An LMI approach to global asymptotic stability of the delayed Cohen-Grossberg neural network via nonsmooth analysis

In this paper, a linear matrix inequality (LMI) to global asymptotic stability of the delayed Cohen-Grossberg neural network is investigated by means of nonsmooth analysis. Several new sufficient conditions are presented to ascertain the uniqueness of the equilibrium point and the global asymptotic stability of the neural network. It is noted that the results herein require neither the smoothness of the behaved function, or the activation function nor the boundedness of the activation function. In addition, from theoretical analysis, it is found that the condition for ensuring the global asymptotic stability of the neural network also implies the uniqueness of equilibrium. The obtained results improve many earlier ones and are easy to apply. Some simulation results are shown to substantiate the theoretical results

Nonlinear analysis of dynamical complex networks

Copyright © 2013 Zidong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Complex networks are composed of a large number of highly interconnected dynamical units and therefore exhibit very complicated dynamics. Examples of such complex networks include the Internet, that is, a network of routers or domains, the World Wide Web (WWW), that is, a network of websites, the brain, that is, a network of neurons, and an organization, that is, a network of people. Since the introduction of the small-world network principle, a great deal of research has been focused on the dependence of the asymptotic behavior of interconnected oscillatory agents on the structural properties of complex networks. It has been found out that the general structure of the interaction network may play a crucial role in the emergence of synchronization phenomena in various fields such as physics, technology, and the life sciences

Stability analysis of impulsive stochastic Cohen–Grossberg 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 - Copyright 2008 Elsevier LtdIn this paper, the problem of stability analysis for a class of impulsive stochastic Cohen–Grossberg neural networks with mixed delays is considered. The mixed time delays comprise both the time-varying and infinite distributed delays. By employing a combination of the M-matrix theory and stochastic analysis technique, a sufficient condition is obtained to ensure the existence, uniqueness, and exponential p-stability of the equilibrium point for the addressed impulsive stochastic Cohen–Grossberg neural network with mixed delays. The proposed method, which does not make use of the Lyapunov functional, is shown to be simple yet effective for analyzing the stability of impulsive or stochastic neural networks with variable and/or distributed delays. We then extend our main results to the case where the parameters contain interval uncertainties. Moreover, the exponential convergence rate index is estimated, which depends on the system parameters. An example is given to show the effectiveness of the obtained results.This work was supported by the Natural Science Foundation of CQ CSTC under grant 2007BB0430, the Scientific Research Fund of Chongqing Municipal Education Commission under Grant KJ070401, an International Joint Project sponsored by the Royal Society of the UK and the National Natural Science Foundation of China, and the Alexander von Humboldt Foundation of Germany

Dynamics of Simple Balancing Models with State Dependent Switching Control

Time-delayed control in a balancing problem may be a nonsmooth function for a variety of reasons. In this paper we study a simple model of the control of an inverted pendulum by either a connected movable cart or an applied torque for which the control is turned off when the pendulum is located within certain regions of phase space. Without applying a small angle approximation for deviations about the vertical position, we see structurally stable periodic orbits which may be attracting or repelling. Due to the nonsmooth nature of the control, these periodic orbits are born in various discontinuity-induced bifurcations. Also we show that a coincidence of switching events can produce complicated periodic and aperiodic solutions.Comment: 36 pages, 12 figure

Non-Euclidean Contraction Analysis of Continuous-Time Neural Networks

Critical questions in dynamical neuroscience and machine learning are related to the study of continuous-time neural networks and their stability, robustness, and computational efficiency. These properties can be simultaneously established via a contraction analysis. This paper develops a comprehensive non-Euclidean contraction theory for continuous-time neural networks. First, for non-Euclidean $\ell_{1}/\ell_{\infty}$ logarithmic norms, we establish quasiconvexity with respect to positive diagonal weights and closed-form worst-case expressions over certain matrix polytopes. Second, for locally Lipschitz maps (e.g., arising as activation functions), we show that their one-sided Lipschitz constant equals the essential supremum of the logarithmic norm of their Jacobian. Third and final, we apply these general results to classes of continuous-time neural networks, including Hopfield, firing rate, Persidskii, Lur'e and other models. For each model, we compute the optimal contraction rate and corresponding weighted non-Euclidean norm via a linear program or, in some special cases, via a Hurwitz condition on the Metzler majorant of the synaptic matrix. Our non-Euclidean analysis establishes also absolute, connective, and total contraction properties

Fixed-time control of delayed neural networks with impulsive perturbations

This paper is concerned with the fixed-time stability of delayed neural networks with impulsive perturbations. By means of inequality analysis technique and Lyapunov function method, some novel fixed-time stability criteria for the addressed neural networks are derived in terms of linear matrix inequalities (LMIs). The settling time can be estimated without depending on any initial conditions but only on the designed controllers. In addition, two different controllers are designed for the impulsive delayed neural networks. Moreover, each controller involves three parts, in which each part has different role in the stabilization of the addressed neural networks. Finally, two numerical examples are provided to illustrate the effectiveness of the theoretical analysis