Indirect stabilization of locally coupled wave-type systems

Fatiha Alabau-Boussouira; Matthieu Léautaud

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 2, page 548-582
  • ISSN: 1292-8119

Abstract

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We study in an abstract setting the indirect stabilization of systems of two wave-like equations coupled by a localized zero order term. Only one of the two equations is directly damped. The main novelty in this paper is that the coupling operator is not assumed to be coercive in the underlying space. We show that the energy of smooth solutions of these systems decays polynomially at infinity, whereas it is known that exponential stability does not hold (see [F. Alabau, P. Cannarsa and V. Komornik, J. Evol. Equ. 2 (2002) 127–150]). We give applications of our result to locally or boundary damped wave or plate systems. In any space dimension, we prove polynomial stability under geometric conditions on both the coupling and the damping regions. In one space dimension, the result holds for arbitrary non-empty open damping and coupling regions, and in particular when these two regions have an empty intersection. Hence, indirect polynomial stability holds even though the feedback is active in a region in which the coupling vanishes and vice versa.

How to cite

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Alabau-Boussouira, Fatiha, and Léautaud, Matthieu. "Indirect stabilization of locally coupled wave-type systems." ESAIM: Control, Optimisation and Calculus of Variations 18.2 (2012): 548-582. <http://eudml.org/doc/276371>.

@article{Alabau2012,
abstract = {We study in an abstract setting the indirect stabilization of systems of two wave-like equations coupled by a localized zero order term. Only one of the two equations is directly damped. The main novelty in this paper is that the coupling operator is not assumed to be coercive in the underlying space. We show that the energy of smooth solutions of these systems decays polynomially at infinity, whereas it is known that exponential stability does not hold (see [F. Alabau, P. Cannarsa and V. Komornik, J. Evol. Equ. 2 (2002) 127–150]). We give applications of our result to locally or boundary damped wave or plate systems. In any space dimension, we prove polynomial stability under geometric conditions on both the coupling and the damping regions. In one space dimension, the result holds for arbitrary non-empty open damping and coupling regions, and in particular when these two regions have an empty intersection. Hence, indirect polynomial stability holds even though the feedback is active in a region in which the coupling vanishes and vice versa. },
author = {Alabau-Boussouira, Fatiha, Léautaud, Matthieu},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Stabilization; indirect damping; hyperbolic systems; wave equation; localized zero order term; boundary damped wave or plate systems; polynomial stability},
language = {eng},
month = {7},
number = {2},
pages = {548-582},
publisher = {EDP Sciences},
title = {Indirect stabilization of locally coupled wave-type systems},
url = {http://eudml.org/doc/276371},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Alabau-Boussouira, Fatiha
AU - Léautaud, Matthieu
TI - Indirect stabilization of locally coupled wave-type systems
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2012/7//
PB - EDP Sciences
VL - 18
IS - 2
SP - 548
EP - 582
AB - We study in an abstract setting the indirect stabilization of systems of two wave-like equations coupled by a localized zero order term. Only one of the two equations is directly damped. The main novelty in this paper is that the coupling operator is not assumed to be coercive in the underlying space. We show that the energy of smooth solutions of these systems decays polynomially at infinity, whereas it is known that exponential stability does not hold (see [F. Alabau, P. Cannarsa and V. Komornik, J. Evol. Equ. 2 (2002) 127–150]). We give applications of our result to locally or boundary damped wave or plate systems. In any space dimension, we prove polynomial stability under geometric conditions on both the coupling and the damping regions. In one space dimension, the result holds for arbitrary non-empty open damping and coupling regions, and in particular when these two regions have an empty intersection. Hence, indirect polynomial stability holds even though the feedback is active in a region in which the coupling vanishes and vice versa.
LA - eng
KW - Stabilization; indirect damping; hyperbolic systems; wave equation; localized zero order term; boundary damped wave or plate systems; polynomial stability
UR - http://eudml.org/doc/276371
ER -

References

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