Discrete-time recurrent neural networks with time-varying delays: Exponential stability analysis

This is the post print version of the article. The official published version can be obtained from the link below - Copyright 2007 Elsevier LtdThis Letter is concerned with the analysis problem of exponential stability for a class of discrete-time recurrent neural networks (DRNNs) with time delays. The delay is of the time-varying nature, and the activation functions are assumed to be neither differentiable nor strict monotonic. Furthermore, the description of the activation functions is more general than the recently commonly used Lipschitz conditions. Under such mild conditions, we first prove the existence of the equilibrium point. Then, by employing a Lyapunov–Krasovskii functional, a unified linear matrix inequality (LMI) approach is developed to establish sufficient conditions for the DRNNs to be globally exponentially stable. It is shown that the delayed DRNNs are globally exponentially stable if a certain LMI is solvable, where the feasibility of such an LMI can be easily checked by using the numerically efficient Matlab LMI Toolbox. A simulation example is presented to show the usefulness of the derived LMI-based stability condition.This work was supported in part by the Engineering and Physical Sciences Research Council (EPSRC) of the UK under Grant GR/S27658/01, the Nuffield Foundation of the UK under Grant NAL/00630/G, the Alexander von Humboldt Foundation of Germany, the Natural Science Foundation of Jiangsu Education Committee of China (05KJB110154), the NSF of Jiangsu Province of China (BK2006064), and the National Natural Science Foundation of China (10471119)

Existence and exponential stability of periodic solution for fuzzy BAM neural networks with periodic coefficient

A class of fuzzy bidirectional associated memory (BAM) networks with periodic coefficients is studied. Some sufficient conditions are established for the existence and global exponential stability of a periodic solution of such fuzzy BAM neural networks by using a continuation theorem based on the coincidence degree and the Lyapunov-function method. The sufficient conditions are easy to verify in pattern recognition and automatic control. Finally, an example is given to show the feasibility and efficiency of our results.Вивчено клас нечiтких нейронних мереж Коско з перiодичним коефiцiєнтом. За допомогою теореми про продовження, що базується на ступенi збiгу та методi функцiй Ляпунова, встановлено достатнi умови для iснування та глобальної експоненцiальної стiйкостi перiодичного розв’язку таких нечiтких нейронних мереж Коско. Цi достатнi умови легко перевiряються при розпiзнаваннi образiв та автоматичному керуваннi. Наведено приклад, що демонструє застосовнiсть та ефективнiсть отриманих результатiв

Multi-almost periodicity and invariant basins of general neural networks under almost periodic stimuli

In this paper, we investigate convergence dynamics of $2^N$ almost periodic encoded patterns of general neural networks (GNNs) subjected to external almost periodic stimuli, including almost periodic delays. Invariant regions are established for the existence of $2^N$ almost periodic encoded patterns under two classes of activation functions. By employing the property of $\mathscr{M}$-cone and inequality technique, attracting basins are estimated and some criteria are derived for the networks to converge exponentially toward $2^N$ almost periodic encoded patterns. The obtained results are new, they extend and generalize the corresponding results existing in previous literature.Comment: 28 pages, 4 figure

Boundedness and stability for Cohen–Grossberg neural network with time-varying delays

AbstractIn this paper, a model is considered to describe the dynamics of Cohen–Grossberg neural network with variable coefficients and time-varying delays. Uniformly ultimate boundedness and uniform boundedness are studied for the model by utilizing the Hardy inequality. Combining with the Halanay inequality and the Lyapunov functional method, some new sufficient conditions are derived for the model to be globally exponentially stable. The activation functions are not assumed to be differentiable or strictly increasing. Moreover, no assumption on the symmetry of the connection matrices is necessary. These criteria are important in signal processing and the design of networks

Non-Convex Multi-species Hopfield models

In this work we introduce a multi-species generalization of the Hopfield model for associative memory, where neurons are divided into groups and both inter-groups and intra-groups pair-wise interactions are considered, with different intensities. Thus, this system contains two of the main ingredients of modern Deep neural network architectures: Hebbian interactions to store patterns of information and multiple layers coding different levels of correlations. The model is completely solvable in the low-load regime with a suitable generalization of the Hamilton-Jacobi technique, despite the Hamiltonian can be a non-definite quadratic form of the magnetizations. The family of multi-species Hopfield model includes, as special cases, the 3-layers Restricted Boltzmann Machine (RBM) with Gaussian hidden layer and the Bidirectional Associative Memory (BAM) model.Comment: This is a pre-print of an article published in J. Stat. Phy

General criteria for asymptotic and exponential stabilities of neural network models with unbounded delays

For a family of differential equations with infinite delay, we give sufficient conditions for the global asymptotic, and global exponential stability of an equilibrium point. This family includes most of the delayed models of neural networks of Cohen-Grossberg type, with both bounded and unbounded distributed delay, for which general asymptotic and exponential stability criteria are derived. As illustrations, the results are applied to several concrete models studied in the literature, and a comparison of results is given.Fundação para a Ciência e a Tecnologia (FCT) - 2009-ISFL-1-209Universidade do Minho. Centro de Matemática (CMAT

Design of exponential state estimators for neural networks with mixed time delays

This is the post print version of the article. The official published version can be obtained from the link below - Copyright 2007 Elsevier Ltd.In this Letter, the state estimation problem is dealt with for a class of recurrent neural networks (RNNs) with mixed discrete and distributed delays. The activation functions are assumed to be neither monotonic, nor differentiable, nor bounded. We aim at designing a state estimator to estimate the neuron states, through available output measurements, such that the dynamics of the estimation error is globally exponentially stable in the presence of mixed time delays. By using the Laypunov–Krasovskii functional, a linear matrix inequality (LMI) approach is developed to establish sufficient conditions to guarantee the existence of the state estimators. We show that both the existence conditions and the explicit expression of the desired estimator can be characterized in terms of the solution to an LMI. A simulation example is exploited to show the usefulness of the derived LMI-based stability conditions.This work was supported in part by the Engineering and Physical Sciences Research Council (EPSRC) of the UK under Grant GR/S27658/01, the Nuffield Foundation of the UK under Grant NAL/00630/G, the Alexander von Humboldt Foundation of Germany, the Natural Science Foundation of Jiangsu Education Committee of China under Grants 05KJB110154 and BK2006064, and the National Natural Science Foundation of China under Grants 10471119 and 10671172

Asymptotic Stability, Orbital Stability of Hopf-Bifurcating Periodic Solution of a Simple Three-Neuron Artificial Neural Network with Distributed Delay

A distributed delay model of a class of three-neuron network has been investigated. Sufficient conditions for existence of unique equilibrium, multiple equilibria and their local stability are derived. A closed interval for a parameter of the system is identified in which Hopf-bifurcating periodic solution occurs for each point of such interval. The orbital stability of such bifurcating periodic solution at the extreme points of the interval is ascertained. Lastly global bifurcation aspect of such periodic solutions is studied. The results are illustrated by numerical simulations