Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-06T08:59:50.284Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  15 August 2002

Kais Ammari  and
Marius Tucsnak
Kais Ammari
Affiliation:
Institut Elie Cartan, Département de Mathématiques, Université de Nancy I, 54506 Vandœuvre-lès-Nancy Cedex, France
Marius Tucsnak
Affiliation:
Institut Elie Cartan, Département de Mathématiques, Université de Nancy I, 54506 Vandœuvre-lès-Nancy Cedex, France
Get access

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.

Keywords

Stabilizationobservability inequalitysecond order evolution equationsunbounded feedbacks.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 6 , 2001 , pp. 361 - 386
Copyright
© EDP Sciences, SMAI, 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ammari, K. and Tucsnak, M., Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force. SIAM. J. Control Optim. 39 (2000) 1160-1181. CrossRef
Ammari, K., Henrot, A. and Tucsnak, M., Optimal location of the actuator for the pointwise stabilization of a string. C. R. Acad. Sci. Paris Sér. I Math. 330 (2000) 275-280. CrossRef
Bamberger, A., Rauch, J. and Taylor, M., A model for harmonics on stringed instruments. Arch. Rational Mech. Anal. 79 (1982) 267-290. CrossRef
Bardos, C., Halpern, L., Lebeau, G., Rauch, J. and Zuazua, E., 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.
Bardos, C., Lebeau, G. and Rauch, J., Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024-1065. CrossRef
A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and control of infinite Dimensional Systems, Vol. I. Birkhauser (1992).
J.W.S. Cassals, An introduction to Diophantine Approximation. Cambridge University Press, Cambridge (1966).
G. Doetsch, Introduction to the theory and application of the Laplace transformation. Springer, Berlin (1974).
Haraux, A., Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Portugal Math. 46 (1989) 245-258.
Ingham, A.E., Some trigonometrical inequalities with applications in the theory of series. Math. Z. 41 (1936) 367-369. CrossRef
Jaffard, S., Tucsnak, M. and Zuazua, E., Singular internal stabilization of the wave equation. J. Differential Equations 145 (1998) 184-215. CrossRef
Komornik, V., Rapid boundary stabilization of linear distributed systems. SIAM J. Control Optim. 35 (1997) 1591-1613. CrossRef
Komornik, V. and Zuazua, E., A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl. 69 (1990) 33-54.
J. Lagnese, Boundary stabilization of thin plates. Philadelphia, SIAM Stud. Appl. Math. (1989).
S. Lang, Introduction to diophantine approximations. Addison Wesley, New York (1966).
J.L. Lions, Contrôlabilité exacte des systèmes distribués. Masson, Paris (1998).
J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. 1. Dunod, Paris (1968).
F.W.J. Olver, Asymptotic and Special Functions. Academic Press, New York.
A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer, New York (1983).
Rebarber, R., Exponential stability of beams with dissipative joints: A frequency approach. SIAM J. Control Optim. 33 (1995) 1-28. CrossRef
Robbiano, L., Fonction de coût et contrôle des solutions des équations hyperboliques. Asymptot. Anal. 10 (1995) 95-115.
Russell, D.L., Decay rates for weakly damped systems in Hilbert space obtained with control theoretic methods. J. Differential Equations 19 (1975) 344-370. CrossRef
Russell, D.L., Controllability and stabilizability theory for linear partial differential equations: Recent and open questions. SIAM Rev. 20 (1978) 639-739. CrossRef
H. Triebel, Interpolation theory, function spaces, differential operators. North Holland, Amsterdam (1978).
Tucsnak, M., Regularity and exact controllability for a beam with piezoelectric actuator. SIAM J. Control Optim. 34 (1996) 922-930. CrossRef
M. Tucsnak and G. Weiss, How to get a conservative well posed linear system out of thin air. Preprint.
G.N. Watson, A treatise on the theory of Bessel functions. Cambridge University Press.
Weiss, G., Regular linear systems with feedback. Math. Control Signals Systems 7 (1994) 23-57. CrossRef