时变时滞的变分数阶神经网络的稳定性分析与控制

The neural network, as a large-scale complex system, exhibits the rich and colorful dynamical behaviors. Because of the finite speeds of transmission between neurons, time-delay is unavoidable. In the actual system, time-delay is not constant but varying with time and space. So, it is important to consider neural networks with time-varying delays. Fractional calculus can provide an excellent instrument to describe the memory and hereditary properties of various materials and processes for the reason that fractional order differentiation takes into account the present state and all the history of its previous states. A lot of results on fractional-order neural networks have been obtained in which the fractional-order is constant. In order to better describe the actual system, the fractional order is taken as a variable of time and space not a constant in the modeling of fractional system. Therefore, it is significant to consider variable-order fractional neural network. In this project, variable-order fractional neural network with time-varying delays is investigated based on fractional calculus theory and control theory. First, the necessary and sufficient stability conditions and global stability of fractional-order neural networks with time delays will be discussed. Especially, considering the fractional order and time delay as parameters, several kinds of stability will be investigated. The robust stability of parameters or structures uncertainties of fractional-order neural networks with time delay is considered. Second, the stability and control of variable-order fractional neural networks with time delays are investigated via improving the existing control methods or developing the novel control methods. The variable-order fractional neural network can be transformed into the constant fractional order neural network by the means of some approximation methods for variable fractional order. Then the constant fractional-order systems are convenient for analysis and control research. Finally, the stability and control of variable-order fractional neural networks with time-varying delays will be analyzed. Furthermore, an effective numerical method for solving variable-order fractional neural networks with time-varying delays will be given to verify the validity of the theoretical conditions. This project combines fractional calculus and nonlinear control theory to study the variable-order fractional neural network with time-varying delays, which not only enriches the theory of fractional calculus, but also broadens the research scope of biological neuroscience.

理论和实践证实，神经元之间信息传输中不可避免的存在随时间变化的延迟，故考虑时变时滞神经网络是有实际意义的。用分数阶微积分描述动力系统时，发现分数阶阶数并非常量，而是与时间或空间相关的变量，因此有必要考虑变分数阶神经网络。本项目结合分数阶微积分和控制理论研究时变时滞的变分数阶神经网络稳定性与控制问题。利用分数阶微积分理论研究常时滞分数阶神经网络的稳定性及全局一致渐近稳定性，给出稳定性的充要条件。研究参数或结构具有不确定性的时滞分数阶神经网络的鲁棒稳定性问题，进一步研究变分数阶神经网络的稳定性，改进现有的控制方法或开发新的控制方法，设计适合变分数阶神经网络的控制器和控制方法实现对其控制；其次，对变分数阶阶数近似逼近，使其转化为有限阶次，便于分析及控制研究。最后，分析时变时滞的变分数阶神经网络稳定性及控制研究，给出求解时变时滞的变分数阶神经网络有效的数值计算方法，以便验证理论条件的有效性。

人工神经网络与人工智能的结合具有改变人类生活方式和思维意识的强大潜力，并将在相当长的一段时间内成为人类发展的主流方向。本项目研究的分数阶神经网络为人工智能和机器学习提供理论依据和新的研究方向。本项目主要研究了时变时滞的变分数阶神经网络的稳定性与控制问题。首先，研究了时滞分数阶忆阻神经网络的稳定性和同步问题。给出了连续不可微的Lyapunov函数的Riemann-Liouville分数阶微分不等式，完善了不连续分数阶系统的理论分析方法。并对参数未知的Riemann-Liouville型分数阶忆阻神经网络，设计了有效的自适应控制器及参数更新律，在实现分数阶神经网络的驱动-响应同步的同时，也实现了对未知参数的准确估计。进一步证明了关于Riemann-Liouville分数阶差分不等式，应用该不等式及构造不同的反馈控制器，基于分数阶Lyapunov比较原理，得到了时滞分数阶离散神经网络的同步条件。其次，研究了分数阶广义反应扩散的时滞神经网络的稳定性和同步问题，得到了全局一致稳定性条件。考虑了神经网络参数不匹配因素，设计了反馈控制器，得到了同步的条件，并分析控制器参数对分数阶神经网络同步的影响。最后，建立了带有时滞的变分数阶经济增长模型。由于分数阶微积分可以有效地捕捉经济运行中的记忆特征，因此用分数阶系统来建立经济增长模型更加的合理。以经济生产，劳动力人口和全要素生产率，并考虑了资本函数具有时滞作用，基于Solow经济增长模型，建立时滞变分数阶经济增长模型。通过对分数阶函数进行优化，用优化后的模型预测了中国未来三十年的GDP变化，找到短期内中国经济中高速增长因素。由结果表明中国正面临人口红利的消失和资本的减速积累，因此中国迫切需要提高全要素生产率和生产率，并将经济增长转变为依赖于全要素生产率的增长。所得结果可以对维持中国经济平稳增长给出政策建议。