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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 274857, 9 pages
http://dx.doi.org/10.1155/2013/274857
Research Article

Exponential Stability of a Linear Distributed Parameter Bioprocess with Input Delay in Boundary Control

1Department of Mathematics, Bohai University, Jinzhou 121013, China
2Beijing Institute of Information and Control, Beijing 100037, China

Received 9 April 2013; Accepted 20 July 2013

Academic Editor: Gen-Qi Xu

Copyright © 2013 Fu Zheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. I. Gumowski and C. Mira, Optimization in Control Theory and Practice, Cambridge University Press, Cambridge, UK, 1968.
  2. R. Datko, J. Lagnese, and M. P. Polis, “An example on the effect of time delays in boundary feedback stabilization of wave equations,” SIAM Journal on Control and Optimization, vol. 24, no. 1, pp. 152–156, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. R. Datko, “Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks,” SIAM Journal on Control and Optimization, vol. 26, no. 3, pp. 697–713, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. I. H. Suh and Z. Bien, “Use of time delay action in the controller design,” IEEE Transactions on Automatic Control, vol. 25, no. 3, pp. 600–603, 1980. View at Google Scholar · View at Scopus
  5. W. H. Kwon, G. W. Lee, and S. W. Kim, “Performance improvement using time delays in multivariable controller design,” International Journal of Control, vol. 52, no. 6, pp. 1455–1473, 1990. View at Google Scholar · View at Scopus
  6. N. Jalili and N. Olgac, “Optimum delayed feedback vibration absorber for MDOF mechanical structures,” in Proceedings of the 37th IEEE Conference on Decision and Control (CDC '98), pp. 4734–4739, Tampa, Fla, USA, December 1998. View at Scopus
  7. W. Aernouts, D. Roose, and R. Sepulchre, “Delayed control of a Moore-Greitzer axial compressor model,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 10, no. 5, pp. 1157–1164, 2000. View at Google Scholar · View at Scopus
  8. H. Logemann, R. Rebarber, and G. Weiss, “Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop,” SIAM Journal on Control and Optimization, vol. 34, no. 2, pp. 572–600, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. B.-Z. Guo and K.-Y. Yang, “Dynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observation,” Automatica, vol. 45, no. 6, pp. 1468–1475, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. B.-Z. Guo, C.-Z. Xu, and H. Hammouri, “Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation,” European Series in Applied and Industrial Mathematics, vol. 18, no. 1, pp. 22–35, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. S. Nicaise and C. Pignotti, “Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks,” SIAM Journal on Control and Optimization, vol. 45, no. 5, pp. 1561–1585, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. G. Q. Xu, S. P. Yung, and L. K. Li, “Stabilization of wave systems with input delay in the boundary control,” European Series in Applied and Industrial Mathematics, vol. 12, no. 4, pp. 770–785, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. S. Bourrel and D. Dochain, “Stability analysis of two linear distributed parameter bioprocess models,” Mathematical and Computer Modelling of Dynamical Systems, vol. 6, no. 3, pp. 267–281, 2000. View at Google Scholar · View at Scopus
  14. H. Sano, “Boundary control of a linear distributed parameter bioprocess,” Journal of the Franklin Institute, vol. 340, no. 5, pp. 293–306, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. F. L. Huang, “Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces,” Annals of Differential Equations, vol. 1, no. 1, pp. 43–56, 1985. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Verlag, Basel, Switzerland, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  17. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983. View at Publisher · View at Google Scholar · View at MathSciNet