Stabilization of second order evolution equations by a class of unbounded feedbacks

Kais Ammari; Marius Tucsnak

ESAIM: Control, Optimisation and Calculus of Variations (2001)

  • Volume: 6, page 361-386
  • ISSN: 1292-8119

Abstract

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In this paper we consider second order evolution equations with unbounded feedbacks. Under a regularity assumption we show that observability properties for the undamped problem imply decay estimates for the damped problem. We consider both uniform and non uniform decay properties.

How to cite

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Ammari, Kais, and Tucsnak, Marius. "Stabilization of second order evolution equations by a class of unbounded feedbacks." ESAIM: Control, Optimisation and Calculus of Variations 6 (2001): 361-386. <http://eudml.org/doc/90598>.

@article{Ammari2001,
abstract = {In this paper we consider second order evolution equations with unbounded feedbacks. Under a regularity assumption we show that observability properties for the undamped problem imply decay estimates for the damped problem. We consider both uniform and non uniform decay properties.},
author = {Ammari, Kais, Tucsnak, Marius},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {stabilization; observability inequality; second order evolution equations; unbounded feedbacks; stabilization by feedback; infinite dimensional systems; controllability; observability; exponential stability; decay rate},
language = {eng},
pages = {361-386},
publisher = {EDP-Sciences},
title = {Stabilization of second order evolution equations by a class of unbounded feedbacks},
url = {http://eudml.org/doc/90598},
volume = {6},
year = {2001},
}

TY - JOUR
AU - Ammari, Kais
AU - Tucsnak, Marius
TI - Stabilization of second order evolution equations by a class of unbounded feedbacks
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2001
PB - EDP-Sciences
VL - 6
SP - 361
EP - 386
AB - In this paper we consider second order evolution equations with unbounded feedbacks. Under a regularity assumption we show that observability properties for the undamped problem imply decay estimates for the damped problem. We consider both uniform and non uniform decay properties.
LA - eng
KW - stabilization; observability inequality; second order evolution equations; unbounded feedbacks; stabilization by feedback; infinite dimensional systems; controllability; observability; exponential stability; decay rate
UR - http://eudml.org/doc/90598
ER -

References

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  1. [1] K. Ammari and M. Tucsnak, Stabilization of Bernoulli–Euler beams by means of a pointwise feedback force. SIAM. J. Control Optim. 39 (2000) 1160-1181. Zbl0983.35021
  2. [2] K. Ammari, A. Henrot and M. Tucsnak, Optimal location of the actuator for the pointwise stabilization of a string. C. R. Acad. Sci. Paris Sér. I Math. 330 (2000) 275-280. Zbl0949.35083MR1753293
  3. [3] A. Bamberger, J. Rauch and M. Taylor, A model for harmonics on stringed instruments. Arch. Rational Mech. Anal. 79 (1982) 267-290. Zbl0534.47027MR656795
  4. [4] C. Bardos, L. Halpern, G. Lebeau, J. Rauch and E. Zuazua, Stabilisation de l’équation des ondes au moyen d’un feedback portant sur la condition aux limites de Dirichlet. Asymptot. Anal. 4 (1991) 285-291. Zbl0764.35055
  5. [5] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024-1065. Zbl0786.93009MR1178650
  6. [6] A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and control of infinite Dimensional Systems, Vol. I. Birkhauser (1992). Zbl0781.93002MR1246331
  7. [7] J.W.S. Cassals, An introduction to Diophantine Approximation. Cambridge University Press, Cambridge (1966). Zbl0077.04801
  8. [8] G. Doetsch, Introduction to the theory and application of the Laplace transformation. Springer, Berlin (1974). Zbl0278.44001MR344810
  9. [9] A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Portugal Math. 46 (1989) 245-258. Zbl0679.93063MR1021188
  10. [10] A.E. Ingham, Some trigonometrical inequalities with applications in the theory of series. Math. Z. 41 (1936) 367-369. Zbl0014.21503MR1545625
  11. [11] S. Jaffard, M. Tucsnak and E. Zuazua, Singular internal stabilization of the wave equation. J. Differential Equations 145 (1998) 184-215. Zbl0920.35029MR1620290
  12. [12] V. Komornik, Rapid boundary stabilization of linear distributed systems. SIAM J. Control Optim. 35 (1997) 1591-1613. Zbl0889.35013MR1466918
  13. [13] V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl. 69 (1990) 33-54. Zbl0636.93064MR1054123
  14. [14] J. Lagnese, Boundary stabilization of thin plates. Philadelphia, SIAM Stud. Appl. Math. (1989). Zbl0696.73034MR1061153
  15. [15] S. Lang, Introduction to diophantine approximations. Addison Wesley, New York (1966). Zbl0144.04005MR209227
  16. [16] J.L. Lions, Contrôlabilité exacte des systèmes distribués. Masson, Paris (1998). MR953547
  17. [17] J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. 1. Dunod, Paris (1968). Zbl0165.10801MR247243
  18. [18] F.W.J. Olver, Asymptotic and Special Functions. Academic Press, New York. Zbl0303.41035
  19. [19] A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer, New York (1983). Zbl0516.47023MR710486
  20. [20] R. Rebarber, Exponential stability of beams with dissipative joints: A frequency approach. SIAM J. Control Optim. 33 (1995) 1-28. Zbl0819.93042MR1311658
  21. [21] L. Robbiano, Fonction de coût et contrôle des solutions des équations hyperboliques. Asymptot. Anal. 10 (1995) 95-115. Zbl0882.35015MR1324385
  22. [22] D.L. Russell, Decay rates for weakly damped systems in Hilbert space obtained with control theoretic methods. J. Differential Equations 19 (1975) 344-370. Zbl0326.93018MR425291
  23. [23] D.L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent and open questions. SIAM Rev. 20 (1978) 639-739. Zbl0397.93001MR508380
  24. [24] H. Triebel, Interpolation theory, function spaces, differential operators. North Holland, Amsterdam (1978). Zbl0387.46032MR503903
  25. [25] M. Tucsnak, Regularity and exact controllability for a beam with piezoelectric actuator. SIAM J. Control Optim. 34 (1996) 922-930. Zbl0853.73051MR1384960
  26. [26] M. Tucsnak and G. Weiss, How to get a conservative well posed linear system out of thin air. Preprint. Zbl1125.93383
  27. [27] G.N. Watson, A treatise on the theory of Bessel functions. Cambridge University Press. Zbl0174.36202MR1349110JFM48.0412.02
  28. [28] G. Weiss, Regular linear systems with feedback. Math. Control Signals Systems 7 (1994) 23-57. Zbl0819.93034MR1359020

Citations in EuDML Documents

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  1. Kaïs Ammari, Mohamed Jellouli, Remark on stabilization of tree-shaped networks of strings
  2. Kaïs Ammari, Mouez Dimassi, Weyl formula with optimal remainder estimate of some elastic networks and applications
  3. Bao-Zhu Guo, Zhi-Xiong Zhang, On the well-posedness and regularity of the wave equation with variable coefficients
  4. Serge Nicaise, Julie Valein, Stabilization of second order evolution equations with unbounded feedback with delay
  5. Romain Joly, Perturbation de la dynamique des équations des ondes amorties
  6. Farah Abdallah, Serge Nicaise, Julie Valein, Ali Wehbe, Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications
  7. Ammar Moulahi, Salsabil Nouira, Stabilisation polynomiale et analytique de l’équation des ondes sur un rectangle

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