分数阶四元数神经网络的稳定性分析与控制

Quaternion, with a real part and three imaginary parts, is regarded as the non-commutative extension of the complex number in the 4-D real space, for which, the 3-D geometrical affine transformation can be represented efficiently and compactly. On the other hand, fractional-order neural networks have attracted wide concerns and applications due to their unique memorability, thus the modelling of fractional-order quaternion neural network can combine both the strengths of fractional-order derivative and quaternion to solve complex problems efficiently in practice. With the aids of some related theories, the fractional-order quaternion neural networks are established, then the dynamics analysis and control for the models will be addressed in this project. The contents of this project include: Firstly, several practical fractional-order models on quaternion field are established; Secondly, the existence and uniqueness of the solutions, the asymptotic stability, Mittag-Leffler stability and finite-time stability of the equilibrium points are respectively exploited to establish corresponding stability criteria; Thirdly, the extensive analyses are launched for the existence and local stability of the multiple equilibrium points on the saturated region; Moreover, the complex dynamical behaviors in unsaturated regions are discussed and some efficient control strategies are proposeed for the stabilization of the considered neural networks; Finally, the appropriate controllers are designed to guarantee the synchronization of the drive-response systems. The implementation of the project not only can promote the improvement of related mathematical theories, enrich the theories of neural network, but also can provide a solid theoretical foundation for the application of quaternion neural networks.

四元数具有一个实部和三个复部，可看作是复数在四维实空间的不可交换延伸。它能有效而简洁地实现3维几何仿射变换。分数阶神经网络因其良好的记忆性而被广泛关注和应用，建立四元数域上的分数阶神经网络有助于同时发挥分数阶导数和四元数的优势，高效准确的解决实际中的复杂问题，本项目拟借助相关理论建立分数阶四元数网络模型，并研究其动力学特征和控制。 具体包括：构建合理的分数阶四元数神经网络模型；探讨解的存在唯一性，平衡点的渐进稳定性、Mittag-Leffler稳定性及有限时间稳定性，建立稳定性判据；深入分析饱和区域上多平衡点的存在性和局部稳定性；探索不饱和区域上的复杂动力学行为，针对不稳定的平衡点，提出有效的镇定控制策略；设计适当的控制器，对驱动-响应系统实现同步控制。 项目的实施可以完善相关的数学理论体系，丰富神经网络的理论成果，为四元数神经网络的应用提供坚实的理论支持。

四元数又称超复数，其特点是可以高效地实现3维几何仿射变换并大大降低复杂度。另外，由于以加权形式积累了函数的全局信息，分数阶神经网络可以更准确地描述具有记忆特性和历史依赖性的物理变化过程和系统状态。本项目结合二者优势，建立了分数阶四元数值神经网络，并综合应用分数阶理论和四元数理论，以及Clifford 代数、Laplace 变换、矩阵代数等知识，围绕分数阶四元数神经网络动力学分析与控制展开系统而深入的研究，目前已公开发表论文10篇，其中SCI论文9篇，顺利完成了各项预期指标。本项目获得的主要研究成果总结如下：利用K类函数研究了一类分数阶离散系统的一致渐进稳定性问题，并构造更加宽松的Lyapunov函数，分别得到了较弱的有界性和吸引性准则； 建立了自适应动态耦合的分数阶网络模型，并实现了该网络的牵引同步控制；建立了高维分数阶耦合反应扩散系统，并针对常数扩散率和变扩散率两种情况，设计基于反步法的边界控制器，实现该系统的镇定控制；在线性耦合、非线性两种耦合拓扑建立了分数阶四元数神经网络，针对前者分析了其多稳定性，并采用脉冲控制器实现了该网络的静态、动态多同步控制，针对后者，设计了基于平均脉冲区间的牵引脉冲策略，实现了该网络的聚类同步控制。项目所得结果不仅丰富了神经网络研究的理论成果，而且具有一定的应用价值。