Skip to main content Accessibility help

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

  • Kais Ammari (a1) and Marius Tucsnak (a1)


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.



Hide All
[1] 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.
[2] 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.
[3] Bamberger, A., Rauch, J. and Taylor, M., A model for harmonics on stringed instruments. Arch. Rational Mech. Anal. 79 (1982) 267-290.
[4] 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.
[5] 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.
[6] A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and control of infinite Dimensional Systems, Vol. I. Birkhauser (1992).
[7] J.W.S. Cassals, An introduction to Diophantine Approximation. Cambridge University Press, Cambridge (1966).
[8] G. Doetsch, Introduction to the theory and application of the Laplace transformation. Springer, Berlin (1974).
[9] Haraux, A., Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Portugal Math. 46 (1989) 245-258.
[10] Ingham, A.E., Some trigonometrical inequalities with applications in the theory of series. Math. Z. 41 (1936) 367-369.
[11] Jaffard, S., Tucsnak, M. and Zuazua, E., Singular internal stabilization of the wave equation. J. Differential Equations 145 (1998) 184-215.
[12] Komornik, V., Rapid boundary stabilization of linear distributed systems. SIAM J. Control Optim. 35 (1997) 1591-1613.
[13] Komornik, V. and Zuazua, E., A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl. 69 (1990) 33-54.
[14] J. Lagnese, Boundary stabilization of thin plates. Philadelphia, SIAM Stud. Appl. Math. (1989).
[15] S. Lang, Introduction to diophantine approximations. Addison Wesley, New York (1966).
[16] J.L. Lions, Contrôlabilité exacte des systèmes distribués. Masson, Paris (1998).
[17] J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. 1. Dunod, Paris (1968).
[18] F.W.J. Olver, Asymptotic and Special Functions. Academic Press, New York.
[19] A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer, New York (1983).
[20] Rebarber, R., Exponential stability of beams with dissipative joints: A frequency approach. SIAM J. Control Optim. 33 (1995) 1-28.
[21] Robbiano, L., Fonction de coût et contrôle des solutions des équations hyperboliques. Asymptot. Anal. 10 (1995) 95-115.
[22] Russell, D.L., Decay rates for weakly damped systems in Hilbert space obtained with control theoretic methods. J. Differential Equations 19 (1975) 344-370.
[23] Russell, D.L., Controllability and stabilizability theory for linear partial differential equations: Recent and open questions. SIAM Rev. 20 (1978) 639-739.
[24] H. Triebel, Interpolation theory, function spaces, differential operators. North Holland, Amsterdam (1978).
[25] Tucsnak, M., Regularity and exact controllability for a beam with piezoelectric actuator. SIAM J. Control Optim. 34 (1996) 922-930.
[26] M. Tucsnak and G. Weiss, How to get a conservative well posed linear system out of thin air. Preprint.
[27] G.N. Watson, A treatise on the theory of Bessel functions. Cambridge University Press.
[28] Weiss, G., Regular linear systems with feedback. Math. Control Signals Systems 7 (1994) 23-57.



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed