• Previous Article
    Global stabilization of a coupled system of two generalized Korteweg-de Vries type equations posed on a finite domain
  • MCRF Home
  • This Issue
  • Next Article
    Global Carleman estimate on a network for the wave equation and application to an inverse problem
September  2011, 1(3): 331-352. doi: 10.3934/mcrf.2011.1.331

Internal stabilization of a Mindlin-Timoshenko model by interior feedbacks

1. 

Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques of Valenciennes, F-59313 - Valenciennes Cedex 9

Received  February 2011 Revised  June 2011 Published  September 2011

A Mindlin-Timoshenko model with non constant and non smooth coefficients set in a bounded domain of $\mathbb{R}^d, d\geq 1$ with some internal dissipations is proposed. It corresponds to the coupling between the wave equation and the dynamical elastic system. If the dissipation acts on both equations, we show an exponential decay rate. On the contrary if the dissipation is only active on the elasticity equation, a polynomial decay is shown; a similar result is proved in one dimension if the dissipation is only active on the wave equation.
Citation: Serge Nicaise. Internal stabilization of a Mindlin-Timoshenko model by interior feedbacks. Mathematical Control & Related Fields, 2011, 1 (3) : 331-352. doi: 10.3934/mcrf.2011.1.331
References:
[1]

F. Alabau-Boussouira, Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control,, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 643.   Google Scholar

[2]

C. Bardos, G. Lebeau and J. Rauch, Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques, Nonlinear hyperbolic equations in applied sciences,, Rend. Sem. Mat. Univ. Politec. Torino, 1988 (1989), 11.   Google Scholar

[3]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary,, SIAM J. Control Optim., 30 (1992), 1024.   Google Scholar

[4]

C. Bardos, T. Masrour and F. Tatout, Observation and control of elastic waves,, in, 91 (1994), 1.   Google Scholar

[5]

A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups,, Math. Nachr., 279 (2006), 1425.   Google Scholar

[6]

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

[7]

C. D. Benchimol, A note on weak stabilizability of contraction semigroups,, SIAM J. Control Optimization, 16 (1978), 373.  doi: 10.1137/0316023.  Google Scholar

[8]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,, Math. Ann., 347 (2010), 455.  doi: 10.1007/s00208-009-0439-0.  Google Scholar

[9]

N. Burq and P. Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes,, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 749.   Google Scholar

[10]

M. Costabel, M. Dauge and S. Nicaise, Singularities of Maxwell interface problems,, M2AN Math. Modél. Numer. Anal., 33 (1999), 627.   Google Scholar

[11]

H. D. Fernández Sare, On the stability of Mindlin-Timoshenko plates,, Quart. Appl. Math., 67 (2009), 249.   Google Scholar

[12]

V. Girault and P.-A. Raviart, "Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms," Springer Series in Computational Mathematics, 5,, Springer-Verlag, (1986).   Google Scholar

[13]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains,", Monographs and Studies in Mathematics, 24 (1985).   Google Scholar

[14]

P. Grisvard, "Singularities in Boundary Value Problems," Research in Applied Mathematics, 22, Masson, Paris,, Springer-Verlag, (1992).   Google Scholar

[15]

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

[16]

J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam,, SIAM J. Control Optim., 25 (1987), 1417.  doi: 10.1137/0325078.  Google Scholar

[17]

J. Lagnese and J.-L. Lions, "Modelling Analysis and Control of Thin Plates," Recherches en Mathématiques Appliquées [Research in Applied Mathematics], 6,, Masson, (1988).   Google Scholar

[18]

J. E. Lagnese, "Boundary Stabilization of Thin Plates," SIAM Studies in Applied Mathematics, 10,, Society for Industrial and Applied Mathematics (SIAM), (1989).   Google Scholar

[19]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation,, Z. Angew. Math. Phys., 56 (2005), 630.  doi: 10.1007/s00033-004-3073-4.  Google Scholar

[20]

S. A. Messaoudi and M. I. Mustafa, On the internal and boundary stabilization of Timoshenko beams,, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 655.   Google Scholar

[21]

J. E. Muñoz Rivera and R. Racke, Global stability for damped Timoshenko systems,, Discrete Contin. Dyn. Syst., 9 (2003), 1625.  doi: 10.3934/dcds.2003.9.1625.  Google Scholar

[22]

J. E. Muñoz Rivera and R. Racke, Timoshenko systems with indefinite damping,, J. Math. Anal. Appl., 341 (2008), 1068.  doi: 10.1016/j.jmaa.2007.11.012.  Google Scholar

[23]

S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks,, SIAM J. Control Optim., 45 (2006), 1561.  doi: 10.1137/060648891.  Google Scholar

[24]

S. Nicaise and A.-M. Sändig, General interface problems I,, Math. Methods in the Appl. Sc., 17 (1994), 395.   Google Scholar

[25]

S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks,, NHM, 2 (2007), 425.  doi: 10.3934/nhm.2007.2.425.  Google Scholar

[26]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Math. Sciences, 44,, Springer-Verlag, (1983).   Google Scholar

[27]

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

[28]

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

[29]

B. Said-Houari and Y. Laskri, A stability result of a Timoshenko system with a delay term in the internal feedback,, Applied Mathematics and Computation, 217 (2010), 2857.  doi: 10.1016/j.amc.2010.08.021.  Google Scholar

[30]

D.-H. Shi and D.-X. Feng, Exponential decay of Timoshenko beam with locally distributed feedback,, IMA J. Math. Control Inform., 18 (2001), 395.  doi: 10.1093/imamci/18.3.395.  Google Scholar

[31]

A. Soufyane, Stabilisation de la poutre de Timoshenko,, C. R. Acad. Sci. Paris Sér. I Math., (1999), 731.   Google Scholar

[32]

A. Soufyane, M. Afilal and T. Aouam, General decay of solutions of a nonlinear Timoshenko system with a boundary control of memory type,, Differential Integral Equations, 22 (2009), 1125.   Google Scholar

[33]

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

[34]

A. Wehbe and W. Youssef, Stabilization of the uniform Timoshenko beam by one locally distributed feedback,, Appl. Anal., 88 (2009), 1067.  doi: 10.1080/00036810903156149.  Google Scholar

[35]

G. Q. Xu and S. P. Yung, Exponential decay rate for a Timoshenko beam with boundary damping,, J. Optim. Theory Appl., 123 (2004), 669.  doi: 10.1007/s10957-004-5728-x.  Google Scholar

[36]

G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control,, ESAIM Control Optim. Calc. Var., 12 (2006), 770.  doi: 10.1051/cocv:2006021.  Google Scholar

show all references

References:
[1]

F. Alabau-Boussouira, Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control,, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 643.   Google Scholar

[2]

C. Bardos, G. Lebeau and J. Rauch, Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques, Nonlinear hyperbolic equations in applied sciences,, Rend. Sem. Mat. Univ. Politec. Torino, 1988 (1989), 11.   Google Scholar

[3]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary,, SIAM J. Control Optim., 30 (1992), 1024.   Google Scholar

[4]

C. Bardos, T. Masrour and F. Tatout, Observation and control of elastic waves,, in, 91 (1994), 1.   Google Scholar

[5]

A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups,, Math. Nachr., 279 (2006), 1425.   Google Scholar

[6]

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

[7]

C. D. Benchimol, A note on weak stabilizability of contraction semigroups,, SIAM J. Control Optimization, 16 (1978), 373.  doi: 10.1137/0316023.  Google Scholar

[8]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,, Math. Ann., 347 (2010), 455.  doi: 10.1007/s00208-009-0439-0.  Google Scholar

[9]

N. Burq and P. Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes,, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 749.   Google Scholar

[10]

M. Costabel, M. Dauge and S. Nicaise, Singularities of Maxwell interface problems,, M2AN Math. Modél. Numer. Anal., 33 (1999), 627.   Google Scholar

[11]

H. D. Fernández Sare, On the stability of Mindlin-Timoshenko plates,, Quart. Appl. Math., 67 (2009), 249.   Google Scholar

[12]

V. Girault and P.-A. Raviart, "Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms," Springer Series in Computational Mathematics, 5,, Springer-Verlag, (1986).   Google Scholar

[13]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains,", Monographs and Studies in Mathematics, 24 (1985).   Google Scholar

[14]

P. Grisvard, "Singularities in Boundary Value Problems," Research in Applied Mathematics, 22, Masson, Paris,, Springer-Verlag, (1992).   Google Scholar

[15]

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

[16]

J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam,, SIAM J. Control Optim., 25 (1987), 1417.  doi: 10.1137/0325078.  Google Scholar

[17]

J. Lagnese and J.-L. Lions, "Modelling Analysis and Control of Thin Plates," Recherches en Mathématiques Appliquées [Research in Applied Mathematics], 6,, Masson, (1988).   Google Scholar

[18]

J. E. Lagnese, "Boundary Stabilization of Thin Plates," SIAM Studies in Applied Mathematics, 10,, Society for Industrial and Applied Mathematics (SIAM), (1989).   Google Scholar

[19]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation,, Z. Angew. Math. Phys., 56 (2005), 630.  doi: 10.1007/s00033-004-3073-4.  Google Scholar

[20]

S. A. Messaoudi and M. I. Mustafa, On the internal and boundary stabilization of Timoshenko beams,, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 655.   Google Scholar

[21]

J. E. Muñoz Rivera and R. Racke, Global stability for damped Timoshenko systems,, Discrete Contin. Dyn. Syst., 9 (2003), 1625.  doi: 10.3934/dcds.2003.9.1625.  Google Scholar

[22]

J. E. Muñoz Rivera and R. Racke, Timoshenko systems with indefinite damping,, J. Math. Anal. Appl., 341 (2008), 1068.  doi: 10.1016/j.jmaa.2007.11.012.  Google Scholar

[23]

S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks,, SIAM J. Control Optim., 45 (2006), 1561.  doi: 10.1137/060648891.  Google Scholar

[24]

S. Nicaise and A.-M. Sändig, General interface problems I,, Math. Methods in the Appl. Sc., 17 (1994), 395.   Google Scholar

[25]

S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks,, NHM, 2 (2007), 425.  doi: 10.3934/nhm.2007.2.425.  Google Scholar

[26]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Math. Sciences, 44,, Springer-Verlag, (1983).   Google Scholar

[27]

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

[28]

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

[29]

B. Said-Houari and Y. Laskri, A stability result of a Timoshenko system with a delay term in the internal feedback,, Applied Mathematics and Computation, 217 (2010), 2857.  doi: 10.1016/j.amc.2010.08.021.  Google Scholar

[30]

D.-H. Shi and D.-X. Feng, Exponential decay of Timoshenko beam with locally distributed feedback,, IMA J. Math. Control Inform., 18 (2001), 395.  doi: 10.1093/imamci/18.3.395.  Google Scholar

[31]

A. Soufyane, Stabilisation de la poutre de Timoshenko,, C. R. Acad. Sci. Paris Sér. I Math., (1999), 731.   Google Scholar

[32]

A. Soufyane, M. Afilal and T. Aouam, General decay of solutions of a nonlinear Timoshenko system with a boundary control of memory type,, Differential Integral Equations, 22 (2009), 1125.   Google Scholar

[33]

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

[34]

A. Wehbe and W. Youssef, Stabilization of the uniform Timoshenko beam by one locally distributed feedback,, Appl. Anal., 88 (2009), 1067.  doi: 10.1080/00036810903156149.  Google Scholar

[35]

G. Q. Xu and S. P. Yung, Exponential decay rate for a Timoshenko beam with boundary damping,, J. Optim. Theory Appl., 123 (2004), 669.  doi: 10.1007/s10957-004-5728-x.  Google Scholar

[36]

G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control,, ESAIM Control Optim. Calc. Var., 12 (2006), 770.  doi: 10.1051/cocv:2006021.  Google Scholar

[1]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[2]

Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020353

[3]

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301

[4]

Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291

[5]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

[6]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045

[7]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[8]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467

[9]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273

[10]

Yuxin Zhang. The spatially heterogeneous diffusive rabies model and its shadow system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020357

[11]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[12]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292

[13]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002

[14]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[15]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[16]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341

[17]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

[18]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[19]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[20]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326

2019 Impact Factor: 0.857

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]