Stability criteria for linear delay differential equations (Q1386292)

Linear delay differential systems of the form \[ \dot x(t)= A(t)x(t-\tau (t)) \] with a continuous \(n\times n\)-matrix-valued function \(A\) defined on \([t_0, \infty)\) and with a continuous \(\tau : [t_0, \infty) \mapsto [0,r]\), \(0 < r =\text{const}\) are considered. Conditions for the stability and asymptotic stability of the zero solution of the given equation are presented. The results are used for discussing the equation \(\dot x(t)= \dfrac {\sin t}{t^{\alpha }}x(t- r)\) in dependence on the parameter \(\alpha \).