Exponential Stabilization of an Overhead Crane with Flexible Cable Via the Cascade Approach 1

Abstract

This paper deals with the uniform exponential stabilization of a hybrid PDE-ODE system which describes an overhead crane with flexible cable. A previous linear boundary feedback law (see [4]), depending on the platform position and velocity and on the angular displacement of the cable at the connection point to the platform, led to asymptotic stabilization but could not provide an exponential decay (see [18, 19]). Exponential stabilization has recently been obtained in [14] (see also [15]). Taking advantage of the "cascade" structure of the hybrid system, we propa5e here a linear boundary feedback which "naturally" depends in addition, on the angular velocity of the cable. We prove that this boundary feedback law leads to uniform exponential stability. Illustrative simulations are displayed.