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

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.

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