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A variational approach to approximate controls for system with essential spectrum: Application to membranal arch
March  2013, 2(1): 153-172. doi: 10.3934/eect.2013.2.153

## Simultaneous stabilization of a system of interacting plate and membrane

 1 Department of Mathematics & Statistics, Florida International University, Miami, FL 33199

Received  July 2012 Revised  September 2012 Published  January 2013

We consider a system consisting of a wave equation and a plate equation, where the rotational forces may be accounted for, in a bounded domain. The system is coupled through the dissipation which is localized in an appropriate portion of the domain under consideration. First, we show that this system is strongly stable; the feedback control region is arbitrarily small when the rotational inertia is greater than or equal to one, but it is required that its boundary intersect a nonnegligible portion of the boundary of the domain under consideration if the rotational inertia is less than one. Next we show that the system accounting for rotational forces is not exponentially stable. Afterwards, using a constructive frequency domain method, we show that the system with no rotational forces is exponentially stable provided that the feedback control region is large enough. New uniqueness and controllability results are derived.
Citation: Louis Tebou. Simultaneous stabilization of a system of interacting plate and membrane. Evolution Equations and Control Theory, 2013, 2 (1) : 153-172. doi: 10.3934/eect.2013.2.153
##### References:
 [1] F. Alabau-Boussouira, Piecewise multiplier method and nonlinear integral inequalities for Petrowsky equation with nonlinear dissipation. J. Evol. Equ. 6 (2006), 95-112. doi: 10.1007/s00028-005-0230-y. [2] F. Alabau-Boussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM J. Control Optim., 41 (2002), 511-541. doi: 10.1137/S0363012901385368. [3] F. Alabau, P. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled systems, J. Evolution Equations, 2 (2002), 127-150. doi: 10.1007/s00028-002-8083-0. [4] W. Arendt, C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups. Trans. Amer. Math. Soc. 306 (1988), 837-852. doi: 10.2307/2000826. [5] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization from the boundary, SIAM J. Control and Opt. 30 (1992), 1024-1065. doi: 10.1137/0330055. [6] A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440. doi: 10.1002/mana.200410429. [7] H. Brezis, "Analyse fonctionnelle," Théorie et Applications. Masson, Paris 1983. [8] G. Chen, Control and stabilization for the wave equation in a bounded domain. SIAM J. Control Optim. 17 (1979), 66-81. doi: 10.1137/0317007. [9] G. Chen, S.A. Fulling, F.J. Narcowich, and S. Sun, Exponential decay of energy of evolution equations with locally distributed damping, SIAM J. Appl. Math. 51 (1991), 266-301. doi: 10.1137/0151015. [10] C.M. Dafermos, Asymptotic behaviour of solutions of evolution equations, in "Nonlinear evolution equations"(M.G. Crandall ed.) pp. 103-123, Academic Press, New-York, 1978. [11] X. Fu, Longtime behavior of the hyperbolic equations with an arbitrary internal damping. Z. Angew. Math. Phys. 62 (2011), 667-680. doi: 10.1007/s00033-010-0113-0. [12] X. Fu, Logarithmic decay of hyperbolic equations with arbitrary small boundary damping. Comm. Partial Differential Equations 34 (2009), 957-975. doi: 10.1080/03605300903116389. [13] J. S. Gibson, A note on stabilization of infinite dimensional linear oscillators by compact linear feedback, SIAM J. Control Optim., 18 (1980), 311-316. doi: 10.1137/0318022. [14] A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations, J. Differential Equations, 59(1985), 145-154. doi: 10.1016/0022-0396(85)90151-2. [15] A. Haraux, On a completion problem in the theory of distributed control of wave equations. Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. X (Paris, 1987-1988), 241-271, Pitman Res. Notes Math. Ser., 220, Longman Sci. Tech., Harlow, 1991. [16] A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Portugal. Math. 46 (1989), 245-258. [17] F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. [18] V. Komornik, Rapid boundary stabilization of the wave equation. SIAM J. Control Optim. 29 (1991), 197-208. doi: 10.1137/0329011. [19] V. Komornik, "Exact controllability and stabilization. The multiplier method," RAM, Masson & John Wiley, Paris, 1994. [20] V. Komornik, Rapid boundary stabilization of linear distributed systems. SIAM J. Control Optim. 35 (1997), 1591-1613. doi: 10.1137/S0363012996301609. [21] V. Komornik, E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl. 69 (1990), 33-54. [22] J. Lagnese, Boundary stabilization of linear elastodynamic systems, S.I.A.M J. Control and Opt., 21 (1983), 968-984. doi: 10.1137/0321059. [23] J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation. J. Differential Equations 50 (1983), 163-182. doi: 10.1016/0022-0396(83)90073-6. [24] J. Lagnese, "Boundary Stabilization of Thin Plates," SIAM Stud. Appl. Math. 10, SIAM, Philadelphia, PA, 1989. doi: 10.1137/1.9781611970821. [25] I. Lasiecka, Exponential decay rates for the solutions of Euler-Bernoulli equations with boundary dissipation occurring in the moments only. J. Differential Equations 95 (1992), 169-182. doi: 10.1016/0022-0396(92)90048-R. [26] I. Lasiecka, Nonlinear boundary feedback stabilization of dynamic elasticity with thermal effects. Shape optimization and optimal design (Cambridge, 1999), 333-354, Lecture Notes in Pure and Appl. Math., 216, Dekker, New York, 2001. [27] I. Lasiecka, D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differential Integral Equations 6 (1993), 507-533. [28] I. Lasiecka, D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms. Nonlinear Anal. 64(2006), 1757-1797. doi: 10.1016/j.na.2005.07.024. [29] I. Lasiecka, R. Triggiani, Uniform exponential energy decay of wave equations in a bounded region with $L^2 (0,\infty;L^2 (\Gamma))$ -feedback control in the Dirichlet boundary conditions. J. Differential Equations 66 (1987), 340-390. doi: 10.1016/0022-0396(87)90025-8. [30] I. Lasiecka, R. Triggiani, Exact controllability of the wave equation with Neumann boundary control. Appl. Math. Optim. 19 (1989), 243-290. doi: 10.1007/BF01448201. [31] I. Lasiecka, R. Triggiani, Exact controllability and uniform stabilization of Kirchoff [Kirchhoff] plates with boundary control only on $\Delta w|_\Sigma$ and homogeneous boundary displacement. J. Differential Equations 93 (1991), 62-101. doi: 10.1016/0022-0396(91)90022-2. [32] I. Lasiecka, R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions. Appl. Math. Optim. 25 (1992), 189-224. doi: 10.1007/BF01182480. [33] I. Lasiecka, R. Triggiani, $L^2 (\Sigma)$-regularity of the boundary to boundary operator $B * L$ for hyperbolic and Petrowski PDEs. Abstr. Appl. Anal. 2003, 19 (2003), 1061-1139. doi: 10.1155/S1085337503305032. [34] G. Lebeau, Equation des ondes amorties. Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), 73-109, Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996. [35] G. Lebeau, L. Robbiano, Stabilisation de l'équation des ondes par le bord. Duke Math. J. 86 (1997), 465-491. doi: 10.1215/S0012-7094-97-08614-2. [36] J.-L. Lions, "Contrôlabilité exacte, Perturbations et Stabilisation des Systèmes Distribués," Vol. 1, RMA 8, Masson, Paris, 1988. [37] K. Liu, Locally distributed control and damping for the conservative systems, S.I.A.M J. Control and Opt. 35 (1997), 1574-1590. doi: 10.1137/S0363012995284928. [38] K. Liu, B. Rao, Exponential stability for the wave equations with local Kelvin-Voigt damping. Z. Angew. Math. Phys. 57 (2006), 419-432. doi: 10.1007/s00033-005-0029-2. [39] P. Martinez, PhD thesis, University of Strasbourg, 1998. [40] P. Martinez, Boundary stabilization of the wave equation in almost star-shaped domains. SIAM J. Control Optim. 37 (1999), 673-94. doi: 10.1137/S0363012997323722. [41] M. Nakao, Decay of solutions of the wave equation with a local degenerate dissipation, Israel J. Math. 95 (1996), 25-42. doi: 10.1007/BF02761033. [42] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [43] K. D. Phung, Polynomial decay rate for the dissipative wave equation. J. Differential Equations 240 (2007), 92-124. doi: 10.1016/j.jde.2007.05.016. [44] K. D. Phung, Boundary stabilization for the wave equation in a bounded cylindrical domain. Discrete Contin. Dyn. Syst. 20 (2008), 1057-1093. doi: 10.3934/dcds.2008.20.1057. [45] J. Prüss, On the spectrum of $C_0$-semigroups. Trans. Amer. Math. Soc. 284 (1984), 847-857. doi: 10.2307/1999112. [46] J. Rauch, M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains. Indiana Univ. Math. J. 24 (1974), 79-86. [47] D.L. Russell, Control theory of hyperbolic equations related to certain questions in harmonic analysis and spectral theory, J. Math. Anal. Appl., 40 (1972), 336-368. [48] D.L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev. 20 (1978), no. 4, 639-739. doi: 10.1137/1020095. [49] D.L. Russell, The Dirichlet-Neumann boundary control problem associated with Maxwell's equations in a cylindrical region. SIAM J. Control Optim. 24 (1986), 199-229. doi: 10.1137/0324012. [50] M. Slemrod, Weak asymptotic decay via a "Relaxed invariance principle" for a wave equation with nonlinear, nonmonotone damping. Proc. Royal Soc. Edinburgh Sect. A 113(1989), 87-97. doi: 10.1017/S0308210500023970. [51] L.R. Tcheugoué Tébou, Sur la stabilisation de l'équation des ondes en dimension 2. C. R. Acad. Sci. Paris Ser. I Math. 319 (1994), no. 6, 585-588. [52] L.R. Tcheugoué Tébou, On the decay estimates for the wave equation with a local degenerate or nondegenerate dissipation, Portugal. Math.,55 (1998), 293-306. [53] L.R. Tcheugoué Tébou, Stabilization of the wave equation with localized nonlinear damping, J.D.E. 145(1998), 502-524. doi: 10.1006/jdeq.1998.3416. [54] L.R. Tcheugoué Tébou, Well-posedness and energy decay estimates for the damped wave equation with L$^r$ localizing coefficient, Comm. in P.D.E., 23 (1998), 1839-1855. doi: 10.1080/03605309808821403. [55] L. R. Tcheugoué Tébou, Energy decay estimates for the damped Euler-Bernoulli equation with an unbounded localizing coefficient, Portugal. Math. 61 (2004), 375-391. [56] L. R. Tcheugoue Tebou, On the stabilization of dynamic elasticity equations with unbounded locally distributed dissipation. Differential Integral Equations 19 (2006), 785-798. [57] L. Tebou, Stabilization of the elastodynamic equations with a degenerate locally distributed dissipation. Systems Control Lett. 56 (2007), 538-545. doi: 10.1016/j.sysconle.2007.03.003. [58] L. Tebou, Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the p-Laplacian. DCDS A, 32 (2012), 2315-2337. doi: 10.3934/dcds.2012.32.2315. [59] L. Tebou, Simultaneous observability and stabilization of some uncoupled wave equations. C. R. Acad. Sci. Paris, Ser. I, 350 (2012), 57-62. doi: 10.1016/j.crma.2011.12.001. [60] L. Tebou, Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms. MCRF, 2 (2012), 45-60. doi: 10.3934/mcrf.2012.2.45. [61] R. Triggiani, Lack of uniform stabilization for noncontractive semigroups under compact perturbation, Proc. Amer. Math. Soc., 105 (1989), 375-383. doi: 10.2307/2046953. [62] E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback. SIAM J. Control Optim. 28 (1990), 466-477. doi: 10.1137/0328025. [63] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Commun. P.D.E., 15 (1990), 205-235. doi: 10.1080/03605309908820684. [64] E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains, J. Math. Pures. Appl., 70 (1991), 513-529.

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##### References:
 [1] F. Alabau-Boussouira, Piecewise multiplier method and nonlinear integral inequalities for Petrowsky equation with nonlinear dissipation. J. Evol. Equ. 6 (2006), 95-112. doi: 10.1007/s00028-005-0230-y. [2] F. Alabau-Boussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM J. Control Optim., 41 (2002), 511-541. doi: 10.1137/S0363012901385368. [3] F. Alabau, P. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled systems, J. Evolution Equations, 2 (2002), 127-150. doi: 10.1007/s00028-002-8083-0. [4] W. Arendt, C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups. Trans. Amer. Math. Soc. 306 (1988), 837-852. doi: 10.2307/2000826. [5] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization from the boundary, SIAM J. Control and Opt. 30 (1992), 1024-1065. doi: 10.1137/0330055. [6] A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440. doi: 10.1002/mana.200410429. [7] H. Brezis, "Analyse fonctionnelle," Théorie et Applications. Masson, Paris 1983. [8] G. Chen, Control and stabilization for the wave equation in a bounded domain. SIAM J. Control Optim. 17 (1979), 66-81. doi: 10.1137/0317007. [9] G. Chen, S.A. Fulling, F.J. Narcowich, and S. Sun, Exponential decay of energy of evolution equations with locally distributed damping, SIAM J. Appl. Math. 51 (1991), 266-301. doi: 10.1137/0151015. [10] C.M. Dafermos, Asymptotic behaviour of solutions of evolution equations, in "Nonlinear evolution equations"(M.G. Crandall ed.) pp. 103-123, Academic Press, New-York, 1978. [11] X. Fu, Longtime behavior of the hyperbolic equations with an arbitrary internal damping. Z. Angew. Math. Phys. 62 (2011), 667-680. doi: 10.1007/s00033-010-0113-0. [12] X. Fu, Logarithmic decay of hyperbolic equations with arbitrary small boundary damping. Comm. Partial Differential Equations 34 (2009), 957-975. doi: 10.1080/03605300903116389. [13] J. S. Gibson, A note on stabilization of infinite dimensional linear oscillators by compact linear feedback, SIAM J. Control Optim., 18 (1980), 311-316. doi: 10.1137/0318022. [14] A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations, J. Differential Equations, 59(1985), 145-154. doi: 10.1016/0022-0396(85)90151-2. [15] A. Haraux, On a completion problem in the theory of distributed control of wave equations. Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. X (Paris, 1987-1988), 241-271, Pitman Res. Notes Math. Ser., 220, Longman Sci. Tech., Harlow, 1991. [16] A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Portugal. Math. 46 (1989), 245-258. [17] F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. [18] V. Komornik, Rapid boundary stabilization of the wave equation. SIAM J. Control Optim. 29 (1991), 197-208. doi: 10.1137/0329011. [19] V. Komornik, "Exact controllability and stabilization. The multiplier method," RAM, Masson & John Wiley, Paris, 1994. [20] V. Komornik, Rapid boundary stabilization of linear distributed systems. SIAM J. Control Optim. 35 (1997), 1591-1613. doi: 10.1137/S0363012996301609. [21] V. Komornik, E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl. 69 (1990), 33-54. [22] J. Lagnese, Boundary stabilization of linear elastodynamic systems, S.I.A.M J. Control and Opt., 21 (1983), 968-984. doi: 10.1137/0321059. [23] J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation. J. Differential Equations 50 (1983), 163-182. doi: 10.1016/0022-0396(83)90073-6. [24] J. Lagnese, "Boundary Stabilization of Thin Plates," SIAM Stud. Appl. Math. 10, SIAM, Philadelphia, PA, 1989. doi: 10.1137/1.9781611970821. [25] I. Lasiecka, Exponential decay rates for the solutions of Euler-Bernoulli equations with boundary dissipation occurring in the moments only. J. Differential Equations 95 (1992), 169-182. doi: 10.1016/0022-0396(92)90048-R. [26] I. Lasiecka, Nonlinear boundary feedback stabilization of dynamic elasticity with thermal effects. Shape optimization and optimal design (Cambridge, 1999), 333-354, Lecture Notes in Pure and Appl. Math., 216, Dekker, New York, 2001. [27] I. Lasiecka, D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differential Integral Equations 6 (1993), 507-533. [28] I. Lasiecka, D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms. Nonlinear Anal. 64(2006), 1757-1797. doi: 10.1016/j.na.2005.07.024. [29] I. Lasiecka, R. Triggiani, Uniform exponential energy decay of wave equations in a bounded region with $L^2 (0,\infty;L^2 (\Gamma))$ -feedback control in the Dirichlet boundary conditions. J. Differential Equations 66 (1987), 340-390. doi: 10.1016/0022-0396(87)90025-8. [30] I. Lasiecka, R. Triggiani, Exact controllability of the wave equation with Neumann boundary control. Appl. Math. Optim. 19 (1989), 243-290. doi: 10.1007/BF01448201. [31] I. Lasiecka, R. Triggiani, Exact controllability and uniform stabilization of Kirchoff [Kirchhoff] plates with boundary control only on $\Delta w|_\Sigma$ and homogeneous boundary displacement. J. Differential Equations 93 (1991), 62-101. doi: 10.1016/0022-0396(91)90022-2. [32] I. Lasiecka, R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions. Appl. Math. Optim. 25 (1992), 189-224. doi: 10.1007/BF01182480. [33] I. Lasiecka, R. Triggiani, $L^2 (\Sigma)$-regularity of the boundary to boundary operator $B * L$ for hyperbolic and Petrowski PDEs. Abstr. Appl. Anal. 2003, 19 (2003), 1061-1139. doi: 10.1155/S1085337503305032. [34] G. Lebeau, Equation des ondes amorties. Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), 73-109, Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996. [35] G. Lebeau, L. Robbiano, Stabilisation de l'équation des ondes par le bord. Duke Math. J. 86 (1997), 465-491. doi: 10.1215/S0012-7094-97-08614-2. [36] J.-L. Lions, "Contrôlabilité exacte, Perturbations et Stabilisation des Systèmes Distribués," Vol. 1, RMA 8, Masson, Paris, 1988. [37] K. Liu, Locally distributed control and damping for the conservative systems, S.I.A.M J. Control and Opt. 35 (1997), 1574-1590. doi: 10.1137/S0363012995284928. [38] K. Liu, B. Rao, Exponential stability for the wave equations with local Kelvin-Voigt damping. Z. Angew. Math. Phys. 57 (2006), 419-432. doi: 10.1007/s00033-005-0029-2. [39] P. Martinez, PhD thesis, University of Strasbourg, 1998. [40] P. Martinez, Boundary stabilization of the wave equation in almost star-shaped domains. SIAM J. Control Optim. 37 (1999), 673-94. doi: 10.1137/S0363012997323722. [41] M. Nakao, Decay of solutions of the wave equation with a local degenerate dissipation, Israel J. Math. 95 (1996), 25-42. doi: 10.1007/BF02761033. [42] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [43] K. D. Phung, Polynomial decay rate for the dissipative wave equation. J. Differential Equations 240 (2007), 92-124. doi: 10.1016/j.jde.2007.05.016. [44] K. D. Phung, Boundary stabilization for the wave equation in a bounded cylindrical domain. Discrete Contin. Dyn. Syst. 20 (2008), 1057-1093. doi: 10.3934/dcds.2008.20.1057. [45] J. Prüss, On the spectrum of $C_0$-semigroups. Trans. Amer. Math. Soc. 284 (1984), 847-857. doi: 10.2307/1999112. [46] J. Rauch, M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains. Indiana Univ. Math. J. 24 (1974), 79-86. [47] D.L. Russell, Control theory of hyperbolic equations related to certain questions in harmonic analysis and spectral theory, J. Math. Anal. Appl., 40 (1972), 336-368. [48] D.L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev. 20 (1978), no. 4, 639-739. doi: 10.1137/1020095. [49] D.L. Russell, The Dirichlet-Neumann boundary control problem associated with Maxwell's equations in a cylindrical region. SIAM J. Control Optim. 24 (1986), 199-229. doi: 10.1137/0324012. [50] M. Slemrod, Weak asymptotic decay via a "Relaxed invariance principle" for a wave equation with nonlinear, nonmonotone damping. Proc. Royal Soc. Edinburgh Sect. A 113(1989), 87-97. doi: 10.1017/S0308210500023970. [51] L.R. Tcheugoué Tébou, Sur la stabilisation de l'équation des ondes en dimension 2. C. R. Acad. Sci. Paris Ser. I Math. 319 (1994), no. 6, 585-588. [52] L.R. Tcheugoué Tébou, On the decay estimates for the wave equation with a local degenerate or nondegenerate dissipation, Portugal. Math.,55 (1998), 293-306. [53] L.R. Tcheugoué Tébou, Stabilization of the wave equation with localized nonlinear damping, J.D.E. 145(1998), 502-524. doi: 10.1006/jdeq.1998.3416. [54] L.R. Tcheugoué Tébou, Well-posedness and energy decay estimates for the damped wave equation with L$^r$ localizing coefficient, Comm. in P.D.E., 23 (1998), 1839-1855. doi: 10.1080/03605309808821403. [55] L. R. Tcheugoué Tébou, Energy decay estimates for the damped Euler-Bernoulli equation with an unbounded localizing coefficient, Portugal. Math. 61 (2004), 375-391. [56] L. R. Tcheugoue Tebou, On the stabilization of dynamic elasticity equations with unbounded locally distributed dissipation. Differential Integral Equations 19 (2006), 785-798. [57] L. Tebou, Stabilization of the elastodynamic equations with a degenerate locally distributed dissipation. Systems Control Lett. 56 (2007), 538-545. doi: 10.1016/j.sysconle.2007.03.003. [58] L. Tebou, Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the p-Laplacian. DCDS A, 32 (2012), 2315-2337. doi: 10.3934/dcds.2012.32.2315. [59] L. Tebou, Simultaneous observability and stabilization of some uncoupled wave equations. C. R. Acad. Sci. Paris, Ser. I, 350 (2012), 57-62. doi: 10.1016/j.crma.2011.12.001. [60] L. Tebou, Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms. MCRF, 2 (2012), 45-60. doi: 10.3934/mcrf.2012.2.45. [61] R. Triggiani, Lack of uniform stabilization for noncontractive semigroups under compact perturbation, Proc. Amer. Math. Soc., 105 (1989), 375-383. doi: 10.2307/2046953. [62] E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback. SIAM J. Control Optim. 28 (1990), 466-477. doi: 10.1137/0328025. [63] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Commun. P.D.E., 15 (1990), 205-235. doi: 10.1080/03605309908820684. [64] E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains, J. Math. Pures. Appl., 70 (1991), 513-529.
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