On considère l’équation des ondes sur un rectangle avec un feedback de type Dirichlet. On se place dans le cas où la condition de contrôle géométrique n’est pas satisfaite (BLR Condition), ce qui implique qu’on n’a pas stabilité exponentielle dans l’espace d’énérgie. On prouve qu’on peut trouver un sous espace de l’espace d’énergie tel qu’on a stabilité exponentielle. De plus, on montre un résultat de décroissance polynomiale pour toute donnée initiale régulière.
the wave equation on a rectangle surface with feedback, that does not satisfy the classical geometric control condition BLR. We prove an exponential stability result for some subspace of the energy space. Moreover, we give a polynomial stability result for all regular initial data.
@article{AMBP_2010__17_2_401_0,
author = {Moulahi, Ammar and Nouira, Salsabil},
title = {Stabilisation polynomiale et analytique de l'\'equation des ondes sur un rectangle},
journal = {Annales math\'ematiques Blaise Pascal},
volume = {17},
year = {2010},
pages = {401-424},
doi = {10.5802/ambp.290},
zbl = {1205.35029},
mrnumber = {2778913},
language = {fr},
url = {http://dml.mathdoc.fr/item/AMBP_2010__17_2_401_0}
}
Moulahi, Ammar; Nouira, Salsabil. Stabilisation polynomiale et analytique de l’équation des ondes sur un rectangle. Annales mathématiques Blaise Pascal, Tome 17 (2010) pp. 401-424. doi : 10.5802/ambp.290. http://gdmltest.u-ga.fr/item/AMBP_2010__17_2_401_0/
[1] Analytic controlability of the wave equation over a cylinder, ESAIM : Control, Optimisation and Calculus of Variations., Tome 4 (1999), pp. 177-207 | Article | Numdam | MR 1816511 | Zbl 0945.93009
[2] Controllability of analytic functions for a wave equaion coupled with a beam, Revista Matematica Iberoamerican, Tome 15 (1999), pp. 547-592 | MR 1742216 | Zbl 0946.35045
[3] Dirichlet boundary stabilization of the wave equation, Asymptoic Analysis, Tome 30 (2002), pp. 117-130 | MR 1919338 | Zbl 1020.35042
[4] Polynomial and analytic stabilization of a wave equation coupled with a Euler–Bernoulli Beam, Math. Appl. Sci, Tome 5 (2009), pp. 556-576 | Article | MR 2500692 | Zbl 1156.35302
[5] Pointwise stabilization of a hybrid system and optimal location of actuator, Appl. Math. Optim., Tome 56 (2007), pp. 105-130 | Article | MR 2334607 | Zbl pre05199502
[6] Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM : Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 361-386 | Article | Numdam | MR 1836048 | Zbl 0992.93039
[7] Sharp sufficient conditions for the observation, control and stabilization of the waves from the boundary, SIAM J. Control Optim., Tome 30 (1992), pp. 1024-1065 | Article | MR 1178650 | Zbl 0786.93009
[8] Interior and analytic stabilization of the wave equation over a cylinder (Soumis)
[9] Condition nécessaire et suffisante pour la contrôllabilité exacte des ondes, Tome 325 (1997), pp. 749-752 | MR 1483711
[10] Contrôllabilité exacte d’une membrane rectangulaire au moyen d’une fonctionnelle analytique, Tome 306 (1988), pp. 125-128 | MR 929104 | Zbl 0643.49029
[11] A generalized internal control for the wave equation in a rectangle, J. Math. Anal. Appl., Tome 154 (1990), pp. 190-216 | Article | MR 1080126 | Zbl 0719.49008
[12] Estimates of the constants in generalized Ingahm’s inequality and applications to the control of the wave equation, Asymptot. Anal., Tome 28 (2001), pp. 181-214 | MR 1878794 | Zbl 1003.93028
[13] Fonctions harmoniques et spectre singulier, Ann. Sci. Ecole Norm. Sup., Tome 13 (1980), pp. 269-291 | Numdam | MR 584087 | Zbl 0446.46035
[14] Control for hyperbolic equations, Journées “Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1992), École Polytech., Palaiseau (1992), pp. 24 | Numdam | MR 1191928 | Zbl 0779.35005
[15] An Ingham type proof for the boundary observability of N-d wave equation, C. R. Acad. Sci. Paris, Ser. I, Tome 347 (2009), pp. 63-68 | MR 2536751 | Zbl 1154.93008
[16] Trigonometric Series, Cambridge University Press, Cambridge (1968) | MR 236587 | Zbl 0085.05601