Partial Differential Equations/Optimal Control
Uniform analyticity and exponential decay of the semigroup associated with a thermoelastic plate equation with perturbed boundary conditions
Comptes Rendus. Mathématique, Volume 351 (2013) no. 13-14, pp. 539-544.

In a bounded domain, we consider an Euler–Bernoulli-type thermoelastic plate equation with perturbed boundary conditions. The boundary conditions are such that when the perturbation parameter goes to infinity, we recover the hinged boundary conditions, while one recovers the clamped boundary conditions when the perturbation parameter goes to zero. Relying on resolvent estimates, we show that the underlying semigroup is uniformly, with respect to the perturbation parameter, analytic and exponentially stable. The main features of our proof are appropriate decompositions of the components of the system and the use of Lionsʼ interpolation inequalities.

Dans un domaine borné, on considère une équation de plaque thermo-elastique de type Euler–Bernoulli avec des conditions aux limites perturbées. Les conditions aux limites utilisées sont telles que lʼon retrouve une plaque simplement posée lorsque le paramètre de perturbation tend vers lʼinfini, alors que lʼon retrouve une plaque encastrée quand le paramètre de perturbation tend vers zéro. En nous appuyant sur des estimations de la résolvante, nous montrons que le semi-groupe associé est analytique et exponentiellement stable, uniformément par rapport au paramètre de perturbation. Les éléments principaux de notre démonstration sont des décompositions appropriées des composantes du système et lʼutilisation dʼinégalités dʼinterpolation de Lions.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2013.07.014
Tebou, Louis 1

1 Department of Mathematics and Statistics, Florida International University, Miami, FL 33199, USA
@article{CRMATH_2013__351_13-14_539_0,
     author = {Tebou, Louis},
     title = {Uniform analyticity and exponential decay of the semigroup associated with a thermoelastic plate equation with perturbed boundary conditions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {539--544},
     publisher = {Elsevier},
     volume = {351},
     number = {13-14},
     year = {2013},
     doi = {10.1016/j.crma.2013.07.014},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2013.07.014/}
}
TY  - JOUR
AU  - Tebou, Louis
TI  - Uniform analyticity and exponential decay of the semigroup associated with a thermoelastic plate equation with perturbed boundary conditions
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 539
EP  - 544
VL  - 351
IS  - 13-14
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2013.07.014/
DO  - 10.1016/j.crma.2013.07.014
LA  - en
ID  - CRMATH_2013__351_13-14_539_0
ER  - 
%0 Journal Article
%A Tebou, Louis
%T Uniform analyticity and exponential decay of the semigroup associated with a thermoelastic plate equation with perturbed boundary conditions
%J Comptes Rendus. Mathématique
%D 2013
%P 539-544
%V 351
%N 13-14
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2013.07.014/
%R 10.1016/j.crma.2013.07.014
%G en
%F CRMATH_2013__351_13-14_539_0
Tebou, Louis. Uniform analyticity and exponential decay of the semigroup associated with a thermoelastic plate equation with perturbed boundary conditions. Comptes Rendus. Mathématique, Volume 351 (2013) no. 13-14, pp. 539-544. doi : 10.1016/j.crma.2013.07.014. http://www.numdam.org/articles/10.1016/j.crma.2013.07.014/

[1] Avalos, G.; Lasiecka, I. Exponential stability of a thermoelastic system without mechanical dissipation, Dedicated to the memory of Pierre Grisvard, Rend. Istit. Mat. Univ. Trieste, Volume 28 (1996) no. suppl., pp. 1-28

[2] Avalos, G.; Lasiecka, I. Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation, SIAM J. Math. Anal., Volume 29 (1998), pp. 155-182

[3] Dafermos, C. On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Ration. Mech. Anal., Volume 29 (1968), pp. 241-271

[4] Huang, F.L. Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, Volume 1 (1985), pp. 43-56

[5] Kim, J.U. On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal., Volume 23 (1992), pp. 889-899

[6] Komornik, V. Exact Controllability and Stabilization: The Multiplier Method, RAM, Masson & John Wiley, Paris, 1994

[7] Lagnese, J. Boundary Stabilization of Thin Plates, SIAM Stud. Appl. Math., vol. 10, SIAM, Philadelphia, PA, 1989

[8] Lasiecka, I.; Triggiani, R. Analyticity of thermo-elastic semigroups with free boundary conditions, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4), Volume 27 (1998), pp. 457-482

[9] Lasiecka, I.; Triggiani, R. Two direct proofs on the analyticity of the s.c. semigroup arising in abstract thermo-elastic equations, Adv. Differential Equations, Volume 3 (1998), pp. 387-416

[10] Lasiecka, I.; Triggiani, R. Analyticity, and lack thereof, of thermo-elastic semigroups, Marseille-Luminy, 1997 (ESAIM Proc.), Volume vol. 4, Soc. Math. Appl. Indust., Paris (1998), pp. 199-222 (electronic)

[11] Lasiecka, I.; Triggiani, R. Analyticity of thermo-elastic semigroups with coupled hinged/Neumann B.C., Abstr. Appl. Anal., Volume 3 (1998), pp. 153-169

[12] Lebeau, G.; Zuazua, E. Decay rates for the three-dimensional linear system of thermoelasticity, Arch. Ration. Mech. Anal., Volume 148 (1999), pp. 179-231

[13] Lions, J.-L. Contrôlabilité exacte, perturbations et stabilisation des systèmes distribués, vol. 2, RMA, Masson, Paris, 1988

[14] Liu, K.; Liu, Z. Exponential stability and analyticity of abstract linear thermoelastic systems, Z. Angew. Math. Phys., Volume 48 (1997), pp. 885-904

[15] Liu, Z.; Renardy, M. A note on the equations of thermoelastic plate, Appl. Math. Lett., Volume 8 (1995), pp. 1-6

[16] Liu, Z.; Zheng, S. Exponential stability of the Kirchhoff plate with thermal or viscoelastic damping, Quart. Appl. Math., Volume 55 (1997), pp. 551-564

[17] Liu, W.J.; Zuazua, E. Uniform stabilization of the higher dimensional system of thermoelasticity with a nonlinear boundary feedback, Quart. Appl. Math., Volume 59 (2001), pp. 269-314

[18] Pazy, A. Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983

[19] Perla-Menzala, G.; Zuazua, E. The energy decay rate for the modified von Kármán system of thermoelastic plates: An improvement, Appl. Math. Lett., Volume 16 (2003), pp. 531-534

[20] Prüss, J. On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., Volume 284 (1984), pp. 847-857

[21] Tebou, L. Stabilization of some coupled hyperbolic/parabolic equations, DCDS B, Volume 14 (2010), pp. 1601-1620

Cited by Sources: