Skip to main content
SpringerLink
Log in

Stability analysis of the Euler-Bernoulli beam with multi-delay controller

  • Research Article
  • Published:
Control Theory and Technology Aims and scope Submit manuscript

Abstract

In this paper, we investigate the stabilization of an Euler-Bernoulli beam with time delays in the boundary controller. The boundary velocity feedback law is applied to obtain the closed-loop system. It is shown that this system generates a \(C_0\)-semigroup of linear operators. Moreover, the stability of the closed-loop system is discussed for different values of the controller constants and time delays via using spectral analysis and a suitable Lyapunov function.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Gugat, M. (2010). Boundary feedback stabilization by time delay for one-dimensional wave equations. IMA Journal of Mathematical Control and Information, 27, 189–203. https://doi.org/10.1093/imamci/dnq007.

    Article  MathSciNet  MATH  Google Scholar 

  2. Mehrkanoon, S., Shardt, Y. A. W., Suykens, J. A. K., & Ding, S. X. (2016). Estimating the unknown time delay in chemical processes. Engineering Applications of Artificial Intelligence, 55, 219–230. https://doi.org/10.1016/j.engappai.2016.06.014.

    Article  Google Scholar 

  3. Morgul, O. (1995). On the stabilization and stability robustness against small delays of some damped wave equations. IEEE Transactions on Automatic Control, 40, 1626–1630. https://doi.org/10.1109/9.412634.

    Article  MathSciNet  MATH  Google Scholar 

  4. Nicaise, S., & Pignotti, C. (2006). Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM Journal on Control and Optimimization, 45, 1561–1585. https://doi.org/10.1137/060648891.

    Article  MathSciNet  MATH  Google Scholar 

  5. Wu, L., Lam, H. K., Zhao, Y., & Shu, Z. (2015). Time-delay systems and their applications in engineering 2014. Mathematical Problems in Engineering. https://doi.org/10.1155/2015/246351.

    Article  Google Scholar 

  6. Chen, G., Krantz, S.G., Ma, D.W., Wayne, C.E., & West, H.H. (1988). The Euler-bernoulli beam equation with boundary energy dissipation. Operator Methods for Optimal Control Problems. https://apps.dtic.mil/sti/citations/ADA189517

  7. Datko, R. (1988). Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM Journal on Control and Optimimization, 26, 697–713. https://doi.org/10.1137/0326040.

    Article  MathSciNet  MATH  Google Scholar 

  8. Datko, R., Lagnese, J., & Polis, M. P. (1986). An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM Journal on Control and Optimimization, 24, 152–156. https://doi.org/10.1137/0324007.

    Article  MathSciNet  MATH  Google Scholar 

  9. Abdallah, C., Dorato, P., Benites-Read, J., Byrne, R. (1993). Delayed positive feedback can stabilize oscillatory systems. In American control conference (pp. 3106–3107). San Francisco, CA, USA. https://doi.org/10.23919/ACC.1993.4793475

  10. Aernouts, W., Roose, D., & Sepulchre, R. (2000). Delayed control of a Moore-Greitzer axial compressor model. International Journal of Bifurcation and Chaos, 10, 1157–1164. https://doi.org/10.1142/S0218127400000827.

    Article  Google Scholar 

  11. Kwon, W. H., Lee, G. W., & Kim, S. W. (1990). Performance improvement using time delays in multivariable controller design. International Journal of Control, 52, 1455–1473. https://doi.org/10.1080/00207179008953604.

    Article  MATH  Google Scholar 

  12. Batkai, A., & Piazzera, S. (2005). Semigroups for Delay Equations. A K Peters Ltd.

  13. Xu, G. Q., & Wang, H. (2013). Stabilisation of Timoshenko beam system with delay in the boundary control. International Journal of Control, 86, 1165–1178. https://doi.org/10.1080/00207179.2013.787494.

    Article  MathSciNet  MATH  Google Scholar 

  14. Shang, Y. F., Xu, G. Q., & Chen, Y. L. (2012). Stability analysis of Euler-bernoulli beam with input delay in the boundary control. Asian Journal of Control, 14, 186–196. https://doi.org/10.1002/asjc.279.

    Article  MathSciNet  MATH  Google Scholar 

  15. Shang, Y. F., & Xu, G. Q. (2012). Stabilization of an Euler-Bernoulli beam with input delay in the boundary control. Systems and Control Letters, 61, 1069–1078. https://doi.org/10.1016/j.sysconle.2012.07.012.

    Article  MathSciNet  MATH  Google Scholar 

  16. Han, Z. J., & Xu, G. Q. (2011). Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks. ESAIM: Control, Optimisation and Calculus of Variations, 17, 552–574. https://doi.org/10.1051/cocv/2010009.

    Article  MathSciNet  MATH  Google Scholar 

  17. Nicaise, S., & Pignotti, C. (2011). Interior feedback stabilization of wave equations with time dependent delay. Electronic Journal of Differential Equations, 41, 1–20. https://ejde.math.txstate.edu/Volumes/2011/41/nicaise.pdf

  18. Park, J. Y., Kang, Y. H., & Kim, J. A. (2008). Existence and exponential stability for a Euler-Bernoulli beam equation with memory and boundary output feedback control term. Acta Applicandae Mathematicae, 104, 287–301. https://doi.org/10.1007/s10440-008-9257-8.

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang, X., Han, Z. J., & Xu, G. Q. (2019). Spectral analysis of Timoshenko beam with time delay in interior damping. Zeitschrift für Angewandte Mathematik und Physik. https://doi.org/10.1007/s00033-019-1109-z.

    Article  MATH  Google Scholar 

  20. Xu, G.Q., Jalili Rahmati, A., & Badpar, F. (2018). Dynamic feedback stabilization of Timoshenko beam with internal input delays. WSEAS Transactions on Mathematics, 17, 101–112. https://www.wseas.org/multimedia/journals/mathematics/2018/a285906-038.php

  21. Yang, K.Y., Li, J.J., & Zhang, J. (2015). Stabilization of an Euler-Bernoulli beam equations with variable coefficients under delayed boundary output feedback. Electronic Journal of Differential Equations,75, 1–14. https://doaj.org/article/44c5c6b686c8446b85f1b0406a8f0630

  22. Zitouni, S., Ardjouni, A., Zennir, K., & Amiar, R. (2018). Existence and stability of a damped wave equation with two delayed terms in boundary. Journal of Nonlinear Analysis and Optimization: Theory and Applications, 9, 49–65. http://www.math.sci.nu.ac.th/ojs302/index.php/jnao/article/view/471

  23. Marzban, H. R., & Razzaghi, M. (2006). Solution of multi-delay systems using hybrid of block-pulse functions and Taylor series. Journal of Sound and Vibration, 292, 954–963. https://doi.org/10.1016/j.jsv.2005.08.007.

    Article  MathSciNet  MATH  Google Scholar 

  24. Mashayekhi, S., Razzaghi, M., & Wattanataweekul, M. (2016). Analysis of multi-delay and piecewise constant delay systems by hybrid functions approximation. Differential Equations Dynamic Systems, 24, 1–20. https://doi.org/10.1007/s12591-014-0203-0.

    Article  MathSciNet  MATH  Google Scholar 

  25. Mohan, B.M., & Kar, S.K. (2010). Optimal control of multi-delay systems via block-pulse functions. 5th International Conference on Industrial and Information Systems, pp. 614–619. Mangalore, India. https://doi.org/10.1109/ICIINFS.2010.5578634

  26. Shah, K. A., & Zada, A. (2021). Controllability and stability analysis of an oscillating system with two delays. Mathematical Methods in the Applied Sciences, 44, 14733–14765. https://doi.org/10.1002/mma.7739.

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhu, Y., Krstic, M., & Su, H. (2018). PDE boundary control of multi-input LTI systems with distinct and uncertain input delays. IEEE Transactions on Automatic Control, 63, 4270–4277. https://doi.org/10.1109/TAC.2018.2810038.

    Article  MathSciNet  MATH  Google Scholar 

  28. Wu, M., He, Y., & She, J. H. (2010). Stability analysis and robust control of time-delay systems. Springer.

  29. Adams, R., & Fournier, J. (2003). Sobolev spaces. Academic Press.

  30. Pazy, A. (1983). Semigroup of linear operator and applications to partial differential equations. Springer.

  31. Lyubich, Y.I., & Phong, V.Q. (1988). Asymptotic stability of linear differential equations in Banach spaces. Studia Mathematica, 88, 37–42. http://eudml.org/doc/218818

Download references

Author information

Author notes

  1. Genqi Xu and Sohrab Effati contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematical Sciences, Sharif University of Technology, Azadi Ave., Tehran, Iran

    Alireza Jalili Rahmtati

  2. Department of Mathematics, Tianjin University, Nankai District, Tianjin, 300072, China

    Genqi Xu

  3. Department of Applied Mathematics, Ferdowsi University of Mashhad, Azadi Square, Mashhad, Iran

    Sohrab Effati

Authors
  1. Alireza Jalili Rahmtati
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Genqi Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sohrab Effati
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alireza Jalili Rahmtati.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jalili Rahmtati, A., Xu, G. & Effati, S. Stability analysis of the Euler-Bernoulli beam with multi-delay controller. Control Theory Technol. 20, 338–348 (2022). https://doi.org/10.1007/s11768-022-00095-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11768-022-00095-x

Keywords

Search

Navigation