# Explicit stability tests for linear neutral delay equations using infinite series

## TL;DR

This paper introduces new explicit exponential stability conditions for linear neutral delay equations with variable delays, using an innovative reduction to equations with infinite non-neutral delay terms, removing previous restrictions.

## Contribution

The authors develop a novel method applying the Bohl-Perron theorem and infinite series reduction to analyze stability without the usual delay restrictions.

## Key findings

- New explicit stability conditions for neutral delay equations
- Method allows variable delays and removes previous restrictions
- Application of infinite series reduction is novel in this context

## Abstract

We obtain new explicit exponential stability conditions for the linear scalar neutral equation with two bounded delays $ (x(t)-a(t)x(g(t)))'+b(t)x(h(t))=0, $ where $|a(t)| \leq A_0 < 1$, $0<b_0\leq b(t)\leq B_0$, assuming that all parameters of the equation are measurable functions. To analyze exponential stability, we apply the Bohl-Perron theorem and a reduction of a neutral equation to an equation with an infinite number of non-neutral delay terms. This method has never been used before for this neutral equation; its application allowed to omit a usual restriction $|a(t)|<\frac{1}{2}$ in known asymptotic stability tests and consider variable delays.