In this paper, we consider the stabilization of an axially moving Kirchhoff-type beam with rotational inertia under controls of force and torque at one boundary. The proposed negative feedbacks of the transverse velocity and angular velocity applied at the control end cover a large class of nonlinear feedback functions. The well-posedness of the resulting closed-loop system is established by means of the nonlinear semigroup theory, where the solution is shown to be depending continuously on the initial value. The asymptotic stability of the closed-loop system is guaranteed by resolving a dissipative ordinary differential equation. The decay rates of the vibration for some special nonlinear feedback functions can be estimated by the dissipative ordinary differential equation provided that growth restrictions on these nonlinear feedbacks near the origin are required. Three types of examples including exponential, polynomial and polynomial-logarithmic decay forms are deduced, and the numerical simulations are presented to verify the proposed control approach.

中文翻译：

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非线性边界反馈控制下具有转动惯量的轴向移动基尔霍夫型梁的稳定​​性和衰减率估计

在本文中，我们考虑在一个边界的力和扭矩控制下，具有转动惯量的轴向移动基尔霍夫型梁的稳定​​性。所提出的在控制端应用的横向速度和角速度的负反馈涵盖了一大类非线性反馈函数。由此产生的闭环系统的适定性是通过非线性半群理论建立的，其中解被证明连续依赖于初始值。通过求解耗散常微分方程保证闭环系统的渐近稳定性。一些特殊非线性反馈函数的振动衰减率可以通过耗散常微分方程来估计，前提是需要对这些非线性反馈在原点附近进行增长限制。推导了指数、多项式和多项式-对数衰减形式等三种类型的例子，并进行了数值模拟以验证所提出的控制方法。