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