A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations

Louis Tebou

ESAIM: Control, Optimisation and Calculus of Variations (2007)

  • Volume: 14, Issue: 3, page 561-574
  • ISSN: 1292-8119

Abstract

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First, we consider a semilinear hyperbolic equation with a locally distributed damping in a bounded domain. The damping is located on a neighborhood of a suitable portion of the boundary. Using a Carleman estimate [Duyckaerts, Zhang and Zuazua, Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear); Fu, Yong and Zhang, SIAM J. Contr. Opt.46 (2007) 1578–1614], we prove that the energy of this system decays exponentially to zero as the time variable goes to infinity. Second, relying on another Carleman estimate [Ruiz, J. Math. Pures Appl.71 (1992) 455–467], we address the same type of problem in an exterior domain for a locally damped semilinear wave equation. For both problems, our method of proof is constructive, and much simpler than those found in the literature. In particular, we improve in some way on earlier results by Dafermos, Haraux, Nakao, Slemrod and Zuazua.

How to cite

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Tebou, Louis. "A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations." ESAIM: Control, Optimisation and Calculus of Variations 14.3 (2007): 561-574. <http://eudml.org/doc/90883>.

@article{Tebou2007,
abstract = { First, we consider a semilinear hyperbolic equation with a locally distributed damping in a bounded domain. The damping is located on a neighborhood of a suitable portion of the boundary. Using a Carleman estimate [Duyckaerts, Zhang and Zuazua, Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear); Fu, Yong and Zhang, SIAM J. Contr. Opt.46 (2007) 1578–1614], we prove that the energy of this system decays exponentially to zero as the time variable goes to infinity. Second, relying on another Carleman estimate [Ruiz, J. Math. Pures Appl.71 (1992) 455–467], we address the same type of problem in an exterior domain for a locally damped semilinear wave equation. For both problems, our method of proof is constructive, and much simpler than those found in the literature. In particular, we improve in some way on earlier results by Dafermos, Haraux, Nakao, Slemrod and Zuazua. },
author = {Tebou, Louis},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Hyperbolic equation; exponential decay; localized damping; Carleman estimates; hyperbolic equation},
language = {eng},
month = {12},
number = {3},
pages = {561-574},
publisher = {EDP Sciences},
title = {A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations},
url = {http://eudml.org/doc/90883},
volume = {14},
year = {2007},
}

TY - JOUR
AU - Tebou, Louis
TI - A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2007/12//
PB - EDP Sciences
VL - 14
IS - 3
SP - 561
EP - 574
AB - First, we consider a semilinear hyperbolic equation with a locally distributed damping in a bounded domain. The damping is located on a neighborhood of a suitable portion of the boundary. Using a Carleman estimate [Duyckaerts, Zhang and Zuazua, Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear); Fu, Yong and Zhang, SIAM J. Contr. Opt.46 (2007) 1578–1614], we prove that the energy of this system decays exponentially to zero as the time variable goes to infinity. Second, relying on another Carleman estimate [Ruiz, J. Math. Pures Appl.71 (1992) 455–467], we address the same type of problem in an exterior domain for a locally damped semilinear wave equation. For both problems, our method of proof is constructive, and much simpler than those found in the literature. In particular, we improve in some way on earlier results by Dafermos, Haraux, Nakao, Slemrod and Zuazua.
LA - eng
KW - Hyperbolic equation; exponential decay; localized damping; Carleman estimates; hyperbolic equation
UR - http://eudml.org/doc/90883
ER -

References

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