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This paper is devoted to stability analysis of continuous-time delay systemsbased on a set of Lyapunov-Krasovskii functionals. New multiple integralinequalities are derived that involve the famous Jensen's and Wirtinger'sinequalities, as well as the recently presented Bessel-Legendre inequalities ofA. Seuret and F. Gouaisbaut, (2015) and the Wirtinger-based multiple-integralinequalities of M. Park et al. (2015) and T.H. Lee et al. (2015).  The present paper aims at showing that the proposed set of sufficientstability conditions can be arranged into a bidirectional hierarchy of LMIsestablishing a rigorous theoretical basis for comparison of conservatism of theinvestigated methods. Numerical examples illustrate the efficiency of themethod.

Multiple integral inequalities and stability analysis of time delay systems