July  2020, 19(7): 3531-3557. doi: 10.3934/cpaa.2020154

General decay for a viscoelastic rotating Euler-Bernoulli beam

Laboratory of SDG, Faculty of Mathematics, University of Science and Technology Houari Boumedienne, P.O. Box 32, El-Alia 16111, Bab Ezzouar, Algiers, Algeria

* Corresponding author

Received  February 2019 Revised  January 2020 Published  April 2020

In this paper, we consider a viscoelastic rotating Euler-Bernoulli beam that has one end fixed to a rotated motor in a horizontal plane and to a tip mass at the other end. For a large class relaxation function $ q $, namely, $ q^{\prime}(t) \leq -\zeta(t)H(q(t)) $, where $ H $ is an increasing and convex function near the origin and $ \zeta $ is a nonincreasing function, we establish optimal explicit and general energy decay results from which we can recover the optimal exponential and polynomial decay.

Citation: Ammar Khemmoudj, Imane Djaidja. General decay for a viscoelastic rotating Euler-Bernoulli beam. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3531-3557. doi: 10.3934/cpaa.2020154
References:
[1]

M. AassilaM. M. Cavalcanti and V. N. Domingos Cavalcanti, Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term, Calc. Var. Partial Differ. Equ., 15 (2002), 155-180.  doi: 10.1007/s005260100096.  Google Scholar

[2]

M. AassilaM. M. Cavalcanti and J. A. Soriano, Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star-shaped domain, SIAM J. Control Optim., 38 (2000), 1581-1602.  doi: 10.1137/S0363012998344981.  Google Scholar

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F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim., 51 (2005), 61-105.  doi: 10.1007/s00245.  Google Scholar

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F. Alabau-Boussouira and P. Cannarsa, A general method for proving sharp energy decay rates for memory dissipative evolution equations, C. R. Acad. Sci. Paris. Ser. I, 347 (2009), 867-872.  doi: 10.1016/j.crma.2009.05.011.  Google Scholar

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A. BerkaniN. E. Tatar and A. Khemmoudj, Control of a viscoelastic translational Euler-Bernoulli beam, Math. Meth. Appl. Sci., 40 (2017), 237-254.  doi: 10.1002/mma.3985.  Google Scholar

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S. Berrimi and S. A. Messaoudi, Existence and decay of solutions of a viscoelastic equation with a nonlinear source, Nonlinear Anal. Theory Meth. Appl., 64 (2006), 2314-2331.  doi: 10.1016/j.na.2005.08.015.  Google Scholar

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W. J. Book, Modeling, design, and control of exible manipulator arms: a tutorial review, in Proceedings ot the 29th Conlnsnce on Decision and Control Honolulu, Hawaii, (1990), 500–506. Google Scholar

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E. L. Cabanillas and J. E. Munoz Rivera, Decay rates of solutions of an anisotropic inhomogeneous n-dimensional viscoelastic equation with polynomial decaying kernels, Commun. Math. Phys., 177 (1996), 583-602.   Google Scholar

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H. CanbolatD. DawsonC. Rahn and P. Vedagarbha, Boundary control of a cantilevered flexible beam with point-mass dynamics at the free end, Mechatronics, 8 (1998), 163-186.   Google Scholar

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P. Cannarsa and D. Sforza, Integro-differential equations of hyperbolic type with positive definite kernels, J. Differ. Equ., 250 (2011), 4289-4335.  doi: 10.1016/j.jde.2011.03.005.  Google Scholar

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R. H. Cannon and E. Schmitz, Initial experiments on the end-point control of a flexible one-link robot, Inter. J. Robotics Res., 3 (1984), 62-75.   Google Scholar

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M. M. Cavalcanti, Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation, Discrete Contin. Dyn. Syst., 8 (2002), 675-695.  doi: 10.3934/dcds.2002.8.675.  Google Scholar

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M. M. CavalcantiV. N. D. Cavalcanti and J. Ferreira, Existence and uniform decay for a nonlinear viscoelastic equation with strong damping, Math. Meth. Appl. Sci., 24 (2001), 1043-1053.  doi: 10.1002/mma.250.  Google Scholar

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M. M. CavalcantiV. N. D. Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, J. Differ. Equ., 236 (2007), 407-459.  doi: 10.1016/j.jde.2007.02.004.  Google Scholar

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M. M. CavalcantiV. N. D. CavalcantiI. Lasiecka and F. A. Nascimento, Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects, Discrete Contin. Dyn. Syst. B, 19 (2014), 1987-2012.  doi: 10.3934/dcdsb.2014.19.1987.  Google Scholar

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M. M. CavalcantiV. N. D. CavalcantiI. Lasiecka and X. Wang, Existence and sharp decay rate estimates for a von Karman system with long memory, Nonlinear Anal. Real World Appl., 22 (2015), 289-306.  doi: 10.1016/j.nonrwa.2014.09.016.  Google Scholar

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M. M. CavalcantiV. N. D. Cavalcanti and P. Martinez, General decay rate estimates for viscoelastic dissipative systems, Nonlinear Anal., 68 (2008), 177-193.  doi: 10.1016/j.na.2006.10.040.  Google Scholar

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M. M. Cavalcanti, V. N. D. Cavalcanti and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electron. J. Differ. Equ., (2002), 14 pp.  Google Scholar

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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.  Google Scholar

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B. D. Coleman and W. Noll, Foundations of linear viscoelasticity, Rev. Modern Phys., 33 (1961), 239-249.  doi: 10.1103/RevModPhys.33.239.  Google Scholar

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F. Conrad and Ö. Morgül, On the stabilization of a flexible beam with a tip mass, SIAM J. Control Optim., 36 (1998), 1962-1986.  doi: 10.1137/S0363012996302366.  Google Scholar

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C. M. Dafermos, On abstract Volterra equations with applications to linear viscoelasticity, J. Differ. Equ., 7 (1970), 554-569.  doi: 10.1016/0022-0396(70)90101-4.  Google Scholar

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M. DaoulatliI. Lasiecka and D. Toundykov, Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions, Discrete Contin. Dyn. Syst., 2 (2009), 67-95.  doi: 10.3934/dcdss.2009.2.67.  Google Scholar

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M. EllerJ. E. Lagnese and S. Nicaise, Decay rates for solutions of a Maxwell system with nonlinear boundary damping, Comput. Appl. Math., 21 (2002), 135-165.   Google Scholar

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B. Z. Guo, Riesz basis approach to the tracking control of a flexible beam with a tip rigid body without dissipativity, Optim. Methods Softw., 17 (2002), 655-681.  doi: 10.1080/1055678021000007288.  Google Scholar

[29]

B. Z. Guo and Q. Song, Tracking control of a flexible beam by nonlinear boundary feedback, J. Appl. Math. Stoch. Anal., 8 (1995), 47-58.  doi: 10.1155/S1048953395000049.  Google Scholar

[30]

B. Z. Guo and Q. Zhang, On harmonic disturbance rejection of an undamped Euler-Bernoulli beam with rigid tip body, ESAIM Control Optim. Calc. Var., 10 (2004), 615-623.  doi: 10.1051/cocv:2004028.  Google Scholar

[31]

Xi. Han and M. Wang, General decay of energy for a viscoelastic equation with nonlinear damping, Math. Meth. Appl. Sci., 32 (2009), 346-358.  doi: 10.1002/mma.1041.  Google Scholar

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J. H. Hassan and S. A. Messaoudi, General decay rate for a class of weakly dissipative second-order systems with memory, Math. Meth. Appl. Sci., (2019), 12 pp. doi: 10.1002/mma.5554.  Google Scholar

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K. P. JinJ. Liang and T. J. Xiao, Coupled second order evolution equations with fading memory: optimal energy decay rate, J. Differ. Equ., 257 (2014), 1501-1528.  doi: 10.1016/j.jde.2014.05.018.  Google Scholar

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A. KellecheN. E. Tatar and A. Khemmoudj, Uniform stabilization of an axially moving Kirchhoff string by a boundary control of memory type, J. Dyn. Control Syst., 23 (2017), 237-247.  doi: 10.1007/s10883-016-9310-2.  Google Scholar

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A. KellecheN. E. Tatar and A. Khemmoudj, Stability of an axially moving viscoelastic beam, J. Dyn. Control Syst., 23 (2017), 283-299.  doi: 10.1007/s10883-016-9317-8.  Google Scholar

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A. Khemmoudj and Y. Mokhtari, General decay of the solution to a nonlinear viscoelastic modified Von-Karman system with delay, Discrete Contin. Dyn. Syst. A, 39 (2019), 3839-3866.  doi: 10.3934/dcds.2019155.  Google Scholar

[38]

I. Lasiecka, Mathematical Control Theory of Coupled PDE's, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA, 2002. doi: 10.1137/1.9780898717099.  Google Scholar

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

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I. Lasiecka and X. Wang, Intrinsic decay rate estimates for semilinear abstract second order equations with memory, in New Prospects in Direct, Inverse and Control Problems for Evolution Equations, Springer INdAM, vol.10, Springer, Cham, (2014), 271–303. doi: 10.1007/978-3-319-11406-4_14.  Google Scholar

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B. Lekdim and A. Khemmoudj, General decay of energy to a nonlinear viscoelastic two dimensional beam, Appl. Math. Mech. Engl. Ed., 39 (2018), 1661-1678.  doi: 10.1007/s10483-018-2389-6.  Google Scholar

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S. LiY. Wang and Z. Liang, Stabilization of vibrating beam with a tip mass controlled by combined feedback forces, J. Math. Anal. Appl., 256 (2001), 13-38.  doi: 10.1006/jmaa.2000.7217.  Google Scholar

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J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites non Lineaires (in French), Dunod, Paris, 1969.  Google Scholar

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W. J. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ricerche Mat., 48 (1999), 61-75.   Google Scholar

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S. A. Messaoudi, General decay of the solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Anal., 69 (2008), 2589-2598.  doi: 10.1016/j.na.2007.08.035.  Google Scholar

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S. A. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl., 341 (2008), 1457-1467.  doi: 10.1016/j.jmaa.2007.11.048.  Google Scholar

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show all references

References:
[1]

M. AassilaM. M. Cavalcanti and V. N. Domingos Cavalcanti, Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term, Calc. Var. Partial Differ. Equ., 15 (2002), 155-180.  doi: 10.1007/s005260100096.  Google Scholar

[2]

M. AassilaM. M. Cavalcanti and J. A. Soriano, Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star-shaped domain, SIAM J. Control Optim., 38 (2000), 1581-1602.  doi: 10.1137/S0363012998344981.  Google Scholar

[3]

F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim., 51 (2005), 61-105.  doi: 10.1007/s00245.  Google Scholar

[4]

F. Alabau-Boussouira and P. Cannarsa, A general method for proving sharp energy decay rates for memory dissipative evolution equations, C. R. Acad. Sci. Paris. Ser. I, 347 (2009), 867-872.  doi: 10.1016/j.crma.2009.05.011.  Google Scholar

[5]

V. I. Arnold, Mathematical Methods of Classical Mechanics, New York, Springer-Verlag, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[6]

A. BerkaniN. E. Tatar and A. Khemmoudj, Control of a viscoelastic translational Euler-Bernoulli beam, Math. Meth. Appl. Sci., 40 (2017), 237-254.  doi: 10.1002/mma.3985.  Google Scholar

[7]

S. Berrimi and S. A. Messaoudi, Existence and decay of solutions of a viscoelastic equation with a nonlinear source, Nonlinear Anal. Theory Meth. Appl., 64 (2006), 2314-2331.  doi: 10.1016/j.na.2005.08.015.  Google Scholar

[8]

W. J. Book, Modeling, design, and control of exible manipulator arms: a tutorial review, in Proceedings ot the 29th Conlnsnce on Decision and Control Honolulu, Hawaii, (1990), 500–506. Google Scholar

[9]

E. L. Cabanillas and J. E. Munoz Rivera, Decay rates of solutions of an anisotropic inhomogeneous n-dimensional viscoelastic equation with polynomial decaying kernels, Commun. Math. Phys., 177 (1996), 583-602.   Google Scholar

[10]

H. CanbolatD. DawsonC. Rahn and P. Vedagarbha, Boundary control of a cantilevered flexible beam with point-mass dynamics at the free end, Mechatronics, 8 (1998), 163-186.   Google Scholar

[11]

P. Cannarsa and D. Sforza, Integro-differential equations of hyperbolic type with positive definite kernels, J. Differ. Equ., 250 (2011), 4289-4335.  doi: 10.1016/j.jde.2011.03.005.  Google Scholar

[12]

R. H. Cannon and E. Schmitz, Initial experiments on the end-point control of a flexible one-link robot, Inter. J. Robotics Res., 3 (1984), 62-75.   Google Scholar

[13]

M. M. Cavalcanti, Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation, Discrete Contin. Dyn. Syst., 8 (2002), 675-695.  doi: 10.3934/dcds.2002.8.675.  Google Scholar

[14]

M. M. CavalcantiV. N. D. Cavalcanti and J. Ferreira, Existence and uniform decay for a nonlinear viscoelastic equation with strong damping, Math. Meth. Appl. Sci., 24 (2001), 1043-1053.  doi: 10.1002/mma.250.  Google Scholar

[15]

M. M. CavalcantiV. N. D. Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, J. Differ. Equ., 236 (2007), 407-459.  doi: 10.1016/j.jde.2007.02.004.  Google Scholar

[16]

M. M. CavalcantiV. N. D. CavalcantiI. Lasiecka and F. A. Nascimento, Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects, Discrete Contin. Dyn. Syst. B, 19 (2014), 1987-2012.  doi: 10.3934/dcdsb.2014.19.1987.  Google Scholar

[17]

M. M. CavalcantiV. N. D. CavalcantiI. Lasiecka and X. Wang, Existence and sharp decay rate estimates for a von Karman system with long memory, Nonlinear Anal. Real World Appl., 22 (2015), 289-306.  doi: 10.1016/j.nonrwa.2014.09.016.  Google Scholar

[18]

M. M. CavalcantiV. N. D. Cavalcanti and P. Martinez, General decay rate estimates for viscoelastic dissipative systems, Nonlinear Anal., 68 (2008), 177-193.  doi: 10.1016/j.na.2006.10.040.  Google Scholar

[19]

M. M. Cavalcanti, V. N. D. Cavalcanti and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electron. J. Differ. Equ., (2002), 14 pp.  Google Scholar

[20]

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.  Google Scholar

[21] R. M. Christensen, Theory of Viscoelasticity: An Introduction, New York/London, Academic Press, 1982.   Google Scholar
[22]

B. D. Coleman and W. Noll, Foundations of linear viscoelasticity, Rev. Modern Phys., 33 (1961), 239-249.  doi: 10.1103/RevModPhys.33.239.  Google Scholar

[23]

F. Conrad and Ö. Morgül, On the stabilization of a flexible beam with a tip mass, SIAM J. Control Optim., 36 (1998), 1962-1986.  doi: 10.1137/S0363012996302366.  Google Scholar

[24]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.  Google Scholar

[25]

C. M. Dafermos, On abstract Volterra equations with applications to linear viscoelasticity, J. Differ. Equ., 7 (1970), 554-569.  doi: 10.1016/0022-0396(70)90101-4.  Google Scholar

[26]

M. DaoulatliI. Lasiecka and D. Toundykov, Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions, Discrete Contin. Dyn. Syst., 2 (2009), 67-95.  doi: 10.3934/dcdss.2009.2.67.  Google Scholar

[27]

M. EllerJ. E. Lagnese and S. Nicaise, Decay rates for solutions of a Maxwell system with nonlinear boundary damping, Comput. Appl. Math., 21 (2002), 135-165.   Google Scholar

[28]

B. Z. Guo, Riesz basis approach to the tracking control of a flexible beam with a tip rigid body without dissipativity, Optim. Methods Softw., 17 (2002), 655-681.  doi: 10.1080/1055678021000007288.  Google Scholar

[29]

B. Z. Guo and Q. Song, Tracking control of a flexible beam by nonlinear boundary feedback, J. Appl. Math. Stoch. Anal., 8 (1995), 47-58.  doi: 10.1155/S1048953395000049.  Google Scholar

[30]

B. Z. Guo and Q. Zhang, On harmonic disturbance rejection of an undamped Euler-Bernoulli beam with rigid tip body, ESAIM Control Optim. Calc. Var., 10 (2004), 615-623.  doi: 10.1051/cocv:2004028.  Google Scholar

[31]

Xi. Han and M. Wang, General decay of energy for a viscoelastic equation with nonlinear damping, Math. Meth. Appl. Sci., 32 (2009), 346-358.  doi: 10.1002/mma.1041.  Google Scholar

[32] G. H. HardyJ. E. Littlewood and G. Polya, Inequalities, Cambridge, U. K., Cambridge Univ Press, 1959.   Google Scholar
[33]

J. H. Hassan and S. A. Messaoudi, General decay rate for a class of weakly dissipative second-order systems with memory, Math. Meth. Appl. Sci., (2019), 12 pp. doi: 10.1002/mma.5554.  Google Scholar

[34]

K. P. JinJ. Liang and T. J. Xiao, Coupled second order evolution equations with fading memory: optimal energy decay rate, J. Differ. Equ., 257 (2014), 1501-1528.  doi: 10.1016/j.jde.2014.05.018.  Google Scholar

[35]

A. KellecheN. E. Tatar and A. Khemmoudj, Uniform stabilization of an axially moving Kirchhoff string by a boundary control of memory type, J. Dyn. Control Syst., 23 (2017), 237-247.  doi: 10.1007/s10883-016-9310-2.  Google Scholar

[36]

A. KellecheN. E. Tatar and A. Khemmoudj, Stability of an axially moving viscoelastic beam, J. Dyn. Control Syst., 23 (2017), 283-299.  doi: 10.1007/s10883-016-9317-8.  Google Scholar

[37]

A. Khemmoudj and Y. Mokhtari, General decay of the solution to a nonlinear viscoelastic modified Von-Karman system with delay, Discrete Contin. Dyn. Syst. A, 39 (2019), 3839-3866.  doi: 10.3934/dcds.2019155.  Google Scholar

[38]

I. Lasiecka, Mathematical Control Theory of Coupled PDE's, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA, 2002. doi: 10.1137/1.9780898717099.  Google Scholar

[39]

I. Lasiecka and D. Doundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and a nonlinear source, Nonlinear Anal., 64 (2006), 1757-1797.  doi: 10.1016/j.na.2005.07.024.  Google Scholar

[40]

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

[41]

I. Lasiecka and X. Wang, Intrinsic decay rate estimates for semilinear abstract second order equations with memory, in New Prospects in Direct, Inverse and Control Problems for Evolution Equations, Springer INdAM, vol.10, Springer, Cham, (2014), 271–303. doi: 10.1007/978-3-319-11406-4_14.  Google Scholar

[42]

B. Lekdim and A. Khemmoudj, General decay of energy to a nonlinear viscoelastic two dimensional beam, Appl. Math. Mech. Engl. Ed., 39 (2018), 1661-1678.  doi: 10.1007/s10483-018-2389-6.  Google Scholar

[43]

S. LiY. Wang and Z. Liang, Stabilization of vibrating beam with a tip mass controlled by combined feedback forces, J. Math. Anal. Appl., 256 (2001), 13-38.  doi: 10.1006/jmaa.2000.7217.  Google Scholar

[44]

J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites non Lineaires (in French), Dunod, Paris, 1969.  Google Scholar

[45]

W. J. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ricerche Mat., 48 (1999), 61-75.   Google Scholar

[46]

S. A. Messaoudi, General decay of the solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Anal., 69 (2008), 2589-2598.  doi: 10.1016/j.na.2007.08.035.  Google Scholar

[47]

S. A. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl., 341 (2008), 1457-1467.  doi: 10.1016/j.jmaa.2007.11.048.  Google Scholar

[48]

Ö Morgül, On a perturbed kernel in viscoelasticity. Dynamic boundary control of a Euler-Bernoulli beam, IEEE Trans. Autom. Control, 37 (1992), 639-642.  doi: 10.1109/9.135504.  Google Scholar

[49]

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