In this Note, we study the indirect boundary stabilization of the Timoshenko system with only one dissipation law. Under the equal speed wave propagation condition, we establish the exponential stability of the system. On the contrary, we show that the decay rate is polynomial.
Nous étudions la stabilité frontière indirecte du système de Timoshenko sous lʼaction dʼune seule loi de dissipation. Sous la condition dʼégalité des vitesses de propagation, nous établissons la stabilité exponentielle du système. Dans le cas contraire, nous montrons que le taux de décroissance est polynomial.
Accepted:
Published online:
Maya Bassam 1, 2; Denis Mercier 1; Serge Nicaise 1; Ali Wehbe 2
@article{CRMATH_2011__349_7-8_379_0, author = {Maya Bassam and Denis Mercier and Serge Nicaise and Ali Wehbe}, title = {Stabilisation fronti\`ere indirecte du syst\`eme de {Timoshenko}}, journal = {Comptes Rendus. Math\'ematique}, pages = {379--384}, publisher = {Elsevier}, volume = {349}, number = {7-8}, year = {2011}, doi = {10.1016/j.crma.2011.03.011}, language = {fr}, }
TY - JOUR AU - Maya Bassam AU - Denis Mercier AU - Serge Nicaise AU - Ali Wehbe TI - Stabilisation frontière indirecte du système de Timoshenko JO - Comptes Rendus. Mathématique PY - 2011 SP - 379 EP - 384 VL - 349 IS - 7-8 PB - Elsevier DO - 10.1016/j.crma.2011.03.011 LA - fr ID - CRMATH_2011__349_7-8_379_0 ER -
Maya Bassam; Denis Mercier; Serge Nicaise; Ali Wehbe. Stabilisation frontière indirecte du système de Timoshenko. Comptes Rendus. Mathématique, Volume 349 (2011) no. 7-8, pp. 379-384. doi : 10.1016/j.crma.2011.03.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.03.011/
[1] Stabilisation frontière indirecte de systèmes faiblement couplés, C. R. Acad. Sci. Paris Sér. I Math., Volume 328 (1999) no. 11, pp. 1015-1020
[2] Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM J. Control Optim., Volume 41 (2002) no. 2, pp. 511-541
[3] Stabilization of the nonuniform Timoshenko beam, J. Math. Anal. Appl., Volume 327 (2007), pp. 525-538
[4] Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., Volume 8 (2008) no. 4, pp. 765-780
[5] A note on weak stabilization of contraction semi-groups, SIAM J. Control Optim., Volume 16 (1978), pp. 373-379
[6] Optimal polynomial decay of functions and operator semigroups, Math. Ann., Volume 347 (2010) no. 2, pp. 455-478
[7] Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. of Diff. Eqs., Volume 1 (1985), pp. 43-56
[8] Boundary control of the Timoshenko beam, SIAM J. Control Optim., Volume 25 (1987) no. 6, pp. 1417-1429
[9] Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., Volume 56 (2005) no. 4, pp. 630-644
[10] On the spectrum of -semigroups, Trans. Amer. Math. Soc., Volume 284 (1984), pp. 847-857
[11] 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
[12] Uniform stabilization for the Timoshenko beam by a locally distributed damping, Electron. J. Differential Equation, Volume 2003 (2003), pp. 1-14
Cited by Sources:
Comments - Policy