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.

Abstract

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”.