Recent Advances on Dynamical Behaviors of Coupled Neural Networks With and Without Reaction-Diffusion Terms

Recently, the dynamical behaviors of coupled neural networks (CNNs) with and without reaction-diffusion terms have been widely researched due to their successful applications in different fields. This article introduces some important and interesting results on this topic. First, synchronization, passivity, and stability analysis results for various CNNs with and without reaction-diffusion terms are summarized, including the results for impulsive, time-varying, time-invariant, uncertain, fuzzy, and stochastic network models. In addition, some control methods, such as sampled-data control, pinning control, impulsive control, state feedback control, and adaptive control, have been used to realize the desired dynamical behaviors in CNNs with and without reaction-diffusion terms. In this article, these methods are summarized. Finally, some challenging and interesting problems deserving of further investigation are discussed.

Global Asymptotic Stability and Stabilization of Neural Networks With General Noise

Neural networks (NNs) in the stochastic environment were widely modeled as stochastic differential equations, which were driven by white noise, such as Brown or Wiener process in the existing papers. However, they are not necessarily the best models to describe dynamic characters of NNs disturbed by nonwhite noise in some specific situations. In this paper, general noise disturbance, which may be nonwhite, is introduced to NNs. Since NNs with nonwhite noise cannot be described by Itô integral equation, a novel modeling method of stochastic NNs is utilized. By a framework in light of random field approach and Lyapunov theory, the global asymptotic stability and stabilization in probability or in the mean square of NNs with general noise are analyzed, respectively. Criteria for the concerned systems based on linear matrix inequality are proposed. Some examples are given to illustrate the effectiveness of the obtained results.

Global Mittag-Leffler Stabilization of Fractional-Order Memristive Neural Networks

According to conventional memristive neural network theories, neurodynamic properties are powerful tools for solving many problems in the areas of brain-like associative learning, dynamic information storage or retrieval, etc. However, as have often been noted in most fractional-order systems, system analysis approaches for integral-order systems could not be directly extended and applied to deal with fractional-order systems, and consequently, it raises difficult issues in analyzing and controlling the fractional-order memristive neural networks. By using the set-valued maps and fractional-order differential inclusions, then aided by a newly proposed fractional derivative inequality, this paper investigates the global Mittag-Leffler stabilization for a class of fractional-order memristive neural networks. Two types of control rules (i.e., state feedback stabilizing control and output feedback stabilizing control) are designed for the stabilization of fractional-order memristive neural networks, while a list of stabilization criteria is established. Finally, two numerical examples are given to show the effectiveness and characteristics of the obtained theoretical results.

Stability Analysis of Distributed Delay Neural Networks Based on Relaxed Lyapunov-Krasovskii Functionals

This paper revisits the problem of asymptotic stability analysis for neural networks with distributed delays. The distributed delays are assumed to be constant and prescribed. Since a positive-definite quadratic functional does not necessarily require all the involved symmetric matrices to be positive definite, it is important for constructing relaxed Lyapunov-Krasovskii functionals, which generally lead to less conservative stability criteria. Based on this fact and using two kinds of integral inequalities, a new delay-dependent condition is obtained, which ensures that the distributed delay neural network under consideration is globally asymptotically stable. This stability criterion is then improved by applying the delay partitioning technique. Two numerical examples are provided to demonstrate the advantage of the presented stability criteria.

Robustness analysis of global exponential stability of recurrent neural networks in the presence of time delays and random disturbances

In recent years, the global stability of recurrent neural networks (RNNs) has been investigated extensively. It is well known that time delays and external disturbances can derail the stability of RNNs. In this paper, we analyze the robustness of global stability of RNNs subject to time delays and random disturbances. Given a globally exponentially stable neural network, the problem to be addressed here is how much time delay and noise the RNN can withstand to be globally exponentially stable in the presence of delay and noise. The upper bounds of the time delay and noise intensity are characterized by using transcendental equations for the RNNs to sustain global exponential stability. Moreover, we prove theoretically that, for any globally exponentially stable RNNs, if additive noises and time delays are smaller than the derived lower bounds arrived at here, then the perturbed RNNs are guaranteed to also be globally exponentially stable. Three numerical examples are provided to substantiate the theoretical results.

Estimating the ultimate bound and positively invariant set for a class of Hopfield networks

In this paper, we investigate the ultimate bound and positively invariant set for a class of Hopfield neural networks (HNNs) based on the Lyapunov stability criterion and Lagrange multiplier method. It is shown that a hyperelliptic estimate of the ultimate bound and positively invariant set for the HNNs can be calculated by solving a linear matrix inequality (LMI). Furthermore, the global stability of the unique equilibrium and the instability region of the HNNs are analyzed, respectively. Finally, the most accurate estimate of the ultimate bound and positively invariant set can be derived by solving the corresponding optimization problems involving the LMI constraints. Some numerical examples are given to illustrate the effectiveness of the proposed results.