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

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 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.

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

“…and is a positive constant sufficiently small so that the norm induced by (2.2) is equivalent to the usual norm of H 0 . The proof of this claim runs on much the same lines as that of the article [3] and hence the details are omitted. Now, one can write the closed-loop system (2.1) as follows…”

Time-Delayed Feedback Control of a Hydraulic Model Governed by a Diffusive Wave System

“…Based on the above discussion, the study of asymptotic behavior of the solutions of system (11) and (12) with input delay is of theoretical as well as practical prominence. e reader can also consult [26][27][28][29][30][31][32][33][34][35], where the effect of time delays in several types of partial differential equations (PDEs) has been studied. e main contribution of this paper is twofold:…”

Boundary stabilization of a flexible structure with dynamic boundary conditions via one time-dependent delayed boundary control

“…Thereafter, a more realistic model has been derived in [34], where the dynamic of m is ignored. In this paper, we take into consideration the dynamics of both m and M as in [2] and for sake of clarity, we will briefly present the derivation of the model.…”

“…represents in our study the boundary time-dependent delayed control F(t). Obviously, the presence of a time-delay in systems control is abundant and arises for numerous reasons and from different sources (see [2,3,17,19,18,25,29,30,31,33]).…”

“…Nevertheless, since our objective is to focus on the dynamics of the flexible cable, we shall only highlight those papers dealing with the cable dynamics in the context of time-delayed infinite-dimensional systems. Otherwise, the reader who is interested in getting some insights about various studies of the model without a delay is referred to [1] and [2]. To the authors' best knowledge, the first attempt to understand the cable dynamics when a delay occurs in the boundary control gave rise to the article [2].…”