基于忆阻的分数阶时滞神经网络的多稳定性分析与控制

In recent years, some valuable results have been obtained about multistability of neural networks. However, it is worth pointing out that most existing multistability results are derived under the assumption that the considered neural networks models are classical integer-order models. Fractional calculus has more advantages than integer-order calculus in describing neurons with memory and hereditary properties. Memristor is an abbreviation for memory and resistor, and is an electronic element that can closely simulate the brain synapses. Based on the theory of fractional calculus, differential inclusion and set-valued map, we study the issues of multistability analysis and control for memristor-based fractional-order delayed neural networks. The main contents of this project include: Firstly, some general classes of memristor-based fractional-order delayed neural networks are established, by making the most of the advantages of fractional calculus and memristor in the neuron modeling. Secondly, the exact number of equilibrium points in the sense of Filippov is given and the positively invariant sets and basins of attraction for these stationary equilibrium points are estimated. Thirdly, the local stability of multiple equilibrium points in saturated regions is completely analyzed, and some testable criteria are presented which ensure the local Mittag-Leffler stability and asymptotic stability of equilibrium points. Moreover, the intrinsic relationships are revealed between the order of fractional derivative, memristive parameters and time delay, and the dynamical behaviors of the networks. Finally, the complex dynamical behaviors in unsaturated regions are discussed and some efficient control strategies are designed for the stabilization control and the finite-time stabilization control of the newly established neural networks model.

近年来，关于神经网络多稳定性的研究已涌现出了许多有价值的成果，但这些成果基本上都是建立在传统的整数阶神经网络模型上。分数阶微积分在描述具有记忆和遗传特性的神经元时比传统整数阶模型更具优势。忆阻是记忆和电阻的简称，是一种能够高度模拟生物神经元突触功能的电子元件。本项目拟运用分数阶微积分理论、微分包含和集值映射理论研究基于忆阻的分数阶时滞神经网络的多稳定性分析与控制。内容包括：充分发挥忆阻和分数阶微积分在神经网络建模上的优势，建立更贴近实际的神经网络模型；确定新建模型微分包含平衡点的精确数目、位置、正向不变集和吸引域；深入分析系统在饱和区域内的局部稳定性，建立可验证的局部Mittag-Leffler稳定性和局部渐近稳定性准则，揭示分数阶次、忆阻参数和时滞与网络动力学行为之间的内在本质联系；探索系统在不饱和区域内的复杂动力学行为，并在网络不稳定的基础上提出有效的镇定控制和有限时间镇定控制策略。

神经网络的多稳定性主要研究网络多个平衡态的存在性及其复杂的动力学行为，包括平衡态的数目、位置、正向不变集、吸引域、局部稳定性和不稳定性等。神经网络的多稳定性可应用于诸多领域，如联想记忆、模式识别、图像处理等。在目前文献中，绝大部分关于神经网络多稳定性的研究工作，都是针对传统的整数阶神经网络。忆阻是一种能够高度模拟生物神经元突触功能的电子元件。分数阶微积分在描述具有记忆和遗传特性的神经元时比传统整数阶模型更具优势。本项目围绕基于忆阻的分数阶时滞神经网络的多稳定性与控制展开了系统深入的研究，研究内容涉及忆阻神经网络的多稳定性分析、分数阶神经网络的多稳定性分析以及分数阶时滞神经网络的稳定性和分岔控制等。已在国际重要学术刊物，如《Nonlinear Analysis: Real World Applications》、《Applied Mathematics and Computation》、《Neural Networks》、《Neural Processing Letters》等发表相关论文7篇，其中包括SCI 论文6篇，顺利完成了各项预期指标。初步统计在《SCI》刊物被他引73次，并且近年来引用有明显增加的趋势，相关工作得到国内外知名学者的高度评价与肯定。项目执行期间，项目负责人聂小兵副教授因在神经网络多稳定性方面的成果获得2019年度江苏省科学技术二等奖，2016年入选江苏高校“青蓝工程”中青年学术带头人。