Verification Methods for the Lyapunov-Krasovskii Functional Inequalities.

We study parameterizations of Lyapunov-Krasovskii functionals (LKFs) to analyze the stability of linear time -delay systems. We discuss the solution to the delay Lyapunov matrix, which constructs an LKF associated with a prescribed time derivative, and relate it to the approaches commonly used in the numerical computation of LKFs. We then compare two approaches for the stability analysis of time -delay systems based on semidefinite programming, namely the method based on integral inequalities and the method based on sum -of -squares programming, which have recently emerged as optimization -based methods to compute LKFs. We discuss their main assumptions and establish connections between both methods. Finally, we formulate a projection -based method allowing us to use general sets of functions to parameterize LKFs, thus encompassing the sets of polynomial functions in the literature. The solutions of the proposed stability conditions and the construction of the corresponding LKFs as stability certificates are illustrated with numerical examples.