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Interior and Analytic Stabilization of the Wave Equation over a Cylinder

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Abstract

We analyze the analytic stabilization of the wave equation on a cylinder subject to an interior dissipation that does not satisfy the classical geometric control condition BLR. For this model, we prove that the space of exponential stabilized functions is lower than the whole energy space. We use spectral properties to find the set of functions that can be stabilized. We define a constant α S which gives an estimate for the analyticity required for an initial data to hold in the space of stabilized functions.

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References

  1. B. Allibert, Analytic controllability of the wave equation over a cylinder. ESAIM: Control Optim. Calc. Var. 4 (1999), 177–207.

    Article  MATH  MathSciNet  Google Scholar 

  2. K. Ammari, Stabilisation d’un modèle d’interaction fluide-structure. Ann. Fac. Sci. Toulouse Math. 10 (2001), 225–254.

    MATH  MathSciNet  Google Scholar 

  3. K. Ammari, A. Henrot, and M. Tucsnak. Asymptotic behaviour of the solution and optimal location of the actuator for the pointwise stabilisation of a string. Asympt. Anal. 28 (2001), 215–240.

    MATH  MathSciNet  Google Scholar 

  4. K. Ammari, Z. Liu, and M. Tucsnak, Decay rates for a beam with pointwize force and moment feedback. Math. Control Signals Systems 15 (2002), 229–255.

    Article  MATH  MathSciNet  Google Scholar 

  5. K. Ammari and S. Nicaise, Polynomial and analytic stabilization of a weave equation coupled with a Euler–Bernoulli beam. To appear in Math. Meth. Appl. Sci.

  6. K. Ammari and A. Saïdi, Decay rates at low and high frequencies for a plates equation with feedback concentrated in interior curves. Z. Angew. Math. Math. Phys. 57 (2006), 832–846.

    Article  MATH  Google Scholar 

  7. _____, Pointwise stabilization of a hybrid system and optimal location of actuator. Appl. Math. Optim. 56 (2007), 105–130.

    Article  MATH  MathSciNet  Google Scholar 

  8. K. Ammari and M. Tucsnak, Stabilisation of Bernoulli–Euler beams by means of pointwise feedback force. SIAM J. Control Optim. 39 (2000), 1160–1181.

    Article  MATH  MathSciNet  Google Scholar 

  9. _____, Stabilization of second order evolution equations by a class of unbounded feedbacks. ESAIM Control Optim. Calc. Var. 6 (2001), 361–386.

    Article  MATH  MathSciNet  Google Scholar 

  10. A. Bamberger, J. Rauch, and M. Taylor, A model for harmonics on stringed instruments. Arch. Rat. Mech. Anal. 79 (1982), 270–290.

    Article  MathSciNet  Google Scholar 

  11. C. Bardos, G. Lebeau, and J. Rauch, Sharp sufficent condition for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim. 305 (1992), 1024–1065.

    Article  MathSciNet  Google Scholar 

  12. C. Dafermos, Asymptotic bihavior of solutions of evolutions equations. In: Nonlinear evolution equations (M. G. Grandall, Ed.), Academic Press, New York (1978), pp. 103–123.

    Google Scholar 

  13. C. Dafermos and M. Selmrod, Asymptotic behavior of nonlinear contraction semi-groups. J. Differ. Eqs. 13 (1973), 97–106.

    MATH  Google Scholar 

  14. A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations. J. Differ. Eqs. 59 (1985), 145–154.

    Article  MATH  MathSciNet  Google Scholar 

  15. A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Portug. Math. 46 (1989), 245–258.

    MATH  MathSciNet  Google Scholar 

  16. S. Jaffard, M. Tucsnak, and E. Zuazua, Singular internal stabilization of the wave equation. J. Differ. Eqs. 145 (1998), 184–215.

    Article  MATH  MathSciNet  Google Scholar 

  17. G. Lebeau, Control for hyperbolic equations. Journées “Èquations aux dèrivèes partialles,” Saint Jean de Monts, 1992. Ecole Polytechnique, Palaiseau (1992).

  18. _____, Fonctions harmoniques et spectre singulier. Ann. Sci. Ecole Norm. Sup. 13 (1980), 269–291.

    MATH  MathSciNet  Google Scholar 

  19. _____, Contrôle analytique 1: estimations a priori. Duke Math. J. 60 (1992), 1–30.

    Article  MathSciNet  Google Scholar 

  20. J.-L. Lions, Contrôlabilitè exacte des systèmes distribuès. Masson, Paris (1988).

    Google Scholar 

  21. F. Macia and E. Zuazua, On the lack of controllability of wave equations: A Gaussian beam approach. Asympt. Anal. 32 (2002), 1–26.

    MATH  MathSciNet  Google Scholar 

  22. S. Micu and E. Zuazua, Boundary controllability of a linear hybrid system arising in the control of noise. SIAM J. Control Optim. 35 (1997), 1614–1638.

    Article  MATH  MathSciNet  Google Scholar 

  23. A. Zygmund, Trignometric series. Cambridge Univ. Press (1968).

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Authors and Affiliations

  1. Institut Supérieur des Sciences Appliquées et de Technologie de Sousse, Cité Taffala, 4000, Sousse, Tunisie

    N. Belghith

  2. Département de Mathématiques, Faculté des Sciences de Monastir, 5019, Monastir, Tunisie

    A. Moulahi

Authors
  1. N. Belghith
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  2. A. Moulahi
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Correspondence to N. Belghith.

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Belghith, N., Moulahi, A. Interior and Analytic Stabilization of the Wave Equation over a Cylinder. J Dyn Control Syst 16, 495–515 (2010). https://doi.org/10.1007/s10883-010-9104-x

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  • DOI: https://doi.org/10.1007/s10883-010-9104-x

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