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Mathematical Problems in Engineering
Volume 2010, Article ID 984927, 12 pages
Research Article

Feedback Controller Stabilizing Vibrations of a Flexible Cable Related to an Overhead Crane

Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, B.P. 2390-40000, Marrakesh, Morocco

Received 30 April 2010; Accepted 27 August 2010

Academic Editor: Horst Ecker

Copyright © 2010 Abdelhadi Elharfi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The problem of stabilizing vibrations of flexible cable related to an overhead crane is considered. The cable vibrations are described by a hyperbolic partial differential equation (HPDE) with an update boundary condition. We provide in this paper a systematic way to derive a boundary feedback law which restores in a closed form the cable vibrations to the desired zero equilibrium. Such a control law is explicitly constructed in terms of the solution of an appropriate kernel PDE. The pursued approach combines the “backstepping method” and “semigroup theory”.