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

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.

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.

Anti-Synchronization in Fixed Time for Discontinuous Reaction-Diffusion Neural Networks With Time-Varying Coefficients and Time Delay

This paper studies the fixed-time anti-synchronization (FTAS) of discontinuous reaction-diffusion neural networks (DRDNNs) with both time-varying coefficients and time delay. First, differential inclusion theory is used to deal with the influence caused by discontinuous activations. In addition, a new fixed-time convergence theorem is used to handle the time-varying coefficients. Second, a novel state-feedback control algorithm and integral state-feedback control algorithm are proposed to realize FTAS of DRDNNs. During the generalized (adaptive) pinning control strategy, a guideline is proposed to select neurons to pin the designed controller. Furthermore, we present several criteria on FTAS by using the generalized Lyapunov function method. Different from the traditional Lyapunov function with negative definite derivative, the derivative of the Lyapunov function can be positive in this paper. Finally, we give two numerical simulations to substantiate the merits of the obtained 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.

Stability Analysis of Genetic Regulatory Networks With Switching Parameters and Time Delays

This paper is concerned with the exponential stability analysis of genetic regulatory networks (GRNs) with switching parameters and time delays. In this paper, a new integral inequality and an improved reciprocally convex combination inequality are considered. By using the average dwell time approach together with a novel Lyapunov-Krasovskii functional, we derived some conditions to ensure the switched GRNs with switching parameters and time delays are exponentially stable. Finally, we give two numerical examples to clarify that our derived results are effective.

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.

Neural-Network-Based Robust Optimal Tracking Control for MIMO Discrete-Time Systems With Unknown Uncertainty Using Adaptive Critic Design

This paper is concerned with the robust optimal tracking control strategy for a class of nonlinear multi-input multi-output discrete-time systems with unknown uncertainty via adaptive critic design (ACD) scheme. The main purpose is to establish an adaptive actor-critic control method, so that the cost function in the procedure of dealing with uncertainty is minimum and the closed-loop system is stable. Based on the neural network approximator, an action network is applied to generate the optimal control signal and a critic network is used to approximate the cost function, respectively. In contrast to the previous methods, the main features of this paper are: 1) the ACD scheme is integrated into the controllers to cope with the uncertainty and 2) a novel cost function, which is not in quadric form, is proposed so that the total cost in the design procedure is reduced. It is proved that the optimal control signals and the tracking errors are uniformly ultimately bounded even when the uncertainty exists. Finally, a numerical simulation is developed to show the effectiveness of the present approach.

Nonfragile Exponential Synchronization of Delayed Complex Dynamical Networks With Memory Sampled-Data Control

This paper considers nonfragile exponential synchronization for complex dynamical networks (CDNs) with time-varying coupling delay. The sampled-data feedback control, which is assumed to allow norm-bounded uncertainty and involves a constant signal transmission delay, is constructed for the first time in this paper. By constructing a suitable augmented Lyapunov function, and with the help of introduced integral inequalities and employing the convex combination technique, a sufficient condition is developed, such that the nonfragile exponential stability of the error system is guaranteed. As a result, for the case of sampled-data control free of norm-bound uncertainties, some sufficient conditions of sampled-data synchronization criteria for the CDNs with time-varying coupling delay are presented. As the formulations are in the framework of linear matrix inequality, these conditions can be easily solved and implemented. Two illustrative examples are presented to demonstrate the effectiveness and merits of the proposed feedback control.

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.

Pinning Impulsive Synchronization of Reaction-Diffusion Neural Networks With Time-Varying Delays

This paper investigates the exponential synchronization of reaction-diffusion neural networks with time-varying delays subject to Dirichlet boundary conditions. A novel type of pinning impulsive controllers is proposed to synchronize the reaction-diffusion neural networks with time-varying delays. By applying the Lyapunov functional method, sufficient verifiable conditions are constructed for the exponential synchronization of delayed reaction-diffusion neural networks with large and small delay sizes. It is shown that synchronization can be realized by pinning impulsive control of a small portion of neurons of the network; the technique used in this paper is also applicable to reaction-diffusion networks with Neumann boundary conditions. Numerical examples are presented to demonstrate the effectiveness of the theoretical results.

Synchronization criteria for generalized reaction-diffusion neural networks via periodically intermittent control

In this paper, the synchronization problem for a class of generalized neural networks with time-varying delays and reaction-diffusion terms is investigated concerning Neumann boundary conditions in terms of p-norm. The proposed generalized neural networks model includes reaction-diffusion local field neural networks and reaction-diffusion static neural networks as its special cases. By establishing a new inequality, some simple and useful conditions are obtained analytically to guarantee the global exponential synchronization of the addressed neural networks under the periodically intermittent control. According to the theoretical results, the influences of diffusion coefficients, diffusion space, and control rate on synchronization are analyzed. Finally, the feasibility and effectiveness of the proposed methods are shown by simulation examples, and by choosing different diffusion coefficients, diffusion spaces, and control rates, different controlled synchronization states can be obtained.

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.

Synchronization for coupled neural networks with interval delay: a novel augmented Lyapunov-Krasovskii functional method

This paper is concerned with the synchronization problems for an array of neural networks with hybrid coupling and interval time-varying delay. First, a novel augmented Lyapunov-Krasovskii functional (LKF) method is proposed to develop delay-dependent synchronization criteria for the networks, which makes use of more relaxed conditions by employing the new type of augmented matrices with Kronecker product operation. The proposed method can handle a multitude of Kronecker product operations in the LKF and alleviates the requirements of the positive definiteness of some conditional matrices which are usually considered in the existing methods for complex networks. This leads to a significant improvement in the performance of the synchronization criteria, i.e., less conservative synchronization results can be obtained. Meanwhile, the case of fast time-varying delay can also be handled by the proposed method. Furthermore, based on the derived criteria, a robust synchronization criterion is obtained for the system with uncertainties both in coefficient and coupling matrix terms. Since an expression based on linear matrix inequality is used, the proposed criteria can be easily checked in practice. Finally, numerical examples are provided to show the effectiveness of the proposed method.