基于忆阻的四元数神经网络的稳定性分析和优化控制

In recent years, some valuable results have been obtained about dynamic behavior of memristive neural networks. However, it is worth pointing out that most of the existing results are derived based on the traditional memristive neural networks models. Quaternion neural networks, which possess numerous advantages of neural network and quaternion, and this kind of neural networks have their distinctive merits when compared with real neural networks and complex neural networks. To take with the target model, the robust analytical methods are imported, i.e., by the selection of measurable functions, the target model is modified as two different systems which contain no identical uncertain parameters, in this way, the corresponding robust analytical methods can be employed to tackle with the dynamic behavior of quaternion memristive neural networks. The main contents of this project include: on the strength of differential inclusion theory, set-valued map theory，Lyapunov stability theory, matrix measure method and linear matrix inequality approach, the corresponding criteria are derived to ensure the quaternion memristive system is exponential stability, fixed-time stability and fixed-time synchronization, respectively. Furthermore, it is able to provide basic theoretical support for the applications in engineering technology fields.

近年来，关于忆阻神经网络的动力学分析已涌现出了许多有价值的结果，但这些结果基本上都是建立在传统的忆阻神经网络模型上。四元数神经网络结合了神经网络和四元数的众多优点，是实值神经网络和复值神经网络都无法比拟的。本项目将采用鲁棒分析方法来处理所给出的忆阻网络模型，即，通过选取适当的可测函数，将目标网络转化为具有不确定参数的鲁棒系统，进而通过鲁棒分析方法相应地探究基于忆阻的四元数神经网络的相关动力学行为。内容包括：利用运用微分包含、集值映射、Lyapunov稳定性、矩阵测度方法和线性矩阵不等式(LMI)等理论，分别获得能分别保证基于忆阻的四元数神经网络的指数稳定、固定时间稳定和固定时间同步等相应的动力学条件与判据，并为四元数神经网络在实际工程技术中的应用提供重要的理论基础。

自2008年惠普公司首次成功研制出基于TiO2的物理器件忆阻器以来，人们掀起了对忆阻器以及忆阻器神经网络的研究热潮。由于四元数神经网络结合了神经网络和四元数的众多优点，是实值和复值神经网络都无法比拟的，近年来，关于四元数忆阻神经网络的动力学分析涌现出了许多有价值的结果。本项目在现有国内外研究成果的基础上，充分运用微分包含、集值映射、Lyapunov稳定性、矩阵测度方法和线性矩阵不等式(LMI) 等理论，分别获得能分别保证基于忆阻的四元数神经网络的指数稳定、固定时间稳定和固定时间同步等相应的动力学条件与判据，并为四元数神经网络在实际工程技术中的应用提供了重要的理论基础。