Exponential asymptotic stability in linear delay-differential equations with variable coefficients

Abstract

In this paper we give some new necessary and sufficient conditions under which the zero solution of the linear delay-differential equations with variable coefficients $$ x'(t)=A(t)x(t-\tau) \tag L $$ is exponentially asymptotically stable. For example, in the case $$ A(t) = -\rho(t)\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}, $$ where $\rho(t) > 0,$ $\lim_{t \to \infty} \int_{t-\tau}^t \rho(s)\,ds = q>0$ and $|\theta| < \frac{\pi}{2},$ the zero solution of (L) is exponentially asymptotically stable if and only if $ q <\frac{\pi}{2}-|\theta|$.