HAL CCSD
Convexity and Weighted Integral Inequalities for Energy Decay Rates of Nonlinear Dissipative Hyperbolic Systems
Alabau-Boussouira, Fatiha
Laboratoire de MathÃ©matiques et Applications de Metz (LMAM) ; UniversitÃ© Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)
Submitted in 2003, accepted August 3, 2004. Communicated by Irena Lasiecka
International audience
ISSN: 0095-4616
EISSN: 1432-0606
Applied Mathematics and Optimization
Springer Verlag (Germany)
hal-03895147
https://cnrs.hal.science/hal-03895147
https://cnrs.hal.science/hal-03895147
Applied Mathematics and Optimization, 2005, 51 (1), pp.61-105. ⟨10.1007/s00245⟩
DOI: 10.1007/s00245
info:eu-repo/semantics/altIdentifier/doi/10.1007/s00245
https://link.springer.com/article/10.1007/s00245
en
Nonlinear dissipation
Boundary damping
Locally distributed feedback
Hyperbolic equations
Optimality
34G10, 35B35, 35B37, 35L90, 93D15, 93D20
[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]
[MATH]Mathematics [math]
info:eu-repo/semantics/article
Journal articles
This work is concerned with the stabilization of hyperbolic systems by a nonlinear feedback which can be localized on a part of the boundary or locally distributed. We show that general weighted integral inequalities together with convexity arguments allow us to produce a general semi-explicit formula which leads to decay rates of the energy 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 also give three other significant examples of nonpolynomial growth at the origin. Our work completes the work of [15] and improves the results of [21] and [22] (see also [23] and [10]). We also prove the optimality of our results for the one-dimensional wave equation with nonlinear boundary dissipation. The key property for obtaining our general energy decay formula is the understanding between convexity properties of an explicit function connected to the feedback and the dissipation of energy.
2005-01