Faculty of Science, Beijing University of Technology, Beijing 100124, China.
Faculty of Information Technology, Beijing University of Technology, Beijing 100124, China.
Neural Netw. 2021 Oct;142:500-508. doi: 10.1016/j.neunet.2021.07.005. Epub 2021 Jul 12.
In this paper, a type of fractional-order quaternion-valued neural networks (FOQVNNs) with leakage and time-varying delays is established to simulate real-world situations, and the global Mittag-Leffler stability of the system is investigated by using the non-decomposition method. First, to avoid decomposing the system into two complex-valued systems or four real-valued systems, a new sign function for quaternion numbers is introduced based on the ones for real and complex numbers. And two novel lemmas for quaternion-valued sign function and Caputo fractional derivative are established in quaternion domain, which are used to investigate the stability of FOQVNNs. Second, a concise and flexible quaternion-valued state feedback controller is directly designed and a novel 1-norm Lyapunov function composed of the absolute values of real and imaginary parts is established. Then, based on the designed quaternion-valued state feedback controller and the proposed lemmas, some sufficient conditions are given to ensure the global Mittag-Leffler stability of the system. Finally, a numerical simulation is given to verify the theoretical results.
本文建立了一类具有泄漏和时变时滞的分数阶四元数神经网络（FOQVNNs），以模拟实际情况，并采用非分解方法研究了系统的全局 Mittag-Leffler 稳定性。首先，为了避免将系统分解为两个复数值系统或四个实数值系统，基于实数和复数的符号函数，引入了一种新的四元数符号函数。并在四元数域中建立了两个新的四元数符号函数和 Caputo 分数导数引理，用于研究 FOQVNNs 的稳定性。其次，直接设计了简洁灵活的四元数状态反馈控制器，并建立了一个由实部和虚部的绝对值组成的新型 1-范数李雅普诺夫函数。然后，基于设计的四元数状态反馈控制器和提出的引理，给出了一些充分条件，以确保系统的全局 Mittag-Leffler 稳定性。最后，进行了数值模拟以验证理论结果。
