On the Stabilization of A Flexible Cable with Boundary Feedback

Touré K. Augustin; Mensah E. Patrice; Taha M. Mathurin

doi:10.26713/cma.v2i2-3.136

Authors

Touré K. Augustin Institut National Polytechnique Houphout-Boigny de Yamoussoukro, BP 1093 Yamoussoukro, Cí´te d'Ivoire
Mensah E. Patrice Institut National Polytechnique Houphout-Boigny de Yamoussoukro, BP 1093 Yamoussoukro, Cí´te d'Ivoire
Taha M. Mathurin Institut National Polytechnique Houphout-Boigny de Yamoussoukro, BP 1093 Yamoussoukro, Cí´te d'Ivoire

DOI:

https://doi.org/10.26713/cma.v2i2-3.136

Keywords:

Hyperbolic boundary value problem, Exponential stability, Asymptotic behavior, Semigroup theory

Abstract

In this paper we study the stability of a flexible cable that is clamped at one end and free at the other. To stabilize this system we apply a control force in position and velocity at the free end of the cable. We prove that the closed-loop system is well-posed and is exponentially stable. We then analyze the spectrum of the system. Using a method due to Shkalikov we prove that the spectrum determines the exponential decay rate of the energy under certain conditions.

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