Stabilization of some elastodynamic systems with localized Kelvin-Voigt damping

Louis Tebou

doi:10.3934/dcds.2016110

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December 2016, 36(12): 7117-7136. doi: 10.3934/dcds.2016110

Stabilization of some elastodynamic systems with localized Kelvin-Voigt damping

Louis Tebou ^1,

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

Received November 2015 Revised August 2016 Published October 2016

Abstract
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We consider the dynamic elasticity equations with a locally distributed damping of Kelvin-Voigt type in a bounded domain. The damping is localized in a suitable open subset, of the domain under consideration, which satisfies the piecewise multipliers condition of Liu. Using multiplier techniques combined with the frequency domain method, we show that: i) the energy of this system decays polynomially when the damping coefficient is only bounded measurable, ii) the energy of this system decays exponentially when the damping coefficient as well as its gradient are bounded measurable, and the damping coefficient further satisfies a structural condition. These results generalize and improve, at the same time, on an earlier result of Liu and Rao involving the wave equation with Kelvin-Voigt damping; those authors proved the exponential decay of the energy provided that the damping region is a neighborhood of the whole boundary, and further restrictions are imposed on the damping coefficient.

Keywords: Kelvin-Voigt damping., Stabilization, wave equation, localized dissipation, dynamic elasticity equations.

Mathematics Subject Classification: Primary: 93D15; Secondary: 74J05, 35L05, 74H4.

Citation: Louis Tebou. Stabilization of some elastodynamic systems with localized Kelvin-Voigt damping. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7117-7136. doi: 10.3934/dcds.2016110

References:

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[2]	G. Alessandrini and A. Morassi, Strong unique continuation for the Lamé system of elasticity,, Comm. Partial Differential Equations, 26 (2001), 1787. doi: 10.1081/PDE-100107459. Google Scholar
[3]	D. D. Ang, M. Ikehata, D. D. Trong and M. Yamamoto, Unique continuation for a stationary isotropic Lamé system with variable coefficients,, Comm. Partial Differential Equations, 23 (1998), 371. doi: 10.1080/03605309808821349. Google Scholar
[4]	W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups,, Trans. Amer. Math. Soc., 306* (1988), 837. doi: 10.1090/S0002-9947-1988-0933321-3. Google Scholar*
[5]	M. A. Astaburuaga and R. Coimbra Charão, Stabilization of the total energy for a system of elasticity with localized dissipation,, Differential Integral Equations, 15 (2002), 1357. Google Scholar
[6]	H. T. Banks, R. C. Smith and Y. Wang, Modeling aspects for piezoelectric patch actuation of shells, plates and beams,, Quart. Appl. Math., 53 (1995), 353. Google Scholar
[7]	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. doi: 10.1137/0330055. Google Scholar
[8]	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
[9]	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
[10]	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
[11]	H. Brezis, Analyse Fonctionnelle. Théorie et Applications,, Masson, (1983). Google Scholar
[12]	N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers,, Asymptot. Anal., 14 (1997), 157. Google Scholar
[13]	N. Burq and H. Christianson, Imperfect geometric control and overdamping for the damped wave equation,, Commun. Math. Phys., 336 (2015), 101. doi: 10.1007/s00220-014-2247-y. Google Scholar
[14]	M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation,, SIAM J. Control Optim., 42 (2003), 1310. doi: 10.1137/S0363012902408010. Google Scholar
[15]	G. Chen, Control and stabilization for the wave equation in a bounded domain,, SIAM J. Control Optim., 17 (1979), 66. doi: 10.1137/0317007. Google Scholar
[16]	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. doi: 10.1137/0151015. Google Scholar
[17]	C. M. Dafermos, Asymptotic behaviour of solutions of evolution equations,, in Nonlinear evolution equations(M.G. Crandall ed.) Academic Press, 40 (1978), 103. Google Scholar
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[19]	G. Duvaut and J. L. Lions, Les Inéquations en Mécanique et en Physique,, Travaux et Recherches Mathématiques, (1972). Google Scholar
[20]	A. Guesmia, On the decay estimates for elasticity systems with some localized dissipations,, Asymptot. Anal., 22 (2000), 1. Google Scholar
[21]	X. Fu, Sharp decay rates for the weakly coupled hyperbolic system with one internal damping,, SIAM J. Control Optim., 50 (2012), 1643. doi: 10.1137/110833051. Google Scholar
[22]	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
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[24]	F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces,, Ann. Differential Equations, (1985), 43. Google Scholar
[25]	V. Komornik, Rapid boundary stabilization of the wave equation,, SIAM J. Control Optim., 29 (1991), 197. doi: 10.1137/0329011. Google Scholar
[26]	V. Komornik, On the nonlinear boundary stabilization of the wave equation,, Chin. Ann. Math., 14 (1993), 153. Google Scholar
[27]	V. Komornik, Exact Controllability and Stabilization. The Multiplier Method,, RAM, (1994). Google Scholar
[28]	V. Komornik, Rapid boundary stabilization of linear distributed systems,, SIAM J. Control and Optimization, 35 (1997), 1591. doi: 10.1137/S0363012996301609. Google Scholar
[29]	V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation,, J.M.P.A., 69 (1990), 33. Google Scholar
[30]	J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation,, J. Differential Equations, 50 (1983), 163. doi: 10.1016/0022-0396(83)90073-6. Google Scholar
[31]	J. Lagnese, Boundary stabilization of linear elastodynamic systems,, SIAM J. Control and Optim., 21 (1983), 968. doi: 10.1137/0321059. Google Scholar
[32]	J. Lagnese, Uniform asymptotic energy estimates for solutions of the equations of dynamic plane elasticity with nonlinear dissipation at the boundary,, Nonlinear Anal., 16 (1991), 35. doi: 10.1016/0362-546X(91)90129-O. Google Scholar
[33]	I. Lasiecka, Nonlinear boundary feedback stabilization of dynamic elasticity with thermal effects,, Shape optimization and optimal design (Cambridge, 216 (2001), 333. Google Scholar
[34]	I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping,, Differential Integral Equations, 6 (1993), 507. Google Scholar
[35]	I. Lasiecka and D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms,, Nonlinear Anal., 64 (2006), 1757. doi: 10.1016/j.na.2005.07.024. Google Scholar
[36]	I. Lasiecka, R. Triggiani and P. F. Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients,, J. Math. Anal. Appl., 235 (1999), 13. doi: 10.1006/jmaa.1999.6348. Google Scholar
[37]	G. Lebeau, Equation des ondes amorties,, Algebraic and geometric methods in mathematical physics (Kaciveli, 19 (1996), 73. Google Scholar
[38]	G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity,, Arch. Ration. Mech. Anal., 148 (1999), 179. doi: 10.1007/s002050050160. Google Scholar
[39]	C.-L. Lin and J.-N. Wang, Strong unique continuation for the Lamé system with Lipschitz coefficients,, Math. Ann., 331 (2005), 611. doi: 10.1007/s00208-004-0597-z. Google Scholar
[40]	J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation Des Systèmes Distribués,, Vol. 1, (1988). Google Scholar
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[42]	K. Liu and Z. Liu, Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping,, S.I.A.M J. Control and Opt, 36 (1998), 1086. doi: 10.1137/S0363012996310703. Google Scholar
[43]	K. Liu and B. Rao, Stabilité exponentielle des équations des ondes avec amortissement local de Kelvin-Voigt,, C. R. Math. Acad. Sci. Paris, 339 (2004), 769. doi: 10.1016/j.crma.2004.09.029. Google Scholar
[44]	K. Liu and B. Rao, Exponential stability for the wave equations with local Kelvin-Voigt damping,, Z. Angew. Math. Phys., 57 (2006), 419. doi: 10.1007/s00033-005-0029-2. Google Scholar
[45]	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
[46]	P. Martinez, PhD Thesis., University of Strasbourg, (1998). Google Scholar
[47]	P. Martinez, Uniform boundary stabilization of elasticity systems of cubic cystals by nonlinear feedbacks,, Nonlinear Analysis 37 (1999), 37 (1999), 719. doi: 10.1016/S0362-546X(98)00068-6. Google Scholar
[48]	G. Nakamura, G. Uhlmann and J.-N. Wang, Unique continuation property for elliptic systems and crack determination in anisotropic elasticity,, Partial differential equations and inverse problems, (2004), 321. doi: 10.1090/conm/362/06621. Google Scholar
[49]	M. Nakao, Decay of solutions of the wave equation with a local degenerate dissipation,, Israel J. Math., 95 (1996), 25. doi: 10.1007/BF02761033. Google Scholar
[50]	M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation,, Math. Ann., 305 (1996), 403. doi: 10.1007/BF01444231. Google Scholar
[51]	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
[52]	J. Prüss, On the spectrum of $C_0$-semigroups,, Trans. Amer. Math. Soc., 284 (1984), 847. doi: 10.2307/1999112. Google Scholar
[53]	J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains,, Indiana Univ. Math. J., 24 (1974), 79. doi: 10.1512/iumj.1975.24.24004. Google Scholar
[54]	M. Renardy, On localized Kelvin-Voigt damping,, ZAMM Z. Angew. Math. Mech., 84 (2004), 280. doi: 10.1002/zamm.200310100. Google Scholar
[55]	D. L. Russell, Controllability and stabilizability theory for linear partial differential equations,, Recent progress and open problems. SIAM Rev., 20 (1978), 639. doi: 10.1137/1020095. Google Scholar
[56]	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. doi: 10.1017/S0308210500023970. Google Scholar
[57]	L. R. Tcheugoué Tébou, Estimations d'énergie pour l'équation des ondes avec un amortissement nonlinéaire localisé,, C. R. Acad. Paris, 325* (1997), 1175. doi: 10.1016/S0764-4442(97)83549-5. Google Scholar*
[58]	L. R. Tcheugoué Tébou, Stabilization of the wave equation with localized nonlinear damping,, J.D.E., 145 (1998), 502. doi: 10.1006/jdeq.1998.3416. Google Scholar
[59]	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
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show all references

References:

[1]	F. Alabau and V. Komornik, Boundary observability, controllability, and stabilization of linear elastodynamic systems,, SIAM J. Control Optim., 37 (1999), 521. doi: 10.1137/S0363012996313835. Google Scholar
[2]	G. Alessandrini and A. Morassi, Strong unique continuation for the Lamé system of elasticity,, Comm. Partial Differential Equations, 26 (2001), 1787. doi: 10.1081/PDE-100107459. Google Scholar
[3]	D. D. Ang, M. Ikehata, D. D. Trong and M. Yamamoto, Unique continuation for a stationary isotropic Lamé system with variable coefficients,, Comm. Partial Differential Equations, 23 (1998), 371. doi: 10.1080/03605309808821349. Google Scholar
[4]	W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups,, Trans. Amer. Math. Soc., 306* (1988), 837. doi: 10.1090/S0002-9947-1988-0933321-3. Google Scholar*
[5]	M. A. Astaburuaga and R. Coimbra Charão, Stabilization of the total energy for a system of elasticity with localized dissipation,, Differential Integral Equations, 15 (2002), 1357. Google Scholar
[6]	H. T. Banks, R. C. Smith and Y. Wang, Modeling aspects for piezoelectric patch actuation of shells, plates and beams,, Quart. Appl. Math., 53 (1995), 353. Google Scholar
[7]	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. doi: 10.1137/0330055. Google Scholar
[8]	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
[9]	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
[10]	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
[11]	H. Brezis, Analyse Fonctionnelle. Théorie et Applications,, Masson, (1983). Google Scholar
[12]	N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers,, Asymptot. Anal., 14 (1997), 157. Google Scholar
[13]	N. Burq and H. Christianson, Imperfect geometric control and overdamping for the damped wave equation,, Commun. Math. Phys., 336 (2015), 101. doi: 10.1007/s00220-014-2247-y. Google Scholar
[14]	M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation,, SIAM J. Control Optim., 42 (2003), 1310. doi: 10.1137/S0363012902408010. Google Scholar
[15]	G. Chen, Control and stabilization for the wave equation in a bounded domain,, SIAM J. Control Optim., 17 (1979), 66. doi: 10.1137/0317007. Google Scholar
[16]	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. doi: 10.1137/0151015. Google Scholar
[17]	C. M. Dafermos, Asymptotic behaviour of solutions of evolution equations,, in Nonlinear evolution equations(M.G. Crandall ed.) Academic Press, 40 (1978), 103. Google Scholar
[18]	B. Dehman and L. Robbiano, La propriété du prolongement unique pour un système elliptique. Le système de Lamé,, J. Math. Pures Appl., 72 (1993), 475. Google Scholar
[19]	G. Duvaut and J. L. Lions, Les Inéquations en Mécanique et en Physique,, Travaux et Recherches Mathématiques, (1972). Google Scholar
[20]	A. Guesmia, On the decay estimates for elasticity systems with some localized dissipations,, Asymptot. Anal., 22 (2000), 1. Google Scholar
[21]	X. Fu, Sharp decay rates for the weakly coupled hyperbolic system with one internal damping,, SIAM J. Control Optim., 50 (2012), 1643. doi: 10.1137/110833051. Google Scholar
[22]	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
[23]	A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps,, Port. Math., 46 (1989), 245. Google Scholar
[24]	F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces,, Ann. Differential Equations, (1985), 43. Google Scholar
[25]	V. Komornik, Rapid boundary stabilization of the wave equation,, SIAM J. Control Optim., 29 (1991), 197. doi: 10.1137/0329011. Google Scholar
[26]	V. Komornik, On the nonlinear boundary stabilization of the wave equation,, Chin. Ann. Math., 14 (1993), 153. Google Scholar
[27]	V. Komornik, Exact Controllability and Stabilization. The Multiplier Method,, RAM, (1994). Google Scholar
[28]	V. Komornik, Rapid boundary stabilization of linear distributed systems,, SIAM J. Control and Optimization, 35 (1997), 1591. doi: 10.1137/S0363012996301609. Google Scholar
[29]	V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation,, J.M.P.A., 69 (1990), 33. Google Scholar
[30]	J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation,, J. Differential Equations, 50 (1983), 163. doi: 10.1016/0022-0396(83)90073-6. Google Scholar
[31]	J. Lagnese, Boundary stabilization of linear elastodynamic systems,, SIAM J. Control and Optim., 21 (1983), 968. doi: 10.1137/0321059. Google Scholar
[32]	J. Lagnese, Uniform asymptotic energy estimates for solutions of the equations of dynamic plane elasticity with nonlinear dissipation at the boundary,, Nonlinear Anal., 16 (1991), 35. doi: 10.1016/0362-546X(91)90129-O. Google Scholar
[33]	I. Lasiecka, Nonlinear boundary feedback stabilization of dynamic elasticity with thermal effects,, Shape optimization and optimal design (Cambridge, 216 (2001), 333. Google Scholar
[34]	I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping,, Differential Integral Equations, 6 (1993), 507. Google Scholar
[35]	I. Lasiecka and D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms,, Nonlinear Anal., 64 (2006), 1757. doi: 10.1016/j.na.2005.07.024. Google Scholar
[36]	I. Lasiecka, R. Triggiani and P. F. Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients,, J. Math. Anal. Appl., 235 (1999), 13. doi: 10.1006/jmaa.1999.6348. Google Scholar
[37]	G. Lebeau, Equation des ondes amorties,, Algebraic and geometric methods in mathematical physics (Kaciveli, 19 (1996), 73. Google Scholar
[38]	G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity,, Arch. Ration. Mech. Anal., 148 (1999), 179. doi: 10.1007/s002050050160. Google Scholar
[39]	C.-L. Lin and J.-N. Wang, Strong unique continuation for the Lamé system with Lipschitz coefficients,, Math. Ann., 331 (2005), 611. doi: 10.1007/s00208-004-0597-z. Google Scholar
[40]	J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation Des Systèmes Distribués,, Vol. 1, (1988). Google Scholar
[41]	K. Liu, Locally distributed control and damping for the conservative systems,, S.I.A.M J. Control and Opt., 35 (1997), 1574. doi: 10.1137/S0363012995284928. Google Scholar
[42]	K. Liu and Z. Liu, Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping,, S.I.A.M J. Control and Opt, 36 (1998), 1086. doi: 10.1137/S0363012996310703. Google Scholar
[43]	K. Liu and B. Rao, Stabilité exponentielle des équations des ondes avec amortissement local de Kelvin-Voigt,, C. R. Math. Acad. Sci. Paris, 339 (2004), 769. doi: 10.1016/j.crma.2004.09.029. Google Scholar
[44]	K. Liu and B. Rao, Exponential stability for the wave equations with local Kelvin-Voigt damping,, Z. Angew. Math. Phys., 57 (2006), 419. doi: 10.1007/s00033-005-0029-2. Google Scholar
[45]	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
[46]	P. Martinez, PhD Thesis., University of Strasbourg, (1998). Google Scholar
[47]	P. Martinez, Uniform boundary stabilization of elasticity systems of cubic cystals by nonlinear feedbacks,, Nonlinear Analysis 37 (1999), 37 (1999), 719. doi: 10.1016/S0362-546X(98)00068-6. Google Scholar
[48]	G. Nakamura, G. Uhlmann and J.-N. Wang, Unique continuation property for elliptic systems and crack determination in anisotropic elasticity,, Partial differential equations and inverse problems, (2004), 321. doi: 10.1090/conm/362/06621. Google Scholar
[49]	M. Nakao, Decay of solutions of the wave equation with a local degenerate dissipation,, Israel J. Math., 95 (1996), 25. doi: 10.1007/BF02761033. Google Scholar
[50]	M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation,, Math. Ann., 305 (1996), 403. doi: 10.1007/BF01444231. Google Scholar
[51]	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
[52]	J. Prüss, On the spectrum of $C_0$-semigroups,, Trans. Amer. Math. Soc., 284 (1984), 847. doi: 10.2307/1999112. Google Scholar
[53]	J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains,, Indiana Univ. Math. J., 24 (1974), 79. doi: 10.1512/iumj.1975.24.24004. Google Scholar
[54]	M. Renardy, On localized Kelvin-Voigt damping,, ZAMM Z. Angew. Math. Mech., 84 (2004), 280. doi: 10.1002/zamm.200310100. Google Scholar
[55]	D. L. Russell, Controllability and stabilizability theory for linear partial differential equations,, Recent progress and open problems. SIAM Rev., 20 (1978), 639. doi: 10.1137/1020095. Google Scholar
[56]	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. doi: 10.1017/S0308210500023970. Google Scholar
[57]	L. R. Tcheugoué Tébou, Estimations d'énergie pour l'équation des ondes avec un amortissement nonlinéaire localisé,, C. R. Acad. Paris, 325* (1997), 1175. doi: 10.1016/S0764-4442(97)83549-5. Google Scholar*
[58]	L. R. Tcheugoué Tébou, Stabilization of the wave equation with localized nonlinear damping,, J.D.E., 145 (1998), 502. doi: 10.1006/jdeq.1998.3416. Google Scholar
[59]	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
[60]	L. R. Tcheugoué Tébou, A direct method for the stabilization of some locally damped semilinear wave equations,, C. R. Acad. Sci. Paris, 342* (2006), 859. doi: 10.1016/j.crma.2006.04.010. Google Scholar*
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