Simple Tests for Uniform Exponential Stability of a Linear Delayed Vector Differential Equation

A linear delayed vector equation
$\dot{x}(t)=\sum _{k=1}^m A_k(t)x(h_{k}(t)), \quad t\in [0,\infty )$
is investigated, where
$x=(x_1,\ldots,x_n)^T$
is an unknown vector function. The system is considered in the most general setting and under weak assumptions about the entries of matrices
$A_k$
and delays
$h_{k}$
. The main result on uniform exponential stability is universal in the sense that it generates a set of
$2^m-1$
independent explicit statements (that can depend on all delays) on uniform exponential stability. The advantages over the existing results are demonstrated. The main tools employed by this article include the Bohl–Perron method,
a priori
estimates of solutions, and transformations of differential equations.