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

Previous Article
Stabilization of turning processes using spindle feedback with state-dependent delay
DCDS-B Home
This Issue
Next Article
Existence and uniqueness of global classical solutions to a two dimensional two species cancer invasion haptotaxis model

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 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.
[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.
[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.
[4]	R. L. Bagley and P. J. Torvik, A different approach to the analysis of viscoelastically damped structures, AIAA J., 21 (1983), 741-748.
[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.
[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.
[7]	J. A. C. Bresse, Cours de Méchanique Appliquée, Mallet Bachelier, Paris, 1859.
[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.
[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.
[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.
[11]	A. Haraux, Two remarks on dissipative hyperbolic problems, Research Notes in Mathematics, Pitman: Boston, MA, 122 (1985), 161–179.
[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.
[13]	V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson-John Wiley, Paris, 1994.
[14]	F. Mainardi and E. Bonetti, The applications of real order derivatives in linear viscoelasticity, Rheol. Acta, 26 (1988), 64-67.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[22]	I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 1999, Academic Press.
[23]	J. Pruss, On the spectrum of C0-semigroups, Transactions of the American Mathematical Society, 284 (1984), 847-857. doi: 10.2307/1999112.
[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.
[25]	C. Wagschal, Fonctions Holomorphes - Equations Différentielles : Exercices Corrigés, Herman, Paris, 2003.

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.
[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.
[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.
[4]	R. L. Bagley and P. J. Torvik, A different approach to the analysis of viscoelastically damped structures, AIAA J., 21 (1983), 741-748.
[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.
[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.
[7]	J. A. C. Bresse, Cours de Méchanique Appliquée, Mallet Bachelier, Paris, 1859.
[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.
[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.
[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.
[11]	A. Haraux, Two remarks on dissipative hyperbolic problems, Research Notes in Mathematics, Pitman: Boston, MA, 122 (1985), 161–179.
[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.
[13]	V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson-John Wiley, Paris, 1994.
[14]	F. Mainardi and E. Bonetti, The applications of real order derivatives in linear viscoelasticity, Rheol. Acta, 26 (1988), 64-67.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[22]	I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 1999, Academic Press.
[23]	J. Pruss, On the spectrum of C0-semigroups, Transactions of the American Mathematical Society, 284 (1984), 847-857. doi: 10.2307/1999112.
[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.
[25]	C. Wagschal, Fonctions Holomorphes - Equations Différentielles : Exercices Corrigés, Herman, Paris, 2003.

[1]	Chun Wang, Tian-Zhou Xu. Stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville fractional derivative. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 505-521. doi: 10.3934/dcdss.2017025
[2]	Fahd Jarad, Sugumaran Harikrishnan, Kamal Shah, Kuppusamy Kanagarajan. Existence and stability results to a class of fractional random implicit differential equations involving a generalized Hilfer fractional derivative. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 723-739. doi: 10.3934/dcdss.2020040
[3]	Nicolas Fourrier, Irena Lasiecka. Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions. Evolution Equations & Control Theory, 2013, 2 (4) : 631-667. doi: 10.3934/eect.2013.2.631
[4]	Guangliang Zhao, Fuke Wu, George Yin. Feedback controls to ensure global solutions and asymptotic stability of Markovian switching diffusion systems. Mathematical Control & Related Fields, 2015, 5 (2) : 359-376. doi: 10.3934/mcrf.2015.5.359
[5]	Ammar Khemmoudj, Taklit Hamadouche. General decay of solutions of a Bresse system with viscoelastic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4857-4876. doi: 10.3934/dcds.2017209
[6]	Christina A. Hollon, Jeffrey T. Neugebauer. Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition. Conference Publications, 2015, 2015 (special) : 615-620. doi: 10.3934/proc.2015.0615
[7]	Nobuyuki Kato. Linearized stability and asymptotic properties for abstract boundary value functional evolution problems. Conference Publications, 1998, 1998 (Special) : 371-387. doi: 10.3934/proc.1998.1998.371
[8]	Masahiro Suzuki. Asymptotic stability of a boundary layer to the Euler--Poisson equations for a multicomponent plasma. Kinetic & Related Models, 2016, 9 (3) : 587-603. doi: 10.3934/krm.2016008
[9]	Kenta Nakamura, Tohru Nakamura, Shuichi Kawashima. Asymptotic stability of rarefaction waves for a hyperbolic system of balance laws. Kinetic & Related Models, 2019, 12 (4) : 923-944. doi: 10.3934/krm.2019035
[10]	Min Ding, Hairong Yuan. Stability of transonic jets with strong rarefaction waves for two-dimensional steady compressible Euler system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2911-2943. doi: 10.3934/dcds.2018125
[11]	Luyi Ma, Hong-Tao Niu, Zhi-Cheng Wang. Global asymptotic stability of traveling waves to the Allen-Cahn equation with a fractional Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2457-2472. doi: 10.3934/cpaa.2019111
[12]	Lei Wang, Zhong-Jie Han, Gen-Qi Xu. Exponential-stability and super-stability of a thermoelastic system of type II with boundary damping. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2733-2750. doi: 10.3934/dcdsb.2015.20.2733
[13]	Marc Chamberland, Anna Cima, Armengol Gasull, Francesc Mañosas. Characterizing asymptotic stability with Dulac functions. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 59-76. doi: 10.3934/dcds.2007.17.59
[14]	Tobias Black. Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1253-1272. doi: 10.3934/dcdsb.2017061
[15]	Ahmed Bchatnia, Aissa Guesmia. Well-posedness and asymptotic stability for the Lamé system with infinite memories in a bounded domain. Mathematical Control & Related Fields, 2014, 4 (4) : 451-463. doi: 10.3934/mcrf.2014.4.451
[16]	Krzysztof Frączek. Polynomial growth of the derivative for diffeomorphisms on tori. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 489-516. doi: 10.3934/dcds.2004.11.489
[17]	Cónall Kelly, Alexandra Rodkina. Constrained stability and instability of polynomial difference equations with state-dependent noise. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 913-933. doi: 10.3934/dcdsb.2009.11.913
[18]	Farah Abdallah, Denis Mercier, Serge Nicaise. Spectral analysis and exponential or polynomial stability of some indefinite sign damped problems. Evolution Equations & Control Theory, 2013, 2 (1) : 1-33. doi: 10.3934/eect.2013.2.1
[19]	Reza Kamyar, Matthew M. Peet. Polynomial optimization with applications to stability analysis and control - Alternatives to sum of squares. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2383-2417. doi: 10.3934/dcdsb.2015.20.2383
[20]	Yuming Qin, Xinguang Yang, Zhiyong Ma. Global existence of solutions for the thermoelastic Bresse system. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1395-1406. doi: 10.3934/cpaa.2014.13.1395

American Institute of Mathematical Sciences