A new method to obtain decay rate estimates for dissipative systems

Patrick Martinez

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 4, page 419-444
  • ISSN: 1292-8119

Abstract

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We consider the wave equation damped with a boundary nonlinear velocity feedback p(u'). Under some geometrical conditions, we prove that the energy of the system decays to zero with an explicit decay rate estimate even if the function ρ has not a polynomial behavior in zero. This work extends some results of Nakao, Haraux, Zuazua and Komornik, who studied the case where the feedback has a polynomial behavior in zero and completes a result of Lasiecka and Tataru. The proof is based on the construction of a special weight function (that depends on the behavior of the function ρ in zero), and on a new nonlinear integral inequality.

How to cite

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Martinez, Patrick. "A new method to obtain decay rate estimates for dissipative systems." ESAIM: Control, Optimisation and Calculus of Variations 4 (2010): 419-444. <http://eudml.org/doc/116559>.

@article{Martinez2010,
abstract = { We consider the wave equation damped with a boundary nonlinear velocity feedback p(u'). Under some geometrical conditions, we prove that the energy of the system decays to zero with an explicit decay rate estimate even if the function ρ has not a polynomial behavior in zero. This work extends some results of Nakao, Haraux, Zuazua and Komornik, who studied the case where the feedback has a polynomial behavior in zero and completes a result of Lasiecka and Tataru. The proof is based on the construction of a special weight function (that depends on the behavior of the function ρ in zero), and on a new nonlinear integral inequality. },
author = {Martinez, Patrick},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Nonlinear stabilization; asymptotic behavior in zero and at infinity.; nonlinear stabilization; asymptotic behavior in zero and at infinity; nonlinear integral inequality},
language = {eng},
month = {3},
pages = {419-444},
publisher = {EDP Sciences},
title = {A new method to obtain decay rate estimates for dissipative systems},
url = {http://eudml.org/doc/116559},
volume = {4},
year = {2010},
}

TY - JOUR
AU - Martinez, Patrick
TI - A new method to obtain decay rate estimates for dissipative systems
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 4
SP - 419
EP - 444
AB - We consider the wave equation damped with a boundary nonlinear velocity feedback p(u'). Under some geometrical conditions, we prove that the energy of the system decays to zero with an explicit decay rate estimate even if the function ρ has not a polynomial behavior in zero. This work extends some results of Nakao, Haraux, Zuazua and Komornik, who studied the case where the feedback has a polynomial behavior in zero and completes a result of Lasiecka and Tataru. The proof is based on the construction of a special weight function (that depends on the behavior of the function ρ in zero), and on a new nonlinear integral inequality.
LA - eng
KW - Nonlinear stabilization; asymptotic behavior in zero and at infinity.; nonlinear stabilization; asymptotic behavior in zero and at infinity; nonlinear integral inequality
UR - http://eudml.org/doc/116559
ER -

References

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