Stability conditions for linear nonautonomous delay differential equations

We derive new sufficient conditions for uniform asymptotic stability of the zero solution of linear non-autonomous delay differential equations. The equations considered include scalar equations of the form \[ x’\left ( t \right ) = - c\left ( t \right )x\left ( t \right ) + \sum \limits _{i = 1}^n {{b_i}\left ( t \right )x\left ( {t - {T_i}} \right )} \] where $c\left ( t \right )$, ${b_i}\left ( t \right )$ are continuous for $t \ge 0$ and ${T_i}$ is a positive number $(i = 1, 2,...,n)$, and also systems of the form \[ x’(t) = B(t)x(t - T) - C(t)x(t)\] where $B(t)$) and $C(t)$ are $n \times n$ matrices. The results are found by using the method of Lyapunov functionals.