时滞神经网络的动力学研究
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摘要
时滞神经网络广泛应用于信号处理、动态图像处理、人工智能和全局优化等领域,但时滞神经网络在运行过程中有可能出现稳定、不稳定、振荡和混沌等动力学行为,近年来时滞神经网络的动力学问题引起了学术界的广泛关注。特别是时滞神经网络平衡点和周期解的全局稳定性(包括绝对稳定性、渐近稳定性、鲁棒稳定性、指数稳定性等)、Hopf分岔和混沌等问题得到了深入地研究,出现了一系列重要的研究成果。
本文主要对时滞神经网络的平衡点的全局稳定性、Hopf分岔以及混沌的控制和同步等方面进行了研究,主要研究内容和取得的创新性成果如下:
1、时滞细胞神经网络的全局鲁棒稳定性
通过构造新的Lyapunov-Krasovskii泛函,利用线性矩阵不等式技术和S-过程,得到了常时滞细胞神经网络的一个比现有文献保守性更小的全局鲁棒稳定性条件。此外,还得到了变时滞细胞神经网络的全局鲁棒稳定性条件。
2、时滞Cohen-Grossberg神经网络的全局指数稳定性
分别针对具常时滞和具变时滞的Cohen-Grossberg神经网络构造适当的Lyapunov-Krasovskii泛函,利用线性矩阵不等式技术,得到了相应的易于验证的全局指数稳定性的充分条件,并根据这些条件分析了模型的指数收敛速度。
3、混合时滞Hopfield神经网络的Hopf分岔
首先将具离散时滞的Hopfield神经网络推广到同时具离散时滞和分布式时滞(即混合时滞)的模型,通过分析其线性系统的超越特征方程得到模型发生Hopf分岔的条件,然后应用中心流形定理和规范形理论分析了模型的Hopf分岔方向以及分岔周期解的稳定性和周期特性。
4、时滞Hopfield神经网络的混沌控制
通过构造一个适当的完全延迟反馈控制器,将时滞Hopfield神经网络的混沌轨道控制到平衡点,在计算反馈增益矩阵的同时能确定出反馈控制器时滞的上界。对于参数未知的时滞Hopfield神经网络,设计了一种自适应控制模型,导出了自适应控制系统渐近稳定的解析判别条件,能有效地将系统的混沌轨道引导到所期望的目标轨道上。
5、时滞Hopfield神经网络的混沌同步
从双向线性耦合控制的角度出发,根据Lyapunov-Krasovskii稳定性理论,基于线性矩阵不等式技术,给出了时滞Hopfield神经网络混沌完全同步的充分条件和控制器设计方法。针对时滞Hopfield神经网络系统参数未知的情况,将自适应技术和系统辨识技术应用于该系统的混沌滞后同步,推导出系统参数未知时混沌滞后同步的充分条件,得到了系统中未知参数的估计公式。
Delayed neural networks are extensively applied in those fields such as signal processing, moving image processing, artificial intelligence, global optimizing, and etc. While the dynamical characteristics of delayed neural networks include stable, unstable, oscillatory and chaotic behaviors, the dynamical issues of delayed neural networks have attracted worldwide attentions in recent years. Recently, many interesting results on global stability (including absolute stability, asymptotic stability, robust stability, and exponential stability, etc.) criteria for the equilibriums or periodic solutions, Hopf bifurcation and chaos of delayed neural networks have been obtained.
This dissertation mainly focuses on the global stability, Hopf bifurcation, chaos control and synchronization of delayed neural networks. Specifically, the main contents are as follows:
1. Global robust stability analysis of delayed cellular neural networks
By constructing a novel Lyapunov-Krasovskii functional, and by applying the linear matrix inequality technique and S-procedure, a less conservative global robust stability criterion for cellular neural networks with constant delay is derived. In addition, global robust stability criterion for cellular neural networks with time-varying delay is also achieved.
2. Global exponential stability analysis of delayed Cohen-Grossberg neural networks
By constructing appropriate Lyapunov-Krasovskii functional for Cohen-Grossberg neural networks with const delay and time-varying delay respectively, based on the linear matrix inequality technique, some easily verified sufficient conditions for global exponential stability are established. In addition, the exponential convergence degrees of the models are discussed in detail.
3. Hopf bifurcation analysis of Hopfield neural networks with mixed delays
The Hopfield neural networks with discrete delay is generalized to a model with both discrete and distributed delays. It is found that this system undergoes a sequence of Hopf bifurcations by analyzing its associated transcendental characteristic equation. By applying the center manifold theorem and the normal form theory, formulae for determining the direction of Hopf bifurcation and the stability and period of bifurcating periodic solutions are derived.
4. Chaos control of delayed Hopfield neural networks
An appropriate full delayed feedback controller is proposed to stabilize the chaotic trajectory of delayed Hopfield neural network to its unstable equilibrium, the feedback gain matrix and an upper bound of the controller time delay are determined simultaneously. For the delayed Hopfield neural networks with uncertain parameters, an adaptive control model is proposed and asymptotic stability condition in analytic form of such model is presented, the chaotic trajectory can be brought to the trajectory of a desired system.
5. Chaos synchronization of delayed Hopfield neural networks
Based on the Lyapunov stability theory and the linear matrix inequality technique, a sufficient condition for chaos complete synchronization of bi-directional coupled delayed Hopfield neural networks is obtained and a control strategy is proposed. By applying the adaptive technique and system identification technique to chaos lag synchronization issue of delayed Hopfield neural networks with uncertain parameters, a sufficient condition for chaos lag synchronization is derived, and the parameter estimation law is obtained.
本文主要对时滞神经网络的平衡点的全局稳定性、Hopf分岔以及混沌的控制和同步等方面进行了研究,主要研究内容和取得的创新性成果如下:
1、时滞细胞神经网络的全局鲁棒稳定性
通过构造新的Lyapunov-Krasovskii泛函,利用线性矩阵不等式技术和S-过程,得到了常时滞细胞神经网络的一个比现有文献保守性更小的全局鲁棒稳定性条件。此外,还得到了变时滞细胞神经网络的全局鲁棒稳定性条件。
2、时滞Cohen-Grossberg神经网络的全局指数稳定性
分别针对具常时滞和具变时滞的Cohen-Grossberg神经网络构造适当的Lyapunov-Krasovskii泛函,利用线性矩阵不等式技术,得到了相应的易于验证的全局指数稳定性的充分条件,并根据这些条件分析了模型的指数收敛速度。
3、混合时滞Hopfield神经网络的Hopf分岔
首先将具离散时滞的Hopfield神经网络推广到同时具离散时滞和分布式时滞(即混合时滞)的模型,通过分析其线性系统的超越特征方程得到模型发生Hopf分岔的条件,然后应用中心流形定理和规范形理论分析了模型的Hopf分岔方向以及分岔周期解的稳定性和周期特性。
4、时滞Hopfield神经网络的混沌控制
通过构造一个适当的完全延迟反馈控制器,将时滞Hopfield神经网络的混沌轨道控制到平衡点,在计算反馈增益矩阵的同时能确定出反馈控制器时滞的上界。对于参数未知的时滞Hopfield神经网络,设计了一种自适应控制模型,导出了自适应控制系统渐近稳定的解析判别条件,能有效地将系统的混沌轨道引导到所期望的目标轨道上。
5、时滞Hopfield神经网络的混沌同步
从双向线性耦合控制的角度出发,根据Lyapunov-Krasovskii稳定性理论,基于线性矩阵不等式技术,给出了时滞Hopfield神经网络混沌完全同步的充分条件和控制器设计方法。针对时滞Hopfield神经网络系统参数未知的情况,将自适应技术和系统辨识技术应用于该系统的混沌滞后同步,推导出系统参数未知时混沌滞后同步的充分条件,得到了系统中未知参数的估计公式。
Delayed neural networks are extensively applied in those fields such as signal processing, moving image processing, artificial intelligence, global optimizing, and etc. While the dynamical characteristics of delayed neural networks include stable, unstable, oscillatory and chaotic behaviors, the dynamical issues of delayed neural networks have attracted worldwide attentions in recent years. Recently, many interesting results on global stability (including absolute stability, asymptotic stability, robust stability, and exponential stability, etc.) criteria for the equilibriums or periodic solutions, Hopf bifurcation and chaos of delayed neural networks have been obtained.
This dissertation mainly focuses on the global stability, Hopf bifurcation, chaos control and synchronization of delayed neural networks. Specifically, the main contents are as follows:
1. Global robust stability analysis of delayed cellular neural networks
By constructing a novel Lyapunov-Krasovskii functional, and by applying the linear matrix inequality technique and S-procedure, a less conservative global robust stability criterion for cellular neural networks with constant delay is derived. In addition, global robust stability criterion for cellular neural networks with time-varying delay is also achieved.
2. Global exponential stability analysis of delayed Cohen-Grossberg neural networks
By constructing appropriate Lyapunov-Krasovskii functional for Cohen-Grossberg neural networks with const delay and time-varying delay respectively, based on the linear matrix inequality technique, some easily verified sufficient conditions for global exponential stability are established. In addition, the exponential convergence degrees of the models are discussed in detail.
3. Hopf bifurcation analysis of Hopfield neural networks with mixed delays
The Hopfield neural networks with discrete delay is generalized to a model with both discrete and distributed delays. It is found that this system undergoes a sequence of Hopf bifurcations by analyzing its associated transcendental characteristic equation. By applying the center manifold theorem and the normal form theory, formulae for determining the direction of Hopf bifurcation and the stability and period of bifurcating periodic solutions are derived.
4. Chaos control of delayed Hopfield neural networks
An appropriate full delayed feedback controller is proposed to stabilize the chaotic trajectory of delayed Hopfield neural network to its unstable equilibrium, the feedback gain matrix and an upper bound of the controller time delay are determined simultaneously. For the delayed Hopfield neural networks with uncertain parameters, an adaptive control model is proposed and asymptotic stability condition in analytic form of such model is presented, the chaotic trajectory can be brought to the trajectory of a desired system.
5. Chaos synchronization of delayed Hopfield neural networks
Based on the Lyapunov stability theory and the linear matrix inequality technique, a sufficient condition for chaos complete synchronization of bi-directional coupled delayed Hopfield neural networks is obtained and a control strategy is proposed. By applying the adaptive technique and system identification technique to chaos lag synchronization issue of delayed Hopfield neural networks with uncertain parameters, a sufficient condition for chaos lag synchronization is derived, and the parameter estimation law is obtained.
引文
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