时滞神经网络渐近稳定性和鲁棒稳定性研究
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摘要
神经网络是一种复杂的大规模动力系统,具有十分丰富的动力学属性。在过去近二十年里,由于其在联想记忆、模式识别和优化等问题中的广泛应用,神经网络的动力学问题得到了深入的研究。为了易于分析和应用,许多神经网络模型忽略了神经元之间信息传输所带来的时间延迟。但是,理论和实践证实,时滞是客观存在的。同时时滞对神经网络的稳定性带来影响,产生振荡行为或其它不稳定现象甚至出现混沌现象。于是,近年来,时滞神经网络的研究吸引了大批的研究人员,并取得了大量较深刻的结果。
本论文主要致力于几类时滞系统的渐近稳定性、指数稳定性和鲁棒稳定性的分析,取得了一些较深刻的结果。其主要内容和创新之处可概述如下:
1.具有时变时滞的递归神经网络的渐近稳定性分析
研究带时变时滞的递归神经网络的全局渐近稳定性。首先将要研究的模型转化为描述系统模型,然后利用Lyapunov-Krasovskii稳定性定理,线性矩阵不等式(LMI)技术,S过程和代数不等式方法,得到了确保时变时滞递归神经网络渐近稳定性的新的充分条件,并将它应用于常时滞神经网络和时滞细胞神经网络模型,分别得到了相应的全局渐近稳定性条件。
2.时变时滞递归神经网络时滞区间相关稳定性分析
通过构建适当的Lyapunov-Krasovskii泛函,研究一类时变时滞广义递归神经网络平衡点的时滞区间相关的稳定性,得到了区间时变时滞神经网络平衡点的全局渐近稳定性的两个充分条件。
3.线性时变系统的时滞相关鲁棒稳定性及其在时滞神经网络中的应用
对于多时滞的不确定线性时变系统,我们通过参数化一阶模型变换,以及从区间不确定性到范数有界的不确定性的变换,得到一些时滞相关的鲁棒渐近稳定性的判定准则,给出了与时滞参数相关的稳定域的量化表示。基于这些结果,我们研究一类时滞神经网络系统的稳定性问题,它能够转化为线性时变系统,并得到了几个新的全局渐近稳定性判定准则。
4.脉冲对时滞双向联想记忆神经网络的镇定影响
研究带有脉冲和时滞的双向联想记忆神经网络平衡点的渐近稳定性,得到了几个判断网络全局指数稳定的充分条件,这些条件刻画了脉冲强度和无脉冲的连续子网络(双向联想记忆神经网络的连续部分)的指数收敛速度对整个网络的全局指数稳定性的混和影响。
5.带脉冲和时滞的广义双向联想记忆神经网络的稳定性分析
利用Lyapunov函数和二维Halanay类型不等式,研究带有脉冲和时滞的广义双向联想记忆神经网络平衡点的全局指数稳定性,得到了几个判断网络全局指数稳定的充分条件。结果表明即使脉冲对网络状态有放大的影响,脉冲时滞双向联想记忆神经网络也可能保持连续子网络的全局指数稳定的性质。
6.混杂脉冲开关神经网络
利用开关Lyapunov函数和广义Halanay不等式,考虑脉冲效应和开关效应的混杂影响,研究一类混杂脉冲开关神经网络模型,得到了一些这类神经网络的渐近稳定性和指数稳定性的判断准则。
The neural network, as a large-scale complex system, exhibits the rich and colorful dynamical behaviors. In the recent twenty years, duo to its important and potential applications in associative memory, pattern recognition, optimizing problems and so on, the dynamical issues of delayed neural networks have been investigated intensively. In order to analyze and apply easily, the transmission delays are ignored in modeling for most of system. But it is demonstrated by theories and practices that time delay is unavoidable. At the same time, time delays may affect the stability of the system, even lead to instability, oscillation or chaos phenomena. Recently, delayed neural networks have attracted a large number of researchers, and a series of significative results have been established.
This dissertation focuses on the exponential and asymptotical and robust stability for several delayed system. The main achievements and originality contained in this dissertation are as follows:
1. Asymptotical stability analysis for recurrent neural networks with time-varying delays
The problem of the globally asymptotical stability of recurrent neural networks with time varying delay is investigated. By transforming the delayed neural model to describer model and then employing Lapunov Krasovskii stability theorem, linear matrix inequality (LMI) technique, S procedure and some algebraic inequality method, a new sufficient condition, which is determined by the coefficients of the model and includes more tuning parameters, for determining the global asymptotical stability of recurrent neural networks with time-varying delay is derived. The proposed result is further applied to two special cases: cellular neural network model with time delay and recurrent neural networks with constant delays. It is shown by theoretical analysis and computer simulations that the presented results provide several new sufficient conditions for the asymptotical stability of the investigated delayed neural network model.
2. Delay-interval dependent stability of recurrent neural networks with time-varying delay
The delay-interval-dependent stability of the equilibrium point of a general class of recurrent neural networks with time-varying delays that may exclude zero has been studied. By constructing the appropriate Lyapunov-Krasovskii functional, two sufficient conditions ensuring the global asymptotic stability of the equilibrium point of such networks with interval time-varying delays are established.
3. Delay-dependent robust stability analysis for interval linear time-variant systems with delays and application to delayed neural networks
Some delay-dependent robust asymptotical stability criteria for uncertain linear time-variant systems with multiple delays are established by means of parameterized first-order model transformation and the transformation of the interval uncertainty into the norm-bounded uncertainty. The stable regions with respect to the delay parameters are also formulated. Based on these results, we investigate the stability issue of a class of delayed neural networks that can be transformed into linear time-variant systems, and then several new global asymptotical stability criteria are exploited. 4. Stabilizing effects of impulses in delayed BAM neural networks
The stabilizing effects of impulses in delayed bidirectional associative memory (DBAM) neural networks when its continuous component does not converge
asymptotically to the equilibrium point have been studied. A general criterion, which characterizes the aggregated effects of the impulse and the deviation of its continuous component from the equilibrium point on the exponential stability of the considered DBAM, is established by using Lyapunov-Razumikhin technique.
5. Stability analysis of BAM neural networks with impulse and delays
We further studies the global exponential stability of the equilibrium point of the delayed bidirectional associative memory (DBAM) neural networks with impulse effects. Several results characterizing the aggregated effects of impulse and dynamical property of the impulse-free DBAM on the exponential stability of the considered DBAM have been established. It is shown that the impulsive DBAM will preserve the global exponential stability of the impulse-free DBAM even if the impulses have enlarging effects on the states of neurons.
6. On hybrid impulsive and switching neural networks
We also formulate and study a model of hybrid impulsive and switching Hopfield neural networks. Using switching Lyapunov functions and a generalized Halanay inequality, some general criteria, which characterize the impulse effect and switching effect in aggregated form, for asymptotic and exponential stability of such neural networks with arbitrary and conditioned impulsive switching are established.
本论文主要致力于几类时滞系统的渐近稳定性、指数稳定性和鲁棒稳定性的分析,取得了一些较深刻的结果。其主要内容和创新之处可概述如下:
1.具有时变时滞的递归神经网络的渐近稳定性分析
研究带时变时滞的递归神经网络的全局渐近稳定性。首先将要研究的模型转化为描述系统模型,然后利用Lyapunov-Krasovskii稳定性定理,线性矩阵不等式(LMI)技术,S过程和代数不等式方法,得到了确保时变时滞递归神经网络渐近稳定性的新的充分条件,并将它应用于常时滞神经网络和时滞细胞神经网络模型,分别得到了相应的全局渐近稳定性条件。
2.时变时滞递归神经网络时滞区间相关稳定性分析
通过构建适当的Lyapunov-Krasovskii泛函,研究一类时变时滞广义递归神经网络平衡点的时滞区间相关的稳定性,得到了区间时变时滞神经网络平衡点的全局渐近稳定性的两个充分条件。
3.线性时变系统的时滞相关鲁棒稳定性及其在时滞神经网络中的应用
对于多时滞的不确定线性时变系统,我们通过参数化一阶模型变换,以及从区间不确定性到范数有界的不确定性的变换,得到一些时滞相关的鲁棒渐近稳定性的判定准则,给出了与时滞参数相关的稳定域的量化表示。基于这些结果,我们研究一类时滞神经网络系统的稳定性问题,它能够转化为线性时变系统,并得到了几个新的全局渐近稳定性判定准则。
4.脉冲对时滞双向联想记忆神经网络的镇定影响
研究带有脉冲和时滞的双向联想记忆神经网络平衡点的渐近稳定性,得到了几个判断网络全局指数稳定的充分条件,这些条件刻画了脉冲强度和无脉冲的连续子网络(双向联想记忆神经网络的连续部分)的指数收敛速度对整个网络的全局指数稳定性的混和影响。
5.带脉冲和时滞的广义双向联想记忆神经网络的稳定性分析
利用Lyapunov函数和二维Halanay类型不等式,研究带有脉冲和时滞的广义双向联想记忆神经网络平衡点的全局指数稳定性,得到了几个判断网络全局指数稳定的充分条件。结果表明即使脉冲对网络状态有放大的影响,脉冲时滞双向联想记忆神经网络也可能保持连续子网络的全局指数稳定的性质。
6.混杂脉冲开关神经网络
利用开关Lyapunov函数和广义Halanay不等式,考虑脉冲效应和开关效应的混杂影响,研究一类混杂脉冲开关神经网络模型,得到了一些这类神经网络的渐近稳定性和指数稳定性的判断准则。
The neural network, as a large-scale complex system, exhibits the rich and colorful dynamical behaviors. In the recent twenty years, duo to its important and potential applications in associative memory, pattern recognition, optimizing problems and so on, the dynamical issues of delayed neural networks have been investigated intensively. In order to analyze and apply easily, the transmission delays are ignored in modeling for most of system. But it is demonstrated by theories and practices that time delay is unavoidable. At the same time, time delays may affect the stability of the system, even lead to instability, oscillation or chaos phenomena. Recently, delayed neural networks have attracted a large number of researchers, and a series of significative results have been established.
This dissertation focuses on the exponential and asymptotical and robust stability for several delayed system. The main achievements and originality contained in this dissertation are as follows:
1. Asymptotical stability analysis for recurrent neural networks with time-varying delays
The problem of the globally asymptotical stability of recurrent neural networks with time varying delay is investigated. By transforming the delayed neural model to describer model and then employing Lapunov Krasovskii stability theorem, linear matrix inequality (LMI) technique, S procedure and some algebraic inequality method, a new sufficient condition, which is determined by the coefficients of the model and includes more tuning parameters, for determining the global asymptotical stability of recurrent neural networks with time-varying delay is derived. The proposed result is further applied to two special cases: cellular neural network model with time delay and recurrent neural networks with constant delays. It is shown by theoretical analysis and computer simulations that the presented results provide several new sufficient conditions for the asymptotical stability of the investigated delayed neural network model.
2. Delay-interval dependent stability of recurrent neural networks with time-varying delay
The delay-interval-dependent stability of the equilibrium point of a general class of recurrent neural networks with time-varying delays that may exclude zero has been studied. By constructing the appropriate Lyapunov-Krasovskii functional, two sufficient conditions ensuring the global asymptotic stability of the equilibrium point of such networks with interval time-varying delays are established.
3. Delay-dependent robust stability analysis for interval linear time-variant systems with delays and application to delayed neural networks
Some delay-dependent robust asymptotical stability criteria for uncertain linear time-variant systems with multiple delays are established by means of parameterized first-order model transformation and the transformation of the interval uncertainty into the norm-bounded uncertainty. The stable regions with respect to the delay parameters are also formulated. Based on these results, we investigate the stability issue of a class of delayed neural networks that can be transformed into linear time-variant systems, and then several new global asymptotical stability criteria are exploited. 4. Stabilizing effects of impulses in delayed BAM neural networks
The stabilizing effects of impulses in delayed bidirectional associative memory (DBAM) neural networks when its continuous component does not converge
asymptotically to the equilibrium point have been studied. A general criterion, which characterizes the aggregated effects of the impulse and the deviation of its continuous component from the equilibrium point on the exponential stability of the considered DBAM, is established by using Lyapunov-Razumikhin technique.
5. Stability analysis of BAM neural networks with impulse and delays
We further studies the global exponential stability of the equilibrium point of the delayed bidirectional associative memory (DBAM) neural networks with impulse effects. Several results characterizing the aggregated effects of impulse and dynamical property of the impulse-free DBAM on the exponential stability of the considered DBAM have been established. It is shown that the impulsive DBAM will preserve the global exponential stability of the impulse-free DBAM even if the impulses have enlarging effects on the states of neurons.
6. On hybrid impulsive and switching neural networks
We also formulate and study a model of hybrid impulsive and switching Hopfield neural networks. Using switching Lyapunov functions and a generalized Halanay inequality, some general criteria, which characterize the impulse effect and switching effect in aggregated form, for asymptotic and exponential stability of such neural networks with arbitrary and conditioned impulsive switching are established.
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