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 \[ x ′ ( t ) = − c ( t ) x ( t ) + ∑ i = 1 n b i ( t ) x ( t − T i ) x’\left ( t \right ) = - c\left ( t \right )x\left ( t \right ) + \sum \limits _{i = 1}^n {{b_i}\left ( t \right )x\left ( {t - {T_i}} \right )} \] where c ( t ) c\left ( t \right ) , b i ( t ) {b_i}\left ( t \right ) are continuous for t ≥ 0 t \ge 0 and T i {T_i} is a positive number ( i = 1 , 2 , . . . , n ) (i = 1, 2,...,n) , and also systems of the form \[ x ′ ( t ) = B ( t ) x ( t − T ) − C ( t ) x ( t ) x’(t) = B(t)x(t - T) - C(t)x(t) \] where B ( t ) B(t) ) and C ( t ) C(t) are n × n n \times n matrices. The results are found by using the method of Lyapunov functionals.