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arXiv · 2508.07268

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

Masato Kimura·Hirotaka Kuma·Yikan Liu·Kazunori Matsui·Masahiro Yamamoto·Zhenxing Yang
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Abstract

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.

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Masato Kimura, Hirotaka Kuma, Yikan Liu, Kazunori Matsui, Masahiro Yamamoto, Zhenxing Yang. 2025-08-10. Threshold dynamics in time-delay systems: polynomial $\beta$-control in a pressing process and connections to blow-up. https://arxiv.org/abs/2508.07268

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