Stability of constant coefficient linear singular systems with delay (Q1971600)

Singular systems of ordinary differential equations with delays \[ Ax'(t)= f(t, x_t),\quad t\geq t_0,\quad x_0= \psi,\tag{\(*\)} \] with a constant matrix \(A\) are considered. In the theory of stability of such systems, fundamental problems on existence and unicity are not satisfactory. Also the application of the Lyapunov method is difficult. Here, some sufficient conditions for stability and instability are derived. The theorems are applied to linear systems of the form \[ Ax'(t)= Bx(t)+ Cx(t- h), \] \(A\), \(B\), \(C\) are constant matrices.