Necessary and sufficient stability condition for time-delay systems arising from Legendre approximation

Recently, necessary conditions of stability for time-delay systems based on the handling Lyapunov-Krasovskii functional have been studied in literature giving rise to a new paradigm. Interestingly, condition developed by Gomez et al. has proven be sufficient. It is presented as simple positivity test matrix issued from Lyapunov matrix. The present paper proposes an extension this result, where uniform discretization state replaced projections first Legendre polynomials. Like al., guaranteed regarding sign eigenvalues matrix, whose size given analytically convergence arguments. Compared them, relying supergeometric rate approximation, required order ensure can remarkably reduced. Thanks significant modification, it possible find outer estimate regions, which converges expected regions with respect number projections, illustrated example section.