Necessary and sufficient stability conditions for linear systems with pointwise and distributed delays

A stability criterion for the exponential stability of systems with multiple pointwise and distributed delays is presented. Conditions in terms of the delay Lyapunov matrix are obtained by evaluating a Lyapunov-Krasovskii functional with prescribed derivative at a pertinent initial function that depends on the system fundamental matrix. The proof relies on properties connecting the delay Lyapunov matrix and the fundamental matrix, which are proven to be valid for both stable and unstable systems. The conditions are applied to the determination of the exact stability region for some examples.