Preservation of stability in a linear neutral differential equation under delay perturbations (Q2753275)

Here, studied the preservation of stability is studied under delay perturbations for the vector neutral functional-differential equation NEWLINE\[NEWLINE \frac{d(x(t)-Cx(t-\tau-\sigma(t))}{dt}=\sum_{i=0}^m A_i x(t-\tau_i-\eta_i(t)). \tag{1}NEWLINE\]NEWLINE It is proved that if the trivial solution to the ``unperturbed'' equation NEWLINE\[NEWLINE \frac d{dt}(x(t)-Cx(t-\tau))=\sum_{i=0}^m A_i(t-\tau_i)\tag{2}NEWLINE\]NEWLINE is asymptotically stable, then the same remains true for the trivial solution to (1), assuming that the delay perturbations are ``small''. As a special case, it is proved that if \(\lim_{t\to\infty}\sigma(t)=0\) and \(\lim_{t\to\infty}\eta_i(t)=0\) for \(i=0,\dots,m\), then the asymptotic stability of the trivial solution to (2) implies that to equation (1). The theoretical results are illustrated by examples.