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

[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]

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.   Google Scholar

[50]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[60]

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

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]

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.   Google Scholar

[50]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[60]

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

[1]

Jong Yeoul Park, Sun Hye Park. On uniform decay for the coupled Euler-Bernoulli viscoelastic system with boundary damping. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 425-436. doi: 10.3934/dcds.2005.12.425

[2]

Maja Miletić, Dominik Stürzer, Anton Arnold. An Euler-Bernoulli beam with nonlinear damping and a nonlinear spring at the tip. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3029-3055. doi: 10.3934/dcdsb.2015.20.3029

[3]

Marcelo Moreira Cavalcanti. Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 675-695. doi: 10.3934/dcds.2002.8.675

[4]

Louis Tebou. Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms. Mathematical Control & Related Fields, 2012, 2 (1) : 45-60. doi: 10.3934/mcrf.2012.2.45

[5]

Fathi Hassine. Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1757-1774. doi: 10.3934/dcdsb.2016021

[6]

Gen Qi Xu, Siu Pang Yung. Stability and Riesz basis property of a star-shaped network of Euler-Bernoulli beams with joint damping. Networks & Heterogeneous Media, 2008, 3 (4) : 723-747. doi: 10.3934/nhm.2008.3.723

[7]

Wenjun Liu, Biqing Zhu, Gang Li, Danhua Wang. General decay for a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, dynamic boundary conditions and a time-varying delay term. Evolution Equations & Control Theory, 2017, 6 (2) : 239-260. doi: 10.3934/eect.2017013

[8]

Denis Mercier. Spectrum analysis of a serially connected Euler-Bernoulli beams problem. Networks & Heterogeneous Media, 2009, 4 (4) : 709-730. doi: 10.3934/nhm.2009.4.709

[9]

Gongwei Liu. The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term. Electronic Research Archive, 2020, 28 (1) : 263-289. doi: 10.3934/era.2020016

[10]

Zhiqing Liu, Zhong Bo Fang. Global solvability and general decay of a transmission problem for kirchhoff-type wave equations with nonlinear damping and delay term. Communications on Pure & Applied Analysis, 2020, 19 (2) : 941-966. doi: 10.3934/cpaa.2020043

[11]

Belkacem Said-Houari, Salim A. Messaoudi. General decay estimates for a Cauchy viscoelastic wave problem. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1541-1551. doi: 10.3934/cpaa.2014.13.1541

[12]

Louis Tebou. Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the $p$-Laplacian. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2315-2337. doi: 10.3934/dcds.2012.32.2315

[13]

Kaïs Ammari, Denis Mercier, Virginie Régnier, Julie Valein. Spectral analysis and stabilization of a chain of serially connected Euler-Bernoulli beams and strings. Communications on Pure & Applied Analysis, 2012, 11 (2) : 785-807. doi: 10.3934/cpaa.2012.11.785

[14]

Valentin Keyantuo, Louis Tebou, Mahamadi Warma. A Gevrey class semigroup for a thermoelastic plate model with a fractional Laplacian: Between the Euler-Bernoulli and Kirchhoff models. Discrete & Continuous Dynamical Systems - A, 2020, 40 (5) : 2875-2889. doi: 10.3934/dcds.2020152

[15]

Qing Chen, Zhong Tan. Time decay of solutions to the compressible Euler equations with damping. Kinetic & Related Models, 2014, 7 (4) : 605-619. doi: 10.3934/krm.2014.7.605

[16]

Qiong Zhang. Exponential stability of a joint-leg-beam system with memory damping. Mathematical Control & Related Fields, 2015, 5 (2) : 321-333. doi: 10.3934/mcrf.2015.5.321

[17]

Tae Gab Ha. On viscoelastic wave equation with nonlinear boundary damping and source term. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1543-1576. doi: 10.3934/cpaa.2010.9.1543

[18]

Haigang Li, Jiguang Bao. Euler-Poisson equations related to general compressible rotating fluids. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1085-1096. doi: 10.3934/dcds.2011.29.1085

[19]

Ammar Khemmoudj, Taklit Hamadouche. General decay of solutions of a Bresse system with viscoelastic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4857-4876. doi: 10.3934/dcds.2017209

[20]

Tae Gab Ha. Global existence and general decay estimates for the viscoelastic equation with acoustic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6899-6919. doi: 10.3934/dcds.2016100

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (180)
  • HTML views (86)
  • Cited by (0)

Other articles
by authors

[Back to Top]