Explicit stability tests for linear neutral delay equations using infinite series

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 $|a(t)| \leq A_0 \lt 1$, $0\lt b_0\leq b(t)\leq B_0$, assuming that all parameters of the equation are measurable functions. To analyze the 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 before been used for this neutral equation; its application allows omitting a usual restriction $|a(t)|\lt {1}/{2}$ in known asymptotic stability tests and the consideration of variable delays.