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On the Stabilization for a Class of Distributed Bilinear Systems using Bounded Controls

Year 2019, Volume: 1 Issue: 1, 28 - 40, 15.06.2019

Abstract

This paper considers the question of the output stabilization for a class of infinite dimensional bilinear system evolving on a spatial domain $\Omega$. Then, we give sufficient conditions for exponential, strong and weak stabilization of the output of such systems. Examples and simualtions illustrate the efficiency of such controls.

References

  • [1] J. M. Ball and M. Slemrod, Feedback stabilization of distributed semilinear control systems, Journal of Applied Mathematics and Optimization. 5 (1979) 169-179.
  • [2] L. Berrahmoune, Stabilization and decay estimate for distributed bilinear systems, Systems Control Letters. 36 (1999) 167-171.
  • [3] H. Bounit and H. Hammouri, Feedback stabilization for a class of distributed semilinear control systems, Nonlinear Analysis. 37 (1999) 953-969.
  • [4] W. Guo, Y. Chen and H. Feng, Output feedback stabilization for a Kirchhoff-type nonlinear beam with general corrupted boundary observation, International Journal of Robust and Nonlinear Control. (2017) doi: 10.1002/rnc.3740.
  • [5] I. Lasiecka and D. Tataru, Uniform boundary stabilisation of semilinear wave equation with nonlinear boundary damping, Journal of Difierential and Integral Equations. 6 (1993) 507- 533.
  • [6] S. Marx and E. Cerpa, Output feedback stabilization of the Korteweg-de Vries equation, Automatica. 87 (2018) 210-217.
  • [7] M. Ouzahra, Partial stabilization of semilinear systems using bounded controls, Interna- tional Journal of Control. 86 (2013) 2253-2262.
  • [8] A. Pazy, Semi-groups of linear operators and applications to partial differential equations, Springer Verlag, New York (1983).
  • [9] E. Zerrik, A. Ait Aadi and R. Larhrissi, Regional stabilization for a class of bilinear systems, IFAC-PapersOnLine. 50 (2017) 4540-4545.
  • [10] E. Zerrik, A. Ait Aadi and R. Larhrissi, On the stabilization of in nite dimensional bilin- ear systems with unbounded control operator, Journal of Nonlinear Dynamics and Systems Theory. 18 (2018) 418-425.
  • [11] E. Zerrik, A. Ait Aadi and R. Larhrissi, On the output feedback stabilization for distributed semilinear systems, Asian Journal of Control. (2019) doi: 10.1002/asjc.2081.
  • [12] E. Zerrik and M. Ouzahra, Regional stabilization for in nite-dimensional systems, Interna- tional Journal of Control. 76 (2003) 73-81.
  • [13] E. Zerrik, M. Ouzahra and K. Ztot, Regional stabilization for in nite bilinear systems, IET Proceeding of Control Theory and Applications. 151 (2004) 109-116.
  • [14] H. C. Zhou and G. Weiss, Output feedback exponential stabilization for one-dimensional un- stable wave equations with boundary control matched disturbance, SIAM Journal on Control and Optimization. 56 (2018) 4098-4129.
Year 2019, Volume: 1 Issue: 1, 28 - 40, 15.06.2019

Abstract

References

  • [1] J. M. Ball and M. Slemrod, Feedback stabilization of distributed semilinear control systems, Journal of Applied Mathematics and Optimization. 5 (1979) 169-179.
  • [2] L. Berrahmoune, Stabilization and decay estimate for distributed bilinear systems, Systems Control Letters. 36 (1999) 167-171.
  • [3] H. Bounit and H. Hammouri, Feedback stabilization for a class of distributed semilinear control systems, Nonlinear Analysis. 37 (1999) 953-969.
  • [4] W. Guo, Y. Chen and H. Feng, Output feedback stabilization for a Kirchhoff-type nonlinear beam with general corrupted boundary observation, International Journal of Robust and Nonlinear Control. (2017) doi: 10.1002/rnc.3740.
  • [5] I. Lasiecka and D. Tataru, Uniform boundary stabilisation of semilinear wave equation with nonlinear boundary damping, Journal of Difierential and Integral Equations. 6 (1993) 507- 533.
  • [6] S. Marx and E. Cerpa, Output feedback stabilization of the Korteweg-de Vries equation, Automatica. 87 (2018) 210-217.
  • [7] M. Ouzahra, Partial stabilization of semilinear systems using bounded controls, Interna- tional Journal of Control. 86 (2013) 2253-2262.
  • [8] A. Pazy, Semi-groups of linear operators and applications to partial differential equations, Springer Verlag, New York (1983).
  • [9] E. Zerrik, A. Ait Aadi and R. Larhrissi, Regional stabilization for a class of bilinear systems, IFAC-PapersOnLine. 50 (2017) 4540-4545.
  • [10] E. Zerrik, A. Ait Aadi and R. Larhrissi, On the stabilization of in nite dimensional bilin- ear systems with unbounded control operator, Journal of Nonlinear Dynamics and Systems Theory. 18 (2018) 418-425.
  • [11] E. Zerrik, A. Ait Aadi and R. Larhrissi, On the output feedback stabilization for distributed semilinear systems, Asian Journal of Control. (2019) doi: 10.1002/asjc.2081.
  • [12] E. Zerrik and M. Ouzahra, Regional stabilization for in nite-dimensional systems, Interna- tional Journal of Control. 76 (2003) 73-81.
  • [13] E. Zerrik, M. Ouzahra and K. Ztot, Regional stabilization for in nite bilinear systems, IET Proceeding of Control Theory and Applications. 151 (2004) 109-116.
  • [14] H. C. Zhou and G. Weiss, Output feedback exponential stabilization for one-dimensional un- stable wave equations with boundary control matched disturbance, SIAM Journal on Control and Optimization. 56 (2018) 4098-4129.
There are 14 citations in total.

Details

Primary Language English
Subjects Software Engineering (Other)
Journal Section Articles
Authors

El Hassan Zerrik 0000-0002-7677-8308

Abderrahman Ait Aadi 0000-0002-8032-1941

Publication Date June 15, 2019
Acceptance Date December 7, 2019
Published in Issue Year 2019 Volume: 1 Issue: 1

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