

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.

[19]

A. Guesmia, Non-exponential and polynomial stability results of a Bresse system with one infinite memory in the vertical displacement, Nonauton. Dyn. Syst., 4 (2017), 78-97.  doi: 10.1515/msds-2017-0008.

[20]

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

[21]

A. Guesmia and M. Kirane, Uniform and weak stability of Bresse system with two infinite memories, Z. Angew. Math. Phys., 67 (2016), Art. 124, 39 pp. doi: 10.1007/s00033-016-0719-y.

[22]

H. HaddarJ. R. Li and D. Matignon, Efficient solution of a wave equation with fractional-order dissipative terms, J. Comput. Appl. Math., 234 (2010), 2003-2010.  doi: 10.1016/j.cam.2009.08.051.

[23]

L. S. Hahn and B. Epstein, Classical Complex Analysis, 1$^{st}$ edition, Jones & Bartlett Learning, Sudbury, 1996.

[24]

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

[25]

A. Khemmoudj and T. Hamadouche, Boundary stabilization of a Bresse-type system, Math. Methods Appl. Sci., 39 (2016), 3282-3293.  doi: 10.1002/mma.3773.

[26]

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

[27]

P. D. Lax and R. D. Richtmyer, Survey of the stability of linear finite difference equations Part I An equivalence theorem, Comm. Pure Appl. Math., 9 (1956), 267-293.  doi: 10.1002/cpa.3160090206.

[28]

C. Li and M. Cai, Theory and Numerical Approximations of Fractional Integrals and Derivatives, 1$^{st}$ edition, Siam, Philadelphia, 2020.

[29]

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.

[30]

D. Matignon and C. Prieur, Asymptotic stability of Webster-Lokshin equation, Math. Control Relat. Fields, 4 (2014), 481-500.  doi: 10.3934/mcrf.2014.4.481.

[31]

B. Mbodje, Wave energy decay under fractional derivative controls, IMA J. Math. Control Inform., 23 (2006), 237-257.  doi: 10.1093/imamci/dni056.

[32]

B. MbodjeG. MontsenyJ. Audounet and P. Benchimol, Optimal control for fractionally damped flexible systems, Proceedings of IEEE International Conference on Control and Applications, 2 (1994), 1329-1333.  doi: 10.1109/CCA.1994.381303.

[33]

S. A. Messaoudi and J. H. Hassan, New general decay results in a finite-memory Bresse system, Commun. Pure Appl. Anal., 18 (2019), 1637-1662.  doi: 10.3934/cpaa.2019078.

[34]

G. Montseny, Diffusive representation of pseudo-differential time-operators, ESAIM: Proceedings, 5 (1998), 159-175.  doi: 10.1051/proc:1998005.

[35]

G. E. B. Moraes and M. A. J. Silva, Arched beams of Bresse type: Observability and application in thermoelasticity, Nonlinear Dynamics, 103 (2021), 2365-2390.  doi: 10.1007/s11071-021-06243-3.

[36]

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.

[37]

J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.

[38]

S. RifoO. V. Villagran and J. E. Muñoz Rivera, The lack of exponential stability of the hybrid Bresse system, J. Math. Anal. Appl., 436 (2016), 1-15.  doi: 10.1016/j.jmaa.2015.11.041.

[39]

B. Said-Houari, Global existence and decay estimates for the solution of a nonlinear Bresse system, Nonlinear Anal., 172 (2018), 180-199.  doi: 10.1016/j.na.2018.03.007.

[40]

M. L. Santos and D. da S. Almeida Júnior, Numerical exponential decay to dissipative Bresse system, J. Appl. Math., 2010 (2010), Art. ID 848620, 17 pp. doi: 10.1155/2010/848620.

[41]

M. de L. SantosA. Soufyane and D. da S. Almeida Júnior, Asymptotic behavior to Bresse system with past history, Quart. Appl. Math., 73 (2015), 23-54.  doi: 10.1090/S0033-569X-2014-01382-4.

[42] G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3$^{rd}$ edition, Clarendon Press, Oxford, 1985. 
[43]

J. A. SorianoJ. E. Muñoz Rivera and L. H. Fatori, Bresse system with indefinite damping, J. Math. Anal. Appl., 387 (2012), 284-290.  doi: 10.1016/j.jmaa.2011.08.072.

[44]

A. Soufyane and B. Said-Houari, The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system, Evol. Equ. Control Theory, 3 (2014), 713-738.  doi: 10.3934/eect.2014.3.713.

[45]

N. E. Tatar, The decay rate for a fractional differential equation, J. Math. Anal. Appl., 295 (2004), 303-314.  doi: 10.1016/j.jmaa.2004.01.047.

[46]

N. E. Tatar, On a boundary controller of fractional type, Nonlinear Anal., 72 (2010), 3209-3215.  doi: 10.1016/j.na.2009.12.017.

[47]

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.  Computational molecules corresponding to $ \psi_{0,j+1} $, $ \left\{ {\psi _{i,j + 1} } \right\}_{0 < i < M + 1} $ and $ \psi_{M+1,j+1} $
Figure 2.  Variation of the absolute error as a function of the time step in logarithmic scales
Figure 3.  Variation of the displacement measures as functions of both time and space variables for experiment 1
Figure 4.  Variation of the displacement measures as functions of both time and space variables for experiment 2
Figure 5.  Variation of the displacement measures as functions of both time and space variables for experiment 3
Figure 6.  Variation of the displacement measures as functions of both time and space variables for experiment 4
Figure 7.  Evolution of the norm of the propagation matrix $ A_j $ versus the number of iterations for experiment 3
Figure 8.  Evolution of the norm of the propagation matrix $ A_j $ versus the number of iterations for experiment 4
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