The necessary and sufficient conditions for the stability of linear systems with an arbitrary delay

A system of linear differential equations with a Hurwitz matrix A and a variable delay is considered. The system is assumed to be stable if it is stable for any delay function τ(t) ≤ h. The necessary and sufficient condition for stability, expressed using the eigenvalues of the matrix A and the quantity h, is found. It is established that the function τ(t), corresponding to the critical value of h, is constant or piecewise-linear depending on to which eigenvalue of matrix A (complex or real respectively) it corresponds. In the first case, the critical values of h in systems with a variable and constant delay are identical and, in the second case, they differ very slightly.