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

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 $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -control method, which replaces conventional linear velocity control with a polynomial of degree $$\beta \ge 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . 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.