Further results on the long-time behavior of a 2D overhead crane with a boundary delay: Exponential convergence

This article is mainly concerned with the exponential convergence of solutions of a 2D overhead crane, which consists of a platform and a flexible cable holding a load mass. While taking into consideration the occurrence of a delay in one boundary condition, the dynamics of the platform and the load mass are taken into consideration. Furthermore, the modulus of the cable is assumed to vary. Besides, the mathematical model of the crane system is assumed to have no displacement term. Then, a distributed (interior) damping feedback law is proposed so that one can exclude any possible negative impact of the delay. Indeed, invoking the frequency domain method, we show that the solutions of the closed-loop system must exponentially converge to a stationary position. This outcome improves the recent result obtained in [3], where the rate of convergence of solutions of the system without the interior damping is at most of polynomial type. The relevance of the theoretical findings is shown through several numerical examples.