Improved results on the nonlinear feedback stabilization of a rotating body-beam system

TL;DR

This paper proves exponential stabilization of a rotating body-beam system with nonlinear boundary and torque controls, extending previous results by handling more general nonlinear functions and weaker conditions on system parameters.

Contribution

It introduces a stabilization method for a rotating beam system using nonlinear controls, improving upon prior work by generalizing nonlinear functions and relaxing conditions.

Findings

Exponential stability under small angular velocity conditions.

Handles more general nonlinear boundary functions.

Weakens assumptions compared to previous studies.

Abstract

This article is dedicated to the investigation of the stabilization problem of a flexible beam attached to the center of a rotating disk. We assume that the feedback law contains a nonlinear torque control applied on the disk and nonlinear boundary controls exerted on the beam. Thereafter, it is proved that the proposed controls guarantee the exponential stability of the system under a realistic smallness condition on the angular velocity of the disk and general assumptions on the nonlinear functions governing the controls. We used here the strategy of Lasiecka and Tataru and Alabau-Boussouira. This permits to improve the stability result shown in \cite{CH:99} in the sense that, on one hand, we deal with a general form of the nonlinear functions involved in the boundary controls. On the other hand, we manage to weaken the conditions on those functions unlike in B. Chentouf and J. F. Couchouron, Nonlinear feedback stabilization of a rotating body-beam without damping, ESAIM: COCV., 4 (1999), 515-535, where the authors consider a special type of functions that are almost linear.