Necessary conditions for the stability of multiple time-delay systems via the delay Lyapunov matrix

Exponential stability necessary conditions for linear systems with multiple delays are presented. An interesting feature is that they depend exclusively on the Lyapunov matrix function of the system, thus extending well known concepts of delay free systems. They are obtained in the framework of Lyapunov-Krasovskii functionals of complete type by enforcing the positivity of the functional for a properly chosen initial condition. A number of new properties of the Lyapunov delay matrix are introduced. The effectiveness of the proposed conditions is shown in illustrative examples.