随机时滞反应扩散Hopfield神经网络的适定性和渐近性研究

In this project we study on the well-posedness and asymptotic behavior for a class of non-autonomous stochastic reaction-diffusion Hopfield neural networks with delays. Firstly, we construct auxiliary equation of network model by using truncation method and study on the global existence of solutions to the auxiliary equations by means of quasi-developing operator theory and stochastic analysis techniques. Secondly, the solutions to auxiliary equations approaches the solutions to network mode. This method can not only prove the well-posedness of the network model, but also the approximate solutions to the mode can be constructed, and it helps to develop simulation software and set up test platform. From the initial research results of the applicant, such research route is feasible. Lastly, based on the global existence of solutions, we study the stability of Mild solutions to the network mode with the help of semimartingale theory and a method of combining Green operator with quasi-developing operator. The combination of the expectation operator, stochastic analysis technique and V- function is adopted, we study the stability of the stochastic strong solutions and discuss stochastic exponential stability by combining stochastic analysis technique with M- function. We shall show the effect of diffusion operator, random disturbance and time delay on the stability of the network mode and pay particular attention to the problem of random attractor of the network model. The purpose of this project provides theoretical supports for the design and development of engineering artificial neural network.

本项目主要研究一类非自治随机时滞反应扩散Hopfild神经网络模型的适定性和渐近性问题。首先，借助截断方法构造网络模型的辅助方程，利用拟发展算子理论和随机分析技巧研究辅助方程解的全局存在性。然后，用辅助方程的解逼近网络模型的解。此方法不仅能证明网络模型的适定性问题，而且能够构造近似解，也有助于研制仿真软件和构建测试平台。申请人的初步成果证实了此研究路线是可行的。其次，基于解的全局存在性的条件下，借助于半鞅理论﹑时滞Green算子与拟发展算子相结合的方法研究网络模型Mild解的稳定性问题。采用期望算子，随机分析技巧与V-函数相结合的方法研究随机强解的稳定性。尝试随机分析技巧与M-函数相结合的方法研究随机指数稳定性。揭示扩散算子，随机扰动和时滞对网络稳定性的影响所产生的效应。特别关注网络模型的随机吸引子问题。为设计和开发工程人工神经网络提供理论保证和支持。

本项目研究一类非自治随机时滞反应扩散Hopfield神经网络模型的适定性和渐近性问题，借助截断方法构造网络模型的辅助方程，进而利用拟发展算子理论和随机分析技巧研究辅助方程解的全局存在性。然后，利用辅助方程的解逼近网络模型的解，这样不仅证明了网络模型的适定性，同时也能够构造该模型的近似解，为研制仿真软件和构造测试平台提供了基础。基于解的全局存在性的条件下，借助半鞅理论，时滞Gren算子与拟发展算子相结合的方法研究网络模型Mild解的稳定性问题。采用期望算子，随机 分析技巧与V-函数项结合的 方法研究随机强解的稳定性。用随机分析技巧与M-函数相结合的方法研究随机指数稳定性。揭示了扩散算子，随机扰动和时滞对网络稳定性的影响所产生的效应。为设计和开发工程人工神经网络提供了理论保证和支持。