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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

1. 

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

Received  November 2015 Revised  August 2016 Published  October 2016

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.
Citation: Louis Tebou. Stabilization of some elastodynamic systems with localized Kelvin-Voigt damping. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7117-7136. doi: 10.3934/dcds.2016110
References:
[1]

F. Alabau and V. Komornik, Boundary observability, controllability, and stabilization of linear elastodynamic systems, SIAM J. Control Optim., 37 (1999), 521-542. doi: 10.1137/S0363012996313835.

[2]

G. Alessandrini and A. Morassi, Strong unique continuation for the Lamé system of elasticity, Comm. Partial Differential Equations, 26 (2001), 1787-1810. doi: 10.1081/PDE-100107459.

[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), 371-385. doi: 10.1080/03605309808821349.

[4]

W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852. doi: 10.1090/S0002-9947-1988-0933321-3.

[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), 1357-1376.

[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), 353-381.

[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), 1024-1065. doi: 10.1137/0330055.

[8]

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.

[9]

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.

[10]

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.

[11]

H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Masson, Paris, 1983.

[12]

N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers, Asymptot. Anal., 14 (1997), 157-191.

[13]

N. Burq and H. Christianson, Imperfect geometric control and overdamping for the damped wave equation, Commun. Math. Phys., 336 (2015), 101-130. doi: 10.1007/s00220-014-2247-y.

[14]

M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42 (2003), 1310-1324. doi: 10.1137/S0363012902408010.

[15]

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.

[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), 266-301. doi: 10.1137/0151015.

[17]

C. M. Dafermos, Asymptotic behaviour of solutions of evolution equations, in Nonlinear evolution equations(M.G. Crandall ed.) Academic Press, New-York, 40 (1978), 103-123.

[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), 475-492.

[19]

G. Duvaut and J. L. Lions, Les Inéquations en Mécanique et en Physique, Travaux et Recherches Mathématiques, No. 21. Dunod, Paris, 1972.

[20]

A. Guesmia, On the decay estimates for elasticity systems with some localized dissipations, Asymptot. Anal., 22 (2000), 1-13.

[21]

X. Fu, Sharp decay rates for the weakly coupled hyperbolic system with one internal damping, SIAM J. Control Optim., 50 (2012), 1643-1660. doi: 10.1137/110833051.

[22]

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.

[23]

A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Port. Math., 46 (1989), 245-258.

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

[25]

V. Komornik, Rapid boundary stabilization of the wave equation, SIAM J. Control Optim., 29 (1991), 197-208. doi: 10.1137/0329011.

[26]

V. Komornik, On the nonlinear boundary stabilization of the wave equation, Chin. Ann. Math., 14 (1993), 153-164.

[27]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM, Masson & John Wiley, Paris, 1994.

[28]

V. Komornik, Rapid boundary stabilization of linear distributed systems, SIAM J. Control and Optimization, 35 (1997), 1591-1613. doi: 10.1137/S0363012996301609.

[29]

V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J.M.P.A.,, 69 (1990), 33-54.

[30]

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.

[31]

J. Lagnese, Boundary stabilization of linear elastodynamic systems, SIAM J. Control and Optim., 21 (1983), 968-984. doi: 10.1137/0321059.

[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), 35-54. doi: 10.1016/0362-546X(91)90129-O.

[33]

I. Lasiecka, Nonlinear boundary feedback stabilization of dynamic elasticity with thermal effects, Shape optimization and optimal design (Cambridge, 1999), Lecture Notes in Pure and Appl. Math., Dekker, New York, 216 (2001), 333-354.

[34]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations, 6 (1993), 507-533.

[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), 1757-1797. doi: 10.1016/j.na.2005.07.024.

[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), 13-57. doi: 10.1006/jmaa.1999.6348.

[37]

G. Lebeau, Equation des ondes amorties, Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), Math. Phys. Stud., Kluwer Acad. Publ., Dordrecht, 19 (1996), 73-109.

[38]

G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity, Arch. Ration. Mech. Anal., 148 (1999), 179-231. doi: 10.1007/s002050050160.

[39]

C.-L. Lin and J.-N. Wang, Strong unique continuation for the Lamé system with Lipschitz coefficients, Math. Ann., 331 (2005), 611-629. doi: 10.1007/s00208-004-0597-z.

[40]

J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation Des Systèmes Distribués, Vol. 1, RMA 8, Masson, Paris, 1988.

[41]

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.

[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), 1086-1098. doi: 10.1137/S0363012996310703.

[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), 769-774. doi: 10.1016/j.crma.2004.09.029.

[44]

K. Liu and 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.

[45]

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

[46]

P. Martinez, PhD Thesis. University of Strasbourg, France, 1998.

[47]

P. Martinez, Uniform boundary stabilization of elasticity systems of cubic cystals by nonlinear feedbacks, Nonlinear Analysis 37 (1999), 719-733. doi: 10.1016/S0362-546X(98)00068-6.

[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, 321-338, Contemp. Math., 362, Amer. Math. Soc., Providence, RI, 2004. doi: 10.1090/conm/362/06621.

[49]

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.

[50]

M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation, Math. Ann., 305 (1996), 403-417. doi: 10.1007/BF01444231.

[51]

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.

[52]

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

[53]

J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J., 24 (1974), 79-86. doi: 10.1512/iumj.1975.24.24004.

[54]

M. Renardy, On localized Kelvin-Voigt damping, ZAMM Z. Angew. Math. Mech., 84 (2004), 280-283. doi: 10.1002/zamm.200310100.

[55]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations, Recent progress and open problems. SIAM Rev., 20 (1978), 639-739. doi: 10.1137/1020095.

[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), 87-97. doi: 10.1017/S0308210500023970.

[57]

L. R. Tcheugoué Tébou, Estimations d'énergie pour l'équation des ondes avec un amortissement nonlinéaire localisé, C. R. Acad. Paris, Série I, 325 (1997), 1175-1179. doi: 10.1016/S0764-4442(97)83549-5.

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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.

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R. Triggiani, Lack of uniform stabilization for noncontractive semigroups under compact perturbation, Proc. Amer. Math. Soc., 105 (1989), 375-383. doi: 10.1090/S0002-9939-1989-0953013-0.

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P. F. Yao, On the observability inequalities for exact controllability of wave equations with variable coefficients, SIAM J. Control Optim., 37 (1999), 1568-1599. doi: 10.1137/S0363012997331482.

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Q. Zhang, Exponential stability of an elastic string with local Kelvin-Voigt damping, ZAMP, 61 (2010), 1009-1015. doi: 10.1007/s00033-010-0064-5.

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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.

[73]

E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains, J. Math. Pures. Appl., 70 (1991), 513-529.

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), 521-542. doi: 10.1137/S0363012996313835.

[2]

G. Alessandrini and A. Morassi, Strong unique continuation for the Lamé system of elasticity, Comm. Partial Differential Equations, 26 (2001), 1787-1810. doi: 10.1081/PDE-100107459.

[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), 371-385. doi: 10.1080/03605309808821349.

[4]

W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852. doi: 10.1090/S0002-9947-1988-0933321-3.

[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), 1357-1376.

[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), 353-381.

[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), 1024-1065. doi: 10.1137/0330055.

[8]

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.

[9]

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.

[10]

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.

[11]

H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Masson, Paris, 1983.

[12]

N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers, Asymptot. Anal., 14 (1997), 157-191.

[13]

N. Burq and H. Christianson, Imperfect geometric control and overdamping for the damped wave equation, Commun. Math. Phys., 336 (2015), 101-130. doi: 10.1007/s00220-014-2247-y.

[14]

M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42 (2003), 1310-1324. doi: 10.1137/S0363012902408010.

[15]

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.

[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), 266-301. doi: 10.1137/0151015.

[17]

C. M. Dafermos, Asymptotic behaviour of solutions of evolution equations, in Nonlinear evolution equations(M.G. Crandall ed.) Academic Press, New-York, 40 (1978), 103-123.

[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), 475-492.

[19]

G. Duvaut and J. L. Lions, Les Inéquations en Mécanique et en Physique, Travaux et Recherches Mathématiques, No. 21. Dunod, Paris, 1972.

[20]

A. Guesmia, On the decay estimates for elasticity systems with some localized dissipations, Asymptot. Anal., 22 (2000), 1-13.

[21]

X. Fu, Sharp decay rates for the weakly coupled hyperbolic system with one internal damping, SIAM J. Control Optim., 50 (2012), 1643-1660. doi: 10.1137/110833051.

[22]

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.

[23]

A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Port. Math., 46 (1989), 245-258.

[24]

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

[25]

V. Komornik, Rapid boundary stabilization of the wave equation, SIAM J. Control Optim., 29 (1991), 197-208. doi: 10.1137/0329011.

[26]

V. Komornik, On the nonlinear boundary stabilization of the wave equation, Chin. Ann. Math., 14 (1993), 153-164.

[27]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM, Masson & John Wiley, Paris, 1994.

[28]

V. Komornik, Rapid boundary stabilization of linear distributed systems, SIAM J. Control and Optimization, 35 (1997), 1591-1613. doi: 10.1137/S0363012996301609.

[29]

V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J.M.P.A.,, 69 (1990), 33-54.

[30]

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.

[31]

J. Lagnese, Boundary stabilization of linear elastodynamic systems, SIAM J. Control and Optim., 21 (1983), 968-984. doi: 10.1137/0321059.

[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), 35-54. doi: 10.1016/0362-546X(91)90129-O.

[33]

I. Lasiecka, Nonlinear boundary feedback stabilization of dynamic elasticity with thermal effects, Shape optimization and optimal design (Cambridge, 1999), Lecture Notes in Pure and Appl. Math., Dekker, New York, 216 (2001), 333-354.

[34]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations, 6 (1993), 507-533.

[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), 1757-1797. doi: 10.1016/j.na.2005.07.024.

[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), 13-57. doi: 10.1006/jmaa.1999.6348.

[37]

G. Lebeau, Equation des ondes amorties, Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), Math. Phys. Stud., Kluwer Acad. Publ., Dordrecht, 19 (1996), 73-109.

[38]

G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity, Arch. Ration. Mech. Anal., 148 (1999), 179-231. doi: 10.1007/s002050050160.

[39]

C.-L. Lin and J.-N. Wang, Strong unique continuation for the Lamé system with Lipschitz coefficients, Math. Ann., 331 (2005), 611-629. doi: 10.1007/s00208-004-0597-z.

[40]

J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation Des Systèmes Distribués, Vol. 1, RMA 8, Masson, Paris, 1988.

[41]

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.

[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), 1086-1098. doi: 10.1137/S0363012996310703.

[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), 769-774. doi: 10.1016/j.crma.2004.09.029.

[44]

K. Liu and 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.

[45]

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

[46]

P. Martinez, PhD Thesis. University of Strasbourg, France, 1998.

[47]

P. Martinez, Uniform boundary stabilization of elasticity systems of cubic cystals by nonlinear feedbacks, Nonlinear Analysis 37 (1999), 719-733. doi: 10.1016/S0362-546X(98)00068-6.

[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, 321-338, Contemp. Math., 362, Amer. Math. Soc., Providence, RI, 2004. doi: 10.1090/conm/362/06621.

[49]

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.

[50]

M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation, Math. Ann., 305 (1996), 403-417. doi: 10.1007/BF01444231.

[51]

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.

[52]

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

[53]

J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J., 24 (1974), 79-86. doi: 10.1512/iumj.1975.24.24004.

[54]

M. Renardy, On localized Kelvin-Voigt damping, ZAMM Z. Angew. Math. Mech., 84 (2004), 280-283. doi: 10.1002/zamm.200310100.

[55]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations, Recent progress and open problems. SIAM Rev., 20 (1978), 639-739. doi: 10.1137/1020095.

[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), 87-97. doi: 10.1017/S0308210500023970.

[57]

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