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<p>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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\beta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation><i>-control</i> method, which replaces conventional linear velocity control with a polynomial of degree <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\beta \ge 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. 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 blowup in DDEs. This study provides a practical control strategy and contributes new insights into threshold dynamics and blowup phenomena in delay systems.</p>

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Threshold dynamics in time-delay systems: polynomial \(\beta \)-control in a pressing process and connections to blowup

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月2日

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  • Masato Kimura,
  • Hirotaka Kuma,
  • Yikan Liu,
  • Kazunori Matsui,
  • Masahiro Yamamoto,
  • Zhenxing Yang

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 \(\beta \) β -control method, which replaces conventional linear velocity control with a polynomial of degree \(\beta \ge 1\) β ≥ 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 blowup in DDEs. This study provides a practical control strategy and contributes new insights into threshold dynamics and blowup phenomena in delay systems.