逆Lipschitz条件下脉冲神经网络稳定性研究
|
摘要
脉冲时滞神经网络的动力学是近几年来神经网络领域的研究热点。在这些研究中,我们常常假定神经网络的激活函数或行为函数满足连续性条件以及全局或局部Lipschitz条件。但这种假定有时是苛刻的,因为在神经计算中,为了表示要解决的实际问题,常常要求激活函数或行为函数满足一些其它条件,例如逆Lipschitz条件。本文研究了逆Lipschitz条件下脉冲时滞神经网络的指数稳定性,得到了一系列较深刻的结果。论文的主要内容如下:
①脉冲时滞神经网络稳定性的数学基础
脉冲时滞神经网络的理论基础是脉冲时滞微分方程。为了论文内容的自包含,介绍了微分方程的相关定性理论,包括解的存在性、唯一性和稳定性等概念以及要用到的多个重要微分不等式和Lyapunov稳定性定理。
②脉冲时滞神经网络的研究现状
介绍了脉冲神经网络和时滞神经网络的基本概念和相应的微分模型,以及稳定性判定的常用定理,并对每一种类型的研究现状作了简要的介绍。
③逆Lipschitz时变脉冲Cohen-Grossberg网络的稳定性
分析了时滞脉冲系统连续部分是发散的情况,但加上脉冲部分后呈现出指数收敛,并且脉冲点的出现跟状态相关。脉冲点的“打击现象”,在文中也作了讨论。对无时滞和有时滞两种情况,分别给出了相应的指数稳定性的充分条件。对后一种情况,根据两种不同的条件,给出了相应的定理,对行为函数是逆Lipschitz的情况也作了讨论。最后给出四个数值仿真例子分别验证了无时滞、有时滞、二阶和三阶时定理的有效性。
④逆Lipschitz连续时间脉冲Cohen-Grossberg网络的稳定性
对激活函数是逆Lipschitz条件的情况作了分析。在这种情况下,解的存在唯一性用拓扑度的理论得到了证明。为了简化拓扑度的计算,构造了一个连续同伦映射并使用了扩展雅可比矩阵集合及相应的引理。提出了全局指数稳定的充分条件,最后用两个数值仿真验证了定理的有效性。
⑤逆Lipschitz离散时间脉冲Cohen-Grossberg网络的稳定性
对行为函数是逆Lipschitz条件的情况作了全局指数稳定性和全局渐近稳定性分析,其中的时间是离散的。又按是否有时滞分别进行了讨论。当时滞有规律时对条件进行了简化。证明中主要用到了离散Halanay不等式。三个数值例子显示了对于无时滞、二阶时滞系统和三阶时滞脉冲系统,即使脉冲是起放大作用,只要它不超出一定的度,定理都是有效的。
In recent years, the dynamics of impulsive time delay neural networks is the study hot of neural networks. In these studies, we often assume that some parts of the neural network, such as the activation function and behavior functions, meet the conditions of continuity, global or local Lipschitz. This assumption is sometimes harsh. For instance, modeling some facts in studing neural computation, we need other condition, such as inverse Lipschitz, to meet. Under the inverse Lipschitz condition, this paper studied the exponential stability of impulsive time delay neural networks, and achieved good results. The main contents are as follows
①The stability mathematical foundation of impulsive time delay neural networks
Impulsive time delay differential equation is the theoretical basis of impulsive time delay neural networks. For the completely, the paper describes the related theorems of differential equations, such as its existence, uniqueness and stability. We also introduce a few of important differential inequalities and Lyapunov stability theory.
②The research advance of impulsive time delay neural networks
We introduce the basic concept of impulse and time delay neural networks, the corresponding differential model, and the stability theorems commonly used to determine. Finally, we give a brief review of the development state for each type.
③The stability of inverse Lipschitz variable-time impulse Cohen-Grossberg neural network
We analyze the time delay neural networks, in which the continous part is divergent, but as a whole it is exponential convergent by mixing impulses. Impulses occur at variable time which depends on states. The paper also discusses the“beating phenomenon”of impulses. We separately propose the sufficient conditions of the exponential stability by whether it has time delay. According to two different conditions, we give its corresponding stability theory for the latter. We also discuss the system, in which behavior functions are in inverse Lipschitz conditions. Four numerical examples demonstrate the effectiveness of the results, which have no time-delay, second-order and third-order time-delay system, respectively.
④The stability of inverse Lipschitz continuous time impulse Cohen-Grossberg neural network
The Cohen-Grossberg neural network, whose activation function is in the inverse Lipschitz condition, is analyzed. In this case, the existence and uniqueness of equations solution is proved by the topological degree theory. To simplify the computing of topological degree, we construct a continuous homotopic map and employ expanded Jocobian matrix set and its theory. The paper proposes some global exponential stability sufficient conditions. Two numerical examples are given to verity the effectiveness of the results.
⑤The stability of inverse Lipschitz discrete time impulse Cohen-Grossberg neural network
The paper analyse the global exponential stability and global asymptotic stability, in which behavior functions are in inverse Lipschitz conditions and their time is discrete. We discuss separately the stability by whether it has time delay. When the time delay is regular, we simplify the condition. We prove it mainly using discrete Halanay inequality. Three numerical examples demonstrate the effectiveness of the results, which have no time-delay, second-order and third-order time-delay system, respectively. And they also demonstrate that those theorems are effective even though the impulse has enlarged effect, provided not exceeding a certain limit.
①脉冲时滞神经网络稳定性的数学基础
脉冲时滞神经网络的理论基础是脉冲时滞微分方程。为了论文内容的自包含,介绍了微分方程的相关定性理论,包括解的存在性、唯一性和稳定性等概念以及要用到的多个重要微分不等式和Lyapunov稳定性定理。
②脉冲时滞神经网络的研究现状
介绍了脉冲神经网络和时滞神经网络的基本概念和相应的微分模型,以及稳定性判定的常用定理,并对每一种类型的研究现状作了简要的介绍。
③逆Lipschitz时变脉冲Cohen-Grossberg网络的稳定性
分析了时滞脉冲系统连续部分是发散的情况,但加上脉冲部分后呈现出指数收敛,并且脉冲点的出现跟状态相关。脉冲点的“打击现象”,在文中也作了讨论。对无时滞和有时滞两种情况,分别给出了相应的指数稳定性的充分条件。对后一种情况,根据两种不同的条件,给出了相应的定理,对行为函数是逆Lipschitz的情况也作了讨论。最后给出四个数值仿真例子分别验证了无时滞、有时滞、二阶和三阶时定理的有效性。
④逆Lipschitz连续时间脉冲Cohen-Grossberg网络的稳定性
对激活函数是逆Lipschitz条件的情况作了分析。在这种情况下,解的存在唯一性用拓扑度的理论得到了证明。为了简化拓扑度的计算,构造了一个连续同伦映射并使用了扩展雅可比矩阵集合及相应的引理。提出了全局指数稳定的充分条件,最后用两个数值仿真验证了定理的有效性。
⑤逆Lipschitz离散时间脉冲Cohen-Grossberg网络的稳定性
对行为函数是逆Lipschitz条件的情况作了全局指数稳定性和全局渐近稳定性分析,其中的时间是离散的。又按是否有时滞分别进行了讨论。当时滞有规律时对条件进行了简化。证明中主要用到了离散Halanay不等式。三个数值例子显示了对于无时滞、二阶时滞系统和三阶时滞脉冲系统,即使脉冲是起放大作用,只要它不超出一定的度,定理都是有效的。
In recent years, the dynamics of impulsive time delay neural networks is the study hot of neural networks. In these studies, we often assume that some parts of the neural network, such as the activation function and behavior functions, meet the conditions of continuity, global or local Lipschitz. This assumption is sometimes harsh. For instance, modeling some facts in studing neural computation, we need other condition, such as inverse Lipschitz, to meet. Under the inverse Lipschitz condition, this paper studied the exponential stability of impulsive time delay neural networks, and achieved good results. The main contents are as follows
①The stability mathematical foundation of impulsive time delay neural networks
Impulsive time delay differential equation is the theoretical basis of impulsive time delay neural networks. For the completely, the paper describes the related theorems of differential equations, such as its existence, uniqueness and stability. We also introduce a few of important differential inequalities and Lyapunov stability theory.
②The research advance of impulsive time delay neural networks
We introduce the basic concept of impulse and time delay neural networks, the corresponding differential model, and the stability theorems commonly used to determine. Finally, we give a brief review of the development state for each type.
③The stability of inverse Lipschitz variable-time impulse Cohen-Grossberg neural network
We analyze the time delay neural networks, in which the continous part is divergent, but as a whole it is exponential convergent by mixing impulses. Impulses occur at variable time which depends on states. The paper also discusses the“beating phenomenon”of impulses. We separately propose the sufficient conditions of the exponential stability by whether it has time delay. According to two different conditions, we give its corresponding stability theory for the latter. We also discuss the system, in which behavior functions are in inverse Lipschitz conditions. Four numerical examples demonstrate the effectiveness of the results, which have no time-delay, second-order and third-order time-delay system, respectively.
④The stability of inverse Lipschitz continuous time impulse Cohen-Grossberg neural network
The Cohen-Grossberg neural network, whose activation function is in the inverse Lipschitz condition, is analyzed. In this case, the existence and uniqueness of equations solution is proved by the topological degree theory. To simplify the computing of topological degree, we construct a continuous homotopic map and employ expanded Jocobian matrix set and its theory. The paper proposes some global exponential stability sufficient conditions. Two numerical examples are given to verity the effectiveness of the results.
⑤The stability of inverse Lipschitz discrete time impulse Cohen-Grossberg neural network
The paper analyse the global exponential stability and global asymptotic stability, in which behavior functions are in inverse Lipschitz conditions and their time is discrete. We discuss separately the stability by whether it has time delay. When the time delay is regular, we simplify the condition. We prove it mainly using discrete Halanay inequality. Three numerical examples demonstrate the effectiveness of the results, which have no time-delay, second-order and third-order time-delay system, respectively. And they also demonstrate that those theorems are effective even though the impulse has enlarged effect, provided not exceeding a certain limit.
引文
[1] J. Cao, D. Zhou. Stability analysis of delayed cellular neural networks [J]. Neural Networks, 1998, 11(9): 1601-5.
[2] J. Cao. On stability of delayed cellular neural networks [J]. Physics Letters A, 1999, 261(5-6): 303-8.
[3] J. Cao. On exponential stability and periodic solutions of CNNs with delays [J]. Physics Letters A, 2000, 267(5-6): 312-8.
[4] J. Cao. Periodic oscillation and exponential stability of delayed CNNs [J]. Physics Letters A, 2000, 270(3-4): 157-63.
[5] J. Cao, Q. Li. On the Exponential Stability and Periodic Solutions of Delayed Cellular Neural Networks [J]. Journal of Mathematical Analysis and Applications, 2000, 252(1): 50-64.
[6] X. Liao, K.-W. Wong, C.-S. Leung, et al. Hopf bifurcation and chaos in a single delayed neuron equation with non-monotonic activation function [J]. Chaos, Solitons & Fractals, 2001, 12(8): 1535-47.
[7] Q. Tao, J. Cao, M. Xue, et al. A high performance neural network for solving nonlinear programming problems with hybrid constraints [J]. Physics Letters A, 2001, 288(2): 88-94.
[8] J. Cao, Q. Li, S. Wan. Periodic solutions of the higher-dimensional non-autonomous systems [J]. Applied Mathematics and Computation, 2002, 130(2-3): 369-82.
[9] X. Liao, G. Chen, E.N. Sanchez. Delay-dependent exponential stability analysis of delayed neural networks: an LMI approach [J]. Neural Networks, 2002, 15(7): 855-66.
[10] X. Liao, Z. Wu, J. Yu. Stability analyses of cellular neural networks with continuous time delay [J]. Journal of Computational and Applied Mathematics, 2002, 143(1): 29-47.
[11] J. Cao. New results concerning exponential stability and periodic solutions of delayed cellular neural networks [J]. Physics Letters A, 2003, 307(2-3): 136-47.
[12] J. Cao, M. Dong. Exponential stability of delayed bi-directional associative memory networks [J]. Applied Mathematics and Computation, 2003, 135(1): 105-12.
[13] A. Chen, J. Cao. Existence and attractivity of almost periodic solutions for cellular neural networks with distributed delays and variable coefficients [J]. Applied Mathematics and Computation, 2003, 134(1): 125-40.
[14] X. Liao, G. Chen, E.N. Sanchez. Delay-dependent exponential stability analysis of delayed neural networks: an LMI approach [J]. Neural Networks, 2003, 16(10): 1401-2.
[15] X. Liao, K.-W. Wong, S. Yang. Convergence dynamics of hybrid bidirectional associative memory neural networks with distributed delays [J]. Physics Letters A, 2003, 316(1-2): 55-64.
[16] J. Cao. An estimation of the domain of attraction and convergence rate for Hopfield continuous feedback neural networks [J]. Physics Letters A, 2004, 325(5-6): 370-4.
[17] J. Cao, Q. Jiang. An analysis of periodic solutions of bi-directional associative memory networks with time-varying delays [J]. Physics Letters A, 2004, 330(3-4): 203-13.
[18] J. Cao, J. Liang, J. Lam. Exponential stability of high-order bidirectional associative memory neural networks with time delays [J]. Physica D: Nonlinear Phenomena, 2004, 199(3-4): 425-36.
[19] C. Li, X. Liao, K.-w. Wong. Bifurcation of the exponentially self-regulating population model [J]. Chaos, Solitons & Fractals, 2004, 20(3): 479-87.
[20] X. Liao, C. Li, K.-w. Wong. Criteria for exponential stability of Cohen-Grossberg neural networks [J]. Neural Networks, 2004, 17(10): 1401-14.
[21] X. Liao, K.-w. Wong. Global exponential stability for a class of retarded functional differential equations with applications in neural networks [J]. Journal of Mathematical Analysis and Applications, 2004, 293(1): 125-48.
[22] X. Liao, K.-w. Wong, C. Li. Global exponential stability for a class of generalized neural networks with distributed delays [J]. Nonlinear Analysis: Real World Applications, 2004, 5(3): 527-47.
[23] J. Cao, A. Chen, X. Huang. Almost periodic attractor of delayed neural networks with variable coefficients [J]. Physics Letters A, 2005, 340(1-4): 104-20.
[24] C. Li, X. Liao. New algebraic conditions for global exponential stability of delayed recurrent neural networks [J]. Neurocomputing, 2005, 64: 319-33.
[25] C. Li, X. Liao, R. Zhang, et al. Global robust exponential stability analysis for interval neural networks with time-varying delays [J]. Chaos, Solitons & Fractals, 2005, 25(3): 751-7.
[26] X. Liao, G. Chen. Hopf bifurcation and chaos analysis of Chen's system with distributed delays [J]. Chaos, Solitons & Fractals, 2005, 25(1): 197-220.
[27] X. Liao, C. Li. An LMI approach to asymptotical stability of multi-delayed neural networks [J]. Physica D: Nonlinear Phenomena, 2005, 200(1-2): 139-55.
[28] Y. Xia, M. Lin, J. Cao. The existence of almost periodic solutions of certain perturbation systems [J]. Journal of Mathematical Analysis and Applications, 2005, 310(1): 81-96.
[29] W. Yu, J. Cao. Hopf bifurcation and stability of periodic solutions for van der Pol equation with time delay [J]. Nonlinear Analysis, 2005, 62(1): 141-65.
[30] H. Zhao, J. Cao. New conditions for global exponential stability of cellular neural networks with delays [J]. Neural Networks, 2005, 18(10): 1332-40.
[31] J. Cao, H.X. Li, L. Han. Novel results concerning global robust stability of delayed neuralnetworks [J]. Nonlinear Analysis: Real World Applications, 2006, 7(3): 458-69.
[32] X. Liao, C. Li. Global attractivity of Cohen-Grossberg model with finite and infinite delays [J]. Journal of Mathematical Analysis and Applications, 2006, 315(1): 244-62.
[33] X. Liao, Q. Liu, W. Zhang. Delay-dependent asymptotic stability for neural networks with distributed delays [J]. Nonlinear Analysis: Real World Applications, 2006, 7(5): 1178-92.
[34] J. Cao, D.W.C. Ho, X. Huang. LMI-based criteria for global robust stability of bidirectional associative memory networks with time delay [J]. Nonlinear Analysis, 2007, 66(7): 1558-72.
[35] C. Li, Q. Chen, T. Huang. Coexistence of anti-phase and complete synchronization in coupled chen system via a single variable [J]. Chaos, Solitons & Fractals, 2008, 38(2): 461-4.
[36] C. Li, Q. Yan. Comments on "Exponential stability of nonlinear differential delay equations" [Systems & Control Letters 28 (1996) 159-165] [J]. Systems & Control Letters, 2008, 57(10): 834-5.
[37] X. Liao, J. Yang, S. Guo. Exponential stability of Cohen-Grossberg neural networks with delays [J]. Communications in Nonlinear Science and Numerical Simulation, 2008, 13(9): 1767-75.
[38] W. Yu, J. Cao, K. Yuan. Synchronization of switched system and application in communication [J]. Physics Letters A, 2008, 372(24): 4438-45.
[39] J. Cao, L. Li. Cluster synchronization in an array of hybrid coupled neural networks with delay [J]. Neural Networks, 2009, 22(4): 335-42.
[40] X. Liao, Y. Liu, S. Guo, et al. Asymptotic stability of delayed neural networks: A descriptor system approach [J]. Communications in Nonlinear Science and Numerical Simulation, 2009, 14(7): 3120-33.
[41] Q. Zhu, J. Cao. Adaptive synchronization under almost every initial data for stochastic neural networks with time-varying delays and distributed delays [J]. Communications in Nonlinear Science and Numerical Simulation, 2011, 16(4): 2139-59.
[42] H. Kong, L. Guan. A neural network adaptive filter for the removal of impulse noise in digital images [J]. Neural Networks, 1996, 9(3): 373-8.
[43] A.L. Orille, N. Khalil, J.A.V. Valencia. A transformer differential protection based on finite impulse response artificial neural network [J]. Computers & Industrial Engineering, 1999, 37(1-2): 399-402.
[44] Y. Li, L. Lu. Global exponential stability and existence of periodic solution of Hopfield-type neural networks with impulses [J]. Physics Letters A, 2004, 333(1-2): 62-71.
[45] Y. Li, W. Xing, L. Lu. Existence and global exponential stability of periodic solution of a class of neural networks with impulses [J]. Chaos, Solitons & Fractals, 2006, 27(2): 437-45.
[46] B. Liu, L. Huang. Global exponential stability of BAM neural networks with recent-history distributed delays and impulses [J]. Neurocomputing, 2006, 69(16-18): 2090-6.
[47] Y. Wang, W. Xiong, Q. Zhou, et al. Global exponential stability of cellular neural networks with continuously distributed delays and impulses [J]. Physics Letters A, 2006, 350(1-2): 89-95.
[48] Z. Chen, J. Ruan. Global dynamic analysis of general Cohen-Grossberg neural networks with impulse [J]. Chaos, Solitons & Fractals, 2007, 32(5): 1830-7.
[49] Z.-T. Huang, X.-S. Luo, Q.-G. Yang. Global asymptotic stability analysis of bidirectional associative memory neural networks with distributed delays and impulse [J]. Chaos, Solitons & Fractals, 2007, 34(3): 878-85.
[50] Y. Xia, J. Cao, S. Sun Cheng. Global exponential stability of delayed cellular neural networks with impulses [J]. Neurocomputing, 2007, 70(13-15): 2495-501.
[51] H. Gu, H. Jiang, Z. Teng. Existence and globally exponential stability of periodic solution of BAM neural networks with impulses and recent-history distributed delays [J]. Neurocomputing, 2008, 71(4-6): 813-22.
[52] K. Li, Q. Song. Exponential stability of impulsive Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms [J]. Neurocomputing, 2008, 72(1-3): 231-40.
[53] H.-F. Huo, W.-T. Li. Dynamics of continuous-time bidirectional associative memory neural networks with impulses and their discrete counterparts [J]. Chaos, Solitons & Fractals, 2009, 42(4): 2218-29.
[54] K. Li. Stability analysis for impulsive Cohen-Grossberg neural networks with time-varying delays and distributed delays [J]. Nonlinear Analysis: Real World Applications, 2009, 10(5): 2784-98.
[55] J. Sun, Q.-G. Wang, H. Gao. Periodic solution for nonautonomous cellular neural networks with impulses [J]. Chaos, Solitons & Fractals, 2009, 40(3): 1423-7.
[56] Z. Wen, J. Sun. Stability analysis of delayed Cohen-Grossberg BAM neural networks with impulses via nonsmooth analysis [J]. Chaos, Solitons & Fractals, 2009, 42(3): 1829-37.
[57] X. Yang, X. Cui, Y. Long. Existence and global exponential stability of periodic solution of a cellular neural networks difference equation with delays and impulses [J]. Neural Networks, 2009, 22(7): 970-6.
[58] J. Zhang, Z. Gui. Existence and stability of periodic solutions of high-order Hopfield neural networks with impulses and delays [J]. Journal of Computational and Applied Mathematics, 2009, 224(2): 602-13.
[59] X. Zhao. Global exponential stability of discrete-time recurrent neural networks withimpulses [J]. Nonlinear Analysis: Theory, Methods & Applications, 2009, 71(12): e2873-e8.
[60] Q. Zhou. Global exponential stability of BAM neural networks with distributed delays and impulses [J]. Nonlinear Analysis: Real World Applications, 2009, 10(1): 144-53.
[61] X. Yang, F. Li, Y. Long, et al. Existence of periodic solution for discrete-time cellular neural networks with complex deviating arguments and impulses [J]. Journal of the Franklin Institute, 2010, 347(2): 559-66.
[62] S. Zhong, C. Li, X. Liao. Global stability of discrete-time Cohen-Grossberg neural networks with impulses [J]. Neurocomputing, 2010, 73(16-18): 3132-8.
[63] X. Mao, C. Yuan, G. Yin. Approximations of Euler-Maruyama type for stochastic differential equations with Markovian switching, under non-Lipschitz conditions [J]. Journal of Computational and Applied Mathematics, 2007, 205(2): 936-48.
[64] H. Wu, X. Xue. Stability analysis for neural networks with inverse Lipschitzian neuron activations and impulses [J]. Appl Math Model, 2008, 32(11): 2347-59.
[65] Q. He, F. Yang. On the approximate solutions of constrained inverse Lipschitz function problems and inverse function theorems [J]. Nonlinear Analysis: Theory, Methods & Applications, 2009, 70(2): 1124-31.
[66] X. Nie, J. Cao. Stability analysis for the generalized Cohen-Grossberg neural networks with inverse Lipschitz neuron activations [J]. Comput Math Appl, 2009, 57(9): 1522-36.
[67] Y. Tan, M. Tan. Global asymptotical stability of continuous-time delayed neural networks without global Lipschitz activation functions [J]. Communications in Nonlinear Science and Numerical Simulation, 2009, 14(11): 3715-22.
[68] H. Wu. Global exponential stability of Hopfield neural networks with delays and inverse Lipschitz neuron activations [J]. Nonlinear Analysis: Real World Applications, 2009, 10(4): 2297-306.
[69] T.D. Chuong, N.Q. Huy, J.C. Yao. Pseudo-Lipschitz property of linear semi-infinite vector optimization problems [J]. European Journal of Operational Research, 2010, 200(3): 639-44.
[70]蒋宗礼.人工神经网络导论[M].北京:高等教育出版社, 2001.8.
[71] W.S. McCulloch, W. Pitts. A logical calculus of the ideas immanent in nervous activity [J]. Bulletin of Mathematical Biology, 1990, 52(1-2): 99-115.
[72] J. Hertz, A. Krogh, R.G. Palmer. Introduction to the theory of neural computation [M]. Redwood: Addison-Wesley, 1991.
[73] T. Kohonen. Correlation Matrix Memories [J]. IEEE Transactions on Computers, 1972, 21(4): 353-9.
[74] J.A. Anderson. A simple neural network generating an interactive memory [M].Neurocomputing: foundations of research. MIT Press. 1988: 181-92.
[75] C.v.d. Malsburg. Self-organization of orientation sensitive cells in the striata cortex [M]. MIT Press, 1988.
[76] P.J. Werbos. Beyond Regression: New Tools for Prediction and Analysis in the Behavioral Sciences [D]. Cambridge: Harvard University, 1974.
[77] W.A. Little, G.L. Shaw. A statistical theory of short and long term memory [M]. MIT Press, 1988.
[78] S. Grossberg. Adaptive pattern classification and universal recoding. I.: parallel development and coding of neural feature detectors [M]. Neurocomputing: foundations of research. MIT Press. 1988: 243-58.
[79] S.I. Amari. Neural theory of association and concept-formation [J]. Biological Cybernetics, 1977, 26(3): 175-85.
[80] J.J. Hopfield. Neural networks and physical systems with emergent collective computational abilities [M]. Neurocomputing: foundations of research. MIT Press. 1988: 457-64.
[81]哈姆,科斯塔尼克.神经计算原理[M].北京:机械工业出版社, 2007.5.
[82] T. Kohonen. Self-organized formation of topologically correct feature maps [J]. Biological Cybernetics, 1982, 43(1): 59-69.
[83] E. Oja. Simplified neuron model as a principal component analyzer [J]. Journal of Mathematical Biology, 1982, 15(3): 267-73.
[84] D.H. Ackley, G.E. Hinton, T.J. Sejnowski. A learning algorithm for Boltzmann machines [M]. Connectionist models and their implications: readings from cognitive science. Ablex Publishing Corp. 1988: 285-307.
[85] D. Parker. Learning-logic. Technical Report TR-47,. Center for Computational Research in Economics and. Management Science [M]. MIT, 1985.
[86] D.E. Rumelhart, G.E. Hinton, R.J. Williams. Learning internal representations by error propagation [M]. Parallel distributed processing: explorations in the microstructure of cognition, vol 1. MIT Press. 1986: 318-62.
[87] D.E. Rumelhart, G.E. Hinton, R.J. Williams. Learning representations by back-propagating errors [M]. Neurocomputing: foundations of research. MIT Press. 1988: 696-9.
[88] G.A. Carpenter, S. Grossberg. A massively parallel architecture for a self-organizing neural pattern recognition machine [J]. Comput Vision Graph Image Process, 1987, 37(1): 54-115.
[89] M.A. Sivilotti, M.A. Mahowald, C.A. Mead. Real-time visual computations using analog CMOS processing arrays; proceedings of the in 1987 Stanford VLSIConference, F, 1987 [C].
[90] T. Yang. Impulsive Control Theory [M]. Springer, 2001.
[91]杨德刚.时滞神经网络稳定性及复杂网络同步的研究[D].重庆:重庆大学, 2007.
[92] S. Grossberg. Nonlinear neural networks: Principles, mechanisms, and architectures [J]. Neural Networks, 1988, 1(1): 17-61.
[93] K. Gopalsamy, B.G. Zhang. On delay differential equations with impulses [J]. Journal of Mathematical Analysis and Applications, 1989, 139(1): 110-22.
[94] A.M.Samoilenko, N.A.Peretyuk. Impulsive Differential Equations [M]. Singapore: World Scientific, 1995.
[95] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov. Theory of Impulsive Differential Equations [M]. Singapore,New York: World Scientific, 1989.
[96] J. Luo. Oscillation for Nonlinear Delay Differential Equations with Impulses [J]. Journal of Mathematical Analysis and Applications, 2000, 250(1): 290-8.
[97] J. Yan, C. Kou. Oscillation of Solutions of Impulsive Delay Differential Equations [J]. Journal of Mathematical Analysis and Applications, 2001, 254(2): 358-70.
[98] J. Yan, A. Zhao. Oscillation and Stability of Linear Impulsive Delay Differential Equations [J]. Journal of Mathematical Analysis and Applications, 1998, 227(1): 187-94.
[99] B. Zhang, Y. Liu. Global attractivity for certain impulsive delay differential equations [J]. Nonlinear Analysis, 2003, 52(3): 725-36.
[100] Z. Chen, J. Ruan. Global stability analysis of impulsive Cohen-Grossberg neural networks with delay [J]. Physics Letters A, 2005, 345(1-3): 101-11.
[101] K. Li, H. Zeng. Stability in impulsive Cohen-Grossberg-type BAM neural networks with time-varying delays: A general analysis [J]. Mathematics and Computers in Simulation, 2010, 80(12): 2329-49.
[102] Q. Song, Z. Wang. Stability analysis of impulsive stochastic Cohen-Grossberg neural networks with mixed time delays [J]. Physica A: Statistical Mechanics and its Applications, 2008, 387(13): 3314-26.
[103] Q. Song, J. Zhang. Global exponential stability of impulsive Cohen-Grossberg neural network with time-varying delays [J]. Nonlinear Analysis: Real World Applications, 2008, 9(2): 500-10.
[104] C. Li, X. Liao, R. Zhang. Impulsive synchronization of nonlinear coupled chaotic systems [J]. Physics Letters A, 2004, 328(1): 47-50.
[105] M.U. Akhmetov, A. Zafer. Stability of the Zero Solution of Impulsive Differential Equations by the Lyapunov Second Method [J]. Journal of Mathematical Analysis and Applications, 2000, 248(1): 69-82.
[106] Z.-H. Guan, J. Lam, G. Chen. On impulsive autoassociative neural networks [J]. NeuralNetworks, 2000, 13(1): 63-9.
[107] S. Mohamad, K. Gopalsamy. Exponential stability of continuous-time and discrete-time cellular neural networks with delays [J]. Applied Mathematics and Computation, 2003, 135(1): 17-38.
[108]张忠.时滞神经网络渐近稳定性和鲁棒稳定性研究[D].重庆:重庆大学, 2008.
[109] W. Fei. Existence and uniqueness of solution for fuzzy random differential equations with non-Lipschitz coefficients [J]. Information Sciences, 2007, 177(20): 4329-37.
[110] W.Y. Fei. A generalization of bihari's inequality and fuzzy random differential equations with non-Lipschitz coefficients [J]. International Journal of Uncertainty Fuzziness and Knowledge-Based Systems, 2007, 15(4): 425-39.
[111] D.V. Khlopin. Stability against small noises in control problems with non-Lipschitz right-hand side of the dynamic equation [J]. Automation and Remote Control, 2008, 69(3): 419-33.
[112] D. Luo. Regularity of solutions to differential equations with non-Lipschitz coefficients [J]. Bulletin Des Sciences Mathematiques, 2008, 132(4): 257-71.
[113] H.J. Qiao, X.C. Zhang. Homeomorphism flows for non-Lipschitz stochastic differential equations with jumps [J]. Stochastic Processes and Their Applications, 2008, 118(12): 2254-68.
[114] Z.D. Wang, X.C. Zhang. Non-Lipschitz backward stochastic volterra type equations with jumps [J]. Stochastics and Dynamics, 2007, 7(4): 479-96.
[115] X.C. Zhang. Exponential ergodicity of non-Lipschitz stochastic differential equations [J]. Proceedings of the American Mathematical Society, 2009, 137(1): 329-37.
[116] S. Albeverio, Z. Brzezniak, J.L. Wu. Existence of global solutions and invariant measures for stochastic differential equations driven by Poisson type noise with non-Lipschitz coefficients [J]. Journal of Mathematical Analysis and Applications, 2010, 371(1): 309-22.
[117] J.H. Bao, Z.T. Hou. Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz coefficients [J]. Comput Math Appl, 2010, 59(1): 207-14.
[118] S.J. Fan, L. Jiang. Finite and infinite time interval BSDEs with non-Lipschitz coefficients [J]. Statistics & Probability Letters, 2010, 80(11-12): 962-8.
[119] M. Ghisi, M. Gobbino. Derivative loss for Kirchhoff equations with non-Lipschitz nonlinear term [J]. Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze, 2009, 8(4): 613-46.
[120] M. Ghisi, M. Gobbino. A uniqueness result for Kirchhoff equations with non-Lipschitznonlinear term [J]. Advances in Mathematics, 2010, 223(4): 1299-315.
[121] J.G. Ren, J. Wu, X.C. Zhang. Exponential ergodicity of non-Lipschitz multivalued stochastic differential equations [J]. Bulletin Des Sciences Mathematiques, 2010, 134(4): 391-404.
[122] X.L. Song, J.G. Peng. Exponential stability of equilibria of differential equations with time-dependent delay and non-Lipschitz nonlinearity [J]. Nonlinear Anal-Real, 2010, 11(5): 3628-38.
[123] T. Taniguchi. The existence and uniqueness of energy solutions to local non-Lipschitz stochastic evolution equations [J]. Journal of Mathematical Analysis and Applications, 2009, 360(1): 245-53.
[124] F.Y. Wang, T.S. Zhang. Gradient estimates for stochastic evolution equations with non-Lipschitz coefficients [J]. Journal of Mathematical Analysis and Applications, 2010, 365(1): 1-11.
[125] S.Y. Xu. Multivalued Stochastic Differential Equations with Non-Lipschitz Coefficients [J]. Chinese Annals of Mathematics Series B, 2009, 30(3): 321-32.
[126] W. Ying, H. Zhen. Backward stochastic differential equations with non-Lipschitz coefficients [J]. Statistics & Probability Letters, 2009, 79(12): 1438-43.
[127] Q. Zhang, H.Z. Zhao. Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients [J]. Journal of Differential Equations, 2010, 248(5): 953-91.
[128] H.Q. Wu, J.Z. Sun, X.Z. Zhong. Analysis of dynamical behaviors for delayed neural networks with inverse Lipschitz neuron activations and impulses [J]. Int J Innov Comput I, 2008, 4(3): 705-15.
[129] C.J. Duan, Q.K. Song. Stability Analysis for Neural Networks with Time-Varying Delays and Inverse Lipschitz Neuron Activations [J]. Proceedings of the 7th Conference on Biological Dynamic System and Stability of Differential Equation, Vols I and Ii, 2010: 588-91,1082.
[130] H.Q. Wu, X.P. Xue. Stability analysis for neural networks with inverse Lipschitzian neuron activations and impulses [J]. Appl Math Model, 2008, 32(11): 2347-59.
[131] Y. Li, H. Wu. Global stability analysis in Cohen-Grossberg neural networks with delays and inverse H?lder neuron activation functions [J]. Information Sciences, 2010, 180(20): 4022-30.
[132] H.Q. Wu. Global exponential stability of Hopfield neural networks with delays and inverse Lipschitz neuron activations [J]. Nonlinear Analysis: Real World Applications, 2009, 10(4): 2297-306.
[133] Q.H. He, F.C. Yang. On the approximate solutions of constrained inverse Lipschitz function problems and inverse function theorems [J]. Nonlinear Anal-Theor, 2009, 70(2): 1124-31.
[134] X.B. Nie, J.D. Cao. Stability analysis for the generalized Cohen-Grossberg neural networkswith inverse Lipschitz neuron activations [J]. Comput Math Appl, 2009, 57(9): 1522-36.
[135] X.B. Nie, J.D. Cao. Dynamics of Competitive Neural Networks with Inverse Lipschitz Neuron Activations [J]. Lect Notes Comput Sc, 2010, 6063: 483-92,757.
[136] C.J. Duan, Q.K. Song. Stability Analysis for Neural Networks with Time-Varying Delays and Inverse Lipschitz Neuron Activations [M]. 2010.
[137] Y.W. Li, H.Q. Wu. Global stability analysis in Cohen-Grossberg neural networks with delays and inverse Holder neuron activation functions [J]. Information Sciences, 2010, 180(20): 4022-30.
[138] X.B. Nie, J.D. Cao. Dynamics of Competitive Neural Networks with Inverse Lipschitz Neuron Activations [M]//L.Q. Zhang, B.L. Lu, J. Kwok. Lect Notes Comput Sc. 2010: 483-92.
[139] Y. Meng, L. Huang, Z. Guo, et al. Stability analysis of Cohen-Grossberg neural networks with discontinuous neuron activations [J]. Appl Math Model, 2010, 34(2): 358-65.
[140] H.Q. Wu. Global stability analysis of a general class of discontinuous neural networks with linear growth activation functions [J]. Information Sciences, 2009, 179(19): 3432-41.
[141] Y.W. Li, H.Q. Wu. Global Stability Analysis for Periodic Solution in Discontinuous Neural Networks with Nonlinear Growth Activations [J]. Advances in Difference Equations, 2009.
[142] H.Q. Wu, C.H. Shan. Stability analysis for periodic solution of BAM neural networks with discontinuous neuron activations and impulses [J]. Appl Math Model, 2009, 33(6): 2564-74.
[143]夏道行等.泛函分析第二教程[M].北京:高等教育出版社, 2008.11.
[144] C. Wing-Sum, M. Qing-Hua, J. Pecaric. Some Discrete Nonlinear Inequalities and Applications to Difference Equations [J]. Acta Mathematica Scientia, 2008, 28(2): 417-30.
[145] W.-S. Wang, X. Zhou. A generalized Gronwall-Bellman-Ou-Iang type inequality for discontinuous functions and applications to BVP [J]. Applied Mathematics and Computation, 2010, 216(11): 3335-42.
[146] E. Magnucka-Blandzi, J. Popenda, R.P. Agarwal. Best possible gronwall inequalities [J]. Mathematical and Computer Modelling, 1997, 26(3): 1-8.
[147] J. Popenda, R.P. Agarwal. Discrete Gronwall inequalities in many variables [J]. Comput Math Appl, 1999, 38(1): 63-70.
[148] X. Mao. Exponential stability of nonlinear differential delay equations [J]. Systems & Control Letters, 1996, 28(3): 159-65.
[149] E.N. Sanchez, J.P. Perez, Ieee. Input-to-state stability (ISS) analysis for dynamic neural networks [M]. 1997.
[150] K.Q. Gu, I.C.S.S.I. Ieee Control Systems Society, S. Control Systems. An integral inequalityin the stability problem of time-delay systems [M]. Proceedings of the 39th Ieee Conference on Decision and Control, Vols 1-5. 2000: 2805-10.
[151] S. Boyd, L.E. Ghaoul, E. Feron, et al. Linear Matrix Inequalities in Systems and Control Theory [M]. Philadephia: SIAM, 1994.
[152]廖晓昕.稳定性的理论、方法和应用[M].武汉:华中科技大学出版社, 1999.4.
[153] K. Gu, V.L. Kharitonov, J. Chen. Stability of Time-Delay Systems (Control Engineering) [M]. Boston: Birkh?user, 2003.
[154] K. Gopalsamy. Stability of artificial neural networks with impulses [J]. Applied Mathematics and Computation, 2004, 154(3): 783-813.
[155] Z.-H. Guan, G. Chen. On delayed impulsive Hopfield neural networks [J]. Neural Networks, 1999, 12(2): 273-80.
[156] H. Ak?a, R. Alassar, V. Covachev, et al. Continuous-time additive Hopfield-type neural networks with impulses [J]. Journal of Mathematical Analysis and Applications, 2004, 290(2): 436-51.
[157] B. Xu, X. Liu, K.L. Teo. Asymptotic stability of impulsive high-order Hopfield type neural networks [J]. Comput Math Appl, 2009, 57(11-12): 1968-77.
[158] M.U. Akhmet, E. Yilmaz. Impulsive Hopfield-type neural network system with piecewise constant argument [J]. Nonlinear Analysis: Real World Applications, 2010, 11(4): 2584-93.
[159] H. Zhang, L. Chen. Asymptotic behavior of discrete solutions to delayed neural networks with impulses [J]. Neurocomputing, 2008, 71(4-6): 1032-8.
[160] X. Fu, X. Li. Global exponential stability and global attractivity of impulsive Hopfield neural networks with time delays [J]. Journal of Computational and Applied Mathematics, 2009, 231(1): 187-99.
[161] Y. Liu, Z. Wang, X. Liu. Global exponential stability of generalized recurrent neural networks with discrete and distributed delays [J]. Neural Networks, 2006, 19(5): 667-75.
[162] B. Xu, X. Liu, K.L. Teo. Global exponential stability of impulsive high-order Hopfield type neural networks with delays [J]. Comput Math Appl, 2009, 57(11-12): 1959-67.
[163] H.-F. Huo, W.-T. Li, S. Tang. Dynamics of high-order BAM neural networks with and without impulses [J]. Applied Mathematics and Computation, 2009, 215(6): 2120-33.
[164] Y. Li, Z. Xing. Existence and global exponential stability of periodic solution of CNNs with impulses [J]. Chaos, Solitons & Fractals, 2007, 33(5): 1686-93.
[165] M.A. Cohen, S. Grossberg. Absolute stability and global pattern formation and parallel memory storage by competitive neural networks [J]. Advances in Psychology, 1987, 42: 288-308.
[166] Y. Li, X. Fan. Existence and globally exponential stability of almost periodic solution for Cohen-Grossberg BAM neural networks with variable coefficients [J]. Appl Math Model, 2009, 33(4): 2114-20.
[167] W. Yu, J. Cao, J. Wang. An LMI approach to global asymptotic stability of the delayed Cohen-Grossberg neural network via nonsmooth analysis [J]. Neural Networks, 2007, 20(7): 810-8.
[168] K. Deimling. Nonlinear Functional Analysis [M]. New York, Heidelberg, Berlin: Springer-Verlag, 1985.
[169] T. Huang, C. Li, G. Chen. Stability of Cohen-Grossberg neural networks with unbounded distributed delays [J]. Chaos, Solitons & Fractals, 2007, 34(3): 992-6.
[170] Y. Li, L. Yang. Anti-periodic solutions for Cohen-Grossberg neural networks with bounded and unbounded delays [J]. Communications in Nonlinear Science and Numerical Simulation, 2009, 14(7): 3134-40.
[171] Y. Li. Existence and stability of periodic solutions for Cohen-Grossberg neural networks with multiple delays [J]. Chaos, Solitons & Fractals, 2004, 20(3): 459-66.
[172] Q. Song, J. Cao. Global exponential robust stability of Cohen-Grossberg neural network with time-varying delays and reaction-diffusion terms [J]. Journal of the Franklin Institute, 2006, 343(7): 705-19.
[173] Q. Song, J. Cao. Stability analysis of Cohen-Grossberg neural network with both time-varying and continuously distributed delays [J]. Journal of Computational and Applied Mathematics, 2006, 197(1): 188-203.
[174] J. Cao, J. Liang. Boundedness and stability for Cohen-Grossberg neural network with time-varying delays [J]. Journal of Mathematical Analysis and Applications, 2004, 296(2): 665-85.
[175] J. Cao, X. Li. Stability in delayed Cohen-Grossberg neural networks: LMI optimization approach [J]. Physica D: Nonlinear Phenomena, 2005, 212(1-2): 54-65.
[176] J. Cao, G. Feng, Y. Wang. Multistability and multiperiodicity of delayed Cohen-Grossberg neural networks with a general class of activation functions [J]. Physica D: Nonlinear Phenomena, 2008, 237(13): 1734-49.
[177] S. Arik. An analysis of exponential stability of delayed neural networks with time varying delays [J]. Neural Networks, 2004, 17(7): 1027-31.
[178] S. Arik, Z. Orman. Global stability analysis of Cohen-Grossberg neural networks with time varying delays [J]. Physics Letters A, 2005, 341(5-6): 410-21.
[179] Y. Li, X. Fan, L. Zhao. Positive periodic solutions of functional differential equations withimpulses and a parameter [J]. Comput Math Appl, 2008, 56(10): 2556-60.
[180] C. Li, X. Liao. Complete and lag synchronization of hyperchaotic systems using small impulses [J]. Chaos, Solitons & Fractals, 2004, 22(4): 857-67.
[181] Y. Li, X. Chen, L. Zhao. Stability and existence of periodic solutions to delayed Cohen-Grossberg BAM neural networks with impulses on time scales [J]. Neurocomputing, 2009, 72(7-9): 1621-30.
[182] M. Liu. Global asymptotic stability analysis of discrete-time Cohen-Grossberg neural networks based on interval systems [J]. Nonlinear Analysis: Theory, Methods & Applications, 2008, 69(8): 2403-11.
[183] H. Zhao, L. Wang. Stability and bifurcation for discrete-time Cohen-Grossberg neural network [J]. Applied Mathematics and Computation, 2006, 179(2): 787-98.
[184] W. Xiong, J. Cao. Global exponential stability of discrete-time Cohen-Grossberg neural networks [J]. Neurocomputing, 2005, 64: 433-46.
[185] T. Zhou, Y. Liu, X. Li, et al. A new criterion to global exponential periodicity for discrete-time BAM neural network with infinite delays [J]. Chaos, Solitons & Fractals, 2009, 39(1): 332-41.
[186] T. Zhou, Y. Liu, Y. Liu. Existence and global exponential stability of periodic solution for discrete-time BAM neural networks [J]. Applied Mathematics and Computation, 2006, 182(2): 1341-54.
[187] X.-G. Liu, M.-L. Tang, R. Martin, et al. Discrete-time BAM neural networks with variable delays [J]. Physics Letters A, 2007, 367(4-5): 322-30.
[188] H. Zhao, L. Wang, C. Ma. Hopf bifurcation and stability analysis on discrete-time Hopfield neural network with delay [J]. Nonlinear Analysis: Real World Applications, 2008, 9(1): 103-13.
[189] S. Guo, L. Huang, L. Wang. Exponential stability of discrete-time Hopfield neural networks [J]. Comput Math Appl, 2004, 47(8-9): 1249-56.
[190] J. Liang, J. Cao, D.W.C. Ho. Discrete-time bidirectional associative memory neural networks with variable delays [J]. Physics Letters A, 2005, 335(2-3): 226-34.
[191] E. Liz, J.B. Ferreiro. A note on the global stability of generalized difference equations [J]. Appl Math Lett, 2002, 15(6): 655-9.
[192] A. Halanay. Differential Equations: Stability, Oscillations, Time Lags [M]. New York: Academic Press, 1966.
[2] J. Cao. On stability of delayed cellular neural networks [J]. Physics Letters A, 1999, 261(5-6): 303-8.
[3] J. Cao. On exponential stability and periodic solutions of CNNs with delays [J]. Physics Letters A, 2000, 267(5-6): 312-8.
[4] J. Cao. Periodic oscillation and exponential stability of delayed CNNs [J]. Physics Letters A, 2000, 270(3-4): 157-63.
[5] J. Cao, Q. Li. On the Exponential Stability and Periodic Solutions of Delayed Cellular Neural Networks [J]. Journal of Mathematical Analysis and Applications, 2000, 252(1): 50-64.
[6] X. Liao, K.-W. Wong, C.-S. Leung, et al. Hopf bifurcation and chaos in a single delayed neuron equation with non-monotonic activation function [J]. Chaos, Solitons & Fractals, 2001, 12(8): 1535-47.
[7] Q. Tao, J. Cao, M. Xue, et al. A high performance neural network for solving nonlinear programming problems with hybrid constraints [J]. Physics Letters A, 2001, 288(2): 88-94.
[8] J. Cao, Q. Li, S. Wan. Periodic solutions of the higher-dimensional non-autonomous systems [J]. Applied Mathematics and Computation, 2002, 130(2-3): 369-82.
[9] X. Liao, G. Chen, E.N. Sanchez. Delay-dependent exponential stability analysis of delayed neural networks: an LMI approach [J]. Neural Networks, 2002, 15(7): 855-66.
[10] X. Liao, Z. Wu, J. Yu. Stability analyses of cellular neural networks with continuous time delay [J]. Journal of Computational and Applied Mathematics, 2002, 143(1): 29-47.
[11] J. Cao. New results concerning exponential stability and periodic solutions of delayed cellular neural networks [J]. Physics Letters A, 2003, 307(2-3): 136-47.
[12] J. Cao, M. Dong. Exponential stability of delayed bi-directional associative memory networks [J]. Applied Mathematics and Computation, 2003, 135(1): 105-12.
[13] A. Chen, J. Cao. Existence and attractivity of almost periodic solutions for cellular neural networks with distributed delays and variable coefficients [J]. Applied Mathematics and Computation, 2003, 134(1): 125-40.
[14] X. Liao, G. Chen, E.N. Sanchez. Delay-dependent exponential stability analysis of delayed neural networks: an LMI approach [J]. Neural Networks, 2003, 16(10): 1401-2.
[15] X. Liao, K.-W. Wong, S. Yang. Convergence dynamics of hybrid bidirectional associative memory neural networks with distributed delays [J]. Physics Letters A, 2003, 316(1-2): 55-64.
[16] J. Cao. An estimation of the domain of attraction and convergence rate for Hopfield continuous feedback neural networks [J]. Physics Letters A, 2004, 325(5-6): 370-4.
[17] J. Cao, Q. Jiang. An analysis of periodic solutions of bi-directional associative memory networks with time-varying delays [J]. Physics Letters A, 2004, 330(3-4): 203-13.
[18] J. Cao, J. Liang, J. Lam. Exponential stability of high-order bidirectional associative memory neural networks with time delays [J]. Physica D: Nonlinear Phenomena, 2004, 199(3-4): 425-36.
[19] C. Li, X. Liao, K.-w. Wong. Bifurcation of the exponentially self-regulating population model [J]. Chaos, Solitons & Fractals, 2004, 20(3): 479-87.
[20] X. Liao, C. Li, K.-w. Wong. Criteria for exponential stability of Cohen-Grossberg neural networks [J]. Neural Networks, 2004, 17(10): 1401-14.
[21] X. Liao, K.-w. Wong. Global exponential stability for a class of retarded functional differential equations with applications in neural networks [J]. Journal of Mathematical Analysis and Applications, 2004, 293(1): 125-48.
[22] X. Liao, K.-w. Wong, C. Li. Global exponential stability for a class of generalized neural networks with distributed delays [J]. Nonlinear Analysis: Real World Applications, 2004, 5(3): 527-47.
[23] J. Cao, A. Chen, X. Huang. Almost periodic attractor of delayed neural networks with variable coefficients [J]. Physics Letters A, 2005, 340(1-4): 104-20.
[24] C. Li, X. Liao. New algebraic conditions for global exponential stability of delayed recurrent neural networks [J]. Neurocomputing, 2005, 64: 319-33.
[25] C. Li, X. Liao, R. Zhang, et al. Global robust exponential stability analysis for interval neural networks with time-varying delays [J]. Chaos, Solitons & Fractals, 2005, 25(3): 751-7.
[26] X. Liao, G. Chen. Hopf bifurcation and chaos analysis of Chen's system with distributed delays [J]. Chaos, Solitons & Fractals, 2005, 25(1): 197-220.
[27] X. Liao, C. Li. An LMI approach to asymptotical stability of multi-delayed neural networks [J]. Physica D: Nonlinear Phenomena, 2005, 200(1-2): 139-55.
[28] Y. Xia, M. Lin, J. Cao. The existence of almost periodic solutions of certain perturbation systems [J]. Journal of Mathematical Analysis and Applications, 2005, 310(1): 81-96.
[29] W. Yu, J. Cao. Hopf bifurcation and stability of periodic solutions for van der Pol equation with time delay [J]. Nonlinear Analysis, 2005, 62(1): 141-65.
[30] H. Zhao, J. Cao. New conditions for global exponential stability of cellular neural networks with delays [J]. Neural Networks, 2005, 18(10): 1332-40.
[31] J. Cao, H.X. Li, L. Han. Novel results concerning global robust stability of delayed neuralnetworks [J]. Nonlinear Analysis: Real World Applications, 2006, 7(3): 458-69.
[32] X. Liao, C. Li. Global attractivity of Cohen-Grossberg model with finite and infinite delays [J]. Journal of Mathematical Analysis and Applications, 2006, 315(1): 244-62.
[33] X. Liao, Q. Liu, W. Zhang. Delay-dependent asymptotic stability for neural networks with distributed delays [J]. Nonlinear Analysis: Real World Applications, 2006, 7(5): 1178-92.
[34] J. Cao, D.W.C. Ho, X. Huang. LMI-based criteria for global robust stability of bidirectional associative memory networks with time delay [J]. Nonlinear Analysis, 2007, 66(7): 1558-72.
[35] C. Li, Q. Chen, T. Huang. Coexistence of anti-phase and complete synchronization in coupled chen system via a single variable [J]. Chaos, Solitons & Fractals, 2008, 38(2): 461-4.
[36] C. Li, Q. Yan. Comments on "Exponential stability of nonlinear differential delay equations" [Systems & Control Letters 28 (1996) 159-165] [J]. Systems & Control Letters, 2008, 57(10): 834-5.
[37] X. Liao, J. Yang, S. Guo. Exponential stability of Cohen-Grossberg neural networks with delays [J]. Communications in Nonlinear Science and Numerical Simulation, 2008, 13(9): 1767-75.
[38] W. Yu, J. Cao, K. Yuan. Synchronization of switched system and application in communication [J]. Physics Letters A, 2008, 372(24): 4438-45.
[39] J. Cao, L. Li. Cluster synchronization in an array of hybrid coupled neural networks with delay [J]. Neural Networks, 2009, 22(4): 335-42.
[40] X. Liao, Y. Liu, S. Guo, et al. Asymptotic stability of delayed neural networks: A descriptor system approach [J]. Communications in Nonlinear Science and Numerical Simulation, 2009, 14(7): 3120-33.
[41] Q. Zhu, J. Cao. Adaptive synchronization under almost every initial data for stochastic neural networks with time-varying delays and distributed delays [J]. Communications in Nonlinear Science and Numerical Simulation, 2011, 16(4): 2139-59.
[42] H. Kong, L. Guan. A neural network adaptive filter for the removal of impulse noise in digital images [J]. Neural Networks, 1996, 9(3): 373-8.
[43] A.L. Orille, N. Khalil, J.A.V. Valencia. A transformer differential protection based on finite impulse response artificial neural network [J]. Computers & Industrial Engineering, 1999, 37(1-2): 399-402.
[44] Y. Li, L. Lu. Global exponential stability and existence of periodic solution of Hopfield-type neural networks with impulses [J]. Physics Letters A, 2004, 333(1-2): 62-71.
[45] Y. Li, W. Xing, L. Lu. Existence and global exponential stability of periodic solution of a class of neural networks with impulses [J]. Chaos, Solitons & Fractals, 2006, 27(2): 437-45.
[46] B. Liu, L. Huang. Global exponential stability of BAM neural networks with recent-history distributed delays and impulses [J]. Neurocomputing, 2006, 69(16-18): 2090-6.
[47] Y. Wang, W. Xiong, Q. Zhou, et al. Global exponential stability of cellular neural networks with continuously distributed delays and impulses [J]. Physics Letters A, 2006, 350(1-2): 89-95.
[48] Z. Chen, J. Ruan. Global dynamic analysis of general Cohen-Grossberg neural networks with impulse [J]. Chaos, Solitons & Fractals, 2007, 32(5): 1830-7.
[49] Z.-T. Huang, X.-S. Luo, Q.-G. Yang. Global asymptotic stability analysis of bidirectional associative memory neural networks with distributed delays and impulse [J]. Chaos, Solitons & Fractals, 2007, 34(3): 878-85.
[50] Y. Xia, J. Cao, S. Sun Cheng. Global exponential stability of delayed cellular neural networks with impulses [J]. Neurocomputing, 2007, 70(13-15): 2495-501.
[51] H. Gu, H. Jiang, Z. Teng. Existence and globally exponential stability of periodic solution of BAM neural networks with impulses and recent-history distributed delays [J]. Neurocomputing, 2008, 71(4-6): 813-22.
[52] K. Li, Q. Song. Exponential stability of impulsive Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms [J]. Neurocomputing, 2008, 72(1-3): 231-40.
[53] H.-F. Huo, W.-T. Li. Dynamics of continuous-time bidirectional associative memory neural networks with impulses and their discrete counterparts [J]. Chaos, Solitons & Fractals, 2009, 42(4): 2218-29.
[54] K. Li. Stability analysis for impulsive Cohen-Grossberg neural networks with time-varying delays and distributed delays [J]. Nonlinear Analysis: Real World Applications, 2009, 10(5): 2784-98.
[55] J. Sun, Q.-G. Wang, H. Gao. Periodic solution for nonautonomous cellular neural networks with impulses [J]. Chaos, Solitons & Fractals, 2009, 40(3): 1423-7.
[56] Z. Wen, J. Sun. Stability analysis of delayed Cohen-Grossberg BAM neural networks with impulses via nonsmooth analysis [J]. Chaos, Solitons & Fractals, 2009, 42(3): 1829-37.
[57] X. Yang, X. Cui, Y. Long. Existence and global exponential stability of periodic solution of a cellular neural networks difference equation with delays and impulses [J]. Neural Networks, 2009, 22(7): 970-6.
[58] J. Zhang, Z. Gui. Existence and stability of periodic solutions of high-order Hopfield neural networks with impulses and delays [J]. Journal of Computational and Applied Mathematics, 2009, 224(2): 602-13.
[59] X. Zhao. Global exponential stability of discrete-time recurrent neural networks withimpulses [J]. Nonlinear Analysis: Theory, Methods & Applications, 2009, 71(12): e2873-e8.
[60] Q. Zhou. Global exponential stability of BAM neural networks with distributed delays and impulses [J]. Nonlinear Analysis: Real World Applications, 2009, 10(1): 144-53.
[61] X. Yang, F. Li, Y. Long, et al. Existence of periodic solution for discrete-time cellular neural networks with complex deviating arguments and impulses [J]. Journal of the Franklin Institute, 2010, 347(2): 559-66.
[62] S. Zhong, C. Li, X. Liao. Global stability of discrete-time Cohen-Grossberg neural networks with impulses [J]. Neurocomputing, 2010, 73(16-18): 3132-8.
[63] X. Mao, C. Yuan, G. Yin. Approximations of Euler-Maruyama type for stochastic differential equations with Markovian switching, under non-Lipschitz conditions [J]. Journal of Computational and Applied Mathematics, 2007, 205(2): 936-48.
[64] H. Wu, X. Xue. Stability analysis for neural networks with inverse Lipschitzian neuron activations and impulses [J]. Appl Math Model, 2008, 32(11): 2347-59.
[65] Q. He, F. Yang. On the approximate solutions of constrained inverse Lipschitz function problems and inverse function theorems [J]. Nonlinear Analysis: Theory, Methods & Applications, 2009, 70(2): 1124-31.
[66] X. Nie, J. Cao. Stability analysis for the generalized Cohen-Grossberg neural networks with inverse Lipschitz neuron activations [J]. Comput Math Appl, 2009, 57(9): 1522-36.
[67] Y. Tan, M. Tan. Global asymptotical stability of continuous-time delayed neural networks without global Lipschitz activation functions [J]. Communications in Nonlinear Science and Numerical Simulation, 2009, 14(11): 3715-22.
[68] H. Wu. Global exponential stability of Hopfield neural networks with delays and inverse Lipschitz neuron activations [J]. Nonlinear Analysis: Real World Applications, 2009, 10(4): 2297-306.
[69] T.D. Chuong, N.Q. Huy, J.C. Yao. Pseudo-Lipschitz property of linear semi-infinite vector optimization problems [J]. European Journal of Operational Research, 2010, 200(3): 639-44.
[70]蒋宗礼.人工神经网络导论[M].北京:高等教育出版社, 2001.8.
[71] W.S. McCulloch, W. Pitts. A logical calculus of the ideas immanent in nervous activity [J]. Bulletin of Mathematical Biology, 1990, 52(1-2): 99-115.
[72] J. Hertz, A. Krogh, R.G. Palmer. Introduction to the theory of neural computation [M]. Redwood: Addison-Wesley, 1991.
[73] T. Kohonen. Correlation Matrix Memories [J]. IEEE Transactions on Computers, 1972, 21(4): 353-9.
[74] J.A. Anderson. A simple neural network generating an interactive memory [M].Neurocomputing: foundations of research. MIT Press. 1988: 181-92.
[75] C.v.d. Malsburg. Self-organization of orientation sensitive cells in the striata cortex [M]. MIT Press, 1988.
[76] P.J. Werbos. Beyond Regression: New Tools for Prediction and Analysis in the Behavioral Sciences [D]. Cambridge: Harvard University, 1974.
[77] W.A. Little, G.L. Shaw. A statistical theory of short and long term memory [M]. MIT Press, 1988.
[78] S. Grossberg. Adaptive pattern classification and universal recoding. I.: parallel development and coding of neural feature detectors [M]. Neurocomputing: foundations of research. MIT Press. 1988: 243-58.
[79] S.I. Amari. Neural theory of association and concept-formation [J]. Biological Cybernetics, 1977, 26(3): 175-85.
[80] J.J. Hopfield. Neural networks and physical systems with emergent collective computational abilities [M]. Neurocomputing: foundations of research. MIT Press. 1988: 457-64.
[81]哈姆,科斯塔尼克.神经计算原理[M].北京:机械工业出版社, 2007.5.
[82] T. Kohonen. Self-organized formation of topologically correct feature maps [J]. Biological Cybernetics, 1982, 43(1): 59-69.
[83] E. Oja. Simplified neuron model as a principal component analyzer [J]. Journal of Mathematical Biology, 1982, 15(3): 267-73.
[84] D.H. Ackley, G.E. Hinton, T.J. Sejnowski. A learning algorithm for Boltzmann machines [M]. Connectionist models and their implications: readings from cognitive science. Ablex Publishing Corp. 1988: 285-307.
[85] D. Parker. Learning-logic. Technical Report TR-47,. Center for Computational Research in Economics and. Management Science [M]. MIT, 1985.
[86] D.E. Rumelhart, G.E. Hinton, R.J. Williams. Learning internal representations by error propagation [M]. Parallel distributed processing: explorations in the microstructure of cognition, vol 1. MIT Press. 1986: 318-62.
[87] D.E. Rumelhart, G.E. Hinton, R.J. Williams. Learning representations by back-propagating errors [M]. Neurocomputing: foundations of research. MIT Press. 1988: 696-9.
[88] G.A. Carpenter, S. Grossberg. A massively parallel architecture for a self-organizing neural pattern recognition machine [J]. Comput Vision Graph Image Process, 1987, 37(1): 54-115.
[89] M.A. Sivilotti, M.A. Mahowald, C.A. Mead. Real-time visual computations using analog CMOS processing arrays; proceedings of the in 1987 Stanford VLSIConference, F, 1987 [C].
[90] T. Yang. Impulsive Control Theory [M]. Springer, 2001.
[91]杨德刚.时滞神经网络稳定性及复杂网络同步的研究[D].重庆:重庆大学, 2007.
[92] S. Grossberg. Nonlinear neural networks: Principles, mechanisms, and architectures [J]. Neural Networks, 1988, 1(1): 17-61.
[93] K. Gopalsamy, B.G. Zhang. On delay differential equations with impulses [J]. Journal of Mathematical Analysis and Applications, 1989, 139(1): 110-22.
[94] A.M.Samoilenko, N.A.Peretyuk. Impulsive Differential Equations [M]. Singapore: World Scientific, 1995.
[95] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov. Theory of Impulsive Differential Equations [M]. Singapore,New York: World Scientific, 1989.
[96] J. Luo. Oscillation for Nonlinear Delay Differential Equations with Impulses [J]. Journal of Mathematical Analysis and Applications, 2000, 250(1): 290-8.
[97] J. Yan, C. Kou. Oscillation of Solutions of Impulsive Delay Differential Equations [J]. Journal of Mathematical Analysis and Applications, 2001, 254(2): 358-70.
[98] J. Yan, A. Zhao. Oscillation and Stability of Linear Impulsive Delay Differential Equations [J]. Journal of Mathematical Analysis and Applications, 1998, 227(1): 187-94.
[99] B. Zhang, Y. Liu. Global attractivity for certain impulsive delay differential equations [J]. Nonlinear Analysis, 2003, 52(3): 725-36.
[100] Z. Chen, J. Ruan. Global stability analysis of impulsive Cohen-Grossberg neural networks with delay [J]. Physics Letters A, 2005, 345(1-3): 101-11.
[101] K. Li, H. Zeng. Stability in impulsive Cohen-Grossberg-type BAM neural networks with time-varying delays: A general analysis [J]. Mathematics and Computers in Simulation, 2010, 80(12): 2329-49.
[102] Q. Song, Z. Wang. Stability analysis of impulsive stochastic Cohen-Grossberg neural networks with mixed time delays [J]. Physica A: Statistical Mechanics and its Applications, 2008, 387(13): 3314-26.
[103] Q. Song, J. Zhang. Global exponential stability of impulsive Cohen-Grossberg neural network with time-varying delays [J]. Nonlinear Analysis: Real World Applications, 2008, 9(2): 500-10.
[104] C. Li, X. Liao, R. Zhang. Impulsive synchronization of nonlinear coupled chaotic systems [J]. Physics Letters A, 2004, 328(1): 47-50.
[105] M.U. Akhmetov, A. Zafer. Stability of the Zero Solution of Impulsive Differential Equations by the Lyapunov Second Method [J]. Journal of Mathematical Analysis and Applications, 2000, 248(1): 69-82.
[106] Z.-H. Guan, J. Lam, G. Chen. On impulsive autoassociative neural networks [J]. NeuralNetworks, 2000, 13(1): 63-9.
[107] S. Mohamad, K. Gopalsamy. Exponential stability of continuous-time and discrete-time cellular neural networks with delays [J]. Applied Mathematics and Computation, 2003, 135(1): 17-38.
[108]张忠.时滞神经网络渐近稳定性和鲁棒稳定性研究[D].重庆:重庆大学, 2008.
[109] W. Fei. Existence and uniqueness of solution for fuzzy random differential equations with non-Lipschitz coefficients [J]. Information Sciences, 2007, 177(20): 4329-37.
[110] W.Y. Fei. A generalization of bihari's inequality and fuzzy random differential equations with non-Lipschitz coefficients [J]. International Journal of Uncertainty Fuzziness and Knowledge-Based Systems, 2007, 15(4): 425-39.
[111] D.V. Khlopin. Stability against small noises in control problems with non-Lipschitz right-hand side of the dynamic equation [J]. Automation and Remote Control, 2008, 69(3): 419-33.
[112] D. Luo. Regularity of solutions to differential equations with non-Lipschitz coefficients [J]. Bulletin Des Sciences Mathematiques, 2008, 132(4): 257-71.
[113] H.J. Qiao, X.C. Zhang. Homeomorphism flows for non-Lipschitz stochastic differential equations with jumps [J]. Stochastic Processes and Their Applications, 2008, 118(12): 2254-68.
[114] Z.D. Wang, X.C. Zhang. Non-Lipschitz backward stochastic volterra type equations with jumps [J]. Stochastics and Dynamics, 2007, 7(4): 479-96.
[115] X.C. Zhang. Exponential ergodicity of non-Lipschitz stochastic differential equations [J]. Proceedings of the American Mathematical Society, 2009, 137(1): 329-37.
[116] S. Albeverio, Z. Brzezniak, J.L. Wu. Existence of global solutions and invariant measures for stochastic differential equations driven by Poisson type noise with non-Lipschitz coefficients [J]. Journal of Mathematical Analysis and Applications, 2010, 371(1): 309-22.
[117] J.H. Bao, Z.T. Hou. Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz coefficients [J]. Comput Math Appl, 2010, 59(1): 207-14.
[118] S.J. Fan, L. Jiang. Finite and infinite time interval BSDEs with non-Lipschitz coefficients [J]. Statistics & Probability Letters, 2010, 80(11-12): 962-8.
[119] M. Ghisi, M. Gobbino. Derivative loss for Kirchhoff equations with non-Lipschitz nonlinear term [J]. Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze, 2009, 8(4): 613-46.
[120] M. Ghisi, M. Gobbino. A uniqueness result for Kirchhoff equations with non-Lipschitznonlinear term [J]. Advances in Mathematics, 2010, 223(4): 1299-315.
[121] J.G. Ren, J. Wu, X.C. Zhang. Exponential ergodicity of non-Lipschitz multivalued stochastic differential equations [J]. Bulletin Des Sciences Mathematiques, 2010, 134(4): 391-404.
[122] X.L. Song, J.G. Peng. Exponential stability of equilibria of differential equations with time-dependent delay and non-Lipschitz nonlinearity [J]. Nonlinear Anal-Real, 2010, 11(5): 3628-38.
[123] T. Taniguchi. The existence and uniqueness of energy solutions to local non-Lipschitz stochastic evolution equations [J]. Journal of Mathematical Analysis and Applications, 2009, 360(1): 245-53.
[124] F.Y. Wang, T.S. Zhang. Gradient estimates for stochastic evolution equations with non-Lipschitz coefficients [J]. Journal of Mathematical Analysis and Applications, 2010, 365(1): 1-11.
[125] S.Y. Xu. Multivalued Stochastic Differential Equations with Non-Lipschitz Coefficients [J]. Chinese Annals of Mathematics Series B, 2009, 30(3): 321-32.
[126] W. Ying, H. Zhen. Backward stochastic differential equations with non-Lipschitz coefficients [J]. Statistics & Probability Letters, 2009, 79(12): 1438-43.
[127] Q. Zhang, H.Z. Zhao. Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients [J]. Journal of Differential Equations, 2010, 248(5): 953-91.
[128] H.Q. Wu, J.Z. Sun, X.Z. Zhong. Analysis of dynamical behaviors for delayed neural networks with inverse Lipschitz neuron activations and impulses [J]. Int J Innov Comput I, 2008, 4(3): 705-15.
[129] C.J. Duan, Q.K. Song. Stability Analysis for Neural Networks with Time-Varying Delays and Inverse Lipschitz Neuron Activations [J]. Proceedings of the 7th Conference on Biological Dynamic System and Stability of Differential Equation, Vols I and Ii, 2010: 588-91,1082.
[130] H.Q. Wu, X.P. Xue. Stability analysis for neural networks with inverse Lipschitzian neuron activations and impulses [J]. Appl Math Model, 2008, 32(11): 2347-59.
[131] Y. Li, H. Wu. Global stability analysis in Cohen-Grossberg neural networks with delays and inverse H?lder neuron activation functions [J]. Information Sciences, 2010, 180(20): 4022-30.
[132] H.Q. Wu. Global exponential stability of Hopfield neural networks with delays and inverse Lipschitz neuron activations [J]. Nonlinear Analysis: Real World Applications, 2009, 10(4): 2297-306.
[133] Q.H. He, F.C. Yang. On the approximate solutions of constrained inverse Lipschitz function problems and inverse function theorems [J]. Nonlinear Anal-Theor, 2009, 70(2): 1124-31.
[134] X.B. Nie, J.D. Cao. Stability analysis for the generalized Cohen-Grossberg neural networkswith inverse Lipschitz neuron activations [J]. Comput Math Appl, 2009, 57(9): 1522-36.
[135] X.B. Nie, J.D. Cao. Dynamics of Competitive Neural Networks with Inverse Lipschitz Neuron Activations [J]. Lect Notes Comput Sc, 2010, 6063: 483-92,757.
[136] C.J. Duan, Q.K. Song. Stability Analysis for Neural Networks with Time-Varying Delays and Inverse Lipschitz Neuron Activations [M]. 2010.
[137] Y.W. Li, H.Q. Wu. Global stability analysis in Cohen-Grossberg neural networks with delays and inverse Holder neuron activation functions [J]. Information Sciences, 2010, 180(20): 4022-30.
[138] X.B. Nie, J.D. Cao. Dynamics of Competitive Neural Networks with Inverse Lipschitz Neuron Activations [M]//L.Q. Zhang, B.L. Lu, J. Kwok. Lect Notes Comput Sc. 2010: 483-92.
[139] Y. Meng, L. Huang, Z. Guo, et al. Stability analysis of Cohen-Grossberg neural networks with discontinuous neuron activations [J]. Appl Math Model, 2010, 34(2): 358-65.
[140] H.Q. Wu. Global stability analysis of a general class of discontinuous neural networks with linear growth activation functions [J]. Information Sciences, 2009, 179(19): 3432-41.
[141] Y.W. Li, H.Q. Wu. Global Stability Analysis for Periodic Solution in Discontinuous Neural Networks with Nonlinear Growth Activations [J]. Advances in Difference Equations, 2009.
[142] H.Q. Wu, C.H. Shan. Stability analysis for periodic solution of BAM neural networks with discontinuous neuron activations and impulses [J]. Appl Math Model, 2009, 33(6): 2564-74.
[143]夏道行等.泛函分析第二教程[M].北京:高等教育出版社, 2008.11.
[144] C. Wing-Sum, M. Qing-Hua, J. Pecaric. Some Discrete Nonlinear Inequalities and Applications to Difference Equations [J]. Acta Mathematica Scientia, 2008, 28(2): 417-30.
[145] W.-S. Wang, X. Zhou. A generalized Gronwall-Bellman-Ou-Iang type inequality for discontinuous functions and applications to BVP [J]. Applied Mathematics and Computation, 2010, 216(11): 3335-42.
[146] E. Magnucka-Blandzi, J. Popenda, R.P. Agarwal. Best possible gronwall inequalities [J]. Mathematical and Computer Modelling, 1997, 26(3): 1-8.
[147] J. Popenda, R.P. Agarwal. Discrete Gronwall inequalities in many variables [J]. Comput Math Appl, 1999, 38(1): 63-70.
[148] X. Mao. Exponential stability of nonlinear differential delay equations [J]. Systems & Control Letters, 1996, 28(3): 159-65.
[149] E.N. Sanchez, J.P. Perez, Ieee. Input-to-state stability (ISS) analysis for dynamic neural networks [M]. 1997.
[150] K.Q. Gu, I.C.S.S.I. Ieee Control Systems Society, S. Control Systems. An integral inequalityin the stability problem of time-delay systems [M]. Proceedings of the 39th Ieee Conference on Decision and Control, Vols 1-5. 2000: 2805-10.
[151] S. Boyd, L.E. Ghaoul, E. Feron, et al. Linear Matrix Inequalities in Systems and Control Theory [M]. Philadephia: SIAM, 1994.
[152]廖晓昕.稳定性的理论、方法和应用[M].武汉:华中科技大学出版社, 1999.4.
[153] K. Gu, V.L. Kharitonov, J. Chen. Stability of Time-Delay Systems (Control Engineering) [M]. Boston: Birkh?user, 2003.
[154] K. Gopalsamy. Stability of artificial neural networks with impulses [J]. Applied Mathematics and Computation, 2004, 154(3): 783-813.
[155] Z.-H. Guan, G. Chen. On delayed impulsive Hopfield neural networks [J]. Neural Networks, 1999, 12(2): 273-80.
[156] H. Ak?a, R. Alassar, V. Covachev, et al. Continuous-time additive Hopfield-type neural networks with impulses [J]. Journal of Mathematical Analysis and Applications, 2004, 290(2): 436-51.
[157] B. Xu, X. Liu, K.L. Teo. Asymptotic stability of impulsive high-order Hopfield type neural networks [J]. Comput Math Appl, 2009, 57(11-12): 1968-77.
[158] M.U. Akhmet, E. Yilmaz. Impulsive Hopfield-type neural network system with piecewise constant argument [J]. Nonlinear Analysis: Real World Applications, 2010, 11(4): 2584-93.
[159] H. Zhang, L. Chen. Asymptotic behavior of discrete solutions to delayed neural networks with impulses [J]. Neurocomputing, 2008, 71(4-6): 1032-8.
[160] X. Fu, X. Li. Global exponential stability and global attractivity of impulsive Hopfield neural networks with time delays [J]. Journal of Computational and Applied Mathematics, 2009, 231(1): 187-99.
[161] Y. Liu, Z. Wang, X. Liu. Global exponential stability of generalized recurrent neural networks with discrete and distributed delays [J]. Neural Networks, 2006, 19(5): 667-75.
[162] B. Xu, X. Liu, K.L. Teo. Global exponential stability of impulsive high-order Hopfield type neural networks with delays [J]. Comput Math Appl, 2009, 57(11-12): 1959-67.
[163] H.-F. Huo, W.-T. Li, S. Tang. Dynamics of high-order BAM neural networks with and without impulses [J]. Applied Mathematics and Computation, 2009, 215(6): 2120-33.
[164] Y. Li, Z. Xing. Existence and global exponential stability of periodic solution of CNNs with impulses [J]. Chaos, Solitons & Fractals, 2007, 33(5): 1686-93.
[165] M.A. Cohen, S. Grossberg. Absolute stability and global pattern formation and parallel memory storage by competitive neural networks [J]. Advances in Psychology, 1987, 42: 288-308.
[166] Y. Li, X. Fan. Existence and globally exponential stability of almost periodic solution for Cohen-Grossberg BAM neural networks with variable coefficients [J]. Appl Math Model, 2009, 33(4): 2114-20.
[167] W. Yu, J. Cao, J. Wang. An LMI approach to global asymptotic stability of the delayed Cohen-Grossberg neural network via nonsmooth analysis [J]. Neural Networks, 2007, 20(7): 810-8.
[168] K. Deimling. Nonlinear Functional Analysis [M]. New York, Heidelberg, Berlin: Springer-Verlag, 1985.
[169] T. Huang, C. Li, G. Chen. Stability of Cohen-Grossberg neural networks with unbounded distributed delays [J]. Chaos, Solitons & Fractals, 2007, 34(3): 992-6.
[170] Y. Li, L. Yang. Anti-periodic solutions for Cohen-Grossberg neural networks with bounded and unbounded delays [J]. Communications in Nonlinear Science and Numerical Simulation, 2009, 14(7): 3134-40.
[171] Y. Li. Existence and stability of periodic solutions for Cohen-Grossberg neural networks with multiple delays [J]. Chaos, Solitons & Fractals, 2004, 20(3): 459-66.
[172] Q. Song, J. Cao. Global exponential robust stability of Cohen-Grossberg neural network with time-varying delays and reaction-diffusion terms [J]. Journal of the Franklin Institute, 2006, 343(7): 705-19.
[173] Q. Song, J. Cao. Stability analysis of Cohen-Grossberg neural network with both time-varying and continuously distributed delays [J]. Journal of Computational and Applied Mathematics, 2006, 197(1): 188-203.
[174] J. Cao, J. Liang. Boundedness and stability for Cohen-Grossberg neural network with time-varying delays [J]. Journal of Mathematical Analysis and Applications, 2004, 296(2): 665-85.
[175] J. Cao, X. Li. Stability in delayed Cohen-Grossberg neural networks: LMI optimization approach [J]. Physica D: Nonlinear Phenomena, 2005, 212(1-2): 54-65.
[176] J. Cao, G. Feng, Y. Wang. Multistability and multiperiodicity of delayed Cohen-Grossberg neural networks with a general class of activation functions [J]. Physica D: Nonlinear Phenomena, 2008, 237(13): 1734-49.
[177] S. Arik. An analysis of exponential stability of delayed neural networks with time varying delays [J]. Neural Networks, 2004, 17(7): 1027-31.
[178] S. Arik, Z. Orman. Global stability analysis of Cohen-Grossberg neural networks with time varying delays [J]. Physics Letters A, 2005, 341(5-6): 410-21.
[179] Y. Li, X. Fan, L. Zhao. Positive periodic solutions of functional differential equations withimpulses and a parameter [J]. Comput Math Appl, 2008, 56(10): 2556-60.
[180] C. Li, X. Liao. Complete and lag synchronization of hyperchaotic systems using small impulses [J]. Chaos, Solitons & Fractals, 2004, 22(4): 857-67.
[181] Y. Li, X. Chen, L. Zhao. Stability and existence of periodic solutions to delayed Cohen-Grossberg BAM neural networks with impulses on time scales [J]. Neurocomputing, 2009, 72(7-9): 1621-30.
[182] M. Liu. Global asymptotic stability analysis of discrete-time Cohen-Grossberg neural networks based on interval systems [J]. Nonlinear Analysis: Theory, Methods & Applications, 2008, 69(8): 2403-11.
[183] H. Zhao, L. Wang. Stability and bifurcation for discrete-time Cohen-Grossberg neural network [J]. Applied Mathematics and Computation, 2006, 179(2): 787-98.
[184] W. Xiong, J. Cao. Global exponential stability of discrete-time Cohen-Grossberg neural networks [J]. Neurocomputing, 2005, 64: 433-46.
[185] T. Zhou, Y. Liu, X. Li, et al. A new criterion to global exponential periodicity for discrete-time BAM neural network with infinite delays [J]. Chaos, Solitons & Fractals, 2009, 39(1): 332-41.
[186] T. Zhou, Y. Liu, Y. Liu. Existence and global exponential stability of periodic solution for discrete-time BAM neural networks [J]. Applied Mathematics and Computation, 2006, 182(2): 1341-54.
[187] X.-G. Liu, M.-L. Tang, R. Martin, et al. Discrete-time BAM neural networks with variable delays [J]. Physics Letters A, 2007, 367(4-5): 322-30.
[188] H. Zhao, L. Wang, C. Ma. Hopf bifurcation and stability analysis on discrete-time Hopfield neural network with delay [J]. Nonlinear Analysis: Real World Applications, 2008, 9(1): 103-13.
[189] S. Guo, L. Huang, L. Wang. Exponential stability of discrete-time Hopfield neural networks [J]. Comput Math Appl, 2004, 47(8-9): 1249-56.
[190] J. Liang, J. Cao, D.W.C. Ho. Discrete-time bidirectional associative memory neural networks with variable delays [J]. Physics Letters A, 2005, 335(2-3): 226-34.
[191] E. Liz, J.B. Ferreiro. A note on the global stability of generalized difference equations [J]. Appl Math Lett, 2002, 15(6): 655-9.
[192] A. Halanay. Differential Equations: Stability, Oscillations, Time Lags [M]. New York: Academic Press, 1966.