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June  2019, 8(2): 423-445. doi: 10.3934/eect.2019021

## Decay rate of the Timoshenko system with one boundary damping

 Laboratoire de Mathématiques et ses Applications de Valenciennes, FR CNRS 2956, Institut des Sciences et Techniques de Valenciennes, Université Polytechnique Hauts-de-France, Le Mont Houy, 59313 VALENCIENNES Cedex 9, FRANCE

* Corresponding author: Virginie Régnier

Received  October 2017 Revised  August 2018 Published  March 2019

In this paper, we study the indirect boundary stabilization of the Timoshenko system with only one dissipation law. This system, which models the dynamics of a beam, is a hyperbolic system with two wave speeds. Assuming that the wave speeds are equal, we prove exponential stability. Otherwise, we show that the decay rate is of exponential or polynomial type. Note that the results hold without the technical assumptions on the coefficients coming from the multiplier method: a sharp analysis of the behaviour of the resolvent operator along the imaginary axis is performed to avoid those artificial restrictions.

Citation: Denis Mercier, Virginie Régnier. Decay rate of the Timoshenko system with one boundary damping. Evolution Equations & Control Theory, 2019, 8 (2) : 423-445. doi: 10.3934/eect.2019021
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