  A Lyapunov-Krasovskii functional with prescribed derivative whose
construction does not require the stability of the system is introduced. It
leads to the presentation of stability/instability theorems. By evaluating the
functional at initial conditions depending on the fundamental matrix we are
able to present necessary and sufficient stability conditions expressed
exclusively in terms of the delay Lyapunov matrix for integral delay systems.
Some examples illustrate and validate the stability conditions.
