A New Approach to Stability of Delay Differential Equations with Time-Varying Delays via Isospectral Reduction

TL;DR

This paper introduces a novel stability criterion for delay differential equations with time-varying delays, utilizing isospectral reduction and graph theory to establish global exponential stability for a broad class of systems.

Contribution

It develops a new stability criterion called intrinsic stability, applicable to equations with bounded, discontinuous, and time-varying delays, using a unique matrix reduction approach.

Findings

Establishes global exponential stability for a large class of systems.

Introduces a novel matrix sequence analysis via isospectral reduction.

Applies the method to delayed reservoir computer stability.

Abstract

Understanding how time delays impact the stability of a delay differential equation is important for modeling many natural and technological systems that experience time delays. Here we introduce a new stability criterion for delay-independent stability of these equations, called intrinsic stability, showing global exponential stability for a large class of nonautonomous nonlinear systems. Our approach is able to incorporate bounded time-varying delays, including those with certain types of discontinuities. The approach we take to prove this result is novel, associating the delay differential equation with a sequence of finite-dimensional matrices of increasing size and using the graph-theoretic technique of isospectral reduction to analyze this sequence. We give an application of these results to the problem of determining consistency for delayed reservoir computers.