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: 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}, pages = {361--386}, publisher = {EDP-Sciences}, volume = {6}, year = {2001}, mrnumber = {1836048}, zbl = {0992.93039}, language = {en}, url = {http://www.numdam.org/item/COCV_2001__6__361_0/} }
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 SP - 361 EP - 386 VL - 6 PB - EDP-Sciences UR - http://www.numdam.org/item/COCV_2001__6__361_0/ LA - en ID - COCV_2001__6__361_0 ER -
%0 Journal Article %A Ammari, Kais %A Tucsnak, Marius %T Stabilization of second order evolution equations by a class of unbounded feedbacks %J ESAIM: Control, Optimisation and Calculus of Variations %D 2001 %P 361-386 %V 6 %I EDP-Sciences %U http://www.numdam.org/item/COCV_2001__6__361_0/ %G en %F 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] Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force. SIAM. J. Control Optim. 39 (2000) 1160-1181. | Zbl
and ,[2] 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 | Zbl
, and ,[3] A model for harmonics on stringed instruments. Arch. Rational Mech. Anal. 79 (1982) 267-290. | MR | Zbl
, and ,[4] 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
, , , and ,[5] Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024-1065. | MR | Zbl
, and ,[6] Representation and control of infinite Dimensional Systems, Vol. I. Birkhauser (1992). | MR | Zbl
, , and ,[7] An introduction to Diophantine Approximation. Cambridge University Press, Cambridge (1966). | Zbl
,[8] Introduction to the theory and application of the Laplace transformation. Springer, Berlin (1974). | MR | Zbl
,[9] Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Portugal Math. 46 (1989) 245-258. | EuDML | MR | Zbl
,[10] Some trigonometrical inequalities with applications in the theory of series. Math. Z. 41 (1936) 367-369. | MR | Zbl
,[11] Singular internal stabilization of the wave equation. J. Differential Equations 145 (1998) 184-215. | MR | Zbl
, and ,[12] Rapid boundary stabilization of linear distributed systems. SIAM J. Control Optim. 35 (1997) 1591-1613. | MR | Zbl
,[13] A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl. 69 (1990) 33-54. | MR | Zbl
and ,[14] Boundary stabilization of thin plates. Philadelphia, SIAM Stud. Appl. Math. (1989). | MR | Zbl
,[15] Introduction to diophantine approximations. Addison Wesley, New York (1966). | MR | Zbl
,[16] Contrôlabilité exacte des systèmes distribués. Masson, Paris (1998). | MR
,[17] Problèmes aux limites non homogènes et applications, Vol. 1. Dunod, Paris (1968). | MR | Zbl
and ,[18] Asymptotic and Special Functions. Academic Press, New York. | Zbl
,[19] Semigroups of linear operators and applications to partial differential equations. Springer, New York (1983). | MR | Zbl
,[20] Exponential stability of beams with dissipative joints: A frequency approach. SIAM J. Control Optim. 33 (1995) 1-28. | MR | Zbl
,[21] Fonction de coût et contrôle des solutions des équations hyperboliques. Asymptot. Anal. 10 (1995) 95-115. | MR | Zbl
,[22] Decay rates for weakly damped systems in Hilbert space obtained with control theoretic methods. J. Differential Equations 19 (1975) 344-370. | MR | Zbl
,[23] Controllability and stabilizability theory for linear partial differential equations: Recent and open questions. SIAM Rev. 20 (1978) 639-739. | MR | Zbl
,[24] Interpolation theory, function spaces, differential operators. North Holland, Amsterdam (1978). | MR | Zbl
,[25] Regularity and exact controllability for a beam with piezoelectric actuator. SIAM J. Control Optim. 34 (1996) 922-930. | MR | Zbl
,[26] How to get a conservative well posed linear system out of thin air. Preprint.
and ,[27] A treatise on the theory of Bessel functions. Cambridge University Press. | JFM | MR | Zbl
,[28] Regular linear systems with feedback. Math. Control Signals Systems 7 (1994) 23-57. | MR | Zbl
,