Stability and Synchronization of Nonautonomous Reaction-Diffusion Neural Networks With General Time-Varying Delays

This article investigates the stability and synchronization of nonautonomous reaction-diffusion neural networks with general time-varying delays. Compared with the existing works concerning reaction-diffusion neural networks, the main innovation of this article is that the network coefficients are time-varying, and the delays are general (which means that fewer constraints are posed on delays; for example, the commonly used conditions of differentiability and boundedness are no longer needed). By Green's formula and some analytical techniques, some easily checkable criteria on stability and synchronization for the underlying neural networks are established. These obtained results not only improve some existing ones but also contain some novel results that have not yet been reported. The effectiveness and superiorities of the established criteria are verified by three numerical examples.

Boundary Stabilization of Stochastic Delayed Cohen-Grossberg Neural Networks With Diffusion Terms

This study considers the boundary stabilization for stochastic delayed Cohen-Grossberg neural networks (SDCGNNs) with diffusion terms by the Lyapunov functional method. In the realization of NNs, sometimes time delays and diffusion phenomenon cannot be ignored, so Cohen-Grossberg NNs with time delays and diffusion terms are studied in this article. Moreover, different from the previously distributed control, the boundary control is used to stabilize the system, which can reduce the spatial cost of the controller and is easy to implement. Boundary controllers are presented for system with Neumann boundary and mixed boundary conditions, and criteria are derived such that the controlled system achieves mean-square exponential stabilization. Based on the criterion, the effects of diffusion matrix, coupling strength, coupling matrix, and time delays on exponentially stability are analyzed. In the process of analysis, two difficulties need to be addressed: 1) how to introduce boundary control into system analysis? and 2) how to analyze the influence of system parameters on stability? We deal with these problems by using Poincaré's inequality and Schur's complement lemma. Moreover, mean-square exponential synchronization of stochastic delayed Hopfield NNs with diffusion terms, as an application of the theoretical result, is considered under the boundary control. Examples are given to illustrate the effectiveness of the theoretical results.

Distributed Adaptive Containment Control for Coupled Reaction-Diffusion Neural Networks With Directed Topology

In this article, we consider the problem of distributed adaptive leader-follower coordination of partial differential systems (i.e., reaction-diffusion neural networks, RDNNs) with directed communication topology in the case of multiple leaders. Different from the dynamical networks with ordinary differential dynamics, the design of adaptive protocols is more difficult due to the existence of spatial variables and nonlinear terms in the model. Under directed networks, a novel adaptive control protocol is proposed to solve the containment control problem of RDNNs. By constructing proper Lyapunov functional and adopting some important prior knowledge, the stability of containment for coupled RDNNs is theoretically proved. Furthermore, a corollary about the leader-follower synchronization with a leader for coupled RDNNs with directed communication topology is given. In the end, two numerical examples are provided to illustrate the obtained theoretical results.

Intermittent boundary stabilization of stochastic reaction-diffusion Cohen-Grossberg neural networks

Cohen-Grossberg neural networks (CGNNs) play an important role in many applications and the stabilization of this system has been well studied. This study considers the exponential stabilization for stochastic reaction-diffusion Cohen-Grossberg neural networks (SRDCGNNs) by means of an aperiodically intermittent boundary control. Both SRDCGNNs without and with time-delays are discussed. By employing the spatial integral functional method and Poincare's inequality, criteria are derived to ensure the controlled systems achieve mean square exponential stabilization. Based on these criteria, the effects of diffusion item, control gains, the minimum control proportion and time-delays on exponential stability are analyzed. Examples are given to illustrate the effectiveness of the obtained theoretical results.

A Finite-Time Convergent and Noise-Rejection Recurrent Neural Network and Its Discretization for Dynamic Nonlinear Equations Solving

The so-called zeroing neural network (ZNN) is an effective recurrent neural network for solving dynamic problems including the dynamic nonlinear equations. There exist numerous unperturbed ZNN models that can converge to the theoretical solution of solvable nonlinear equations in infinity long or finite time. However, when these ZNN models are perturbed by external disturbances, the convergence performance would be dramatically deteriorated. To overcome this issue, this paper for the first time proposes a finite-time convergent ZNN with the noise-rejection capability to endure disturbances and solve dynamic nonlinear equations in finite time. In theory, the finite-time convergence and noise-rejection properties of the finite-time convergent and noise-rejection ZNN (FTNRZNN) are rigorously proved. For potential digital hardware realization, the discrete form of the FTNRZNN model is established based on a recently developed five-step finite difference rule to guarantee a high computational accuracy. The numerical results demonstrate that the discrete-time FTNRZNN can reject constant external noises. When perturbed by dynamic bounded or unbounded linear noises, the discrete-time FTNRZNN achieves the smallest steady-state errors in comparison with those generated by other discrete-time ZNN models that have no or limited ability to handle these noises. Discrete models of the FTNRZNN and the other ZNNs are comparatively applied to redundancy resolution of a robotic arm with superior positioning accuracy of the FTNRZNN verified.

An improved result on synchronization control for memristive neural networks with inertial terms and reaction-diffusion items

This paper investigates the synchronization issue of the memristive neural networks (MNNs) with inertial terms and reaction-diffusion items. In order to smoothly derive the controller gains and obtain an excellent control effect, the desired controller that contains a discontinuous function is proposed. Moreover, by constructing a novel Lyapunov-Krasovskii functional and combining the inequality techniques, several sufficient conditions in terms of algebraic inequalities are obtained to guarantee the synchronization of the proposed drive and response systems. Finally, three numerical simulations are exploited to support the acquired theoretical results.

Synchronization of Multiple Reaction-Diffusion Neural Networks With Heterogeneous and Unbounded Time-Varying Delays

The synchronization problem of multiple/coupled reaction-diffusion neural networks with time-varying delays is investigated. Differing from the existing considerations, state delays among distinct neurons and coupling delays among different subnetworks are included in the proposed model, the assumptions posed on the arisen delays are very weak, time-varying, heterogeneous, even unbounded delays are permitted. To overcome the difficulties from this kind of delay as well as diffusion effects, a comparison-based approach is applied to this model and a series of algebraic criteria are successfully obtained to verify the global asymptotical synchronization. By specifying the existing delays, some M -matrix-based criteria are derived to justify the power-rate synchronization and exponential synchronization. In addition, new criterion on synchronization of general connected neural networks without diffusion effects is also given. Finally, two simulation examples are given to verify the effectiveness of the obtained theoretical results and provide a comparison with the existing criterion.

Distributed Adaptive Tracking Synchronization for Coupled Reaction-Diffusion Neural Network

This paper considers the tracking synchronization problem for a class of coupled reaction-diffusion neural networks (CRDNNs) with undirected topology. For the case where the tracking trajectory has identical individual dynamic as that of the network nodes, the edge-based and vertex-based adaptive strategies on coupling strengths as well as adaptive controllers, which demand merely the local neighbor information, are proposed to synchronize the CRDNNs to the tracking trajectory. To reduce the control costs, an adaptive pinning control technique is employed. For the case where the tracking trajectory has different individual dynamic from that of the network nodes, the vertex-based adaptive strategy is proposed to drive the synchronization error to a relatively small area, which is adjustable according to the parameters of the adaptive strategy. This kind of adaptive design can enhance the robustness of the network against the external disturbance posed on the tracking trajectory. The obtained theoretical results are verified by two representative examples.

Synchronization of Coupled Reaction-Diffusion Neural Networks With Directed Topology via an Adaptive Approach

This paper investigates the synchronization issue of coupled reaction-diffusion neural networks with directed topology via an adaptive approach. Due to the complexity of the network structure and the presence of space variables, it is difficult to design proper adaptive strategies on coupling weights to accomplish the synchronous goal. Under the assumptions of two kinds of special network structures, that is, directed spanning path and directed spanning tree, some novel edge-based adaptive laws, which utilized the local information of node dynamics fully are designed on the coupling weights for reaching synchronization. By constructing appropriate energy function, and utilizing some analytical techniques, several sufficient conditions are given. Finally, some simulation examples are given to verify the effectiveness of the obtained theoretical results.

A New Local Bipolar Autoassociative Memory Based on External Inputs of Discrete Recurrent Neural Networks With Time Delay

In this paper, local bipolar auto-associative memories are presented based on discrete recurrent neural networks with a class of gain type activation function. The weight parameters of neural networks are acquired by a set of inequalities without the learning procedure. The global exponential stability criteria are established to ensure the accuracy of the restored patterns by considering time delays and external inputs. The proposed methodology is capable of effectively overcoming spurious memory patterns and achieving memory capacity. The effectiveness, robustness, and fault-tolerant capability are validated by simulated experiments.In this paper, local bipolar auto-associative memories are presented based on discrete recurrent neural networks with a class of gain type activation function. The weight parameters of neural networks are acquired by a set of inequalities without the learning procedure. The global exponential stability criteria are established to ensure the accuracy of the restored patterns by considering time delays and external inputs. The proposed methodology is capable of effectively overcoming spurious memory patterns and achieving memory capacity. The effectiveness, robustness, and fault-tolerant capability are validated by simulated experiments.

Synchronization of Reaction-Diffusion Neural Networks With Dirichlet Boundary Conditions and Infinite Delays

This paper is concerned with synchronization for a class of reaction-diffusion neural networks with Dirichlet boundary conditions and infinite discrete time-varying delays. By utilizing theories of partial differential equations, Green's formula, inequality techniques, and the concept of comparison, algebraic criteria are presented to guarantee master-slave synchronization of the underlying reaction-diffusion neural networks via a designed controller. Additionally, sufficient conditions on exponential synchronization of reaction-diffusion neural networks with finite time-varying delays are established. The proposed criteria herein enhance and generalize some published ones. Three numerical examples are presented to substantiate the validity and merits of the obtained theoretical results.

Impulsive Synchronization of Reaction-Diffusion Neural Networks With Mixed Delays and Its Application to Image Encryption

This paper presents a new impulsive synchronization criterion of two identical reaction-diffusion neural networks with discrete and unbounded distributed delays. The new criterion is established by applying an impulse-time-dependent Lyapunov functional combined with the use of a new type of integral inequality for treating the reaction-diffusion terms. The impulse-time-dependent feature of the proposed Lyapunov functional can capture more hybrid dynamical behaviors of the impulsive reaction-diffusion neural networks than the conventional impulse-time-independent Lyapunov functions/functionals, while the new integral inequality, which is derived from Wirtinger's inequality, overcomes the conservatism introduced by the integral inequality used in the previous results. Numerical examples demonstrate the effectiveness of the proposed method. Later, the developed impulsive synchronization method is applied to build a spatiotemporal chaotic cryptosystem that can transmit an encrypted image. The experimental results verify that the proposed image-encrypting cryptosystem has the advantages of large key space and high security against some traditional attacks.

Dynamical Behavior of Delayed Reaction-Diffusion Hopfield Neural Networks Driven by Infinite Dimensional Wiener Processes

In this paper, we focus on the long time behavior of the mild solution to delayed reaction-diffusion Hopfield neural networks (DRDHNNs) driven by infinite dimensional Wiener processes. We analyze the existence, uniqueness, and stability of this system under the local Lipschitz function by constructing an appropriate Lyapunov-Krasovskii function and utilizing the semigroup theory. Some easy-to-test criteria affecting the well-posedness and stability of the networks, such as infinite dimensional noise and diffusion effect, are obtained. The criteria can be used as theoretic guidance to stabilize DRDHNNs in practical applications when infinite dimensional noise is taken into consideration. Meanwhile, considering the fact that the standard Brownian motion is a special case of infinite dimensional Wiener process, we undertake an analysis of the local Lipschitz condition, which has a wider range than the global Lipschitz condition. Two samples are given to examine the availability of the results in this paper. Simulations are also given using the MATLAB.

Mode-dependent stochastic stability criteria of fuzzy Markovian jumping neural networks with mixed delays

This paper investigates the stochastic stability of fuzzy Markovian jumping neural networks with mixed delays in mean square. The mixed delays include time-varying delay and continuously distributed delay. By using the Lyapunov functional method, Jensen integral inequality, the generalized Jensen integral inequality, linear convex combination technique and the free-weight matrix method, several novel sufficient conditions are derived to ensure the global asymptotic stability of the equilibrium point of the considered networks in mean square. The proposed results, which do not require the differentiability of the activation functions, can be easily checked via Matlab software. Finally, two numerical examples are given to demonstrate the effectiveness and less conservativeness of our theoretical results over existing literature.

Modeling of batch processes using explicitly time-dependent artificial neural networks

A neural network architecture incorporating time dependency explicitly, proposed recently, for modeling nonlinear nonstationary dynamic systems is further developed in this paper, and three alternate configurations are proposed to represent the dynamics of batch chemical processes. The first configuration consists of L subnets, each having M inputs representing the past samples of process inputs and output; each subnet has a hidden layer with polynomial activation function; the outputs of the hidden layer are combined and acted upon by an explicitly time-dependent modulation function. The outputs of all the subnets are summed to obtain the output prediction. In the second configuration, additional weights are incorporated to obtain a more generalized model. In the third configuration, the subnets are eliminated by incorporating an additional hidden layer consisting of L nodes. Backpropagation learning algorithm is formulated for each of the proposed neural network configuration to determine the weights, the polynomial coefficients, and the modulation function parameters. The modeling capability of the proposed neural network configuration is evaluated by employing it to represent the dynamics of a batch reactor in which a consecutive reaction takes place. The results show that all the three time-varying neural networks configurations are able to represent the batch reactor dynamics accurately, and it is found that the third configuration is exhibiting comparable or better performance over the other two configurations while requiring much smaller number of parameters. The modeling ability of the third configuration is further validated by applying to modeling a semibatch polymerization reactor challenge problem. This paper illustrates that the proposed approach can be applied to represent dynamics of any batch/semibatch process.

Delay-dependent stability criteria for reaction–diffusion neural networks with time-varying delays

This paper studies the global asymptotic stability problem of a class of reaction–diffusion neural networks with time-varying delays. To overcome the difficulty caused by the partial differential term, a novel Lyapunov–Krasovskii functional is proposed, and a partial differential equation technique together with a linear operator approach are also applied to obtain the delay-dependent stability criteria, which are less conservative than the existing results. Finally, simulation examples are given to verify and illustrate the theoretical analysis.

Robust exponential stability of uncertain stochastic neural networks with distributed delays and reaction-diffusions

This paper considers the problem of stability analysis for uncertain stochastic neural networks with distributed delays and reaction-diffusions. Two sufficient conditions for the robust exponential stability in the mean square of the given network are developed by using a Lyapunov-Krasovskii functional, an integral inequality, and some analysis techniques. The conditions, which are expressed by linear matrix inequalities, can be easily checked. Two simulation examples are given to demonstrate the reduced conservatism of the proposed conditions.

Convergence dynamics of stochastic Cohen-Grossberg neural networks with unbounded distributed delays

This paper addresses the issue of the convergence dynamics of stochastic Cohen-Grossberg neural networks (SCGNNs) with white noise, whose state variables are described by stochastic nonlinear integro-differential equations. With the help of Lyapunov functional, semi-martingale theory, and inequality techniques, some novel sufficient conditions on pth moment exponential stability and almost sure exponential stability for SCGNN are given. Furthermore, as byproducts of our main results, some sufficient conditions for checking stability of deterministic CGNNs with unbounded distributed delays have been established. Especially, even when the spectral radius of the coefficient matrix is greater than 1, in some cases our theory is also effective.