Necessary stability conditions for linear delay systems

We present necessary conditions for the exponential stability of linear systems with multiple delays. They are expressed in terms of the delay Lyapunov matrix of the Lyapunov–Krasovskii functionals of complete type approach. New properties of independent interest, that establish connections of the system fundamental matrix with its Lyapunov matrix, are crucial elements of our proof. We illustrate our work with a number of examples.