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Stabilization of bilinear control systems in Hilbert space with nonquadratic feedback

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Abstract

We consider bilinear control systems of the form y′(t) = Ay(t) + u(t)By(t) where A generates a strongly continuous semigroup of contraction (e t A) t⩾0 on an infinite-dimensional Hilbert space Y whose scalar product is denoted by 〈.,.〉. The function u denotes the scalar control. We suppose that B is a linear bounded operator from the state Y into itself. Tacking into account the control saturation, we study the problem of stabilization by feedback of the form u(t)=−f(〈By(t), y(t)〉). Application to the heat equation is considered.

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References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Benchimol, C.D,: A note on weak stabilizability of contraction semigroups, SIAM J. Control Optim., 16 (1978), 373–379

    Article  MATH  MathSciNet  Google Scholar 

  3. Berrahmoune, L.: Stabilization of linear control systems in Hilbert space with nonlinear feedback, Rend. Circ. Mat. Palermo, 54 (2005), 273–290

    Article  MATH  MathSciNet  Google Scholar 

  4. Berrahmoune, L., Elboukfaoui, Y., Erraoui, M.: Remarks on the feedback stabilization of systems affine in control, European Journal of Control, 7 (2001), 17–28

    Google Scholar 

  5. Bounit, H., Hammouri, H.: Feedback stabilization for a class of distributed semilinear control systems, Nonlinear Anal., 37 (1999), 953–969

    Article  MathSciNet  Google Scholar 

  6. Brezis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. Mathematics Studies 5, North-Holland (1973)

  7. Dafermos, C., Slemrod, M.: Asymptotic behaviour of nonlinear contraction semigroups, J. Funct. Anal., 13 (1973), 97–106

    Article  MATH  MathSciNet  Google Scholar 

  8. Nagy, B., Foias, C.: Analyse harmonique des opérateurs de l’espace de Hilbert. Masson et Cie, Akadémai Kiadó, Budapest, (French edition, Masson et Cie (1967)

    MATH  Google Scholar 

  9. Slemrod, M.: Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control, Math. Control Signals Systems, 2 1989), 265–285

    Article  MATH  MathSciNet  Google Scholar 

  10. Sontag, E.D., Sussman, H.J., Yang, Y.:A general result on the stabilization of linear systems using bounded controls, IEEE Trans. Automat. Control, 39 (1994), 2411–2425

    Article  MATH  MathSciNet  Google Scholar 

  11. Web, G.F.: Continuous nonlinear perturbations of linear accretive operators in Banach spaces, J. Funct. Anal., 10 (1972), 191–203

    Article  Google Scholar 

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  1. Département de Mathématiques, Ecole Normale Supérieure de Rabat, BP 5118, Rabat, Morocco

    Larbi Berrahmoune

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  1. Larbi Berrahmoune
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Correspondence to Larbi Berrahmoune.

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Berrahmoune, L. Stabilization of bilinear control systems in Hilbert space with nonquadratic feedback. Rend. Circ. Mat. Palermo 58, 275–282 (2009). https://doi.org/10.1007/s12215-009-0021-3

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  • DOI: https://doi.org/10.1007/s12215-009-0021-3

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