Instability of systems with linear delay reducible to singularly perturbed ones (Q847876)

The stability of the following system of linear delay differential equations is considered in the case when one of the subsystems is singular: \[ \begin{aligned} {dx(t)\over dt}&= {1\over t} (A_1 x(t)+A_2 x(\mu t)+B_1 y(t)+B_2 y(\mu t))\,, \\ {dy(t)\over dt}&= (A_3 x(t)+A_4 x(\mu t)+B_3 y(t)+B_4 y(\mu t))\,, \end{aligned} \] where \(x=x(t)\) and \(y=y(t)\) are \(m\)-dimensional vector functions of time \(t\geq t_0>0\), \(\mu=\text{const}\), \(0<\mu< 1\), and the matrices \(A_j\), \(B_j\) are constant. Sufficient conditions are established for the instability of the solutions of such systems. The solution is based on the use of the Laplace transformation.