Reaction-diffusion Neural Networks Research Articles

Intermittent Boundary Control for Synchronization of Fractional Delay Neural Networks With Diffusion Terms

This article studies the synchronization of new coupled fractional delayed reaction–diffusion neural networks with reaction terms satisfying the global Lipschitz condition via time-continuous and time-discontinuous boundary controllers. The realization of neural networks inevitably involves diffusion phenomena and time delays, and all the neurons of neural networks are interrelated. Considering these aspects, this study focuses on coupled fractional neural networks with time-delay and diffusion terms. A state-dependent boundary control (BC) is designed for when the state information is available, and a criterion is presented to ensure the synchronization of the considered systems. Considering the advantages of a time-discontinuous controller, an intermittent BC and a criterion of synchronization are given. When the state information cannot be fully obtained, a boundary-output-based observer is provided for estimating the states. Then, an observer-based intermittent boundary controller is given to ensure the synchronization. From the given criteria, the effects of time delay and the control time length on synchronization are analyzed. This research involves two key challenges: 1) consideration of the BC and intermittent control parameters in the system performance analysis and 2) clarification of the influence of system parameters on synchronization. These challenges are addressed using Poincaré’s inequality, the fractional Razumikhin-type theorem, and several properties of the Mittag-Leffler function are used to deal with the above difficulties. Examples show that our results are valid.

Event-triggered boundary feedback synchronisation control of nonlinear coupling reaction–diffusion neural networks

This paper aims to deal with Robin boundary synchronisation control for a kind of nonlinear coupling reaction–diffusion neural networks (RDNNs) with non-identical nodes. Some novel sufficient conditions involving the parameters of event-triggered mechanism are proposed to achieve synchronisation of this RDNNs. By utilising Lyapunov's method and Wiringer inequality, the error closed-loop systems with the designed event-triggered controller are proved to be well-posed and the RDNNs to be exponentially synchronised with target system. Moreover, Zeno behaviour of event-triggered control is avoided. Finally, the effectiveness of our theoretical results is verified by one numerical example.

Stability Analysis of Second-Order Linear PDEs on Time Scales

This paper presents the stability analysis of a class of second-order linear partial differential equations (PDEs) on time scales with diffusion operator and first-order partial derivative. According to the time scale theory, the Lyapunov functional method, and some inequality techniques, sufficient conditions for exponential stability are strictly obtained, and the results are generalized for that where both the discrete-time and continuous-time cases are considered jointly. In addition, the theoretical results are applied to exponential synchronization of reaction-diffusion neural networks (RDNNs). Simulation examples are given to verify the feasibility of our results.

Synchronization of fractional-order reaction-diffusion neural networks via mixed boundary control

This article studies the synchronization issue of fractional reaction-diffusion neural networks (FRDNNs) with time delay and mixed boundary condition. First, a novel boundary controller with constant-valued gain is designed, which only relies on the boundary state information. Subsequently, by virtue of Lyapunov direct technique and LMI approach, the Mittag–Leffler synchronization conditions are established. Besides, to effectively regulate the control gain, a fractional-order adaptive boundary controller is developed and the adaptive synchronization of FRDNNs is rigorously analyzed. Note that, the above control strategies are also workable for traditional integer-order reaction-diffusion neural networks. The developed theoretical analysis is supported eventually via a numerical example.

Sampling-Based Event-Triggered Exponential Synchronization for Reaction-Diffusion Neural Networks.

In this article, the exponential synchronization control issue of reaction-diffusion neural networks (RDNNs) is mainly resolved by the sampling-based event-triggered scheme under Dirichlet boundary conditions. Based on the sampled state information, the event-triggered control protocol is updated only when the triggering condition is met, which effectively reduces the communication burden and saves energy. In addition, the proposed control algorithm is combined with sampled-data control, which can effectively avoid the Zeno phenomenon. By thinking of the proper Lyapunov-Krasovskii functional and using some momentous inequalities, a sufficient condition is obtained for RDNNs to achieve exponential synchronization. Finally, some simulation results are shown to demonstrate the validity of the algorithm.

Synchronization of reaction‐diffusion neural networks with distributed delay via quantized boundary control

SummaryThis paper investigates the synchronization of reaction‐diffusion neural networks (RDNNs) with distributed delay via quantized boundary control. To reduce the communication burden, a novel control strategy combined boundary control and logarithmic quantizer is proposed, and two controllers respectively subject to constant and adaptive coefficients are carried out. Worth mentioning that the adaptive feedback gain is a matrix in this paper rather than a one‐dimensional variable in most of the existing literatures. Using the Lyapunov functional, the sufficient conditions for delay‐dependent synchronization are obtained through linear matrix inequalities. The effectiveness of the proposed control strategy is illustrated via two examples.

Finite time passivity analysis for Caputo fractional BAM reaction–diffusion delayed neural networks

The issues of finite time passivity are explored for BAM reaction–diffusion neural networks including discrete delayed and Caputo fractional partial differential operator. With the help of fractional Lyapunov function and existing passivity definition (integral form or fractional integral form), four finite time passivity definitions are given out in Caputo-type fractional derivative form under the double-layer network structure. In this way, the fractional derivative of Lyapunov function can be directly obtained to judge whether the system is passive. Constructing a suitable mixed controller with sign function, several passivity conditions are established as matrix inequality form by utilizing the Green’s theorem, Jensen’s integral inequality, Lyapunov functional approach and four novel definitions. At last, one example is appeared to better illustrate the obtained conditions.

Cluster synchronization for directed coupled inertial reaction-diffusion neural networks with nonidentical nodes via non-reduced order method

The cluster synchronization issues are investigated for directed coupled inertial reaction-diffusion neural networks (CIRDNNs) with nonidentical nodes by imposing two effective pinning control. A novel Lyapunov-Krasovskii functional (LKF) is established to directly analyze the dynamic behavior of CIRDNNs and deal with reaction-diffusion term, inertia term and coupling term. Moreover, based on different desired cluster synchronization states including a set of un-decoupled trajectories and the particular solutions of the decoupled node systems, two class of synchronization criteria in view of algebraic inequalities are derived under two different communication topologies, respectively. Finally, two typical examples are given to verify the theoretical results.

Asymptotic Behavior of Delayed Reaction-Diffusion Neural Networks Modeled by Generalized Proportional Caputo Fractional Partial Differential Equations

In this paper, a delayed reaction-diffusion neural network model of fractional order and with several constant delays is considered. Generalized proportional Caputo fractional derivatives with respect to the time variable are applied, and this type of derivative generalizes several known types in the literature for fractional derivatives such as the Caputo fractional derivative. Thus, the obtained results additionally generalize some known models in the literature. The long term behavior of the solution of the model when the time is increasing without a bound is studied and sufficient conditions for approaching zero are obtained. Lyapunov functions defined as a sum of squares with their generalized proportional Caputo fractional derivatives are applied and a comparison result for a scalar linear generalized proportional Caputo fractional differential equation with several constant delays is presented. Lyapunov functions and the comparison principle are then combined to establish our main results.

Synchronization of reaction–diffusion neural networks with sampled-data control via a new two-sided looped-functional

In this paper, the exponential synchronization problem of reaction–diffusion neural networks with sampled-data control is addressed via looped-functional approach. Considering the transmission delay of sampled-data controller, a two-sided looped-functional is designed to obtain the synchronization conditions, which are less conservative than those with the traditional Lyapunov method. The research results are applied to Takagi–Sugeno (T–S) fuzzy models with reaction–diffusion terms. Three numerical examples are presented to show the feasibility and effectiveness of our methods.

A review of dynamics analysis of neural networks and applications in creation psychology

<abstract><p>The synchronization problem and the dynamics analysis of neural networks have been thoroughly explored, and there have been many interesting results. This paper presents a review of the issues of synchronization problem, the periodic solution and the stability/stabilization with emphasis on the memristive neural networks and reaction-diffusion neural networks. First, this paper introduces the origin and development of neural networks. Then, based on different types of neural networks, some synchronization problems and the design of the controllers are introduced and summarized in detail. Some results of the periodic solution are discussed according to different neural networks, including bi-directional associative memory (BAM) neural networks and cellular neural networks. From the perspective of memristive neural networks and reaction-diffusion neural networks, some results of stability and stabilization are reviewed comprehensively with latest progress. Based on a review of dynamics analysis of neural networks, some applications in creation psychology are also introduced. Finally, the conclusion and the future research directions are provided.</p></abstract>

Temporal-Intermittent-Spatio-Sampling Control for Stochastic Switched RDNNs With Impulsive Effects

This brief mainly investigates the stabilization problem for a class of state-dependent switched reaction-diffusion neural networks (RDNNs) with stochastic disturbance and impulsive effects. First, a hybrid control strategy of time domain intermittent-spatial domain point sampling is proposed, which can greatly reduce the control cost. Then, by using the constructed multi-partition Lyapunov function and the convex combination method, two different stabilization criteria are derived for the target system with stabilizing impulse and destabilizing impulse, which are shown in the form of easy-to-calculate LMIs. Moreover, the multi-gain phenomenon is effectively avoided through the single decision-multivariable technique, which facilitates the operation of the actual system. Note that our results are more practical than those established only on pure stabilizing impulse and destabilizing impulse. Particularly, one of the results shows that even if the switched RDNN with destabilizing impulse is unstable, it can be stabilized by the hybrid intermittent sampling control strategy proposed in this brief. Finally, numerical examples show the rationality and validity of the new results.

Quasisynchronization of Reaction–Diffusion Neural Networks Under Deception Attacks

This study focuses on the quasisynchronization problem for reaction–diffusion neural networks (RDNNs) in the presence of deception attacks. Under deception attacks, a time–space sampled-data (TSSD) control mechanism is proposed for RDNNs. Compared with traditional control strategies, the proposed control mechanism can not only save network bandwidth but also improve the cybersecurity of communications. Inspired by Halanay’s inequality, a new inequality is proposed, which can be effectively applied to the quasisynchronization problem for dynamical systems. Then, by using this inequality and the Lyapunov functional approach, quasisynchronization criteria are set for RDNNs. The desired control gain is gained from solving a group of linear matrix inequalities. Moreover, in the absence of deception attacks, the exponential synchronization problem is studied for RDNNs. In the end, simulation results are given to demonstrate the usefulness of the theoretical analysis.

Robust Composite H∞ Synchronization of Markov Jump Reaction-Diffusion Neural Networks via a Disturbance Observer-Based Method.

This article focuses on the composite H∞ synchronization problem for jumping reaction-diffusion neural networks (NNs) with multiple kinds of disturbances. Due to the existence of disturbance effects, the performance of the aforementioned system would be degraded; therefore, improving the control performance of closed-loop NNs is the main goal of this article. Notably, for these disturbances, one of them can be described as a norm-bounded, and the other is generated by an exogenous model. In order to reject the above one kind of disturbance, a disturbance observer is developed. Furthermore, combining the disturbance observer approach and conventional state-feedback control scheme, a composite disturbance rejection controller is specifically designed to compensate for the influences of the disturbances. Then, some criteria are established based on the general Lyapunov stability theory, which can ensure that the synchronization error system is stochastically stable and satisfies a fixed H∞ performance level. A simulation example is finally presented to verify the availability of our developed method.

Synchronization and adaptive control for coupled fractional-order reaction–diffusion neural networks with multiple couplings

In this paper, two kinds of coupled fractional-order reaction–diffusion neural networks (CFORDNNs) with multiple state couplings or spatial diffusion couplings are proposed. By resorting to the Laplace transform and the properties of Mittag-Leffler functions, sufficient synchronization conditions are derived for the concerned network models. In addition, to guarantee the synchronization of these two networks, several appropriate adaptive control schemes are also developed. Ultimately, the validity of the devised adaptive strategies are verified by adopting some numerical examples with simulation results.

Global Mittag-Leffler Stability of the Delayed Fractional-Coupled Reaction-Diffusion System on Networks Without Strong Connectedness.

In this article, we mainly consider the existence of solutions and global Mittag-Leffler stability of delayed fractional-order coupled reaction-diffusion neural networks without strong connectedness. Using the Leary-Schauder's fixed point theorem and the Lyapunov method, some criteria for the existence of solutions and global Mittag-Leffler stability are given. Finally, the correctness of the theory is verified by a numerical example.

Lag [formula omitted] synchronization in coupled reaction–diffusion neural networks with multiple state or derivative couplings

This paper mainly attempts to discuss lag H∞ synchronization in multiple state or derivative coupled reaction–diffusion neural networks without and with parameter uncertainties. Firstly, we respectively propose two types of reaction–diffusion neural networks with multiple state and derivative couplings subject to parameter uncertainties. Secondly, by exploiting designed state feedback controllers, several criteria of the lag H∞ synchronization for these two networks are developed based on Lyapunov functional and inequality techniques. Thirdly, lag H∞ synchronization issues of these two networks are also coped with by virtue of devised adaptive control strategies. Finally, we provide two numerical examples to verify the obtained lag H∞ synchronization criteria.

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.

Spatiotemporal dynamics on a class of [formula omitted]-dimensional reaction–diffusion neural networks with discrete delays and a conical structure

In this paper, the stability and Hopf bifurcation problems on a class of high-order reaction–diffusion neural networks are investigated, where the conical structure and signal transmission delays are introduced to make the model more accurate. By the linear stability analysis, Hopf bifurcation theory and Turing instability theorem, some criteria on the local stability and Hopf bifurcation of this system are established by selecting the time delay as bifurcation parameter. Especially, the Coates formula and the holistic element method are taken to determine the critical value, at which the purely imaginary root appears in the associated high-degree characteristic equation. On the other hand, the effects of diffusion terms on the spatial distribution of periodic solutions are explored. New sufficient conditions are derived to guarantee the occurrence of spatially homogeneous periodic solutions. Finally, some numerical examples are provided to validate the main results. It is found that the critical value can also be affected by the network size and the self-feedback rate.

Output synchronization of reaction–diffusion neural networks under random packet losses via event-triggered sampled–data control

For space-varying reaction–diffusion neural networks (RDNNs), this article mainly studies the output synchronization via an event-triggered sampled-data (ETSD) control under spatially point measurements (SPMs) with random packet losses. To reduce the communication burden, an ETSD control scheme is adopted to decrease the unnecessary SD and the update frequency of the controller. Meanwhile, the problem of random packet losses in the communication channels from controller to actuator is considered. Some synchronization criteria based on spatial linear matrix inequalities (SLMIs) are established through an ETSD controller under SPMs with random packet losses to guarantee the mean square exponential stability of synchronization error system with drive and response dynamics via utilizing inequality techniques and Lyapunov functional. Furthermore, we express SLMIs as LMIs for solving the ETSD control design problem for output synchronization of space-varying RDNNs. Finally, the effectiveness of the proposed method is demonstrated by one numerical example.