Stability test for some classes of linear time-delay systems: A Legendre polynomial approximation-based approach

In this paper, we present necessary and sufficient stability tests for two classes of linear delay systems: neutral-type linear time-delay systems and linear time-delay systems with distributed delays. In both cases, they are based on the Lyapunov-Krasovskii functionals with prescribed derivative expressed in terms of the delay Lyapunov matrix. The stability tests amount to verify the semi-positivity of a matrix resulting from the substitution of Legendre polynomial approximation of the functional argument. As illustrated in two examples, the low matrix dimension for establishing sufficiency makes the test numerically efficient.