时标上复数神经网络的单稳定性和多稳定性研究

复数神经网络已广泛应用于优化计算、图像处理、模式识别等诸多领域，稳定性是其应用的先决条件。本项目采用时标理论，把连续时间型和离散时间型两类复数神经网络模型统一成时标上复数神经网络模型，研究其单稳定性和多稳定性问题。通过构造时标上时滞复数神经网络的比较系统，建立比较原理；基于Hermite二次型的能量函数方法，建立时标上时滞复数神经网络单稳定性的一系列判定条件。关于时标上时滞复数神经网络多稳定性的研究，我们将根据网络激励函数的几何特征，把网络状态空间分割成多个子集，并严格分析每个子集中平衡点的存在性、唯一性和稳定性，探究网络不稳定点的数目，给出网络不稳定的判定条件。基于建立的时标上时滞复数神经网络稳定性理论，研究两个时标上时滞复数神经网络的同步性问题，通过设计控制器，建立一系列同步性判定条件。本项目的完成，将丰富和发展神经网络动力学理论，为复数神经网络的进一步应用提供理论支持。

The complex-valued neural networks have been widely applied within many fields such as optimization computation, images processing, pattern recognition. In such applications, it is of prime importance to ensure that the designed neural networks are stable. In this project, we will consider the complex-valued neural networks on time scales in order to unify the study of the discrete-time and contiguous-time complex-valued neural networks, and investigate the problems on the monostability and multistabilit for the complex-valued neural networks on time scales. By constructing the comparison system of the complex-valued neural networks on time scales, we will establish the comparison principle of stability for the complex-valued neural networks on time scales and its comparison system. Based on the energy function method of Hermite's quadric form, we will establish some criteria to ensure the monostability for the delayed complex-valued neural networks on time scales. On the study of multistabilit for the complex-valued neural networks on time scales, we will divide the whole state space into some subsets according to the geometrical configurations of activation functions, and strictly analyze the existence, uniqueness, and stability of equilibrium point in each subset. We will also probe into the numbers of the unstable equilibrium points, and establish the criteria for these unstable equilibrium points. By employing the established stability theory for the complex-valued neural networks on time scales, we will investigate the synchronization of the delayed complex-valued neural networks on time scales, and establish some criteria for checking two delayed complex-valued neural networks on time scales by designing proper controllers. The obtained achievements in this project will enrich and develop stability theory for neural networks, and provide rationale in application of complex-valued neural networks.