Assessing the exact stability region of the single-delay scalar equation via its Lyapunov function

It is well known that one can determine the stability of a delay-free linear system either by verifying that all the roots of its characteristic polynomial are in the left half plane or by checking if the solution of the Lyapunov equation is positive definite. For linear systems with delays, many extensions of the first approach are reported in the literature. On the contrary, there exist no publications on extending the second approach to delay systems. In this note, it is shown that the second approach is possible for one of the simplest linear delay systems: stability conditions in terms of the Lyapunov function for the scalar delay equation, that match the frequency domain well-known result, are presented.