Stabilization of the nonuniform Timoshenko beam

Abstract

In this paper we consider a Timoshenko beam with variable physical parameters, we prove that the model can be stabilize by one control force for both internal and boundary cases.

Stabilisation frontière indirecte du système de Timoshenko

“…Kim and Renardy in [8] proved the exponential stability of the system under two boundary controls. In [3], Ammar-Khodja and his co-authors studied the decay rate of the energy of the nonuniform Timoshenko beam with two boundary controls acting in the rotation-angle equation. Under the equal speed wave propagation condition, they established exponential decay results up to an unknown finite dimensional space of initial data.…”

“…According to [3,12] we know that the energy E(t) of the system (1)-(2) loses the exponential decay rate obtained when a = 1. Nevertheless we prove the following polynomial type decay rate:…”

“…Dans [8], Kim et Renardy ont établi la stabilisation exponentielle du système moyennant deux contrôles frontières. Dans [3], Ammar-Khodja et al ont étu-dié la stabilisation frontière du système de Timoshenko non uniforme en appliquant deux contrôles frontières sur l'équation de rotation angulaire. Sous la condition d'égalité de vitesse de propagation (a = 1), ils ont établi un taux de décroissance exponentiel de l'énergie dans un sous-espace orthogonal à un sous espace de dimension finie mais non précisé.…”

On the Internal and Boundary Stabilization of Timoshenko Beams

“…This result has been generalized by Messaoudi and Soufyane [8], where they considered a multidimensional Timoshenko-type system with boundary conditions of memory type and proved energy decay results, for which the usual exponential and polynomial decay rates are only special cases. For more results concerning the controllability of Timoshenko systems, we refer the reader to [2,3,14,17], and [19].…”

“…The proofs of our results are done basically in two steps. In the first step, we use the multiplier method and benefit from [2] and [8] to choose the right multipliers. In the second step, we follow, with necessary modifications dictated by the nature of our systems, the method introduced and used by Martinez [7] to study the wave equations.…”