# 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

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topAlabau-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 -

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