Legendre Approximation-Based Stability Test for Distributed Delay Systems

Abstract

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 641-660, April 2025.
Abstract. This contribution presents an exponential stability criterion for linear systems with multiple pointwise and distributed delays. This result is obtained in the Lyapunov–Krasovskii framework via the approximations of the argument of the functional by projection on the first Legendre polynomials. The reduction of the number of mathematical operations in the stability test is a benefit of the supergeometric convergence of Legendre polynomials approximation. For a single-delay linear system with a constant distributed kernel, a new computational procedure for the solution of the integrals involved in the stability test is developed considering the case of Jordan nilpotent blocks. This strategy is the basis for developing new procedures that allow the numerical construction of the stability test for different classes of kernels, such as polynomial, exponential, or [math] distribution.