Advertisement

Stabilization for distributed semilinear systems governed by optimal feedback control

  • A. TsouliEmail author
  • Y. Benslimane
Article
  • 16 Downloads

Abstract

This paper focuses on the problem of polynomial and weak stabilization of abstract distributed semilinear systems in a real Hilbert space governed by an optimal multiplicative feedback control. A new proposed feedback control is constructed to achieves the two kinds of stabilization. Necessary and sufficient conditions for stabilization problems are investigated as well. Furthermore, the used feedback control is the unique solution of an appropriate minimization problem. Some examples of hyperbolic and parabolic partial differential equations are provided. Finally, simulations are given.

Keywords

Semilinear systems Weak stabilization Polynomial stabilization Optimal feedback control 

References

  1. 1.
    Ball J, Slemrod M (1979) Feedback stabilization of distributed semilinear control systems. J Appl Math Opt 5:169–179MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Jurdjevic V, Quinn J (1978) Controllability and stability. J Differ Equ 28:289–381MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ouzahra M (2008) Strong stabilization with decay estimate of semilinear systems. Syst Control Lett 57:813–815MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Slemrod M (1978) Stabilization of bilinear control systems with applications to nnonconservative problems in elasticity. SIAMJ Control 16:131–141CrossRefzbMATHGoogle Scholar
  5. 5.
    Bounit H, Hammouri H (1999) Feedback stabilization for a class of distributed semilinear control systems. Nonlinear Anal 37:953–969MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Tsouli A, Ouzahra M, Boutoulout A (2014) A decay estimate for constrained semilinear systems. Inf Sci Lett 3(2):77–83CrossRefzbMATHGoogle Scholar
  7. 7.
    Balakrishnan AV (1976) Applied functional analysis, applications of mathematics, vol 3. Springer, New YorkGoogle Scholar
  8. 8.
    Pazy A (1983) Semi-groups of linear operators and applications to partial differential equations. Springer, New YorkCrossRefzbMATHGoogle Scholar
  9. 9.
    Tsouli A, Boutoulout A (2015) Polynomial decay rate estimate for bilinear parabolic systems under weak observability condition. Rend Circ Mat Palermo 64:347–364MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Engel KJ, Nagel R (2005) One-parameter semigroups for linear evolution equations. Springer, BerlinzbMATHGoogle Scholar
  11. 11.
    Khapalov AY, Nag P (2003) Energy decay estimates for Lienard’s equation with quadratic viscous feedback. Electron J Differ Equ 70:1–12MathSciNetzbMATHGoogle Scholar
  12. 12.
    Quinn JP (1980) Stabilization of bilinear systems by quadratic feedback control. J Math Anal Appl 75:66–80MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Triggiani R (1992) Counterexamples to some stability questions for dissipative generators. J Math Anal Appl 170:49–64MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratory of Analysis Modeling and SimulationHassan II University ENSAM CasablancaCasablancaMorocco
  2. 2.Laboratory of Mathematics and ApplicationsHassan II University ENSAM CasablancaCasablancaMorocco

Personalised recommendations