On exponential stability of linear and nonlinear delay differential equations: a review and new results

TL;DR

This paper reviews existing criteria and introduces new uniform exponential stability tests for scalar delay differential equations, including nonlinear models, using linearization, solution estimates, and the Bohl-Perron theorem, supported by numerical examples.

Contribution

It provides new uniform exponential stability tests for delay differential equations, extending existing criteria to nonlinear models with a comprehensive theoretical and numerical analysis.

Findings

New stability criteria for scalar delay equations

Extension of stability analysis to nonlinear models

Numerical examples illustrating theoretical results

Abstract

An extensive overview of existing criteria, as well as some new uniform exponential stability tests are included for a scalar delay equation $$ \dot{x}(t)+ \sum_{j=1}^n a_j(t)x(h_j(t))=0. $$ Both cases of continuous and measurable parameters $h_j$, $a_j$ are explored. We apply the global linearisation approach and employ linear results to explore global exponential stability for nonlinear models of the form $$ \dot{x}(t)+\sum_{j=1}^n f_j\left( t,x(h_j(t)) \right) =0. $$ The proofs are based on solution estimations. Further, the Bohl-Perron theorem on exponential dichotomy is instrumental for establishing global exponential stability for nonlinear models. Conclusions are illustrated with numerical examples.