326
Views
52
CrossRef citations to date
0
Altmetric
Original Articles

Stabilisation of Timoshenko beam system with delay in the boundary control

&
Pages 1165-1178 | Received 05 Aug 2012, Accepted 09 Mar 2013, Published online: 28 Apr 2013

References

  • Ait Benhassi, E. M., Ammari, K., Boulite, S., & Maniar, L. (2009). Feedback stabilization of a class of evolution equations with delay. Journal of Evolution Equations, 9, 103–121.
  • Datko, R. (1988). Not all feedback stabilized hyperbolic systems are robust with respect to small time delay in their feedbacks. SIAM Journal on Control and Optimization, 26, 697–713.
  • Datko, R. (1993). Two examples of ill-posedness with respect to small delays in stabilized elastic systems. IEEE Transactions on Automatic Control, 38, 163–166.
  • Datko, R., Lagnese, J., & Polis, M. P. (1986). An example of the effect of time delays in boundary feedback stabilization of wave equations. SIAM Journal on Control and Optimization, 24, 152–156.
  • Guiver, C., & Opmeer, M. R. (2010). Non-dissipative boundary feedback for Rayleigh and Timoshenko beams. Systems and Control Letters, 59, 578–586.
  • Guo, B. Z., & Luo, Y. H. (2002). Controllability and stability of a second-order hyperbolic system with collocated sensor/actuator. Systems and Control Letters, 46, 45–65.
  • Han, Z. J., & Xu, G. Q. (2010a). Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks. ESAIM: Control, Optimization and Calculus of Variations, 17, 552–574.
  • Han, Z. J., & Xu, G. Q. (2010b). Stabilization and Riesz basis of a star-shaped network of Timoshenko beams. Journal of Dynamical and Control Systems, 16(2), 227–258.
  • Han, Z. J., & Xu, G. Q. (2011). Dynamical behavior of networks of non-uniform Timoshenko beams system with boundary time-delay inputs. Networks and Heterogeneous Media, 6, 297–327.
  • Kim, J. U., & Renardy, Y. (1987). Boundary control of the Timoshenko beam. SIAM Journal on Control and Optimization, 25, 1417–1429.
  • Kirane, M., Said-Houari, B., & Anwar, M. N. (2011). Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks. Communications on Pure and Applied Analysis, 10(2), 667–686.
  • Lagnese, J. E., Leugering, G., & Schmidt, E. J. P. G. (1993). Control of planar networks of Timoshenko beams. SIAM Journal on Control and Optimization, 31, 780–811.
  • Leugering, G. (1999). Dynamic domain decomposition of optimal control problems for networks of strings and Timoshenko beams. SIAM Journal on Control and Optimization, 37(6), 1649–1675.
  • 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 Optimization, 45(5), 1561–1585.
  • Nicaise, S., & Pignotti, C. (2008). Stabilization of the wave equation with boundary or internal distributed delay. Differential and Integral Equations, 21(9–10), 935–958.
  • Nicaise, S., & Valein, J. (2007). Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks. Networks and Heterogeneous Media, 2(3), 425–479.
  • Nicaise, S., & Valein, J. (2010). Stabilization of second order evolution equations with unbounded feedback with delay. ESAIM: Control, Optimization and Calculus of Variations, 16, 420–456.
  • Said-Houari, B., & Laskri, Y. (2010). A stability result of a Timoshenko system with a delay term in the internal feedback. Applied Mathematics and Computation, 217, 2857–2869.
  • Said-Houari, B., & Soufyane, A. (2012). Stability result of the Timoshenko system with delay and boundary feedback. IMA Journal of Mathematical Control and Information, 29(3): 383–398.
  • 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.
  • 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.
  • Shubov, M. A. (2002). Asymptotic and spectral analysis of the spatially nonhomogeneous Timoshenko beam model. Mathematische Nachrichten, 241, 125–162.
  • Xu, G. Q. (in preparation). Exponential family, Riesz basis sequence and its applications in control theory.
  • Xu, G. Q. (2005). Boundary feedback exponential stabilization of a Timoshenko beam with both ends free. International Journal of Control, 78, 286–297.
  • Xu, G. Q., & Feng, D. X. (2002). The Riesz basis property of a Timoshenko beam with boundary feedback and application. IMA Journal of Applied Mathematics, 67, 357–370.
  • Xu, G. Q., Han, Z. J., & Yung, S. P. (2007). Riesz basis property of serially connected Timoshenko beams. International Journal of Control, 80, 470–485.
  • Xu, G. Q., & Yung, S. P. (2004). Exponential decay rate for a Timoshenko beam with boundary damping. Journal of Optimization Theory and Applications, 123, 669–693.
  • Xu, G. Q., Yung, S. P., & Li, L. K. (2006). Stabilization of wave systems with input delay in the boundary control. ESAIM: Control, Optimization and Calculus of Variations, 12, 770–785.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.