Stability analysis of dissipative systems subject to nonlinear damping via Lyapunov techniques

TL;DR

This paper develops Lyapunov-based methods to analyze the global asymptotic stability of infinite-dimensional systems with nonlinear damping, extending stability characterizations to nonlinear cases and applying results to PDE models.

Contribution

It introduces a Lyapunov functional approach for nonlinear damping in infinite-dimensional systems, generalizing stability analysis beyond linear damping cases.

Findings

Characterization of exponential and polynomial stability via Lyapunov functionals.

Construction of Lyapunov functionals for nonlinear systems based on linear system functionals.

Application of the method to Korteweg-de Vries and wave equations.

Abstract

In this article, we provide a general strategy based on Lyapunov functionals to analyse global asymptotic stability of linear infinite-dimensional systems subject to nonlinear dampings under the assumption that the origin of the system is globally asymp-totically stable with a linear damping. To do so, we first characterize, in terms of Lyapunov functionals, several types of asymptotic stability for linear infinite-dimensional systems, namely the exponential and the polynomial stability. Then, we derive a Lyapunov functional for the nonlinear system, which is the sum of a Lyapunov functional coming from the linear system and another term with compensates the nonlinearity. Our results are then applied to the linearized Korteweg-de Vries equation and some wave equations.