Abstract

This paper is concerned with the global Mittag-Leffler stability (GMLS) and global finite-time stability (GFTS) for fractional Hopfield neural networks (FHNNs) with Hölder neuron activation functions subject to nonlinear growth. Firstly, four functions possessing convexity are proposed, which can guarantee that four formulas with respect to the fractional derivative are established. Correspondingly, a novel principle of convergence in finite-time for FHNNs is developed based on the proposed formulas. In addition, by applying the Brouwer topological degree theory and inequality analysis techniques, the proof of the existence and uniqueness of equilibrium point is addressed. Subsequently, by means of the Lur’e-type Postnikov Lyapunov functional approach, and the presented principle of convergence in finite-time, the GMLS and GFTS conditions are achieved in terms of linear matrix inequalities (LMIs). Moreover, the upper bound of the setting time for the GFTS is calculated accurately. Finally, three numerical examples are given to verify the validity of the theoretical results.

中文翻译：

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分数阶神经网络的全局有限时间稳定性

摘要

本文涉及具有跳变神经的Hölder神经元激活函数的分数Hopfield神经网络（FHNN）的全局Mittag-Leffler稳定性（GMLS）和全局有限时间稳定性（GFTS）。首先，提出了四个具有凸性的函数，它们可以保证建立关于分数导数的四个公式。相应地，基于提出的公式，提出了一种新的FHNNs有限时间收敛的原理。此外，通过应用布劳维尔拓扑度理论和不等式分析技术，解决了平衡点存在和唯一性的证明。随后，借助Lur'e型Postnikov Lyapunov函数方法以及所提出的有限时间收敛原理，GMLS和GFTS条件是根据线性矩阵不等式（LMI）实现的。此外，可以精确计算出GFTS设置时间的上限。最后，通过三个数值例子验证了理论结果的正确性。