On the Contractivity and Asymptotic Stability of Systems of Delay Differential Equations of Neutral Type

Abstract

In this paper we investigate both the contractivity and the asymptotic stability of the solutions of linear systems of delay differential equations of neutral type (NDDEs) of the form y′(t) = Ly(t) + M(t)y(t − τ(t)) + N(t)y′(t − τ(t)). Asymptotic stability properties of numerical methods applied to NDDEs have been recently studied by numerous authors. In particular, most of the obtained results refer to the constant coefficient version of the previous system and are based on algebraic analysis of the associated characteristic polynomials. In this work, instead, we play on the contractivity properties of the solutions and determine sufficient conditions for the asymptotic stability of the zero solution by considering a suitable reformulation of the given system. Furthermore, a class of numerical methods preserving the above-mentioned stability properties is also presented.