hal-03895147
https://cnrs.hal.science/hal-03895147
doi:10.1007/s00245
[CNRS] CNRS - Centre national de la recherche scientifique
[UNIV-METZ] Université Paul Verlaine - Metz
[IECN] IECL - Institut Elie Cartan de Lorraine
[INSMI] CNRS-INSMI - INstitut des Sciences Mathématiques et de leurs Interactions
[UNIV-LORRAINE] Université de Lorraine
[TDS-MACS] Réseau de recherche en Théorie des Systèmes Distribués, Modélisation, Analyse et Contrôle des Systèmes
Convexity and Weighted Integral Inequalities for Energy Decay Rates of Nonlinear Dissipative Hyperbolic Systems
Alabau-Boussouira, Fatiha
[MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC]
[MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP]
[MATH] Mathematics [math]
ART
Nonlinear dissipation
Boundary damping
Locally distributed feedback
Hyperbolic equations
Optimality
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
en
Applied Mathematics and Optimization
Springer Verlag (Germany)