# Asymptotic analysis of a 2D overhead crane with input delays in the boundary control

## TL;DR

This paper analyzes the long-term behavior of a 2D overhead crane system with boundary control that includes input delays, demonstrating well-posedness and polynomial convergence to a stationary position.

## Contribution

It introduces a boundary control dependent on velocity with delays, proving asymptotic stability and polynomial convergence using semigroup and resolvent methods.

## Key findings

- System is well-posed in the semigroup sense
- Solutions converge polynomially to a stationary position
- Control based on velocity with delays achieves stability

## Abstract

The paper investigates the asymptotic behavior of a 2D overhead crane with input delays in the boundary control. A linear boundary control is proposed. The main feature of such a control lies in the facts that it solely depends on the velocity but under the presence of time-delays. We end-up with a closed-loop system where no displacement term is involved. It is shown that the problem is well-posed in the sense of semigroups theory. LaSalle's invariance principle is invoked in order to establish the asymptotic convergence for the solutions of the system to a stationary position which depends on the initial data. Using a resolvent method it is proved that the convergence is indeed polynomial.