Optimal Control
A general formula for decay rates of nonlinear dissipative systems
Comptes Rendus. Mathématique, Volume 338 (2004) no. 1, pp. 35-40.

This work is concerned with stabilization of hyperbolic systems by a nonlinear feedback which can be localized on part of the boundary or locally distributed. We present here a general formula which gives the energy decay rates in terms of the behavior of the nonlinear feedback close to the origin. This formula allows us to unify for instance the cases where the feedback has a polynomial growth at the origin, with the cases where it goes exponentially fast to zero at the origin. We give also two other significant examples of nonpolynomial growth at the origin. We also show that we either obtain or improve significantly the decay rates of Lasiecka and Tataru (Differential Integral Equations 8 (1993) 507–533) and Martinez (Rev. Mat. Comput. 12 (1999) 251–283).

On étudie le problème de la stabilisation des équations de type hyperbolique par un feedback qui peut être frontière ou bien localement distribué. L'objet de cette Note est de montrer qu'il existe une formule générale qui permet d'obtenir un taux de décroissance de l'énergie en fonction du comportement au voisinage de zéro du terme de dissipation non linéaire. Cette formule permet d'unifier tous les cas et notamment ceux pour lesquels le feedback croı̂t polynomialement et ceux pour lesquels il s'écrase exponentiellement en zéro. On donne aussi deux autres exemples significatifs de croissance non polynomiale. On montre pour tous ces exemples que l'on retrouve ou obtient de meilleurs taux de décroissance que ceux de Lasiecka et Tataru (Differential Integral Equations 8 (1993) 507–533) et Martinez (Rev. Mat. Comput. 12 (1999) 251–283).

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2003.10.024
Alabau-Boussouira, Fatiha 1

1 LMAM, CNRS-UMR 7122, Université de Metz, Ile du Saulcy, 57045 Metz cedex 01, France
@article{CRMATH_2004__338_1_35_0,
     author = {Alabau-Boussouira, Fatiha},
     title = {A general formula for decay rates of nonlinear dissipative systems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {35--40},
     publisher = {Elsevier},
     volume = {338},
     number = {1},
     year = {2004},
     doi = {10.1016/j.crma.2003.10.024},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2003.10.024/}
}
TY  - JOUR
AU  - Alabau-Boussouira, Fatiha
TI  - A general formula for decay rates of nonlinear dissipative systems
JO  - Comptes Rendus. Mathématique
PY  - 2004
SP  - 35
EP  - 40
VL  - 338
IS  - 1
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2003.10.024/
DO  - 10.1016/j.crma.2003.10.024
LA  - en
ID  - CRMATH_2004__338_1_35_0
ER  - 
%0 Journal Article
%A Alabau-Boussouira, Fatiha
%T A general formula for decay rates of nonlinear dissipative systems
%J Comptes Rendus. Mathématique
%D 2004
%P 35-40
%V 338
%N 1
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2003.10.024/
%R 10.1016/j.crma.2003.10.024
%G en
%F CRMATH_2004__338_1_35_0
Alabau-Boussouira, Fatiha. A general formula for decay rates of nonlinear dissipative systems. Comptes Rendus. Mathématique, Volume 338 (2004) no. 1, pp. 35-40. doi : 10.1016/j.crma.2003.10.024. http://www.numdam.org/articles/10.1016/j.crma.2003.10.024/

[1] Bardos, C.; Lebeau, G.; Rauch, J. Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., Volume 30 (1992), pp. 1024-1065

[2] Dafermos, C.M. Asymptotic behavior of solutions of evolution equations, Nonlinear Evolution Equations, Publ. Math. Res. Center Univ. Wisconsin, vol. 40, Academic Press, New York, 1978, pp. 103-123

[3] Haraux, A. Nonlinear Evolution Equations – Global Behavior of Solutions, Lecture Notes in Math., vol. 841, Springer-Verlag, Berlin, 1981

[4] Haraux, A.; Zuazua, E. Decay estimates for some semilinear damped hyperbolic problems, Arch. Rational Mech. Anal., Volume 100 (1988), pp. 191-206

[5] Komornik, V. Exact Controllability and Stabilization. The Multiplier Method, Collect. RMA, vol. 36, Masson-Wiley, Paris–Chicester, 1994

[6] Lasiecka, I.; Tataru, D. Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential Integral Equations, Volume 8 (1993), pp. 507-533

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

[8] Liu, K. Locally distributed control and damping for the conservative systems, SIAM J. Control Optim., Volume 35 (1997)

[9] Martinez, P. A new method to obtain decay rate estimates for dissipative systems with localized damping, Rev. Mat. Complut., Volume 12 (1999), pp. 251-283

[10] Nakao, M. Decay of solutions of the wave equation with a local nonlinear dissipation, Math. Ann., Volume 305 (1996), pp. 403-417

[11] Zuazua, E. Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations, Volume 15 (1990), pp. 205-235

[12] Zuazua, E. Uniform stabilization of the wave equation by nonlinear feedbacks, SIAM J. Control Optim., Volume 28 (1989), pp. 265-268

Cited by Sources: