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 $|a(t)| \leq A_0 < 1$, $0<b_0\leq b(t)\leq B_0$, 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.