Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Regional Weak and Strong Stabilization of Semilinear Systems with Decay Estimate

Discontinuity, Nonlinearity, and Complexity 8(4) (2019) 353--367 | DOI:10.5890/DNC.2019.12.001

A. El Alami, A. Boutoulout

Laboratory of Modeling Analysis & Control systems (MACS), Department of Mathematics, Moulay Ismail University, Faculty of Sciences Meknes, Morocco

Download Full Text PDF



The aim of this paper is to develop the question of the regional stabilization for infinite-semilinear parabolic systems on a spatial domain Ω. precisely the study of such system on a subregion ω of Ω. We consider a decomposition of the state space via the spectral properties of the system. Then we apply this approach to regional strong and weak stabilization problem using bounded feedback. Some applications and simulations are presented.


The authors wish to thank the referees for their helpful comments and suggestions. This work has been carried out with a grant from Hassan II Academy of Sciences and Technology.


  1. [1]  EL jai, A., Simon, M.C., Zerrikm E., and Pritchard, A.J. (1995), Regional controllability of distributed parameter systems, International Journal of Control, 62(6), 1351-1365, DOI: 10.1080/00207179508921603.
  2. [2]  Ball, J. and Slemrod, M. (1979), Feedback stabilization of distributed semilinear control systems, Appl. Math. Opt., 5, 169-179.
  3. [3]  Berrahmoune, L. (1999), Stabilization and decay estimate for distributed bilinear systems, Systems and Control Letters, 36, 167-171.
  4. [4]  Berrahmoune, L., Elboukfaoui, Y., and Erraoui, M. (2001), Remarks on the feedback stabilization of systems affine in control, Eur. J. Control, 7, 17-28.
  5. [5]  Ouzahra, M. (2008), Strong stabilization with decay estimate of semilinear systems, Systems and Control Letters, 57, 813-815.
  6. [6]  Zerrik, E. and Ouzahra, M. (2003), Regional stabilization for infinite dimensional systems, International Journal of Control, 76, 73-81.
  7. [7]  Zerrik, E., Ouzahra, M., and Ztot, K, (2004), Regional stabilization for infinite bilinear, IEE Proc.-Control Theory Appl, 151, 109-116.
  8. [8]  EL Alami, A., El harrakim I., and Boutoulout, A. (2017), Regional feedback stabilization for infinite semilinear systems, J Dyn Control Syst,
  9. [9]  El Harraki, I., El Alami, A., Boutoulout, A., and Serhani, M. (2016), Regional stabilization for semilinear parabolic systems, IMA Journal of Mathematical Control and Information, 2015-197.
  10. [10]  Chen, M. (1998), Exponential stabilization of a constrained bilinear system, Automatica, 34, 989-992.
  11. [11]  Kato, T. (1980), Perturbation theory for linear operators, New York., Springer.
  12. [12]  Quinn, J.P. (1980), Stabilization of bilinear systems by quadratic feedback control, J. Math. Anal. Appl, 75, 66-80.
  13. [13]  Pazy, A. (1983), Semi-groups of linear operators and applications to partial differential equations, Springer Verlag, New York.
  14. [14]  Lasiecka, I. and Tataru, D. (1993), Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential and Integral Equations, 6, 507-533.
  15. [15]  Tsouli, A., Boutouloutm A., and El Alami, A. (2015), Constrained Feedback stabilization for bilinear parabolic systems, Intelligent Control and Automation, 6, 103-115.
  16. [16]  Ouzahra, M. (2010), Exponential and weak stabilization of constrained bilinear systems, SIAM J. Control Optim., 48(6), 3962-3974.
  17. [17]  Berrahmoune, L. (2009), Stabilization of bilinear control systems in Hilbert space with nonquadratic feedback, Rend. Circ. Mat. Palermo., 58, 275-82.
  18. [18]  Ouzahra,M. (2009), Stabilization of infinite-dimensional bilinear systems using a quadratic feedback control, International Journal of Control, 82, 1657-1664.
  19. [19]  Luesink, R. and Nijmeijer, H. (1989), On the stabilization of bilinear systems via constant feedback, Linear Algebra and its Applications, 122/123/124, 457-474.
  20. [20]  Triggiani, R. (1975), On the stabilizability problem in Banach space, J. Math. Anal. Appl., 52, 383-403.
  21. [21]  Seidman, T.I. and Li, H. (2001), A note on stabilization with saturating feedback, Discrete Contin. Dyn. Syst., 7, 319-28.