Stability of Linear Systems With Time-Varying Delays Using Bessel–Legendre Inequalities

This paper addresses the stability problem of linear systems with a time-varying delay. Hierarchical stability conditions based on linear matrix inequalities are obtained from an extensive use of the Bessel inequality applied to Legendre polynomials of arbitrary orders. While this inequality has been only used for constant discrete and distributed delays, this paper generalizes the same methodology to time-varying delays. We take advantages of the dependence of the stability criteria on both the delay and its derivative to propose a new definition of allowable delay sets. A light and smart modification in their definition leads to relevant conclusions on the numerical results.