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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 & 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), 6 (2006), 95.  doi: 10.1007/s00028-005-0230-y.  Google Scholar [2] F. Alabau-Boussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems,, SIAM J. Control Optim., 41 (2002), 511.  doi: 10.1137/S0363012901385368.  Google Scholar [3] F. Alabau, P. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled systems,, J. Evolution Equations, 2 (2002), 127.  doi: 10.1007/s00028-002-8083-0.  Google Scholar [4] W. Arendt, C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups., Trans. Amer. Math. Soc. 306 (1988), 306 (1988), 837.  doi: 10.2307/2000826.  Google Scholar [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), 30 (1992), 1024.  doi: 10.1137/0330055.  Google Scholar [6] A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups,, Math. Nachr., 279 (2006), 1425.  doi: 10.1002/mana.200410429.  Google Scholar [7] H. Brezis, "Analyse fonctionnelle,", Théorie et Applications. Masson, (1983).   Google Scholar [8] G. Chen, Control and stabilization for the wave equation in a bounded domain., SIAM J. Control Optim. 17 (1979), 17 (1979), 66.  doi: 10.1137/0317007.  Google Scholar [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), 51 (1991), 266.  doi: 10.1137/0151015.  Google Scholar [10] C.M. Dafermos, Asymptotic behaviour of solutions of evolution equations,, in, (1978), 103.   Google Scholar [11] X. Fu, Longtime behavior of the hyperbolic equations with an arbitrary internal damping., Z. Angew. Math. Phys. 62 (2011), 62 (2011), 667.  doi: 10.1007/s00033-010-0113-0.  Google Scholar [12] X. Fu, Logarithmic decay of hyperbolic equations with arbitrary small boundary damping., Comm. Partial Differential Equations 34 (2009), 34 (2009), 957.  doi: 10.1080/03605300903116389.  Google Scholar [13] J. S. Gibson, A note on stabilization of infinite dimensional linear oscillators by compact linear feedback,, SIAM J. Control Optim., 18 (1980), 311.  doi: 10.1137/0318022.  Google Scholar [14] A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations,, J. Differential Equations, 59 (1985), 145.  doi: 10.1016/0022-0396(85)90151-2.  Google Scholar [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, (1991), 1987.   Google Scholar [16] A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps,, Portugal. Math. 46 (1989), 46 (1989), 245.   Google Scholar [17] F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces,, Ann. Differential Equations, 1 (1985), 43.   Google Scholar [18] V. Komornik, Rapid boundary stabilization of the wave equation., SIAM J. Control Optim. 29 (1991), 29 (1991), 197.  doi: 10.1137/0329011.  Google Scholar [19] V. Komornik, "Exact controllability and stabilization. The multiplier method,", RAM, (1994).   Google Scholar [20] V. Komornik, Rapid boundary stabilization of linear distributed systems., SIAM J. Control Optim. 35 (1997), 35 (1997), 1591.  doi: 10.1137/S0363012996301609.  Google Scholar [21] V. Komornik, E. Zuazua, A direct method for the boundary stabilization of the wave equation,, J. Math. Pures Appl. 69 (1990), 69 (1990), 33.   Google Scholar [22] J. Lagnese, Boundary stabilization of linear elastodynamic systems,, S.I.A.M J. Control and Opt., 21 (1983), 968.  doi: 10.1137/0321059.  Google Scholar [23] J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation., J. Differential Equations 50 (1983), 50 (1983), 163.  doi: 10.1016/0022-0396(83)90073-6.  Google Scholar [24] J. Lagnese, "Boundary Stabilization of Thin Plates,", SIAM Stud. Appl. Math. 10, (1989).  doi: 10.1137/1.9781611970821.  Google Scholar [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), 95 (1992), 169.  doi: 10.1016/0022-0396(92)90048-R.  Google Scholar [26] I. Lasiecka, Nonlinear boundary feedback stabilization of dynamic elasticity with thermal effects., Shape optimization and optimal design (Cambridge, (1999), 333.   Google Scholar [27] I. Lasiecka, D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping., Differential Integral Equations 6 (1993), 6 (1993), 507.   Google Scholar [28] I. Lasiecka, D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms., Nonlinear Anal. 64(2006), 64 (2006), 1757.  doi: 10.1016/j.na.2005.07.024.  Google Scholar [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), 66 (1987), 340.  doi: 10.1016/0022-0396(87)90025-8.  Google Scholar [30] I. Lasiecka, R. Triggiani, Exact controllability of the wave equation with Neumann boundary control., Appl. Math. Optim. 19 (1989), 19 (1989), 243.  doi: 10.1007/BF01448201.  Google Scholar [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), 93 (1991), 62.  doi: 10.1016/0022-0396(91)90022-2.  Google Scholar [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), 25 (1992), 189.  doi: 10.1007/BF01182480.  Google Scholar [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.  doi: 10.1155/S1085337503305032.  Google Scholar [34] G. Lebeau, Equation des ondes amorties., Algebraic and geometric methods in mathematical physics (Kaciveli, 19 (1993), 73.   Google Scholar [35] G. Lebeau, L. Robbiano, Stabilisation de l'équation des ondes par le bord., Duke Math. J. 86 (1997), 86 (1997), 465.  doi: 10.1215/S0012-7094-97-08614-2.  Google Scholar [36] J.-L. Lions, "Contrôlabilité exacte, Perturbations et Stabilisation des Systèmes Distribués,", Vol. 1, (1988).   Google Scholar [37] K. Liu, Locally distributed control and damping for the conservative systems,, S.I.A.M J. Control and Opt. 35 (1997), 35 (1997), 1574.  doi: 10.1137/S0363012995284928.  Google Scholar [38] K. Liu, B. Rao, Exponential stability for the wave equations with local Kelvin-Voigt damping., Z. Angew. Math. Phys. 57 (2006), 57 (2006), 419.  doi: 10.1007/s00033-005-0029-2.  Google Scholar [39] P. Martinez, PhD thesis,, University of Strasbourg, (1998).   Google Scholar [40] P. Martinez, Boundary stabilization of the wave equation in almost star-shaped domains., SIAM J. Control Optim. 37 (1999), 37 (1999), 673.  doi: 10.1137/S0363012997323722.  Google Scholar [41] M. Nakao, Decay of solutions of the wave equation with a local degenerate dissipation,, Israel J. Math. 95 (1996), 95 (1996), 25.  doi: 10.1007/BF02761033.  Google Scholar [42] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar [43] K. D. Phung, Polynomial decay rate for the dissipative wave equation., J. Differential Equations 240 (2007), 240 (2007), 92.  doi: 10.1016/j.jde.2007.05.016.  Google Scholar [44] K. D. Phung, Boundary stabilization for the wave equation in a bounded cylindrical domain., Discrete Contin. Dyn. Syst. 20 (2008), 20 (2008), 1057.  doi: 10.3934/dcds.2008.20.1057.  Google Scholar [45] J. Prüss, On the spectrum of $C_0$-semigroups., Trans. Amer. Math. Soc. 284 (1984), 284 (1984), 847.  doi: 10.2307/1999112.  Google Scholar [46] J. Rauch, M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains., Indiana Univ. Math. J. 24 (1974), 24 (1974), 79.   Google Scholar [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.   Google Scholar [48] D.L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions., SIAM Rev. 20 (1978), 20 (1978), 639.  doi: 10.1137/1020095.  Google Scholar [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), 24 (1986), 199.  doi: 10.1137/0324012.  Google Scholar [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), (1989), 87.  doi: 10.1017/S0308210500023970.  Google Scholar [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), 319 (1994), 585.   Google Scholar [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.   Google Scholar [53] L.R. Tcheugoué Tébou, Stabilization of the wave equation with localized nonlinear damping,, J.D.E. 145(1998), 145 (1998), 502.  doi: 10.1006/jdeq.1998.3416.  Google Scholar [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.  doi: 10.1080/03605309808821403.  Google Scholar [55] L. R. Tcheugoué Tébou, Energy decay estimates for the damped Euler-Bernoulli equation with an unbounded localizing coefficient,, Portugal. Math. 61 (2004), 61 (2004), 375.   Google Scholar [56] L. R. Tcheugoue Tebou, On the stabilization of dynamic elasticity equations with unbounded locally distributed dissipation., Differential Integral Equations 19 (2006), 19 (2006), 785.   Google Scholar [57] L. Tebou, Stabilization of the elastodynamic equations with a degenerate locally distributed dissipation., Systems Control Lett. 56 (2007), 56 (2007), 538.  doi: 10.1016/j.sysconle.2007.03.003.  Google Scholar [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.  doi: 10.3934/dcds.2012.32.2315.  Google Scholar [59] L. Tebou, Simultaneous observability and stabilization of some uncoupled wave equations., C. R. Acad. Sci. Paris, 350 (2012), 57.  doi: 10.1016/j.crma.2011.12.001.  Google Scholar [60] L. Tebou, Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms., MCRF, 2 (2012), 45.  doi: 10.3934/mcrf.2012.2.45.  Google Scholar [61] R. Triggiani, Lack of uniform stabilization for noncontractive semigroups under compact perturbation,, Proc. Amer. Math. Soc., 105 (1989), 375.  doi: 10.2307/2046953.  Google Scholar [62] E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback., SIAM J. Control Optim. 28 (1990), 28 (1990), 466.  doi: 10.1137/0328025.  Google Scholar [63] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping,, Commun. P.D.E., 15 (1990), 205.  doi: 10.1080/03605309908820684.  Google Scholar [64] E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains,, J. Math. Pures. Appl., 70 (1991), 513.   Google Scholar

show all references

##### References:
 [1] F. Alabau-Boussouira, Piecewise multiplier method and nonlinear integral inequalities for Petrowsky equation with nonlinear dissipation., J. Evol. Equ. 6 (2006), 6 (2006), 95.  doi: 10.1007/s00028-005-0230-y.  Google Scholar [2] F. Alabau-Boussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems,, SIAM J. Control Optim., 41 (2002), 511.  doi: 10.1137/S0363012901385368.  Google Scholar [3] F. Alabau, P. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled systems,, J. Evolution Equations, 2 (2002), 127.  doi: 10.1007/s00028-002-8083-0.  Google Scholar [4] W. Arendt, C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups., Trans. Amer. Math. Soc. 306 (1988), 306 (1988), 837.  doi: 10.2307/2000826.  Google Scholar [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), 30 (1992), 1024.  doi: 10.1137/0330055.  Google Scholar [6] A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups,, Math. Nachr., 279 (2006), 1425.  doi: 10.1002/mana.200410429.  Google Scholar [7] H. Brezis, "Analyse fonctionnelle,", Théorie et Applications. Masson, (1983).   Google Scholar [8] G. Chen, Control and stabilization for the wave equation in a bounded domain., SIAM J. Control Optim. 17 (1979), 17 (1979), 66.  doi: 10.1137/0317007.  Google Scholar [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), 51 (1991), 266.  doi: 10.1137/0151015.  Google Scholar [10] C.M. Dafermos, Asymptotic behaviour of solutions of evolution equations,, in, (1978), 103.   Google Scholar [11] X. Fu, Longtime behavior of the hyperbolic equations with an arbitrary internal damping., Z. Angew. Math. Phys. 62 (2011), 62 (2011), 667.  doi: 10.1007/s00033-010-0113-0.  Google Scholar [12] X. Fu, Logarithmic decay of hyperbolic equations with arbitrary small boundary damping., Comm. Partial Differential Equations 34 (2009), 34 (2009), 957.  doi: 10.1080/03605300903116389.  Google Scholar [13] J. S. Gibson, A note on stabilization of infinite dimensional linear oscillators by compact linear feedback,, SIAM J. Control Optim., 18 (1980), 311.  doi: 10.1137/0318022.  Google Scholar [14] A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations,, J. Differential Equations, 59 (1985), 145.  doi: 10.1016/0022-0396(85)90151-2.  Google Scholar [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, (1991), 1987.   Google Scholar [16] A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps,, Portugal. Math. 46 (1989), 46 (1989), 245.   Google Scholar [17] F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces,, Ann. Differential Equations, 1 (1985), 43.   Google Scholar [18] V. Komornik, Rapid boundary stabilization of the wave equation., SIAM J. Control Optim. 29 (1991), 29 (1991), 197.  doi: 10.1137/0329011.  Google Scholar [19] V. Komornik, "Exact controllability and stabilization. The multiplier method,", RAM, (1994).   Google Scholar [20] V. Komornik, Rapid boundary stabilization of linear distributed systems., SIAM J. Control Optim. 35 (1997), 35 (1997), 1591.  doi: 10.1137/S0363012996301609.  Google Scholar [21] V. Komornik, E. Zuazua, A direct method for the boundary stabilization of the wave equation,, J. Math. Pures Appl. 69 (1990), 69 (1990), 33.   Google Scholar [22] J. Lagnese, Boundary stabilization of linear elastodynamic systems,, S.I.A.M J. Control and Opt., 21 (1983), 968.  doi: 10.1137/0321059.  Google Scholar [23] J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation., J. Differential Equations 50 (1983), 50 (1983), 163.  doi: 10.1016/0022-0396(83)90073-6.  Google Scholar [24] J. Lagnese, "Boundary Stabilization of Thin Plates,", SIAM Stud. Appl. Math. 10, (1989).  doi: 10.1137/1.9781611970821.  Google Scholar [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), 95 (1992), 169.  doi: 10.1016/0022-0396(92)90048-R.  Google Scholar [26] I. Lasiecka, Nonlinear boundary feedback stabilization of dynamic elasticity with thermal effects., Shape optimization and optimal design (Cambridge, (1999), 333.   Google Scholar [27] I. Lasiecka, D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping., Differential Integral Equations 6 (1993), 6 (1993), 507.   Google Scholar [28] I. Lasiecka, D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms., Nonlinear Anal. 64(2006), 64 (2006), 1757.  doi: 10.1016/j.na.2005.07.024.  Google Scholar [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), 66 (1987), 340.  doi: 10.1016/0022-0396(87)90025-8.  Google Scholar [30] I. Lasiecka, R. Triggiani, Exact controllability of the wave equation with Neumann boundary control., Appl. Math. Optim. 19 (1989), 19 (1989), 243.  doi: 10.1007/BF01448201.  Google Scholar [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), 93 (1991), 62.  doi: 10.1016/0022-0396(91)90022-2.  Google Scholar [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), 25 (1992), 189.  doi: 10.1007/BF01182480.  Google Scholar [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.  doi: 10.1155/S1085337503305032.  Google Scholar [34] G. Lebeau, Equation des ondes amorties., Algebraic and geometric methods in mathematical physics (Kaciveli, 19 (1993), 73.   Google Scholar [35] G. Lebeau, L. Robbiano, Stabilisation de l'équation des ondes par le bord., Duke Math. J. 86 (1997), 86 (1997), 465.  doi: 10.1215/S0012-7094-97-08614-2.  Google Scholar [36] J.-L. Lions, "Contrôlabilité exacte, Perturbations et Stabilisation des Systèmes Distribués,", Vol. 1, (1988).   Google Scholar [37] K. Liu, Locally distributed control and damping for the conservative systems,, S.I.A.M J. Control and Opt. 35 (1997), 35 (1997), 1574.  doi: 10.1137/S0363012995284928.  Google Scholar [38] K. Liu, B. Rao, Exponential stability for the wave equations with local Kelvin-Voigt damping., Z. Angew. Math. Phys. 57 (2006), 57 (2006), 419.  doi: 10.1007/s00033-005-0029-2.  Google Scholar [39] P. Martinez, PhD thesis,, University of Strasbourg, (1998).   Google Scholar [40] P. Martinez, Boundary stabilization of the wave equation in almost star-shaped domains., SIAM J. Control Optim. 37 (1999), 37 (1999), 673.  doi: 10.1137/S0363012997323722.  Google Scholar [41] M. Nakao, Decay of solutions of the wave equation with a local degenerate dissipation,, Israel J. Math. 95 (1996), 95 (1996), 25.  doi: 10.1007/BF02761033.  Google Scholar [42] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar [43] K. D. Phung, Polynomial decay rate for the dissipative wave equation., J. Differential Equations 240 (2007), 240 (2007), 92.  doi: 10.1016/j.jde.2007.05.016.  Google Scholar [44] K. D. Phung, Boundary stabilization for the wave equation in a bounded cylindrical domain., Discrete Contin. Dyn. Syst. 20 (2008), 20 (2008), 1057.  doi: 10.3934/dcds.2008.20.1057.  Google Scholar [45] J. Prüss, On the spectrum of $C_0$-semigroups., Trans. Amer. Math. Soc. 284 (1984), 284 (1984), 847.  doi: 10.2307/1999112.  Google Scholar [46] J. Rauch, M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains., Indiana Univ. Math. J. 24 (1974), 24 (1974), 79.   Google Scholar [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.   Google Scholar [48] D.L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions., SIAM Rev. 20 (1978), 20 (1978), 639.  doi: 10.1137/1020095.  Google Scholar [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), 24 (1986), 199.  doi: 10.1137/0324012.  Google Scholar [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), (1989), 87.  doi: 10.1017/S0308210500023970.  Google Scholar [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), 319 (1994), 585.   Google Scholar [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.   Google Scholar [53] L.R. Tcheugoué Tébou, Stabilization of the wave equation with localized nonlinear damping,, J.D.E. 145(1998), 145 (1998), 502.  doi: 10.1006/jdeq.1998.3416.  Google Scholar [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.  doi: 10.1080/03605309808821403.  Google Scholar [55] L. R. Tcheugoué Tébou, Energy decay estimates for the damped Euler-Bernoulli equation with an unbounded localizing coefficient,, Portugal. Math. 61 (2004), 61 (2004), 375.   Google Scholar [56] L. R. 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