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 and 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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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. 

[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. 

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

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

[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.

[24]

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

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

[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.

[40]

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

[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.

[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.

[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.

[44]

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

[45]

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

[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.

[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.

[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.

[49]

J. E. Munoz Rivera JE and F. P. Quispe Gomez, Existence and decay in non linear viscoelasticity, Boll. Unione Mat. Ital., 6-B (2003), 1-37. 

[50]

M. I. Mustafa, Optimal decay rates for the viscoelastic wave equation, Math. Meth. Appl. Sci., (2017), 13 pp. doi: 10.1002/mma.4604.

[51]

M. I. Mustafa and S. A. Messaoudi, General stability result for viscoelastic wave equations, J. Math. Phys., 53 (2012), Art. 053702. doi: 10.1063/1.4711830.

[52]

M. I. Mustapha, General decay result for nonlinear viscoelastic equation, J. Math. Anal. Appl., 457 (2018), 134-152.  doi: 10.1016/j.jmaa.2017.08.019.

[53]

M. I. Mustapha, Laminated Timoshenko beams with viscoelastic damping, J. Math. Anal. Appl., 466 (2018), 619-641.  doi: 10.1016/j.jmaa.2018.06.016.

[54]

S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.  doi: 10.1137/060648891.

[55]

T. D. Nguyen and O. Egeland, Tracking and observer design for a motorized Euler-Bernoulli Beam, in Proc. IEEE International Conference on Decision and Control, Maui, Hawaii, (2003), 3325–3330.

[56]

P. Y. ParkK. H. Kang and J. A. Kim, Existence and exponential stability for a Euler-Bernoulli beam equation with memory and boundary output feedback control term, Acta Appl. Math., 104 (2008), 287-301.  doi: 10.1007/s10440-008-9257-8.

[57]

P. Y. Park and J. A. Kim, Existence and uniform decay for Euler-Bernoulli beam equation with memory term, Math. Meth. Appl. Sci., 27 (2004), 1629-1640.  doi: 10.1002/mma.512.

[58]

J. Y. Park and S. H. Park, General Decay for Quasilinear Viscoelastic Equations with Nonlinear Weak Damping, J. Math. Phys., 50 (2009), Art. 083505. doi: 10.1063/1.3187780.

[59]

L. SeghourA. Khemmoudj and N. E. Tatar, Control of a riser through the dynamic of the vessel, Appl. Anal., 95 (2016), 1957-1973.  doi: 10.1080/00036811.2015.1080249.

[60]

N. E. Tatar, Arbitrary decays in linear viscoelasticity, J. Math. Phys., 52 (2011), Art. 013502, 12 pp. doi: 10.1063/1.3533766.

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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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. 

[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. 

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

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

[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.

[24]

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

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

[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.

[40]

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

[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.

[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.

[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.

[44]

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

[45]

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

[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.

[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.

[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.

[49]

J. E. Munoz Rivera JE and F. P. Quispe Gomez, Existence and decay in non linear viscoelasticity, Boll. Unione Mat. Ital., 6-B (2003), 1-37. 

[50]

M. I. Mustafa, Optimal decay rates for the viscoelastic wave equation, Math. Meth. Appl. Sci., (2017), 13 pp. doi: 10.1002/mma.4604.

[51]

M. I. Mustafa and S. A. Messaoudi, General stability result for viscoelastic wave equations, J. Math. Phys., 53 (2012), Art. 053702. doi: 10.1063/1.4711830.

[52]

M. I. Mustapha, General decay result for nonlinear viscoelastic equation, J. Math. Anal. Appl., 457 (2018), 134-152.  doi: 10.1016/j.jmaa.2017.08.019.

[53]

M. I. Mustapha, Laminated Timoshenko beams with viscoelastic damping, J. Math. Anal. Appl., 466 (2018), 619-641.  doi: 10.1016/j.jmaa.2018.06.016.

[54]

S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.  doi: 10.1137/060648891.

[55]

T. D. Nguyen and O. Egeland, Tracking and observer design for a motorized Euler-Bernoulli Beam, in Proc. IEEE International Conference on Decision and Control, Maui, Hawaii, (2003), 3325–3330.

[56]

P. Y. ParkK. H. Kang and J. A. Kim, Existence and exponential stability for a Euler-Bernoulli beam equation with memory and boundary output feedback control term, Acta Appl. Math., 104 (2008), 287-301.  doi: 10.1007/s10440-008-9257-8.

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