Skip to main content
SpringerLink
Log in

Polynomial decay rate estimate for bilinear parabolic systems under weak observability condition

  • Published:
Rendiconti del Circolo Matematico di Palermo (1952 -) Aims and scope Submit manuscript

Abstract

In this paper, we shall study the stability for distributed bilinear systems on a Hilbert state space that can be decomposed in two subspaces: unstable finite-dimensional and stable infinite-dimensional with respect to the evolution generator. Then, we shall show under a weaker observability assumption that stabilizing such a system with a feedback control of the form \(p_{r}(t)=-\Vert y(t)\Vert ^{-r}\langle y(t),By(t)\rangle \) for \(r<2\), reduces stabilizing only its projection on the finite-dimension subspace which make the whole system stable. To this end, we shall propose a new family of continuous feedback controls that guarantee the uniform stabilizability with an explicit optimal decay rate estimate of the stabilized state. Two illustrating examples and simulations are provided.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Ammari, K., Tucsnak, M.: Stabilization of second order evolution equations by a class of unbounded feedbacks. ESAIM Control Calc. Var. 6, 361–386 (2001)

  2. Ball, J., Slemrod, M.: Feedback stabilization of distributed semilinear control systems. Appl. Math. Optim. 5, 169–179 (1979)

  3. Bruni, C., Dipillo, G., Koch, G.: Bilinear systems: an appealing class of nearly linear systems in theory and applications. IEEE Trans. Autom. Control 19, 334–348 (1974)

  4. Chen, M.S.: Exponential stabilization of a constrained bilinear system. Automatica. 34, 989–992 (1998)

    Article  MATH  Google Scholar 

  5. Chueshov, I.D.: Introduction to the theory of infinite-dimensional dissipative systems. ACTA Scientific Publishing House, ISBN 966-7021-64-5 (in Russian) (1999)

  6. Guo, L., Schwarz, H.: A control scheme for bilinear systems and application to a secondary controlled hydraulic rotary drive. In: Proceedings of the 28th IEEE Conference on Decision and Control. Tampa (1989)

  7. Kato, T.: Perturbation theory for linear operators. Springer, New York (1980)

    MATH  Google Scholar 

  8. Mohler, R.R.: Bilinear Control Processe. Academic Press, New York (1973)

    Google Scholar 

  9. Mohler, R.R., Khapalov, A.Y.: Bilinear control and application to flexible a.c. transmission systems. J. Optim. Theory Appl. 105(3), 621–637 (2000)

  10. Orlov, Y., Lou, Y., Christtfides, P.D.: Robust stabilization of infinite-dimensional systems using sliding mode output feedback control. Int. J. Control 77(12), 1115–1136 (2004)

  11. Ouzahra, M.: Strong stabilization with decay estimate of semilinear systems. Syst. Control Lett. 57, 813–815 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ouzahra, M.: Stabilisation of infinite-dimensional bilinear systems using a quadratic feedback control. Int. J. Control 82, 1657–1664 (2009)

  13. Ouzahra, M.: Exponential and weak stabilization of constrained bilinear systems. SIAM J. Control Optim. 48(6), 3962–3974 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ouzahra, M.: Feedback stabilization of parabolic systems with bilinear controls. Electron. J. Differ. Equ. 38, 1–10 (2011)

  15. Ouzahra, M., Tsouli, A., Boutoulout, A.: Stabilisation and polynomial decay estimate for distributed semilinear systems. Int. J. Control 85(4), 451–456 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Pazy, A.: Semi-groups of linear operators and applications to partial differential equations. Springer, New York (1983)

  17. Quinn, J.P.: Stabilization of bilinear systems by quadratic feedback control. J. Math. Anal. Appl. 75, 66–80 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  18. Triggiani, R.: On the stabilizability problem in Banach space. J. Math. Anal. Appl. 52, 383–403 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  19. Tsouli, A., Boutoulout, A.: Controllability of the parabolic system via bilinear control. J. Dyn. Control Syst. 20(3) (2014). doi:10.1007/s10883-014-9247-2

Download references

Acknowledgments

The authors would like to thank the anonymous referee for the very thorough reading of the paper and many helpful and constructive comments.

Author information

Authors and Affiliations

  1. TSI Team MACS Laboratory, Faculty of Sciences, Moulay Ismail University, Box 11201, Zitoune, Meknes, Morocco

    A. Tsouli & A. Boutoulout

Authors
  1. A. Tsouli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Boutoulout
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Tsouli.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tsouli, A., Boutoulout, A. Polynomial decay rate estimate for bilinear parabolic systems under weak observability condition. Rend. Circ. Mat. Palermo 64, 347–364 (2015). https://doi.org/10.1007/s12215-015-0204-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12215-015-0204-z

Keywords

Mathematics Subject Classification

Search

Navigation