[Stabilisation locale indirecte d'un système N–d de deux équations d'ondes couplées sous conditions géométriques]
Nous nous intéressons à la stabilisation d'un système multidimensionnel de deux équations d'ondes couplées par les termes de vitesse, et dont l'une seulement est localement amortie. La principale nouveauté contenue dans cette note est que les ondes ne se propagent pas forcément à la même vitesse et que le coefficient de couplage n'est pas supposé être positif et petit. Nous supposons que la zone de couplage et la zone d'amortissement s'intersectent. D'abord, nous montrons que notre système est fortement stable sans conditions géométriques. Puis nous étudions le taux de décroissance de l'énergie en distinguant deux cas. Dans le premier cas, nous supposons que les ondes se propagent à la même vitesse ; nous établissons alors, sous certaines conditions géométriques, un taux de décroissance exponentiel de l'énergie du système pour des données initiales usuelles. Dans le second cas, nous montrons d'abord que l'énergie ne décroît pas exponentiellement vers 0. Ensuite, sous les mêmes conditions géométriques, nous établissons un taux de décroissance polynomial de type pour des données initiales régulières. Finalement, dans le cas particulier où la dimension de l'espace est égale à 1, en utilisant la partie réelle du développement asymptotique des valeurs propres de système, nous montrons, de plus, que le taux polynomial obtenu est optimal.
The purpose of this note is to investigate the stabilization of a system of two wave equations coupled by velocities with only one localized damping. The main novelty in this note is that the waves are not necessarily propagating at same speed and the coupling coefficient is not assumed to be positive and small. Assume that the coupling region and the damping region intersect. We prove that our system is strongly stable without geometric conditions. We then study the energy decay rate by distinguishing two cases. The first one is when the waves propagate at the same speed. In this case, under appropriate geometric conditions, we establish an exponential energy decay estimate for usual initial data. For the other case, we first show that our system is not uniformly stable. Next, under the same geometric conditions, we establish a polynomial energy decay of type for smooth initial data. Finally, in one space dimension, using the real part of the asymptotic expansion of eigenvalues of the system, we prove that the obtained polynomial decay rate is optimal.
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@article{CRMATH_2019__357_6_494_0,
author = {Kassem, Chiraz and Mortada, Amina and Toufayli, Layla and Wehbe, Ali},
title = {Local indirect stabilization of {N{\textendash}d} system of two coupled wave equations under geometric conditions},
journal = {Comptes Rendus. Math\'ematique},
pages = {494--512},
publisher = {Elsevier},
volume = {357},
number = {6},
year = {2019},
doi = {10.1016/j.crma.2019.06.002},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2019.06.002/}
}
TY - JOUR AU - Kassem, Chiraz AU - Mortada, Amina AU - Toufayli, Layla AU - Wehbe, Ali TI - Local indirect stabilization of N–d system of two coupled wave equations under geometric conditions JO - Comptes Rendus. Mathématique PY - 2019 SP - 494 EP - 512 VL - 357 IS - 6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2019.06.002/ DO - 10.1016/j.crma.2019.06.002 LA - en ID - CRMATH_2019__357_6_494_0 ER -
%0 Journal Article %A Kassem, Chiraz %A Mortada, Amina %A Toufayli, Layla %A Wehbe, Ali %T Local indirect stabilization of N–d system of two coupled wave equations under geometric conditions %J Comptes Rendus. Mathématique %D 2019 %P 494-512 %V 357 %N 6 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2019.06.002/ %R 10.1016/j.crma.2019.06.002 %G en %F CRMATH_2019__357_6_494_0
Kassem, Chiraz; Mortada, Amina; Toufayli, Layla; Wehbe, Ali. Local indirect stabilization of N–d system of two coupled wave equations under geometric conditions. Comptes Rendus. Mathématique, Tome 357 (2019) no. 6, pp. 494-512. doi : 10.1016/j.crma.2019.06.002. http://www.numdam.org/articles/10.1016/j.crma.2019.06.002/
[1] Stabilisation frontière indirecte de systèmes faiblement couplés, C. R. Acad. Sci. Paris, Ser. I, Volume 328 (1999) no. 11, pp. 1015-1020
[2] Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equ., Volume 2 (2002) no. 2, pp. 127-150
[3] Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM J. Control Optim., Volume 41 (2002) no. 2, pp. 511-541
[4] A general formula for decay rates of nonlinear dissipative systems, C. R. Acad. Sci. Paris, Ser. I, Volume 338 (2004) no. 1, pp. 35-40
[5] Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim., Volume 51 (Jan 2005) no. 1, pp. 61-105
[6] Indirect stabilization of locally coupled wave-type systems, ESAIM Control Optim. Calc. Var., Volume 18 (2012) no. 2, pp. 548-582
[7] Indirect controllability of locally coupled wave-type systems and applications, J. Math. Pures Appl., Volume 99 (2013) no. 5, pp. 544-576
[8] A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities, ESAIM Control Optim. Calc. Var., Volume 23 (2017) no. 2, pp. 721-749
[9] Stabilization of the nonuniform Timoshenko beam, J. Math. Anal. Appl., Volume 327 (2007) no. 1, pp. 525-538
[10] Stabilization of coupled systems, Acta Math. Hung., Volume 123 (2009), pp. 1-10
[11] Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., Volume 30 (Sept. 1992) no. 5, pp. 1024-1065
[12] Polynomial stability of the Timoshenko system by one boundary damping, J. Math. Anal. Appl., Volume 425 (2015) no. 2, pp. 1177-1203
[13] Stability results of some distributed systems involving Mindlin-Timoshenko plates in the plane, Z. Angew. Math. Mech., Volume 96 (2016) no. 8, pp. 916-938
[14] Optimal polynomial decay of functions and operator semigroups, Math. Ann., Volume 347 (2010) no. 2, pp. 455-478
[15] Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differ. Equ., Volume 1 (1985) no. 1, pp. 43-56
[16] Uniform stabilization and exact controllability for a class of coupled hyperbolic systems, Mat. Apl. Comput., Volume 15 (1996) no. 3, pp. 199-212
[17] Stabilizability of systems of one-dimensional wave equations by one internal or boundary control force, SIAM J. Control Optim., Volume 39 (2001) no. 6, pp. 1833-1851
[18] Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems, J. Funct. Anal., Volume 258 (2010) no. 8, pp. 2739-2778
[19] , Springer, Dordrecht, The Netherlands (1996), pp. 73-109
[20] G. Lebeau, L. Robbiano, Contrôle exact de l'équation de la chaleur, in: Séminaire Équations aux dérivées partielles (Polytechnique), pp. 1–11, 1994–1995. talk:7.
[21] Stabilisation de l'équation des ondes par le bord, Duke Math. J., Volume 861 (1997), pp. 465-491
[22] Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués, Paris, Masson (1988)
[23] Locally distributed control and damping for the conservative systems, SIAM J. Control Optim., Volume 35 (Sept. 1997) no. 5, pp. 1574-1590
[24] Frequency domain approach for the polynomial stability of a system of partially damped wave equations, J. Math. Anal. Appl., Volume 335 (2007) no. 2, pp. 860-881
[25] Optimal energy decay rate for partially damped systems by spectral compensation, SIAM J. Control Optim., Volume 45 (2006) no. 5, pp. 1612-1632 (electronic)
[26] Étude de la stabilisation exponentielle et polynomiale de certains systèmes d'équations couplées par des contrôles indirects bornés ou non bornés, 2016 http://ged.univ-valenciennes.fr/nuxeo/site/esupversions/aaac617d-95a5-4b80-8240-0dd043f20ee5 (PhD thesis, université de Valenciennes, France)
[27] Weakly locally thermal stabilization of Bresse systems, Electron. J. Differ. Equ. (2014)
[28] On the spectrum of -semigroups, Trans. Amer. Math. Soc., Volume 284 (1984) no. 2, pp. 847-857
[29] A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl., Volume 173 (1993) no. 2, pp. 339-358
[30] Uniform stabilization for the Timoshenko beam by a locally distributed damping, Electron. J. Differ. Equ. (2003)
[31] Stabilization of the uniform Timoshenko beam by one locally distributed feedback, Appl. Anal., Volume 88 (2009) no. 7, pp. 1067-1078
[32] Exponential and polynomial stability of an elastic Bresse system with two locally distributed feedbacks, J. Math. Phys., Volume 51 (2010) no. 10
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