Convexity and Weighted Integral Inequalities for Energy Decay Rates of Nonlinear Dissipative Hyperbolic Systems

Journal Article · Sat Jan 15 04:00:00 EST 2005 · Applied Mathematics and Optimization

DOI:https://doi.org/10.1007/S00245· OSTI ID:21064218

Alabau-Boussouira, Fatiha ^[1]

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1. L.M.A.M., CNRS-UMR 7122, Universite de Metz, Ile du Saulcy, 57045 Metz Cedex 01 (France)

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. 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.

• OSTI ID: 21064218
• Journal Information: Applied Mathematics and Optimization, Journal Name: Applied Mathematics and Optimization Journal Issue: 1 Vol. 51; ISSN 0095-4616
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
FEEDBACK
INTEGRALS
NONLINEAR PROBLEMS
ONE-DIMENSIONAL CALCULATIONS
POLYNOMIALS
WAVE EQUATIONS