Stabilization of second order evolution equations by a class of unbounded feedbacks
ESAIM: Control, Optimisation and Calculus of Variations, Volume 6  (2001), p. 361-386

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.

Classification:  93B52,  93D15,  93B07
Keywords: stabilization, observability inequality, second order evolution equations, unbounded feedbacks
@article{COCV_2001__6__361_0,
author = {Ammari, Kais and Tucsnak, Marius},
title = {Stabilization of second order evolution equations by a class of unbounded feedbacks},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {6},
year = {2001},
pages = {361-386},
zbl = {0992.93039},
mrnumber = {1836048},
language = {en},
url = {http://www.numdam.org/item/COCV_2001__6__361_0}
}

Ammari, Kais; Tucsnak, Marius. Stabilization of second order evolution equations by a class of unbounded feedbacks. ESAIM: Control, Optimisation and Calculus of Variations, Volume 6 (2001) , pp. 361-386. http://www.numdam.org/item/COCV_2001__6__361_0/

[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. | Zbl 0983.35021

[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. | MR 1753293 | Zbl 0949.35083

[3] A. Bamberger, J. Rauch and M. Taylor, A model for harmonics on stringed instruments. Arch. Rational Mech. Anal. 79 (1982) 267-290. | MR 656795 | Zbl 0534.47027

[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. | Zbl 0764.35055

[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. | MR 1178650 | Zbl 0786.93009

[6] A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and control of infinite Dimensional Systems, Vol. I. Birkhauser (1992). | MR 1246331 | Zbl 0781.93002

[7] J.W.S. Cassals, An introduction to Diophantine Approximation. Cambridge University Press, Cambridge (1966). | Zbl 0077.04801

[8] G. Doetsch, Introduction to the theory and application of the Laplace transformation. Springer, Berlin (1974). | MR 344810 | Zbl 0278.44001

[9] A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Portugal Math. 46 (1989) 245-258. | MR 1021188 | Zbl 0679.93063

[10] A.E. Ingham, Some trigonometrical inequalities with applications in the theory of series. Math. Z. 41 (1936) 367-369. | MR 1545625 | Zbl 0014.21503

[11] S. Jaffard, M. Tucsnak and E. Zuazua, Singular internal stabilization of the wave equation. J. Differential Equations 145 (1998) 184-215. | MR 1620290 | Zbl 0920.35029

[12] V. Komornik, Rapid boundary stabilization of linear distributed systems. SIAM J. Control Optim. 35 (1997) 1591-1613. | MR 1466918 | Zbl 0889.35013

[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. | MR 1054123 | Zbl 0636.93064

[14] J. Lagnese, Boundary stabilization of thin plates. Philadelphia, SIAM Stud. Appl. Math. (1989). | MR 1061153 | Zbl 0696.73034

[15] S. Lang, Introduction to diophantine approximations. Addison Wesley, New York (1966). | MR 209227 | Zbl 0144.04005

[16] J.L. Lions, Contrôlabilité exacte des systèmes distribués. Masson, Paris (1998). | MR 953547

[17] J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. 1. Dunod, Paris (1968). | MR 247243 | Zbl 0165.10801

[18] F.W.J. Olver, Asymptotic and Special Functions. Academic Press, New York. | Zbl 0303.41035

[19] A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer, New York (1983). | MR 710486 | Zbl 0516.47023

[20] R. Rebarber, Exponential stability of beams with dissipative joints: A frequency approach. SIAM J. Control Optim. 33 (1995) 1-28. | MR 1311658 | Zbl 0819.93042

[21] L. Robbiano, Fonction de coût et contrôle des solutions des équations hyperboliques. Asymptot. Anal. 10 (1995) 95-115. | MR 1324385 | Zbl 0882.35015

[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. | MR 425291 | Zbl 0326.93018

[23] D.L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent and open questions. SIAM Rev. 20 (1978) 639-739. | MR 508380 | Zbl 0397.93001

[24] H. Triebel, Interpolation theory, function spaces, differential operators. North Holland, Amsterdam (1978). | MR 503903 | Zbl 0387.46032

[25] M. Tucsnak, Regularity and exact controllability for a beam with piezoelectric actuator. SIAM J. Control Optim. 34 (1996) 922-930. | MR 1384960 | Zbl 0853.73051

[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. | JFM 48.0412.02 | MR 1349110 | Zbl 0174.36202

[28] G. Weiss, Regular linear systems with feedback. Math. Control Signals Systems 7 (1994) 23-57. | MR 1359020 | Zbl 0819.93034