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December  2018, 23(10): 4361-4395. doi: 10.3934/dcdsb.2018168

Well-posedeness and energy decay of solutions to a bresse system with a boundary dissipation of fractional derivative type

Abbes Benaissa and  Abderrahmane Kasmi

Laboratory of Analysis and Control of PDEs, Djillali Liabes University, P. O. Box 89, Sidi Bel Abbes 22000, ALGERIA

Received  March 2017 Revised  January 2018 Published  December 2018 Early access  June 2018

We consider the Bresse system with three control boundary conditions of fractional derivative type. We prove the polynomial decay result with an estimation of the decay rates. Our result is established using the semigroup theory of linear operators and a result obtained by Borichev and Tomilov.

Keywords: Bresse system, boundary fractional derivative controls, strong asymptotic stability, polynomial stability.
Mathematics Subject Classification: Primary: 93D15, 35B40, 47D06, 35P20, 74D05.
Citation: Abbes Benaissa, Abderrahmane Kasmi. Well-posedeness and energy decay of solutions to a bresse system with a boundary dissipation of fractional derivative type. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4361-4395. doi: 10.3934/dcdsb.2018168
References:
[1]

M. S. Alves, O. Vera, J. M. Rivera and A. Rambaudm, Exponential stability to the bresse system with boundary dissipation conditions, arXiv: 150601657A. Google Scholar

[2]

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.  Google Scholar

[3]

R. L. Bagley and P. J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheology., 27 (1983), 201-210.   Google Scholar

[4]

R. L. Bagley and P. J. Torvik, A different approach to the analysis of viscoelastically damped structures, AIAA J., 21 (1983), 741-748.   Google Scholar

[5]

R. L. Bagley and P. J. Torvik, On the appearance of the fractional derivative in the behavior of real material, J. Appl. Mech., 51 (1983), 294-298.  doi: 10.1115/1.3167615.  Google Scholar

[6]

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

[7]

J. A. C. Bresse, Cours de Méchanique Appliquée, Mallet Bachelier, Paris, 1859. Google Scholar

[8]

H. Brézis, Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces De Hilbert, Notas de Matemática (50), Universidade Federal do Rio de Janeiro and University of Rochester, North-Holland, Amsterdam, 1973.  Google Scholar

[9]

M. M. Cavalcanti, V. D. Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, J. Diff. Equa., 236 (2007), 407-459.  doi: 10.1016/j.jde.2007.02.004.  Google Scholar

[10]

J. U. 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.  Google Scholar

[11]

A. Haraux, Two remarks on dissipative hyperbolic problems, Research Notes in Mathematics, Pitman: Boston, MA, 122 (1985), 161–179.  Google Scholar

[12]

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

[13]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson-John Wiley, Paris, 1994.  Google Scholar

[14]

F. Mainardi and E. Bonetti, The applications of real order derivatives in linear viscoelasticity, Rheol. Acta, 26 (1988), 64-67.   Google Scholar

[15]

B. Mbodje, Wave energy decay under fractional derivative controls, IMA Journal of Mathematical Control and Information., 23 (2006), 237-257.  doi: 10.1093/imamci/dni056.  Google Scholar

[16]

B. Mbodje and G. Montseny, Boundary fractional derivative control of the wave equation, IEEE Transactions on Automatic Control., 40 (1995), 368-382.  doi: 10.1109/9.341815.  Google Scholar

[17]

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

[18]

S. A. Messaoudi and M. I. Mustapha, On the stabilization of the Timochenko system by a weak nonlinear dissipation, Math. Meth. Appl. Sci., 32 (2009), 454-469.  doi: 10.1002/mma.1047.  Google Scholar

[19]

J. H. Park and J. R. Kang, Energy decay of solutions for Timoshenko beam with a weak non-linear dissipation, IMA J. Appl. Math., 76 (2011), 340-350.  doi: 10.1093/imamat/hxq040.  Google Scholar

[20]

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

[21]

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

[22]

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 1999, Academic Press.  Google Scholar

[23]

J. Pruss, On the spectrum of C0-semigroups, Transactions of the American Mathematical Society, 284 (1984), 847-857.  doi: 10.2307/1999112.  Google Scholar

[24]

J. A. Soriano, W. 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.  Google Scholar

[25]

C. Wagschal, Fonctions Holomorphes - Equations Différentielles : Exercices Corrigés, Herman, Paris, 2003. Google Scholar

show all references

References:
[1]

M. S. Alves, O. Vera, J. M. Rivera and A. Rambaudm, Exponential stability to the bresse system with boundary dissipation conditions, arXiv: 150601657A. Google Scholar

[2]

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.  Google Scholar

[3]

R. L. Bagley and P. J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheology., 27 (1983), 201-210.   Google Scholar

[4]

R. L. Bagley and P. J. Torvik, A different approach to the analysis of viscoelastically damped structures, AIAA J., 21 (1983), 741-748.   Google Scholar

[5]

R. L. Bagley and P. J. Torvik, On the appearance of the fractional derivative in the behavior of real material, J. Appl. Mech., 51 (1983), 294-298.  doi: 10.1115/1.3167615.  Google Scholar

[6]

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

[7]

J. A. C. Bresse, Cours de Méchanique Appliquée, Mallet Bachelier, Paris, 1859. Google Scholar

[8]

H. Brézis, Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces De Hilbert, Notas de Matemática (50), Universidade Federal do Rio de Janeiro and University of Rochester, North-Holland, Amsterdam, 1973.  Google Scholar

[9]

M. M. Cavalcanti, V. D. Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, J. Diff. Equa., 236 (2007), 407-459.  doi: 10.1016/j.jde.2007.02.004.  Google Scholar

[10]

J. U. 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.  Google Scholar

[11]

A. Haraux, Two remarks on dissipative hyperbolic problems, Research Notes in Mathematics, Pitman: Boston, MA, 122 (1985), 161–179.  Google Scholar

[12]

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

[13]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson-John Wiley, Paris, 1994.  Google Scholar

[14]

F. Mainardi and E. Bonetti, The applications of real order derivatives in linear viscoelasticity, Rheol. Acta, 26 (1988), 64-67.   Google Scholar

[15]

B. Mbodje, Wave energy decay under fractional derivative controls, IMA Journal of Mathematical Control and Information., 23 (2006), 237-257.  doi: 10.1093/imamci/dni056.  Google Scholar

[16]

B. Mbodje and G. Montseny, Boundary fractional derivative control of the wave equation, IEEE Transactions on Automatic Control., 40 (1995), 368-382.  doi: 10.1109/9.341815.  Google Scholar

[17]

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

[18]

S. A. Messaoudi and M. I. Mustapha, On the stabilization of the Timochenko system by a weak nonlinear dissipation, Math. Meth. Appl. Sci., 32 (2009), 454-469.  doi: 10.1002/mma.1047.  Google Scholar

[19]

J. H. Park and J. R. Kang, Energy decay of solutions for Timoshenko beam with a weak non-linear dissipation, IMA J. Appl. Math., 76 (2011), 340-350.  doi: 10.1093/imamat/hxq040.  Google Scholar

[20]

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

[21]

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

[22]

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 1999, Academic Press.  Google Scholar

[23]

J. Pruss, On the spectrum of C0-semigroups, Transactions of the American Mathematical Society, 284 (1984), 847-857.  doi: 10.2307/1999112.  Google Scholar

[24]

J. A. Soriano, W. 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.  Google Scholar

[25]

C. Wagschal, Fonctions Holomorphes - Equations Différentielles : Exercices Corrigés, Herman, Paris, 2003. Google Scholar

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