随机时滞神经网络的P阶矩指数稳定性研究
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摘要
随机时滞神经网络一直是非线性系统领域研究的热点问题,因为这种系统把时间的延迟即时滞考虑了进来,还把突触之间信息的传递用随机噪声过程来刻划,这样既包含时滞又有随机扰动的系统更接近于现实中的神经网络,具有更加丰富的动力学行为和更实用的研究价值.除了时滞和随机扰动,系统建模时有时还要考虑一些不可避免的脉冲和马尔可夫跳参数.在随机系统的分析中,稳定性是非常重要的一个特性.时滞、随机扰动、脉冲和马尔可夫跳参数都将在较大程度上影响系统的稳定性.近年来,关于随机时滞神经网络的稳定性,因为随机系统神经网络的广泛适用性而成为一大批研究人员的感兴趣的课题,并取得了很多有意义的结果.
本文致力于几类随机时滞神经网络模型的矩指数稳定性研究.以Lyapunov第二研究方法为基础,以重要的不等式为工具,借助随机分析的方法和Razumikhin等定理,在随机时滞神经网络的矩指数稳定性方面,给出了一些适用范围更广、对系数的要求更弱的新的判定条件.具体研究内容包括以下几个方面：
(1)我们研究了带有非线性脉冲的随机递归神经网络.通过同胚映射的相关结果证明了这类系统均衡解的存在唯一性,并借助随机分析和不等式技巧得到了这类系统p阶矩指数稳定的条件,这些结果是已有文献结论的更一般的形式.
(2)我们研究了随机时滞细胞神经网络.通过构造合适的Lyapunov函数和Razumikhin定理得到了带脉冲的随机时变时滞细胞神经网络和带有混合时滞(有界的时变时滞和无界的分布时滞)的随机细胞神经网络的p阶矩指数稳定的条件,这些条件改善和推广了已有文献的结果.
(3)研究了带有逐段常数滞后变量的随机网络.我们首次把逐段常数滞后变量引入随机神经网络模型中,利用Picard迭代的方法得到了系统均衡解的存在唯一性,并用均衡解的积分形式以及构造的Lyapunov函数得到了有脉冲和无脉冲的带有逐段常数滞后变量的随机神经网络的矩指数稳定的结果.这些结果在参数上对确定性的带有逐段常数滞后变量的神经网络有所放松,而确定性的神经网络作为随机系统的一个特例,其有关结果也被包含到我们的结果中.
(4)最后,我们研究的是随机模糊Cohen-Grossberg神经系统.利用Lyapunov函数和Halanay不等式建立了保证随机时滞模糊Cohen-Grossberg神经网络的矩指数稳定性条件,并将马尔可夫调制引入随机模糊Cohen-Grossberg神经网络中,也得到了该模型矩指数稳定的判断条件.参考文献107篇.
Stochastic neural networks with delays have been the popular research topics in the nonlinear field. These kind of systems not only consider the transmission delays but also take the synaptic transmission as a noisy process. A model of stochastic neural network with delays is more convenient to express practical plant, more complex in the dynamic network behaviors and more practical in applications. In addition to the time delays and stochastic disturbances, some inevitable impulses and Markov jump parameters should be taken into account. The stability is an important character in the analysis of a stochastic dynamic system. The time delays, stochastic disturbances, impulses and Markov jump parameters may affect the stability to a large degree. In recent years, a large number of researchers have been attracted by the stability of stochastic neural networks, with a series of significant results achieved.
This dissertation focuses on the study of moment exponential stability of several stochastic neural networks with delays. On the bases of stochastic Lyapunov functional theory, stochastic analysis, Razumikhin theory and Halanay inequality, some new conditions of p-th moment exponential stability for the stochastic neural networks are proposed. The concrete research contents include:
(1)Stochastic recurrent neural networks with nonlinear impulses are investigated. We prove that there exists an unique equilibrium point of this system by using homomorphism. The results of moment exponential stability are obtained by stochastic analysis and inequality technique. These conditions are more applicable than some existing results.
(2) Stochastic cellular neural networks are studied. By means of correct Lyapunov functional and Razumikhin theory, the results of p-th moment exponential stability of stochastic cellular neural networks with time-varying delays and impulses are obtained. We also derived some stability conditions of stochastic cellular neural networks with bounded time-varying delays and unbounded distributed delays. All these results improve and generalize some existing ones.
(3) Stochastic neural networks with piecewise constant argument are concerned. It is the first time that piecewise constant arguments are introduced into stochastic neural networks. The existence and uniqueness of equilibrium point are proved by Picard iteration. Criteria on p-th moment exponential stability of equilibrium point of this system are obtained by constructing a suitable Lyapunov functional. The results of global stability of deterministic system, as a special situation of our results, are concluded in the results, and the restrictions of coefficients are found more loosen.
(4)Finally, stochastic fuzzy Cohen-Grossberg neural networks with delays are discussed. By constructing suitable Lyapunov functional and using Halanay inequality, some sufficient conditions ensuring p-th moment exponential stability are proposed. P-th moment exponential stability for stochastic fuzzy Cohen-Grossberg neural networks with Markov switching are obtained also.
本文致力于几类随机时滞神经网络模型的矩指数稳定性研究.以Lyapunov第二研究方法为基础,以重要的不等式为工具,借助随机分析的方法和Razumikhin等定理,在随机时滞神经网络的矩指数稳定性方面,给出了一些适用范围更广、对系数的要求更弱的新的判定条件.具体研究内容包括以下几个方面：
(1)我们研究了带有非线性脉冲的随机递归神经网络.通过同胚映射的相关结果证明了这类系统均衡解的存在唯一性,并借助随机分析和不等式技巧得到了这类系统p阶矩指数稳定的条件,这些结果是已有文献结论的更一般的形式.
(2)我们研究了随机时滞细胞神经网络.通过构造合适的Lyapunov函数和Razumikhin定理得到了带脉冲的随机时变时滞细胞神经网络和带有混合时滞(有界的时变时滞和无界的分布时滞)的随机细胞神经网络的p阶矩指数稳定的条件,这些条件改善和推广了已有文献的结果.
(3)研究了带有逐段常数滞后变量的随机网络.我们首次把逐段常数滞后变量引入随机神经网络模型中,利用Picard迭代的方法得到了系统均衡解的存在唯一性,并用均衡解的积分形式以及构造的Lyapunov函数得到了有脉冲和无脉冲的带有逐段常数滞后变量的随机神经网络的矩指数稳定的结果.这些结果在参数上对确定性的带有逐段常数滞后变量的神经网络有所放松,而确定性的神经网络作为随机系统的一个特例,其有关结果也被包含到我们的结果中.
(4)最后,我们研究的是随机模糊Cohen-Grossberg神经系统.利用Lyapunov函数和Halanay不等式建立了保证随机时滞模糊Cohen-Grossberg神经网络的矩指数稳定性条件,并将马尔可夫调制引入随机模糊Cohen-Grossberg神经网络中,也得到了该模型矩指数稳定的判断条件.参考文献107篇.
Stochastic neural networks with delays have been the popular research topics in the nonlinear field. These kind of systems not only consider the transmission delays but also take the synaptic transmission as a noisy process. A model of stochastic neural network with delays is more convenient to express practical plant, more complex in the dynamic network behaviors and more practical in applications. In addition to the time delays and stochastic disturbances, some inevitable impulses and Markov jump parameters should be taken into account. The stability is an important character in the analysis of a stochastic dynamic system. The time delays, stochastic disturbances, impulses and Markov jump parameters may affect the stability to a large degree. In recent years, a large number of researchers have been attracted by the stability of stochastic neural networks, with a series of significant results achieved.
This dissertation focuses on the study of moment exponential stability of several stochastic neural networks with delays. On the bases of stochastic Lyapunov functional theory, stochastic analysis, Razumikhin theory and Halanay inequality, some new conditions of p-th moment exponential stability for the stochastic neural networks are proposed. The concrete research contents include:
(1)Stochastic recurrent neural networks with nonlinear impulses are investigated. We prove that there exists an unique equilibrium point of this system by using homomorphism. The results of moment exponential stability are obtained by stochastic analysis and inequality technique. These conditions are more applicable than some existing results.
(2) Stochastic cellular neural networks are studied. By means of correct Lyapunov functional and Razumikhin theory, the results of p-th moment exponential stability of stochastic cellular neural networks with time-varying delays and impulses are obtained. We also derived some stability conditions of stochastic cellular neural networks with bounded time-varying delays and unbounded distributed delays. All these results improve and generalize some existing ones.
(3) Stochastic neural networks with piecewise constant argument are concerned. It is the first time that piecewise constant arguments are introduced into stochastic neural networks. The existence and uniqueness of equilibrium point are proved by Picard iteration. Criteria on p-th moment exponential stability of equilibrium point of this system are obtained by constructing a suitable Lyapunov functional. The results of global stability of deterministic system, as a special situation of our results, are concluded in the results, and the restrictions of coefficients are found more loosen.
(4)Finally, stochastic fuzzy Cohen-Grossberg neural networks with delays are discussed. By constructing suitable Lyapunov functional and using Halanay inequality, some sufficient conditions ensuring p-th moment exponential stability are proposed. P-th moment exponential stability for stochastic fuzzy Cohen-Grossberg neural networks with Markov switching are obtained also.
引文
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