Multiple integral inequalities and stability analysis of time delay systems

• New multiple integral inequalities are derived.
• A set of sufficient LMI stability conditions for time delay systems are derived.
• The LMI conditions are arranged into a bidirectional hierarchy.

This paper is devoted to stability analysis of continuous-time delay systems based on a set of Lyapunov–Krasovskii functionals. New multiple integral inequalities are derived that involve the famous Jensen’s and Wirtinger’s inequalities, as well as the recently presented Bessel–Legendre inequalities of Seuret and Gouaisbaut (2015) and the Wirtinger-based multiple-integral inequalities of Park et al. (2015) and Lee et al. (2015). The present paper aims at showing that the proposed set of sufficient stability conditions can be arranged into a bidirectional hierarchy of LMIs establishing a rigorous theoretical basis for comparison of conservatism of the investigated methods. Numerical examples illustrate the efficiency of the method.