doi: 10.3934/dcdsb.2021230
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Exponential stability of axially moving Kirchhoff-beam systems with nonlinear boundary damping and disturbance

1. 

School of Mathematical Sciences, Bohai University, Jinzhou 121013, China

2. 

School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

* Corresponding author: Yi Cheng

Received  March 2021 Revised  July 2021 Early access September 2021

Fund Project: This work were partially supported by Natural Science Foundation of Liaoning Province (No. 2020-MS-290), Liaoning Natural Fund Guidance Plan (No. 2019-ZD-0508) and Young Science and Technology Talents "Nursery Seedling" Project of Liaoning Provincial Department of Education (No, LQ2019008)

This paper examines the stabilization problem of the axially moving Kirchhoff beam. Under the nonlinear damping criterion established by the slope-restricted condition, the existence and uniqueness of solutions of the closed-loop system equipped with nonlinear time-delay disturbance at the boundary is investigated via the Faedo-Galerkin approximation method. Furthermore, the solution is continuously dependent on initial conditions. Then the exponential stability of the closed-loop system is established by the direct Lyapunov method, where a novel energy function is constructed.

Citation: Yi Cheng, Zhihui Dong, Donal O' Regan. Exponential stability of axially moving Kirchhoff-beam systems with nonlinear boundary damping and disturbance. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021230
References:
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J. M. Ball, Initial boundary value problem for an extensible beam, J. Math. Analys. Appl., 42 (1973), 61-90.  doi: 10.1016/0022-247X(73)90121-2.  Google Scholar

[2]

L.-Q. Chen and W. Zhang, Adaptive vibration reduction of an axially moving string via a tensioner, Internat. J. Mechan. Sci., 48 (2006), 1409-1415.  doi: 10.1016/j.ijmecsci.2006.07.004.  Google Scholar

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R.-F. FungJ.-H. Chou and Y.-L. Kuo, Optimal boundary control of an axially moving material system, J. Dyn. Syst., 124 (2002), 55-61.  doi: 10.1115/1.1435364.  Google Scholar

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S. S. GeS. Zhang and W. He, Vibration control of an Euler-Bernoulli beam under unknown spatiotemporally varying disturbance, Internat. J. Contr., 84 (2011), 947-960.  doi: 10.1080/00207179.2011.584197.  Google Scholar

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B.-Z. Guo and W. Guo, Adaptive stabilization for a Kirchhoff-type nonlinear beam under boundary output feedback control, Nonlinear Anal., 66 (2007), 427-441.  doi: 10.1016/j.na.2005.11.037.  Google Scholar

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B.-Z. Guo and K.-Y. Yang, Dynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observation, Automatica J. IFAC, 45 (2009), 1468-1475.  doi: 10.1016/j.automatica.2009.02.004.  Google Scholar

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W. GuoY. Chen and H. Feng, Output feedback stabilization for a Kirchhoff-type nonlinear beam with general corrupted boundary observation, Internat. J. Robust Nonlinear Control, 27 (2017), 3280-3295.  doi: 10.1002/rnc.3740.  Google Scholar

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W. M. Haddad and V. Kapila, Absolute stability criteria for multiple slope-restricted monotonic nonlinearities, IEEE Trans. Automat. Control, 40 (1995), 361-365.  doi: 10.1109/9.341811.  Google Scholar

[10]

Z.-J. Han and G.-Q. Xu, Output-based stabilization of Euler-Bernoulli beam with time-delay in boundary input, IMA J. Math. Control Inform., 31 (2013), 533-550.  doi: 10.1093/imamci/dnt030.  Google Scholar

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S. W. Hansen and B.-Y. Zhang, Boundary control of a linear Thermoelastic beam, J. Math. Analys. Appl., 210 (1997), 182-205.  doi: 10.1006/jmaa.1997.5437.  Google Scholar

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H. K. Khalil, Nonlinear Systems, Upper Saddle River, 2002. Google Scholar

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T. KobayashiM. Oya and N. Takagi, Adaptive stabilization of a Kirchhoff's nonlinear beam with output disturbances, Nonlinear Anal., 71 (2009), 4798-4812.  doi: 10.1016/j.na.2009.03.056.  Google Scholar

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Y. LiG. Xu and Z. Han, Feedback stabilisation of an Euler-Bernoulli beam with the boundary time-delay disturbance, Internat. J. Control, 91 (2018), 1835-1847.  doi: 10.1080/00207179.2017.1333154.  Google Scholar

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J. Liang, Y. Chen and B. Guo, A new boundary control method for beam equation with delayed boundary measurement using modified smith predictors, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475), (2003), 809–814. doi: 10.1109/CDC.2003.1272665.  Google Scholar

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[17]

A. Mokhtari and H. R. Mirdamadi, Study on vibration and stability of an axially translating viscoelastic Timoshenko beam: Non-transforming spectral element analysis, Appl. Math. Mod., 56 (2018), 342-358.  doi: 10.1016/j.apm.2017.12.007.  Google Scholar

[18]

Ö. Morgül, On the stabilization and stability robustness against small delays of some damped wave equations, IEEE Trans. Automat. Control, 40 (1995), 1626-1630.  doi: 10.1109/9.412634.  Google Scholar

[19]

S. P. NagarkattiF. ZhangB. T. CosticD. M. Dawson and C. D. Rahn, Speed tracking and transverse vibration control of an axially accelerating web, Mech. Syst. Signal Proc., 16 (2002), 337-356.  doi: 10.1006/mssp.2000.1342.  Google Scholar

[20]

Y. F. Shang and G. Q. Xu, Stabilization of an Euler-Bernoulli beam with input delay in the boundary control, Syst. Control Lett., 61 (2012), 1069-1078.  doi: 10.1016/j.sysconle.2012.07.012.  Google Scholar

[21]

S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech. ASME, 17 (1950), 35-36.  doi: 10.1115/1.4010053.  Google Scholar

[22]

K.-Y. Yang, J.-J. Li and J. Zhang, Stabilization of Euler-Bernoulli beam equations with variable coefficients under delayed boundary output feedback, Electron. J. Differential Equations, 75 (2015), 14 pp.  Google Scholar

show all references

References:
[1]

J. M. Ball, Initial boundary value problem for an extensible beam, J. Math. Analys. Appl., 42 (1973), 61-90.  doi: 10.1016/0022-247X(73)90121-2.  Google Scholar

[2]

L.-Q. Chen and W. Zhang, Adaptive vibration reduction of an axially moving string via a tensioner, Internat. J. Mechan. Sci., 48 (2006), 1409-1415.  doi: 10.1016/j.ijmecsci.2006.07.004.  Google Scholar

[3]

H. Ding and L.-Q. Chen, Galerkin methods for natural frequencies of high-speed axially moving beams, Journal of Sound and Vibration, 329 (2010), 3484-3494.  doi: 10.1016/j.jsv.2010.03.005.  Google Scholar

[4]

R.-F. FungJ.-H. Chou and Y.-L. Kuo, Optimal boundary control of an axially moving material system, J. Dyn. Syst., 124 (2002), 55-61.  doi: 10.1115/1.1435364.  Google Scholar

[5]

S. S. GeS. Zhang and W. He, Vibration control of an Euler-Bernoulli beam under unknown spatiotemporally varying disturbance, Internat. J. Contr., 84 (2011), 947-960.  doi: 10.1080/00207179.2011.584197.  Google Scholar

[6]

B.-Z. Guo and W. Guo, Adaptive stabilization for a Kirchhoff-type nonlinear beam under boundary output feedback control, Nonlinear Anal., 66 (2007), 427-441.  doi: 10.1016/j.na.2005.11.037.  Google Scholar

[7]

B.-Z. Guo and K.-Y. Yang, Dynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observation, Automatica J. IFAC, 45 (2009), 1468-1475.  doi: 10.1016/j.automatica.2009.02.004.  Google Scholar

[8]

W. GuoY. Chen and H. Feng, Output feedback stabilization for a Kirchhoff-type nonlinear beam with general corrupted boundary observation, Internat. J. Robust Nonlinear Control, 27 (2017), 3280-3295.  doi: 10.1002/rnc.3740.  Google Scholar

[9]

W. M. Haddad and V. Kapila, Absolute stability criteria for multiple slope-restricted monotonic nonlinearities, IEEE Trans. Automat. Control, 40 (1995), 361-365.  doi: 10.1109/9.341811.  Google Scholar

[10]

Z.-J. Han and G.-Q. Xu, Output-based stabilization of Euler-Bernoulli beam with time-delay in boundary input, IMA J. Math. Control Inform., 31 (2013), 533-550.  doi: 10.1093/imamci/dnt030.  Google Scholar

[11]

S. W. Hansen and B.-Y. Zhang, Boundary control of a linear Thermoelastic beam, J. Math. Analys. Appl., 210 (1997), 182-205.  doi: 10.1006/jmaa.1997.5437.  Google Scholar

[12]

H. K. Khalil, Nonlinear Systems, Upper Saddle River, 2002. Google Scholar

[13]

T. KobayashiM. Oya and N. Takagi, Adaptive stabilization of a Kirchhoff's nonlinear beam with output disturbances, Nonlinear Anal., 71 (2009), 4798-4812.  doi: 10.1016/j.na.2009.03.056.  Google Scholar

[14]

Y. LiG. Xu and Z. Han, Feedback stabilisation of an Euler-Bernoulli beam with the boundary time-delay disturbance, Internat. J. Control, 91 (2018), 1835-1847.  doi: 10.1080/00207179.2017.1333154.  Google Scholar

[15]

J. Liang, Y. Chen and B. Guo, A new boundary control method for beam equation with delayed boundary measurement using modified smith predictors, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475), (2003), 809–814. doi: 10.1109/CDC.2003.1272665.  Google Scholar

[16]

Z.-H. Luo, B.-Z. Guo and O. Morgül, Stability and Stabilization of Infinite Dimensional Systems with Applications, Springer-Verlag London, Ltd., London, 1999. doi: 10.1007/978-1-4471-0419-3.  Google Scholar

[17]

A. Mokhtari and H. R. Mirdamadi, Study on vibration and stability of an axially translating viscoelastic Timoshenko beam: Non-transforming spectral element analysis, Appl. Math. Mod., 56 (2018), 342-358.  doi: 10.1016/j.apm.2017.12.007.  Google Scholar

[18]

Ö. Morgül, On the stabilization and stability robustness against small delays of some damped wave equations, IEEE Trans. Automat. Control, 40 (1995), 1626-1630.  doi: 10.1109/9.412634.  Google Scholar

[19]

S. P. NagarkattiF. ZhangB. T. CosticD. M. Dawson and C. D. Rahn, Speed tracking and transverse vibration control of an axially accelerating web, Mech. Syst. Signal Proc., 16 (2002), 337-356.  doi: 10.1006/mssp.2000.1342.  Google Scholar

[20]

Y. F. Shang and G. Q. Xu, Stabilization of an Euler-Bernoulli beam with input delay in the boundary control, Syst. Control Lett., 61 (2012), 1069-1078.  doi: 10.1016/j.sysconle.2012.07.012.  Google Scholar

[21]

S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech. ASME, 17 (1950), 35-36.  doi: 10.1115/1.4010053.  Google Scholar

[22]

K.-Y. Yang, J.-J. Li and J. Zhang, Stabilization of Euler-Bernoulli beam equations with variable coefficients under delayed boundary output feedback, Electron. J. Differential Equations, 75 (2015), 14 pp.  Google Scholar

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