标题： 时间延迟系统中的阈值动力学：在压紧过程中的多项式$β$-控制以及与爆破的联系 显示英文标题

标题： Threshold dynamics in time-delay systems: polynomial $β$-control in a pressing process and connections to blow-up

摘要： 本文解决了由于系统通信导致的小时间延迟在矫直机中的压力控制问题。 为了解决这个问题，我们提出了一种广义的$\beta$-控制方法，该方法用次数为$\beta \ge 1$的多项式代替了传统的线性速度控制。 得到的模型是一个时滞微分方程（DDE），我们通过无量纲化和分析推导出其基本性质。 数值实验表明存在一个阈值初始速度，可以区分超调和非超调动力学，我们将这一现象表述为一个猜想。 基于此，我们在速度约束下设计了一个控制算法并验证了其有效性。 我们还强调了阈值行为与DDE中的有限时间爆破之间的联系。 本研究提供了一种实用的控制策略，并为时滞系统中的阈值动力学和爆破现象提供了新的见解。 显示英文摘要

摘要： This paper addresses a press control problem in straightening machines with small time delays due to system communication. To handle this, we propose a generalized $\beta$-control method, which replaces conventional linear velocity control with a polynomial of degree $\beta \ge 1$. The resulting model is a delay differential equation (DDE), for which we derive basic properties through nondimensionalization and analysis. Numerical experiments suggest the existence of a threshold initial velocity separating overshoot and non-overshoot dynamics, which we formulate as a conjecture. Based on this, we design a control algorithm under velocity constraints and confirm its effectiveness. We also highlight a connection between threshold behavior and finite-time blow-up in DDEs. This study provides a practical control strategy and contributes new insights into threshold dynamics and blow-up phenomena in delay systems.