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

TL;DR

This paper introduces a polynomial $eta$-control method for delay differential equations in press systems, analyzing threshold behaviors and their relation to blow-up phenomena, supported by numerical experiments and control algorithm design.

Contribution

It proposes a novel polynomial $eta$-control approach for delay systems, linking threshold dynamics to blow-up phenomena and providing a practical control strategy.

Findings

Existence of a threshold initial velocity separating overshoot behaviors

Development of a control algorithm effective under velocity constraints

Connection established between threshold behavior and finite-time blow-up in DDEs

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.