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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 & 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.  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, Special Issue, 1988 (1989), 11-31.  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), 1024-1065.  Google Scholar [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.  Google Scholar [5] A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440.  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), 765-780. 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), 373-379. doi: 10.1137/0316023.  Google Scholar [8] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478. 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), 749-752.  Google Scholar [10] M. Costabel, M. Dauge and S. Nicaise, Singularities of Maxwell interface problems, M2AN Math. Modél. Numer. Anal., 33 (1999), 627-649.  Google Scholar [11] H. D. Fernández Sare, On the stability of Mindlin-Timoshenko plates, Quart. Appl. Math., 67 (2009), 249-263.  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, Berlin, 1986.  Google Scholar [13] P. Grisvard, "Elliptic Problems in Nonsmooth Domains," Monographs and Studies in Mathematics, 24, Pitman (Advanced Publishing Program), Boston, MA, 1985.  Google Scholar [14] P. Grisvard, "Singularities in Boundary Value Problems," Research in Applied Mathematics, 22, Masson, Paris, Springer-Verlag, Berlin, 1992.  Google Scholar [15] F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56.  Google Scholar [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.  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, Paris, 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), Philadelphia, PA, 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), 630-644. 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), 655-671.  Google Scholar [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.  Google Scholar [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.  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), 1561-1585 (electronic). 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), 395-429, 1994.  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), 425-479. 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, New York, 1983.  Google Scholar [27] J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. 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), 535-541. 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), 2857-2869. 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), 395-403. 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., 328 (1999), 731-734.  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), 1125-1139.  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), 1067-1078. 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), 669-693. 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), 770-785 (electronic). 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), 643-669.  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, Special Issue, 1988 (1989), 11-31.  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), 1024-1065.  Google Scholar [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.  Google Scholar [5] A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440.  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), 765-780. 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), 373-379. doi: 10.1137/0316023.  Google Scholar [8] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478. 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), 749-752.  Google Scholar [10] M. Costabel, M. Dauge and S. Nicaise, Singularities of Maxwell interface problems, M2AN Math. Modél. Numer. Anal., 33 (1999), 627-649.  Google Scholar [11] H. D. Fernández Sare, On the stability of Mindlin-Timoshenko plates, Quart. Appl. Math., 67 (2009), 249-263.  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, Berlin, 1986.  Google Scholar [13] P. Grisvard, "Elliptic Problems in Nonsmooth Domains," Monographs and Studies in Mathematics, 24, Pitman (Advanced Publishing Program), Boston, MA, 1985.  Google Scholar [14] P. Grisvard, "Singularities in Boundary Value Problems," Research in Applied Mathematics, 22, Masson, Paris, Springer-Verlag, Berlin, 1992.  Google Scholar [15] F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56.  Google Scholar [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.  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, Paris, 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), Philadelphia, PA, 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), 630-644. 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), 655-671.  Google Scholar [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.  Google Scholar [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.  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), 1561-1585 (electronic). 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), 395-429, 1994.  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), 425-479. 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, New York, 1983.  Google Scholar [27] J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. 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), 535-541. 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), 2857-2869. 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), 395-403. 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., 328 (1999), 731-734.  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), 1125-1139.  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), 1067-1078. 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), 669-693. 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), 770-785 (electronic). doi: 10.1051/cocv:2006021.  Google Scholar
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