Stability analysis of Cohen-Grossberg neural networks

Without assuming boundedness and differentiability of the activation functions and any symmetry of interconnections, we employ Lyapunov functions to establish some sufficient conditions ensuring existence, uniqueness, global asymptotic stability, and even global exponential stability of equilibria for the Cohen-Grossberg neural networks with and without delays. Our results are not only presented in terms of system parameters and can be easily verified and also less restrictive than previously known criteria and can be applied to neural networks, including Hopfield neural networks, bidirectional association memory neural networks, and cellular neural networks.     

Global robust stability of delayed neural networks with a class of general activation functions

In this paper, the global robust stability is discussed for delayed neural networks with a class of general activation functions. By constructing new Lyapunov functionals, several novel conditions are derived to guarantee the existence, uniqueness and global robust stability of the equilibrium of neural networks with time delays. These conditions do not require the activation functions to be differentiable, bounded or monotonically nondecreasing. The results obtained here are generalizations of some earlier results reported in the literature for neural networks with time delays. In addition, two examples are given to illustrate our proposed results.     

Global exponential periodicity of three-unit neural networks in a ring with time-varying delays

This paper formulates and studies a model of three-unit neural networks in a ring. The model can well describe many practical architectures of delayed neural networks, which is generalization of some existing neural networks under a time-varying environment. Without assuming the boundedness, monotonicity, and differentiability of activation functions and any symmetry of interconnections, we establish some sufficient conditions for checking the existence of periodic solution and global exponential stability for the neural networks. A continuation theorem of the coincidence degree and inequality analysis are employed. Our results are all independent of the delays and maybe more convenient to design a circuit network.     

On the global stability of delayed neural networks

Lyapunov functional methods, combining with some inequality techniques, are employed to study the global asymptotic stability of delayed neural networks. Without assuming Lipschitz conditions on the activation functions, a new sufficient condition is established. Such criteria allows us to include non-Lipschitzian activation functions in the design of delayed neural networks. The result presented here is also discussed from the point of view of its relationship to some previous results.     

Multistability analysis for recurrent neural networks with unsaturating piecewise linear transfer functions

Multistability is a property necessary in neural networks in order to enable certain applications (e.g., decision making), where monostable networks can be computationally restrictive. This article focuses on the analysis of multistability for a class of recurrent neural networks with unsaturating piecewise linear transfer functions. It deals fully with the three basic properties of a multistable network: boundedness, global attractivity, and complete convergence. This article makes the following contributions: conditions based on local inhibition are derived that guarantee boundedness of some multistable networks, conditions are established for global attractivity, bounds on global attractive sets are obtained, complete convergence conditions for the network are developed using novel energy-like functions, and simulation examples are employed to illustrate the theory thus developed.     

On Robust Exponential Periodicity of Interval Neural Networks with Delays

In this Letter, based on globally Lipschitz continous activation functions, new conditions ensuring existence, uniqueness and global robust exponential stability of the periodic solution of interval-delayed neural networks with periodic input are obtained. All the results obtained are generalizations of some resent results reported in the literature for neural networks with constant input.     

Analysis and design of associative memories based on recurrent neural networks with linear saturation activation functions and time-varying delays

In this letter, some sufficient conditions are obtained to guarantee recurrent neural networks with linear saturation activation functions, and time-varying delays have multiequilibria located in the saturation region and the boundaries of the saturation region. These results on pattern characterization are used to analyze and design autoassociative memories, which are directly based on the parameters of the neural networks. Moreover, a formula for the numbers of spurious equilibria is also derived. Four design procedures for recurrent neural networks with linear saturation activation functions and time-varying delays are developed based on stability results. Two of these procedures allow the neural network to be capable of learning and forgetting. Finally, simulation results demonstrate the validity and characteristics of the proposed approach.     

泛函网络神经元构造理论与方法

泛函网络是近年提出的一种对神经网络的有效推广.与神经网络不同,它处理的是一般的泛函模型,它在各个神经元之间的连接没有权值,并且神经元函数不固定的,往往是一给定的基函数的组合,泛函网络学习的目的就是求出神经元函数的精确表达式或近似表达式. 迄今关于泛函网络神经元基函数的存在性和选取方法缺乏理论依据.文中基于Banach空间中偏序理论,分析了泛函网络神经元基函数的存在性,给出了泛函网络神经元基函数选取方法,对于完善泛函网络的基础理论具有参考价值.     

Exponential periodicity and stability of neural networks with reaction-diffusion terms and both variable and unbounded delays

In this paper, the exponential periodicity and stability of neural networks with Lipschitz continuous activation functions are investigated, without assuming the boundedness of the activation functions and the differentiability of time-varying delays, as needed in most other papers. The neural networks contain reaction-diffusion terms and both variable and unbounded delays. Some sufficient conditions ensuring the existence and uniqueness of periodic solution and stability of neural networks with reaction-diffusion terms and both variable and unbounded delays are obtained by analytic methods and inequality technique. Furthermore, the exponential converging index is also estimated. The methods, which does not make use of Lyapunov functional, is simple and valid for the periodicity and stability analysis of neural networks with variable and/or unbounded delays. The results extend some previous results. Two examples are given to show the effectiveness of the obtained results.     

Recurrent neural networks for solving second-order cone programs

This paper proposes using the neural networks to efficiently solve the second-order cone programs (SOCP). To establish the neural networks, the SOCP is first reformulated as a second-order cone complementarity problem (SOCCP) with the Karush-Kuhn-Tucker conditions of the SOCP. The SOCCP functions, which transform the SOCCP into a set of nonlinear equations, are then utilized to design the neural networks. We propose two kinds of neural networks with the different SOCCP functions. The first neural network uses the Fischer-Burmeister function to achieve an unconstrained minimization with a merit function. We show that the merit function is a Lyapunov function and this neural network is asymptotically stable. The second neural network utilizes the natural residual function with the cone projection function to achieve low computation complexity. It is shown to be Lyapunov stable and converges globally to an optimal solution under some condition. The SOCP simulation results demonstrate the effectiveness of the proposed neural networks.     

Global Exponential Stability and Global Convergence in Finite Time of Neural Networks with Discontinuous Activations

In this paper, we consider a general class of neural networks, which have arbitrary constant delays in the neuron interconnections, and neuron activations belonging to the set of discontinuous monotone increasing and (possibly) unbounded functions. Based on the topological degree theory and Lyapunov functional method, we provide some new sufficient conditions for the global exponential stability and global convergence in finite time of these delayed neural networks. Under these conditions the uniqueness of initial value problem (IVP) is proved. The exponential convergence rate can be quantitatively estimated on the basis of the parameters defining the neural network. These conditions are easily testable and independent of the delay. In the end some remarks and examples are discussed to compare the present results with the existing ones.     

Exponential periodicity and stability of delayed neural networks

In this paper, exponential periodicity and stability of delayed neural networks is investigated. Without assuming the boundedness and differentiability of the activation functions, some new sufficient conditions ensuring existence and uniqueness of periodic solution for a general class of neural systems are obtained. The delayed Hopfield network, bidirectional associative memory network, and cellular neural network are special cases of the neural system model considered.     

一类分布参数动态系统的周期性分析

通过构造适当的Lyapunov泛函、利用M矩阵性质和不等式技巧, 在不要求神经网络激励函数的有界性、单调性和可微性弱保守条件下, 探讨了一类具有分布参数和分布时滞的Cohen-Grossberg动态神经网络周期解的存在性和指数稳定性问题, 提出了一系列充分性判据来确保这类同时具有分布参数和分布时滞神经网络周期解的存在性和指数稳定性, 并通过几个注解以及与其他文献结果进行比较说明了该方法的优越性. 最后, 给出了数值例子和计算机仿真来验证这一理论的有效性.     

New criteria for globally exponential stability of delayed Cohen–Grossberg neural network

This paper is concerned with analysis problem for the global exponential stability of the Cohen–Grossberg neural networks with discrete delays and with distributed delays. We first prove the existence and uniqueness of the equilibrium point under mild conditions, assuming neither differentiability nor strict monotonicity for the activation function. Then, we employ Lyapunov functions to establish some sufficient conditions ensuring global exponential stability of equilibria for the Cohen–Grossberg neural networks with discrete delays and with distributed delays. Our results are not only presented in terms of system parameters and can be easily verified and also less restrictive than previously known criteria. A comparison between our results and the previous results admits that our results establish a new set of stability criteria for delayed neural networks.     

Global asymptotic stability of bidirectional associative memory neural networks with S-type distributed delays

Bidirectional associative memory (BAM) models are two-layer heteroassociative networks. This paper is devoted to the investigation of the global asymptotic stability for BAM neural networks with S-type distributed signal transmission delays along the axon of a neuron. A theorem and corollary was obtained in which the boundedness and differentiability of the signal functions in some papers are deleted. Some sufficient conditions for the existence of global asymtotic stable equilibrium of the networks in this paper are better than the sufficient conditions in the quoted literature.     

Exponential stability of impulsive high-order Hopfield-type neural networks with time-varying delays

This paper considers the problems of global exponential stability and exponential convergence rate for impulsive high-order Hopfield-type neural networks with time-varying delays. By using the method of Lyapunov functions, some sufficient conditions for ensuring global exponential stability of these networks are derived, and the estimated exponential convergence rate is also obtained. As an illustration, an numerical example is worked out using the results obtained.     

Convergence of discrete delayed Hopfield neural networks

The discrete delayed Hopfield neural networks is an extension of the discrete Hopfield neural networks. In this paper, the convergence of discrete delayed Hopfield neural networks is mainly studied, and some results on the convergence are obtained by using Lyapunov function. Several new sufficient conditions for the delayed networks converging towards a limit cycle with period at most 2 are proved in parallel updating mode. Also, some conditions for the delayed networks converging towards a limit cycle with 2-period are investigated in parallel updating mode. All results established in this paper extend the previous results on the convergence of both the discrete Hopfield neural networks, and the discrete delayed Hopfield neural networks in parallel updating mode.     

Stability results for Takagi–Sugeno fuzzy uncertain BAM neural networks with time delays in the leakage term

This paper deals with the delay-dependent asymptotic stability analysis problem for a class of fuzzy bidirectional associative memory (BAM) neural networks with time delays in the leakage term by Takagi–Sugeno (T–S) fuzzy model. The nonlinear delayed BAM neural networks are first established as a modified T–S fuzzy model in which the consequent parts are composed of a set of BAM neural networks with time-varying delays. The parameter uncertainties are assumed to be norm bounded. Some new delay-dependent stability conditions are derived in terms of linear matrix inequality by constructing a new Lyapunov–Krasovskii functional and introducing some free-weighting matrices. Even there is no leakage delay, the obtained results are also less restrictive than some recent works. It can be applied to BAM neural networks with activation functions without assuming their boundedness, monotonicity, or differentiability. Numerical examples are given to demonstrate the effectiveness of the proposed methods.

     

Neural Network Method for Solving Partial Differential Equations

A method is presented to solve partial differential equations (pde's) and its boundary and/or initial conditions by using neural networks. It uses the fact that multiple input, single output, single hidden layer feedforward networks with a linear output layer with no bias are capable of arbitrarily well approximating arbitrary functions and its derivatives, which is proven by a number of authors and well known in literature. Knowledge about the pde and its boundary and/or initial conditions is incorporated into the structures and the training sets of several neural networks. In this way we obtain networks of which some are specifically structured. To find the solution of the pde and its boundary and/or initial conditions we have to train all obtained networks simultaneously. Therefore we use an evolutionary algorithm to train the networks. We demonstrate the working of our method by applying it to two problems.     

Global asymptotic stability of delayed neural networks with discontinuous neuron activations

This paper investigates a class of delayed neural networks whose neuron activations are modeled by discontinuous functions. By utilizing the Leray–Schauder fixed point theorem of multivalued version, the properties of M-matrix and generalized Lyapunov approach, we present some sufficient conditions to ensure the existence and global asymptotic stability of the state equilibrium point. Furthermore, the global convergence of the output solutions are also discussed. The assumptive conditions imposed on activation functions are allowed to be unbounded and nonmonotonic, which are less restrictive than previews works on the discontinuous or continuous neural networks. Hence, we improve and extend some existing results of other researchers. Finally, one numerical example is given to illustrate the effectiveness of the criteria proposed in this paper.