Estimating the Evolution of Solution Norms in Vector Delay Nonlinear Systems: Stability and Boundedness

TL;DR

This paper introduces a novel framework for analyzing the stability and boundedness of vector delay differential equations by reducing the problem to scalar systems, enabling easier assessment and more accurate criteria.

Contribution

It develops a method to construct scalar bounds for vector DDEs, improving stability and boundedness analysis with new criteria validated through simulations.

Findings

New scalar bounds for vector DDEs established

Enhanced stability criteria derived and validated

Simulations confirm the effectiveness of the approach

Abstract

Existing methods rarely capture the temporal evolution of solution norms in vector nonlinear DDEs with variable delays and coefficients, often leading to overly conservative boundedness and stability criteria. We develop a framework that constructs scalar counterparts of vector DDEs whose solutions upper-bound the evolution of the original solution norms when the corresponding history functions are matched. This reduction enables boundedness and stability assessment of vector DDEs through the dynamics of their scalar counterparts, using straightforward simulations or simplified analytical reasoning. New boundedness and stability criteria and the estimates of the radii of balls containing history functions that yield bounded or stable solutions for the original vector systems were derived and validated through representative simulations.