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

Abstract

AbstractThe 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 fact 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 of polynomial type as long as the delay term satisfies a smallness condition. Lastly, non‐convergence results are put forward in the case when such a condition on the delay term is not fulfilled.