泛函微分系统的定性分析及其在神经网络中的应用
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摘要
本文研究了几类泛函微分系统的耗散性，周期解，稳定性，不变集和吸引性等定性问题及其在神经网络上的应用。
第一章，主要介绍了几类泛函微分系统的研究背景，同时简单介绍了关于确定型泛函微分系统和随机泛函微分系统定性分析中已有的相关结果。
第二章，首先，利用转移函数的性质和马尔可夫不等式，建立了一个周期马尔可夫过程的存在性定理。然后，利用此定理，得到了It(?)型随机泛函微分系统周期解存在的充分条件。最后，利用It(?)公式和随机分析技巧，获得了一类随机时滞神经网络存在不变集，吸引集以及周期吸引子的充分条件。
第三章，利用Leray-Schauder原理，算术几何平均不等式和向量型时滞微分不等式，研究了一类具有混合时滞的神经网络系统，获得了此系统存在唯一全局渐近稳定平衡点的充分条件。
第四章，研究了一类非自治变时滞神经网络系统的耗散性和周期吸引子。首先利用建立的时滞微分不等式和M-矩阵的性质，讨论了该系统的一致耗散性。进而，利用Banach不动点定理得到了一类非线性泛函微分系统存在全局稳定周期解的充分条件。最后，获得了非自治变时滞神经网络存在周期吸引子的充分条件，并对周期吸引子的范围作出了估计，改进和推广了已有的一些结果。
第五章，利用M-矩阵的性质和不等式分析技巧对一类变时滞模糊神经网络进行了研究，得到了系统的耗散性，不变集和吸引集。
This paper is concerned with the qualitative analysis of functional differentialsystems and stochastic functional differential systems, such as dissipativity, periodicsolutions, invariant set and attractivity, and its Applications in neural networks.
Chapter 1 offers the research background for this paper, and some known re-suits for the qualitative analysis of deterministic functional differential equations andstochastic functional differential equations are introduced.
In Chapter 2, the existence theorem for periodic Markov process is firstly de-veloped by employing the properties of transition functions and Markov inequality.Furthermore, the existence theorem for periodic solution of It(?) stochastic functionaldifferential equations is given. Lastly, by using stochastic analysis technique and It(?)formula, sufficient conditions for the existence of invariant set, attracting set and peri-odic attractor for stochastic neural networks with delays are obtained.
In Chapter 3, a class of neural networks with mixed delays is considered. Theexistence, uniqueness and global asymptotic stability of the equilibrium for the neuralnetworks are established by means of Leray-Schauder principle, Arithmetic-mean-geometric-mean inequality and a vector delay differential inequality.
In Chapter 4, the dissipativity and periodic attractor of a class of non-autonomousneural networks with time-varying delays are investigated. Firstly, by employing theproperties of M-matrix and a delayed differential inequality, the uniformly dissipativ-ity of the neural networks is obtained. Then, by using the Banach fixed point theory, the sufficient conditions for the existence and uniqueness of the periodic solutions ofa class of nonlinear functional differential equations are given. Lastly, some sufficientconditions for the existence range of the periodic attractor are obtained by using theabove results.
In Chapter 5, a class of fuzzy neural networks with delays is considered. Byemploying the properties of M-matrix and the techniques of inequality, dissipativity,invariant sets and attracting sets of the system are obtained.
第一章，主要介绍了几类泛函微分系统的研究背景，同时简单介绍了关于确定型泛函微分系统和随机泛函微分系统定性分析中已有的相关结果。
第二章，首先，利用转移函数的性质和马尔可夫不等式，建立了一个周期马尔可夫过程的存在性定理。然后，利用此定理，得到了It(?)型随机泛函微分系统周期解存在的充分条件。最后，利用It(?)公式和随机分析技巧，获得了一类随机时滞神经网络存在不变集，吸引集以及周期吸引子的充分条件。
第三章，利用Leray-Schauder原理，算术几何平均不等式和向量型时滞微分不等式，研究了一类具有混合时滞的神经网络系统，获得了此系统存在唯一全局渐近稳定平衡点的充分条件。
第四章，研究了一类非自治变时滞神经网络系统的耗散性和周期吸引子。首先利用建立的时滞微分不等式和M-矩阵的性质，讨论了该系统的一致耗散性。进而，利用Banach不动点定理得到了一类非线性泛函微分系统存在全局稳定周期解的充分条件。最后，获得了非自治变时滞神经网络存在周期吸引子的充分条件，并对周期吸引子的范围作出了估计，改进和推广了已有的一些结果。
第五章，利用M-矩阵的性质和不等式分析技巧对一类变时滞模糊神经网络进行了研究，得到了系统的耗散性，不变集和吸引集。
This paper is concerned with the qualitative analysis of functional differentialsystems and stochastic functional differential systems, such as dissipativity, periodicsolutions, invariant set and attractivity, and its Applications in neural networks.
Chapter 1 offers the research background for this paper, and some known re-suits for the qualitative analysis of deterministic functional differential equations andstochastic functional differential equations are introduced.
In Chapter 2, the existence theorem for periodic Markov process is firstly de-veloped by employing the properties of transition functions and Markov inequality.Furthermore, the existence theorem for periodic solution of It(?) stochastic functionaldifferential equations is given. Lastly, by using stochastic analysis technique and It(?)formula, sufficient conditions for the existence of invariant set, attracting set and peri-odic attractor for stochastic neural networks with delays are obtained.
In Chapter 3, a class of neural networks with mixed delays is considered. Theexistence, uniqueness and global asymptotic stability of the equilibrium for the neuralnetworks are established by means of Leray-Schauder principle, Arithmetic-mean-geometric-mean inequality and a vector delay differential inequality.
In Chapter 4, the dissipativity and periodic attractor of a class of non-autonomousneural networks with time-varying delays are investigated. Firstly, by employing theproperties of M-matrix and a delayed differential inequality, the uniformly dissipativ-ity of the neural networks is obtained. Then, by using the Banach fixed point theory, the sufficient conditions for the existence and uniqueness of the periodic solutions ofa class of nonlinear functional differential equations are given. Lastly, some sufficientconditions for the existence range of the periodic attractor are obtained by using theabove results.
In Chapter 5, a class of fuzzy neural networks with delays is considered. Byemploying the properties of M-matrix and the techniques of inequality, dissipativity,invariant sets and attracting sets of the system are obtained.
引文
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