Further results on the asymptotic behavior of a 2D overhead crane with input delays: Exponential convergence

TL;DR

This paper demonstrates that a 2D overhead crane system with boundary delays can be stabilized exponentially using a distributed damping feedback law, improving upon previous polynomial convergence results.

Contribution

The paper introduces a boundary delay compensation method that achieves exponential convergence for the 2D crane system, advancing prior polynomial rate results.

Findings

Solutions converge exponentially to a stationary position

The proposed feedback law effectively compensates boundary delays

Improves convergence rate from polynomial to exponential

Abstract

This article is concerned with the asymptotic behavior of a 2D overhead crane. Taking into account the presence of a delay in the boundary, and assuming that no displacement term appears in the system, a distributed (interior) damping feedback law is proposed in order to compensate the effect of the delay. Then, invoking the frequency domain method, the solutions of the closed-loop system are proved to converge exponentially to a stationary position. This improves the recent result obtained by Al-Musallam-Ammari-Chentouf, where the rate of convergence is at most of polynomial type.