A Lyapunov Stability Test for Neutral Type Delay Systems: A Discretized Functional Approach

Necessary and sufficient stability conditions for neutral type linear time delay systems are presented. Our approach relies on discretizing functionals with prescribed derivative based on the delay Lyapunov matrix via the discretized Lyapunov functional method introduced in Gu (1997). As a result, the discretized functional is expressed as a quadratic form, whose inner block matrix involves the delay Lyapunov matrix valued at discrete points. Remarkably, this matrix is connected with those presented recently in Gomez et al. (2019). This fact, along with the estimation of the functional approximation error on a special set of functions, provides a stability criterion expressed through the positive definiteness of the above-mentioned matrix. The use of a simpler structure of the functional, which involves derivatives of the function argument instead of derivatives of the delay Lyapunov matrix, is a key factor to simplify developments. It is worth emphasizing that the discretization of the functional derivative is eliminated thanks to the use of functionals with prescribed derivative. Numerical examples show a considerable reduction of the approximation order required to test stability.