Variable-delay feedback control of unstable steady states in retarded time-delayed systems

TL;DR

This paper investigates how variable-delay feedback control can stabilize unstable steady states in retarded time-delayed systems, showing larger control domains and dependence on modulation parameters.

Contribution

It introduces an analytic and numerical framework for analyzing stability under variable-delay feedback, expanding control domain understanding in retarded time-delayed systems.

Findings

Control domains are significantly larger than in constant-delay feedback.

High-frequency modulation relates stability to distributed delay systems.

Domain structure complexity depends on delay modulation form and frequency.

Abstract

We study the stability of unstable steady states in scalar retarded time-delayed systems subjected to a variable-delay feedback control. The important aspect of such a control problem is that time-delayed systems are already infinite-dimensional before the delayed feedback control is turned on. When the frequency of the modulation is large compared to the system's dynamics, the analytic approach consists of relating the stability properties of the resulting variable-delay system with those of an analogous distributed delay system. Otherwise, the stability domains are obtained by a numerical integration of the linearized variable-delay system. The analysis shows that the control domains are significantly larger than those in the usual time-delayed feedback control, and that the complexity of the domain structure depends on the form and the frequency of the delay modulation.