Exponential asymptotic stability in linear delay-differential equations with variable coefficients (Q1978199)

The authors give sufficient and also necessary conditions for the exponential stability of a system \( \frac {dx}{dt}(t) = A(t) x(t-\tau) \) where \(\tau >0 \) is a delay and \(A(t)\) is a \(2 \times 2\)-matrix function defined by \[ A(t) = \rho (t) \left (\begin{matrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{matrix} \right) \] where \(\theta \) is a constant and \(\rho (t)\) is a positive function. The conditions for the exponential stability are expressed by means of the integral \( \int _{t-\tau }^t \rho (s) ds \) and values of \( \limsup _{t\rightarrow \infty } \int _{t-\tau }^t \rho (s) ds\), \(\liminf _{t\rightarrow \infty } \int _t^{t+ \Delta } \rho (s) ds\) with \(\Delta >0\).