{"ID":2884104,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.07268","arxiv_id":"2508.07268","title":"Threshold dynamics in time-delay systems: polynomial $β$-control in a pressing process and connections to blow-up","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 $β$-control method, which replaces conventional linear velocity control with a polynomial of degree $β\\ge 1$. 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.","short_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 $β$-control method, which replaces conventional linear velocity control with a polynomial of degree $β\\ge 1$. The resulting model is a delay differential eq...","url_abs":"https://arxiv.org/abs/2508.07268","url_pdf":"https://arxiv.org/pdf/2508.07268v2","authors":"[\"Masato Kimura\",\"Hirotaka Kuma\",\"Yikan Liu\",\"Kazunori Matsui\",\"Masahiro Yamamoto\",\"Zhenxing Yang\"]","published":"2025-08-10T09:48:27Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"eess.SY\",\"math.DS\"]","methods":"[]","has_code":false}