Exponential Stability of laminated beam with constant delay feedback
Abstract
In this article, we consider a system of laminated beams with an internal constant delay term in the transverse displacement. We prove that the dissipation through structural damping at the interface is strong enough to exponentially stabilize the system under suitable assumptions on delay feedback and coefficients of wave propagation speed.
Keyword : laminated beam, interfacial slip, constant delay, exponential decay
How to Cite
Mpungu, K., & Apalara, T. A. (2021). Exponential Stability of laminated beam with constant delay feedback. Mathematical Modelling and Analysis, 26(4), 566-581. https://doi.org/10.3846/mma.2021.13759
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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T.A. Apalara. Uniform stability of a laminated beam with structural damping and second sound. Z. Angew. Math. Phys., 68(2):41, 2017. https://doi.org/10.1007/s00033-017-0784-x
T.A. Apalara. On the stability of a thermoelastic laminated beam. Acta Math. Sci., 39(6):1517–1524, 2019. https://doi.org/10.1007/s10473-019-0604-9
T.A. Apalara. Exponential stability of laminated beams with interfacial slip. Mech. Solids, 56(1):131–137, 2021.
T.A. Apalara and S.A. Messaoudi. An exponential stability result of a Timoshenko system with thermoelasticity with second sound and in the presence of delay. Appl. Math. Optim., 71(3):449–472, 2015. https://doi.org/10.1007/s00245014-9266-0
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T.A. Apalara, C.A. Raposo and C.A. Nonato. Exponential stability for laminated beams with a frictional damping. Arch. Math. (Basel), 114(4):471–480, 2020.
X.G. Cao, D.Y. Liu and G.Q. Xu. Easy test for stability of laminated beams with structural damping and boundary feedback controls. J. Dyn. Control Syst., 13(3):313–336, 2007. https://doi.org/10.1007/s10883-007-9022-8
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R. Datko, J. Lagnese and M.P. Polis. An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim., 24(1):152–156, 1986. https://doi.org/10.1137/0324007
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S.W. Hansen and R.D. Spies. Structural damping in laminated beams due to interfacial slip. J. Sound Vib., 204(2):183–202, 1997. https://doi.org/10.1006/jsvi.1996.0913
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I. Lasiecka. Global uniform decay rates for the solutions to wave equation with nonlinear boundary conditions. Appl. Anal., 47(1-4):191–212, 1992. https://doi.org/10.1080/00036819208840140
W. Liu, X. Kong and G. Li. Asymptotic stability for a laminated beam with structural damping and infinite memory. Math. Mech. Solids, 25(10):1979–2004, 2020. https://doi.org/10.1177/1081286520917440
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A. Lo and N. E Tatar. Uniform stability of a laminated beam with structural memory. Qual. Theory Dyn. Syst., 15(2):517–540, 2016. https://doi.org/10.1007/s12346-015-0147-y
E. Moyer and M. Miraglia. Peridynamic solutions for Timoshenko beams. Engineering, 6(6):304–317, 2014. https://doi.org/10.4236/eng.2014.66034
M.I. Mustafa. Uniform stability for thermoelastic systems with boundary time-varying delay. J. Math. Anal. Appl., 383(2):490–498, 2011. https://doi.org/10.1016/j.jmaa.2011.05.066
M.I. Mustafa. Boundary control of laminated beams with interfacial slip. J. Math. Phys., 59(5):051508, 2018. https://doi.org/10.1063/1.5017923
M.I. Mustafa. Laminated Timoshenko beams with viscoelastic damping. J. Math. Anal. Appl., 466(1):619–641, 2018. https://doi.org/10.1016/j.jmaa.2018.06.016
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(5):1561–1585, 2006. https://doi.org/10.1137/060648891
C. Pignotti. A note on stabilization of locally damped wave equations with time delay. Syst. Control Lett., 61(1):92–97, 2012. https://doi.org/10.1016/j.sysconle.2011.09.016
C.A. Raposo. Exponential stability for a structure with interfacial slip and frictional damping. Appl. Math. Lett., 53:85–91, 2016. https://doi.org/10.1016/j.aml.2015.10.005
B. Said-Houari and Y. Laskri. A stability result of a Timoshenko system with a delay term in the internal feedback. Appl. Math. Comput., 217(6):2857–2869, 2010. https://doi.org/10.1016/j.amc.2010.08.021
L. Seghour, N.E. Tatar and A. Berkani. Stability of a thermoelastic laminated system subject to a neutral delay. Math. Methods Appl. Sci., 43(1):281–304, 2020. https://doi.org/10.1002/mma.5878
N.E. Tatar. Stabilization of a laminated beam with interfacial slip by boundary controls. Bound. Value Probl., 2015(1):169, 2015. https://doi.org/10.1186/s13661-015-0432-3
J.M. Wang, G.Q. Xu and S.P Yung. Exponential stabilization of laminated beams with structural damping and boundary feedback controls. SIAM J. Control Optim., 44(5):1575–1597, 2005. https://doi.org/10.1137/040610003
G.Q. Xu, S.P. Yung and L.K. Li. Stabilization of wave systems with input delay in the boundary control. ESAIM: COCV, 12(4):770–785, 2006. https://doi.org/10.1051/cocv:2006021
E. Zuazua. Uniform stabilization of the wave equation by nonlinear boundary feedback. SIAM J. Control Optim., 28(2):466–477, 1990. https://doi.org/10.1137/0328025
M.S. Alves and R.N. Monteiro. Exponential stability of laminated Timoshenko beams with boundary/internal controls. J. Math. Anal. Appl., 482(1):123516, 2020. https://doi.org/10.1016/j.jmaa.2019.123516
T.A. Apalara. Asymptotic behavior of weakly dissipative Timoshenko system with internal constant delay feedbacks. Appl. Anal., 95(1):187–202, 2016. https://doi.org/10.1080/00036811.2014.1000314
T.A. Apalara. Uniform stability of a laminated beam with structural damping and second sound. Z. Angew. Math. Phys., 68(2):41, 2017. https://doi.org/10.1007/s00033-017-0784-x
T.A. Apalara. On the stability of a thermoelastic laminated beam. Acta Math. Sci., 39(6):1517–1524, 2019. https://doi.org/10.1007/s10473-019-0604-9
T.A. Apalara. Exponential stability of laminated beams with interfacial slip. Mech. Solids, 56(1):131–137, 2021.
T.A. Apalara and S.A. Messaoudi. An exponential stability result of a Timoshenko system with thermoelasticity with second sound and in the presence of delay. Appl. Math. Optim., 71(3):449–472, 2015. https://doi.org/10.1007/s00245014-9266-0
T.A. Apalara, A.M. Nass and H. Al Sulaimani. On a laminated Timoshenko beam with nonlinear structural damping. Math. Comput. Appl., 25(2):35, 2020. https://doi.org/10.3390/mca25020035
T.A. Apalara, C.A. Raposo and C.A. Nonato. Exponential stability for laminated beams with a frictional damping. Arch. Math. (Basel), 114(4):471–480, 2020.
X.G. Cao, D.Y. Liu and G.Q. Xu. Easy test for stability of laminated beams with structural damping and boundary feedback controls. J. Dyn. Control Syst., 13(3):313–336, 2007. https://doi.org/10.1007/s10883-007-9022-8
Z. Chen, W. Liu and D. Chen. General decay rates for a laminated beam with memory. Taiwan. J. Math., 23(5):1227–1252, 2019. https://doi.org/10.11650/tjm/181109
R. Datko, J. Lagnese and M.P. Polis. An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim., 24(1):152–156, 1986. https://doi.org/10.1137/0324007
B. Feng. Well-posedness and exponential decay for laminated Timoshenko beams with time delays and boundary feedbacks. Math. Methods Appl. Sci., 41(3):1162– 1174, 2018. https://doi.org/10.1002/mma.4655
B. Feng. On a thermoelastic laminated Timoshenko beam: Well posedness and stability. Complexity, Art. 5139419, 2020. https://doi.org/10.1155/2020/5139419
S.W. Hansen and R.D. Spies. Structural damping in laminated beams due to interfacial slip. J. Sound Vib., 204(2):183–202, 1997. https://doi.org/10.1006/jsvi.1996.0913
V. Komornik and E. Zuazua. A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl., 69(1):33–54, 1990.
I. Lasiecka. Global uniform decay rates for the solutions to wave equation with nonlinear boundary conditions. Appl. Anal., 47(1-4):191–212, 1992. https://doi.org/10.1080/00036819208840140
W. Liu, X. Kong and G. Li. Asymptotic stability for a laminated beam with structural damping and infinite memory. Math. Mech. Solids, 25(10):1979–2004, 2020. https://doi.org/10.1177/1081286520917440
W. Liu, Y. Luan, Y. Liu and G. Li. Well-posedness and asymptotic stability to a laminated beam in thermoelasticity of type III. Math. Meth. Appl. Sci., 43(6):3148–3166, 2020. https://doi.org/10.1002/mma.6108
W. Liu and W. Zhao. Stabilization of a thermoelastic laminated beam with past history. Appl. Math. Optim., 80(1):103–133, 2019. https://doi.org/10.1007/s00245-017-9460-y
A. Lo and N. E Tatar. Uniform stability of a laminated beam with structural memory. Qual. Theory Dyn. Syst., 15(2):517–540, 2016. https://doi.org/10.1007/s12346-015-0147-y
E. Moyer and M. Miraglia. Peridynamic solutions for Timoshenko beams. Engineering, 6(6):304–317, 2014. https://doi.org/10.4236/eng.2014.66034
M.I. Mustafa. Uniform stability for thermoelastic systems with boundary time-varying delay. J. Math. Anal. Appl., 383(2):490–498, 2011. https://doi.org/10.1016/j.jmaa.2011.05.066
M.I. Mustafa. Boundary control of laminated beams with interfacial slip. J. Math. Phys., 59(5):051508, 2018. https://doi.org/10.1063/1.5017923
M.I. Mustafa. Laminated Timoshenko beams with viscoelastic damping. J. Math. Anal. Appl., 466(1):619–641, 2018. https://doi.org/10.1016/j.jmaa.2018.06.016
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(5):1561–1585, 2006. https://doi.org/10.1137/060648891
C. Pignotti. A note on stabilization of locally damped wave equations with time delay. Syst. Control Lett., 61(1):92–97, 2012. https://doi.org/10.1016/j.sysconle.2011.09.016
C.A. Raposo. Exponential stability for a structure with interfacial slip and frictional damping. Appl. Math. Lett., 53:85–91, 2016. https://doi.org/10.1016/j.aml.2015.10.005
B. Said-Houari and Y. Laskri. A stability result of a Timoshenko system with a delay term in the internal feedback. Appl. Math. Comput., 217(6):2857–2869, 2010. https://doi.org/10.1016/j.amc.2010.08.021
L. Seghour, N.E. Tatar and A. Berkani. Stability of a thermoelastic laminated system subject to a neutral delay. Math. Methods Appl. Sci., 43(1):281–304, 2020. https://doi.org/10.1002/mma.5878
N.E. Tatar. Stabilization of a laminated beam with interfacial slip by boundary controls. Bound. Value Probl., 2015(1):169, 2015. https://doi.org/10.1186/s13661-015-0432-3
J.M. Wang, G.Q. Xu and S.P Yung. Exponential stabilization of laminated beams with structural damping and boundary feedback controls. SIAM J. Control Optim., 44(5):1575–1597, 2005. https://doi.org/10.1137/040610003
G.Q. Xu, S.P. Yung and L.K. Li. Stabilization of wave systems with input delay in the boundary control. ESAIM: COCV, 12(4):770–785, 2006. https://doi.org/10.1051/cocv:2006021
E. Zuazua. Uniform stabilization of the wave equation by nonlinear boundary feedback. SIAM J. Control Optim., 28(2):466–477, 1990. https://doi.org/10.1137/0328025