MATIGNON, Denis (ENST & CNRS)

Asymptotic Stability of the Webster-Lokshin Model

Scheduled for presentation during the Mini-Symposium "Distributed Parameter Systems-IV: Frequency Domain Approaches" (ThA06), Thursday, July 27, 2006, 12:05−12:30, Room G

17th International Symposium on Mathematical Theory of Networks and Systems, July 24-28, 2006, Kyoto, Japan

This information is tentative and subject to change. Compiled on May 19, 2024

Abstract

The aim of this paper is to study the asymptotic stability of some wave equation with fractional damping accounting for visco-thermal losses at the walls of a flared pipe; moreover, the radiation boundary condition at the end of the pipe is described by a positive real impedance.

The difficulty of this model is twofold: first the fractional differential operator is non-local in time and must be transformed into a diffusive realization in the sense of systems theory; second, although a global energy can be built for this system, made of the wave energy and the diffusive energy, LaSalle's invariance principle does not apply, since a lack of compactness is to be found in this model. In this case, a refined analysis of the spectrum of the generator of the semigroup is needed, in order to apply Arendt--Batty stability theorem. This has already been carried out on the ODE corresponding to the projection on only one mode in [Matignon and Prieur, ESAIM:~COCV, 2005], but the question is even more difficult to tackle on the whole PDE.