On the stabilization of an overhead crane system with dynamic and delayed boundary conditions

This article is dedicated to the investigation of long-Time behavior of an overhead crane with input delays in the boundary control. Contrary to previous works, the compensating terms in the proposed feedback control law are not of the same type as the delayed term. Next, the closed-loop system is shown to be well-posed in the sense of semigroups theory. Furthermore, the asymptotic convergence for the solutions of the system to a stationary position, which depends on the initial data, is obtained by applying LaSalle's invariance principle. Then, the convergence rate is shown to be exponential by means of the frequency-domain method. The asymptotic distribution of the eigenvalues, as well as the eigenfunctions of the system, are also provided, based on which, the differentiability of its semigroup is proved, and hence, the spectrum-determined growth condition holds. Finally, the outcomes are illustrated by a set of numerical simulations.