This paper investigates the stability problem of linear systems with a time-varying delay that the delay’s derivative has an upper bound or no constraint. Based on the state vectors and multiple integral state vectors involved in the delay related Bessel–Legendre inequalities (BLIs), the hierarchical Lyapunov–Krasovskii functionals (LKFs) are proposed. In order to deal with the fractions of the delay introduced by the BLIs, a novel iterative reciprocally high-order polynomial (IRHP) combination lemma is developed, which encompasses the existing reciprocally convex inequalities as special cases. Then, by introducing the specific matrix valued negative definite conditions (NDCs) of the odd number high degree polynomials, the hierarchical stability conditions are expressed in the form of linear matrix inequalities (LMIs). Eventually, the validity and advantages of the proposed conditions are illustrated through some classical numerical examples.

中文翻译：

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两类时变时滞系统的层次稳定条件和迭代倒数高阶多项式不等式

本文研究时变导数有上限或无约束的时变线性系统的稳定性问题。基于延迟相关的 Bessel-Legendre 不等式 (BLI) 中涉及的状态向量和多个积分状态向量，提出了分层 Lyapunov-Krasovskii 泛函 (LKF)。为了处理 BLI 引入的延迟分数，开发了一种新颖的迭代倒数高阶多项式 (IRHP) 组合引理，其中包含现有的倒数凸不等式作为特殊情况。然后，通过引入奇数高次多项式的特定矩阵值负定条件（NDC），将层次稳定性条件表达为线性矩阵不等式（LMI）的形式。最后，通过一些经典的数值例子说明了所提出条件的有效性和优点。