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Exponential stability of a geometric nonlinear beam with a nonlinear delay term in boundary feedbacks

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Abstract

This paper is concerned with the stabilization of a geometric nonlinear beam with a nonlinear delay term in boundary control. The well-posedness of the closed-loop system where a nonlinear damping and a nonlinear delay damping are applied at the boundary is examined using the Faedo–Galerkin approximation method. Constructing a novel energy-like function to handle the nonlinear delay, the explicit exponential decay rate of the closed-loop system is established with a generalized Gronwall-type integral inequality and the integral-type multiplier method.

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References

  1. Krstic, M., Smyshlyaev, A.: Boundary Control of PDEs: A Course on Backstepping Designs. Society for Industrial and Applied Mathematics, Philadelphia, PA (2008)

    Book  MATH  Google Scholar 

  2. Luo, Z.-H., Guo, B.-Z., Morgül, Ö.: Stability and Stabilization of Infinite Dimensional Systems with Applications. Springer, London (1999)

    Book  MATH  Google Scholar 

  3. Mokhtari, A., Mirdamadi, H.R.: Study on vibration and stability of an axially translating viscoelastic Timoshenko beam: non-transforming spectral element analysis. Appl. Math. Model. 56, 342–358 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  4. Hansen, S.W., Zhang, B.-Y.: Boundary control of a linear thermoelastic beam. J. Math. Anal. Appl. 210(1), 182–205 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Yang, K.J., Hong, K.S.: Robust boundary control of an axially moving steel strip. IFAC Proc. Vol. 35(1), 7–12 (2002)

    Article  Google Scholar 

  6. Ding, H., Chen, L.-Q.: Galerkin methods for natural frequencies of high-speed axially moving beams. J. Sound Vib. 329(17), 3484–3494 (2010)

    Article  Google Scholar 

  7. Kelleche, A., Tatar, N.-E.: Control and exponential stabilization for the equation of an axially moving viscoelastic strip. Math. Methods Appl. Sci. 40(18), 6239–6253 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Li, T., Hou, Z.: Exponential stabilization of an axially moving string with geometrical nonlinearity by linear boundary feedback. J. Sound Vib. 296(4–5), 861–870 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Yang, K.J., Hong, K.S., Matsuno, F.: Robust boundary control of an axially moving string by using a PR transfer function. IEEE Trans. Autom. Control 50(12), 2053–2058 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Datko, R.: Two examples of ill-posedness with respect to small time delays in stabilized elastic systems. IEEE Trans. Autom. Control 38(1), 163–166 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Nicaise, S., Pignotti, C.: 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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Suh, H., Bien, Z.: Use of time-delay actions in the controller design. IEEE Trans. Autom. Control 25(3), 600–603 (1980)

    Article  MATH  Google Scholar 

  13. Morgül, Ö.: On the stabilization and stability robustness against small delays of some damped wave equations. IEEE Trans. Autom. Control 40(9), 1626–1630 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Liang, J.-S., Chen, Y.-Q., Guo, B.-Z.: A new boundary control method for beam equation with delayed boundary measurement using modified smith predictors. In: Proceedings of 42nd IEEE Conference on Decision and Control, 2003 (2004)

  15. Nicaise, S., Pignotti, C.: Stabilization of the wave equation with boundary or internal distributed delay. Differ. Integral Equ. 21(9–10), 935–958 (2008)

    MathSciNet  MATH  Google Scholar 

  16. Han, Z.-J., Xu, G.-Q.: Output-based stabilization of Euler-Bernoulli beam with time-delay in boundary input. IMA J. Math. Control. Inf. 31(4), 533–550 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, Y.-F., Xu, G.-Q., Han, Z.-J.: Feedback stabilisation of an Euler–Bernoulli beam with the boundary time-delay disturbance. Int. J. Control 91(8), 1835–1847 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  18. Shang, Y.-F., Xu, G.-Q.: Stabilization of an Euler-Bernoulli beam with input delay in the boundary control. Syst. Control Lett. 61(11), 1069–1078 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Xu, G.-Q., Yung, S.-P., Li, L.-K.: Stabilization of wave systems with input delay in the boundary control. ESAIM: Control Optim. Cal. Variat. 12, 770–785 (2006)

    MathSciNet  MATH  Google Scholar 

  20. Nicaise, S., Valein, J.: Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks. Networks Heterog. Media 2(3), 425–479 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Cheng, Y., Dong, Z.-H., O’Regan, D.: Exponential stability of axially moving Kirchhoff-beam systems with nonlinear boundary damping and disturbance. Discrete Cont. Dyn. Syst.-B 27(8), 4331–4346 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ding, H., Chen, L.-Q.: On two transverse nonlinear models of axially moving beams. Sci. China 52(3), 743–751 (2009)

    Article  MATH  Google Scholar 

  23. Ding, H., Chen, L.-Q.: Nonlinear models for transverse forced vibration of axially moving viscoelastic beams. Shock Vib. 18(1), 281–287 (2011)

    Article  Google Scholar 

  24. Li, T.-C., Hou, Z.-C., Li, J.-F.: Stabilization analysis of a generalized nonlinear axially moving string by boundary velocity feedback. Automatica 44(2), 498–503 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Haddad, W.M., Kapila, V.: Absolute stability criteria for multiple slope-restricted monotonic nonlinearities. IEEE Trans. Autom. Control 40(2), 361–365 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  26. Khalil, H.K.: Nonlinear Systems. Prentice Hall, Upper Saddle River (2002)

    MATH  Google Scholar 

  27. Shang, Y.-F., Xu, G.-Q., Chen, Y.-L.: Stability analysis of Euler-Bernoulli beam with input delay in the boundary control. Asian J. Control 14(4), 186–196 (2010)

    MathSciNet  MATH  Google Scholar 

  28. Jin, F.-F., Guo, B.-Z.: Lyapunov approach to output feedback stabilization for the Euler–Bernoulli beam equation with boundary input disturbance. Automatica 52, 95–102 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Miletić, M., Stürzer, D., Arnold, A., et al.: Stability of an Euler–Bernoulli beam with a nonlinear dynamic feedback system. IEEE Trans. Autom. Control 61(10), 2782–2795 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Nicaise, S., Pignotti, C.: 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)

    Article  MathSciNet  MATH  Google Scholar 

  31. Komornik, V.: Exact Controllability and Stabilization: The Multiplier Method. John Wiley Ltd, Chichester (1994)

    MATH  Google Scholar 

  32. Kim, J.U., Renardy, Y.: Boundary Control of the Timoshenko Beam. SIAM J. Control Optim. 25(6), 1417–1429 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  33. Guo, B.-Z., Guo, W.: Adaptive stabilization for a Kirchhoff-type nonlinear beam under boundary output feedback control. Nonlinear Anal. Theory Methods Appl. 66(2), 427–441 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are in debt to the anonymous referees whose comments helped them to improve the final version of this article. This work was partially supported by the Natural Science Foundation of Liaoning Province (No. 2020-MS-290) and the Basic Project of Bohai University.

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Authors and Affiliations

  1. School of Mathematical Sciences, Bohai University, Jinzhou, 121013, People’s Republic of China

    Cuiying Li & Yi Cheng

  2. School of Mathematical and Statistical Sciences, University of Galway, Galway, Ireland

    Donal O’Regan

Authors
  1. Cuiying Li
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  2. Yi Cheng
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  3. Donal O’Regan
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Contributions

C.L. contributed to writing–review and editing; Y.C. was involved in writing–original draft preparation; and D.O’ contributed to formal analysis and editing. All authors reviewed the manuscript.

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Correspondence to Yi Cheng.

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Li, C., Cheng, Y. & O’Regan, D. Exponential stability of a geometric nonlinear beam with a nonlinear delay term in boundary feedbacks. Z. Angew. Math. Phys. 74, 125 (2023). https://doi.org/10.1007/s00033-023-02018-5

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  • DOI: https://doi.org/10.1007/s00033-023-02018-5

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