Boundary stabilizability of the  linearized   viscous  Saint-Venant  system

Hassen Arfaoui; Faker Ben Belgacem; Henda El Fekih; Jean-Pierre Raymond

doi:10.3934/dcdsb.2011.15.491

Article Contents

2011, Volume 15, Issue 3: 491-511. Doi: 10.3934/dcdsb.2011.15.491

This issue Previous Article Next Article Mathematical models for strongly magnetized plasmas with mass disparate particles

Boundary stabilizability of the linearized viscous Saint-Venant system

1.
LAMSIN, Ecole Nationale d'Ingénieurs de Tunis, B.P. 37, 1002 Tunis Le Belvédère
2.
Université de Technologie de Compiègne, BP 20529, 60205 Compiègne cedex
3.
Université de Toulouse & CNRS, Institut de Mathématiques, UMR 5219, 31062 Toulouse Cedex 9

Received: February 28, 2009

Revised: April 30, 2010

Published: September 2011

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Abstract

We consider a shallow water flow in a channel modeled by the Saint-Venant equations with a viscous term. We are interested in the stabilization of the flow at a steady state. We establish that the semi-group of the linearized system is exponentially stable. However when the convection coefficient is dominant, the natural stabilization turns out to be very slow. One way to enhance the stabilization of the system is to use boundary controls by means of a moving device located at the extremities of the channel. We determine, by an extension method due to Fursikov, boundary Dirichlet controls able to accelerate the stabilization of the flow. Numerical experiments illustrate the efficiency of the control.

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Mathematics Subject Classification: 76D55, 93C20.

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