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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 and 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), 643-669.

[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, Special Issue, 1988 (1989), 11-31.

[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), 1024-1065.

[4]

C. Bardos, T. Masrour and F. Tatout, Observation and control of elastic waves, in "Singularities and Oscillations" (Minneapolis, MN, 1994/1995), 1-16, IMA Vol. Math. Appl., 91, Springer, New York, 1997.

[5]

A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440.

[6]

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.

[7]

C. D. Benchimol, A note on weak stabilizability of contraction semigroups, SIAM J. Control Optimization, 16 (1978), 373-379. doi: 10.1137/0316023.

[8]

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.

[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), 749-752.

[10]

M. Costabel, M. Dauge and S. Nicaise, Singularities of Maxwell interface problems, M2AN Math. Modél. Numer. Anal., 33 (1999), 627-649.

[11]

H. D. Fernández Sare, On the stability of Mindlin-Timoshenko plates, Quart. Appl. Math., 67 (2009), 249-263.

[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, Berlin, 1986.

[13]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains," Monographs and Studies in Mathematics, 24, Pitman (Advanced Publishing Program), Boston, MA, 1985.

[14]

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

[15]

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

[16]

J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam, SIAM J. Control Optim., 25 (1987), 1417-1429. doi: 10.1137/0325078.

[17]

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

[18]

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

[19]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644. doi: 10.1007/s00033-004-3073-4.

[20]

S. A. Messaoudi and M. I. Mustafa, On the internal and boundary stabilization of Timoshenko beams, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 655-671.

[21]

J. E. Muñoz Rivera and R. Racke, Global stability for damped Timoshenko systems, Discrete Contin. Dyn. Syst., 9 (2003), 1625-1639. doi: 10.3934/dcds.2003.9.1625.

[22]

J. E. Muñoz Rivera and R. Racke, Timoshenko systems with indefinite damping, J. Math. Anal. Appl., 341 (2008), 1068-1083. doi: 10.1016/j.jmaa.2007.11.012.

[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), 1561-1585 (electronic). doi: 10.1137/060648891.

[24]

S. Nicaise and A.-M. Sändig, General interface problems I, Math. Methods in the Appl. Sc., 17 (1994), 395-429, 1994.

[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), 425-479. doi: 10.3934/nhm.2007.2.425.

[26]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Math. Sciences, 44, Springer-Verlag, New York, 1983.

[27]

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

[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), 535-541. doi: 10.1016/j.aml.2004.03.017.

[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), 2857-2869. doi: 10.1016/j.amc.2010.08.021.

[30]

D.-H. Shi and D.-X. Feng, Exponential decay of Timoshenko beam with locally distributed feedback, IMA J. Math. Control Inform., 18 (2001), 395-403. doi: 10.1093/imamci/18.3.395.

[31]

A. Soufyane, Stabilisation de la poutre de Timoshenko, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 731-734.

[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), 1125-1139.

[33]

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

[34]

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.

[35]

G. Q. Xu and S. P. Yung, Exponential decay rate for a Timoshenko beam with boundary damping, J. Optim. Theory Appl., 123 (2004), 669-693. doi: 10.1007/s10957-004-5728-x.

[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), 770-785 (electronic). doi: 10.1051/cocv:2006021.

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), 643-669.

[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, Special Issue, 1988 (1989), 11-31.

[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), 1024-1065.

[4]

C. Bardos, T. Masrour and F. Tatout, Observation and control of elastic waves, in "Singularities and Oscillations" (Minneapolis, MN, 1994/1995), 1-16, IMA Vol. Math. Appl., 91, Springer, New York, 1997.

[5]

A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440.

[6]

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.

[7]

C. D. Benchimol, A note on weak stabilizability of contraction semigroups, SIAM J. Control Optimization, 16 (1978), 373-379. doi: 10.1137/0316023.

[8]

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.

[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), 749-752.

[10]

M. Costabel, M. Dauge and S. Nicaise, Singularities of Maxwell interface problems, M2AN Math. Modél. Numer. Anal., 33 (1999), 627-649.

[11]

H. D. Fernández Sare, On the stability of Mindlin-Timoshenko plates, Quart. Appl. Math., 67 (2009), 249-263.

[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, Berlin, 1986.

[13]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains," Monographs and Studies in Mathematics, 24, Pitman (Advanced Publishing Program), Boston, MA, 1985.

[14]

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

[15]

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

[16]

J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam, SIAM J. Control Optim., 25 (1987), 1417-1429. doi: 10.1137/0325078.

[17]

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

[18]

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

[19]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644. doi: 10.1007/s00033-004-3073-4.

[20]

S. A. Messaoudi and M. I. Mustafa, On the internal and boundary stabilization of Timoshenko beams, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 655-671.

[21]

J. E. Muñoz Rivera and R. Racke, Global stability for damped Timoshenko systems, Discrete Contin. Dyn. Syst., 9 (2003), 1625-1639. doi: 10.3934/dcds.2003.9.1625.

[22]

J. E. Muñoz Rivera and R. Racke, Timoshenko systems with indefinite damping, J. Math. Anal. Appl., 341 (2008), 1068-1083. doi: 10.1016/j.jmaa.2007.11.012.

[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), 1561-1585 (electronic). doi: 10.1137/060648891.

[24]

S. Nicaise and A.-M. Sändig, General interface problems I, Math. Methods in the Appl. Sc., 17 (1994), 395-429, 1994.

[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), 425-479. doi: 10.3934/nhm.2007.2.425.

[26]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Math. Sciences, 44, Springer-Verlag, New York, 1983.

[27]

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

[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), 535-541. doi: 10.1016/j.aml.2004.03.017.

[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), 2857-2869. doi: 10.1016/j.amc.2010.08.021.

[30]

D.-H. Shi and D.-X. Feng, Exponential decay of Timoshenko beam with locally distributed feedback, IMA J. Math. Control Inform., 18 (2001), 395-403. doi: 10.1093/imamci/18.3.395.

[31]

A. Soufyane, Stabilisation de la poutre de Timoshenko, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 731-734.

[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), 1125-1139.

[33]

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

[34]

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.

[35]

G. Q. Xu and S. P. Yung, Exponential decay rate for a Timoshenko beam with boundary damping, J. Optim. Theory Appl., 123 (2004), 669-693. doi: 10.1007/s10957-004-5728-x.

[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), 770-785 (electronic). doi: 10.1051/cocv:2006021.

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