Reaction-diffusion Terms Research Articles

Stability and bipartite synchronization of fractional-order coupled reaction–diffusion neural networks under unbalanced graph

This paper delves into the analysis of stability and bipartite synchronization (BS) in fractional-order coupled neural networks with reaction–diffusion terms (FORDNNs) under structurally unbalanced signed graphs. Central to our investigation is the novel observation in structurally unbalanced graphs without cycles, where specific nodes under particular structures are capable of achieving BS, while others maintain stability. Subsequently, for unbalanced graphs with negative rooted cycles, we present criteria that guarantee the stability of FORDNNs. Moreover, under structurally balanced graphs, sufficient conditions for achieving BS are established. A novel approach is introduced, wherein synchronization is attained without the presence of a leader node, marking a significant departure from conventional models. Finally, illustrative examples are provided, demonstrating the effectiveness and relevance of our findings.

Exponential Synchronization Control of Reaction-Diffusion Fuzzy Memristive Neural Networks: Hardy-Poincarè Inequality.

This article is devoted to solving the exponential synchronization problem of a new type of fuzzy memristive neural network with reaction-diffusion terms. By introducing adaptive laws, two controllers are designed. After combining the inequality technique with the Lyapunov function approach, some easily verified sufficient conditions are established to ensure the exponential synchronization of the reaction-diffusion fuzzy memristive system under the proposed adaptive scheme. In addition, by using the Hardy-Poincarè inequality, the diffusion terms are estimated associated with the information of the reaction-diffusion coefficients and the regional feature, which improves some existing conclusions. Finally, an illustrative example is presented to demonstrate the validity of the theoretical results.

Finite-time quasi-projective synchronization of fractional-order reaction-diffusion delayed neural networks

This paper investigates the finite-time quasi-projective synchronization (FTQPS) issue of fractional-order reaction-diffusion neural networks (FORDNNs). To the best of our knowledge, this paper introduces the concept of FTQPS for the first time. First, an integral-type Lyapunov function is constructed relying on the characterization of the reaction-diffusion term and some inequality methods. Subsequently, the nonlinear feedback control strategy is designed to achieve the FTQPS goal and some sufficient conditions are obtained to guarantee FTQPS of FORDNNs. Further, the system's synchronization speed is measured by estimating the settling time. It should be noted that the above control strategy is also applicable to conventional integer-order reaction-diffusion neural networks with time delays. Finally, a numerical example is used to illustrate the validity of the theoretical analysis presented.

Global Exponential Stability of Impulsive Delayed Neural Networks with Parameter Uncertainties and Reaction–Diffusion Terms

Global Exponential Stability of Impulsive Delayed Neural Networks with Parameter Uncertainties and Reaction–Diffusion Terms

New synchronization results for fractional neural networks with parameter uncertainty and reaction–diffusion terms

New synchronizations of fractional neural networks (FNNs) with uncertain parameters and reaction diffusion terms are investigated in this paper. Firstly, considering a system with mixed boundary conditions, a mixed boundary controller is designed. Sufficient criteria for the global exponential synchronization of the designed FNNs are given through Lyapunov function, inequality techniques and Razumikhin theorem. Furthermore, the exponential synchronization of the system can be explored under Neumann boundary conditions. An example is provided to verify the theoretical part and present the synchronization results for different fractional orders and diffusion coefficients.

Novel Caputo fractional differential approach and hybrid control scheme for finite‐time synchronization of nonidentical delayed reaction–diffusion neural networks

AbstractThis work focuses on the finite‐time synchronization (F‐TS) for nonidentical delayed neural networks (DNNs) including Caputo fractional partial differential operator and reaction–diffusion terms. A novel F‐TS lemma is proposed by constructing a new Caputo differential inequality. Applying the new lemma and designing two hybrid controllers with time delay and the sign function, the F‐TS criteria on the introduced Caputo reaction–diffusion nonidentical DNNs under the Neumann boundary condition are derived by making use of the Filippov differential inclusion, Green's theorem, and fractional Razumikhin theorem. The conditions are expressed through the algebraic inequality which can greatly decrease the computation in checking the F‐TS performance. Moreover, the validity and correctness of the F‐TS results are illustrated by selecting various orders, spatial positions, and diffusion parameters.

Global exponential synchronization of BAM memristive neural networks with mixed delays and reaction–diffusion terms

Based on m-norm, this paper investigates global exponential synchronization (GES) for BAM memristive neural networks (BAMMNNs) with mixed delays and reaction–diffusion (RD) terms. Different from the existing literatures, this paper discusses the GES of the NNs based on a new integral inequality with infinite distributed delay. This method is based on inequality technique and comparison principle, which makes the form of Lyapunov function and controller more simple. Next, by introducing two different control strategies and the concept of driven response, two sufficient conditions are got to ensure GES of the proposed system. It is noteworthy that the results obtained by algebraic inequality are extension of the previous conclusions. Finally, two instances verify the correctness of the conclusions.

General Decay Synchronization of State and Spatial Diffusion Coupled Delayed Memristive Neural Networks With Reaction-diffusion Terms

General Decay Synchronization of State and Spatial Diffusion Coupled Delayed Memristive Neural Networks With Reaction-diffusion Terms

Event-Triggered Synchronization of Coupled Neural Networks with Reaction–Diffusion Terms

This paper focuses on the event-triggered synchronization of coupled neural networks with reaction–diffusion terms. At first, an effective event-triggered controller was designed based on time sampling. It is worth noting that the data of the controller for this type can be updated only when corresponding triggering conditions are satisfied, which can significantly reduce the communication burden of the control systems compared to other control strategies. Furthermore, some sufficient criteria were obtained to ensure the event-triggered synchronization of the considered systems through the use of an inequality techniques as well as the designed controller. Finally, the validity of the theoretical results was confirmed using numerical examples.

Synchronization of delayed stochastic reaction–diffusion Hopfield neural networks via sliding mode control

Synchronization of stochastic reaction–diffusion Hopfield neural networks with s-delays via sliding mode control is investigated in this article. To begin with, we choose suitable functional space for state variables, then the system is transformed into a functional differential equation in an infinite-dimensional Hilbert space by using appropriate functional analysis technique. Based on above preliminary preparation, sliding mode control (SMC) is constructed to drive the error trajectory into the designed switching surface. Specifically, the switching surface is constructed as linear combination of state variables, which is related to control gains. Then novel SMC law is designed which involving delay, reaction diffusion term, and reaching law. Furthermore, the criterion of mean-square exponential synchronization for stochastic delayed reaction–diffusion Hopfield neural networks with s-delays is given in the form of matrix form. This criterion is less restrictive and easy to check in computer. Meanwhile, a different novel Lyapunov–Krasovskii functional (LKF) mixed with Itô’s formula, Young inequality, Hanalay inequality is employed in this proof procedure. At last, a numerical example is presented to validate the availability of theoretical result. The simulation is based on the finite difference method, and numerical result coincides with the theoretical result proposed.

Distributed sliding mode control for a class of impulsive uncertain delayed partial differential equations via sliding mode compensator approach

The distributed sliding mode control (SMC) method of delayed partial differential equations (PDEs)with information of periodical impulse and reaction–diffusion terms (RDTs) is discussed in the article. Firstly, by adopting the sliding mode compensator approach, the switching function including impulsive effects is constructed, which is continuous along the system states. The sliding motion equation is then derived, and its parameters depend on that of sliding mode compensator.Via the direct linear matrix inequality method(DLIM), the exponential stability of sliding motion is analyzed. The proposed distributed SMC strategy can eliminate the impulsive effects and drive the closed-loop state onto a specific sliding manifold in limited time. Finally, a numerical simulation is presented to certify the availability of the proposed scheme.

Fuzzy adaptive event-triggered synchronization control mechanism for T–S fuzzy RDNNs under deception attacks

In this paper, a fuzzy-dependent adaptive event-triggered mechanism (FAETM) for synchronizing Takagi–Sugeno (T–S) fuzzy reaction–diffusion neural networks (RDNNs) is developed while considering deception attacks. Firstly, a general neural network model considering both fuzzy logic rules and reaction–diffusion terms is established. Secondly, a FAETM based on an aperiodic sampling period is presented under deception attacks to alleviate the communication burden, wherein the adaptive threshold function is dependent on membership functions. Moreover, a membership-function-dependent asymmetric Lyapunov functional (LF) is constructed, and some positive-definite and symmetric constraints of matrices are removed as a result. Based on the LF method and integral inequality techniques, the H∞ synchronization criteria for the T–S fuzzy RDNNs are presented by linear matrix inequalities (LMIs). Finally, one simulation example is exploited to demonstrate the feasibility and validity of the established method, and the outcome is applied to image secure communication.

Stability of Fractional Reaction-Diffusion Memristive Neural Networks Via Event-Based Hybrid Impulsive Controller

This article explores the asymptotic stability of fractional delayed memristive neural networks with reaction-diffusion terms. A novel hybrid impulsive controller triggered by a specific event is proposed to stabilize the network, thereby replacing the conventional approach of modifying network parameters. The proposed controller is proven to prevent Zeno behavior. Sufficient conditions for the asymptotic stability of fractional delayed memristive neural networks with reaction-diffusion terms are established through Lyapunov direct method, inequality techniques, Green’s theorem and impulse analysis. Furthermore, the proposed controller is theoretically shown to be more resource-efficient than the conventional one, and our work extends existing research to make it more suitable for practical application such as pattern recognition, image processing and so on. Finally, an example is provided to illustrate the validity of the findings.

Boundary Disturbance Rejection Control for Fractional-order Multi-agent Systems with Reaction-diffusion

This paper focuses on boundary disturbance rejection control (BDRC) for fractional-order multi-agent systems (FMASs) with reaction diffusion terms. Firstly, based on the output information, the corresponding observer is designed to obtain information about all the agents and the unknown external perturbations. Then, a novel distributed boundary control protocol is designed based on the observed values. Thus, the convergence analysis of the closed-loop system is accomplished through the Mittag-Leffler stability theory and the linear matrix inequality condition. Finally, a simulation example is used to verify the correctness of the results.

Prefixed-Time Local Intermittent Sampling Synchronization of Stochastic Multicoupling Delay Reaction-Diffusion Dynamic Networks.

This article focuses on the problem of prefixed-time synchronization for stochastic multicoupled delay dynamic networks with reaction-diffusion terms and discontinuous activation by means of local intermittent sampling control. Notably, unlike the existing common fixed-time synchronization, this article puts forward a new synchronization concept, prefixed-time synchronization, based on the fact that stochastic noise and discontinuous activation can be seen everywhere in practical engineering, which can effectively perfect and improve the existing works. Specifically, a local intermittent in the time domain and point sampling control strategy in the spatial domain is proposed instead of a simple single intermittent control approach, which greatly reduces the control cost. In addition, by some effective means, including the famous Young's inequality, Jensen's inequality, and Hölder's inequality, we obtain two different synchronization criteria of the networks without delay and with multicoupling delays and deeply reveal the quantitative relationship among control period, point sampling length, and network scale. Finally, a numerical example is given to verify the effectiveness of the developed method and the practicability by Chua's circuit model.

Synchronization of fractional-order delayed coupled networks with reaction–diffusion terms and Neumann boundary value conditions

This paper investigates the adaptive pinning strategies for achieving synchronization of fractional-order delayed coupled networks with reaction–diffusion terms and a digraph topology by incorporating Neumann boundary value conditions. By employing the inf–sup method, a novel fractional-order inequality is proved. The classical Poincaré inequality is also extended by utilizing the Hölder inequality. Two types of control laws are developed to achieve synchronization: one with control gains dependent solely on time, and another with control gains dependent on both space and time. For each case, adaptive control laws and synchronization criteria based on matrix inequalities are proposed. Finally, the effectiveness of the synchronization results is demonstrated through two numerical examples.

Event-triggered impulsive cluster synchronization of coupled reaction–diffusion neural networks and its application to image encryption

This paper investigates the cluster synchronization of coupled neural networks with reaction–diffusion terms. With the help of impulsive control strategies, some cluster synchronization criteria are proposed by an appropriate event-triggered mechanism. A numerical example is given to verify the validity of the theoretical results. Additionally, the proposed event-triggered impulsive synchronization is successfully applied to image encryption with encouraging cryptanalysis results demonstrating its strong ability to efficiently encrypt images.

Preassigned-time synchronization for complex-valued memristive neural networks with reaction–diffusion terms and Markov parameters

This study addresses the preassigned-time synchronization for complex-valued memristive neural networks with reaction–diffusion terms and Markov parameters. Employing a preassigned-time stable control strategy, two distinct controllers with varying power exponent parameters are designed to ensure that synchronization can be achieved within a predefined time frame. Unlike existing finite/fixed-time results, a priori specification of the settling time is addressed. Furthermore, Green’s formula and boundary conditions are efficiently applied to overcome potential symmetry loss. Additionally, the activation function’s constraint range is more lenient compared to existing constraints. Finally, the effectiveness of the presented methods are demonstrated through two examples.

H∞ Bipartite Synchronization Control of Markov Jump Cooperation-Competition Networks With Reaction-Diffusions.

This article is concerned with the bipartite synchronization problem of coupled switching neural networks with cooperative-competitive interactions and reaction-diffusion terms. Different from the existing literature, the networked systems under investigation possess the relationship of cooperation and competition among nodes. Notably, the switching topology is described by a signed graph subject to the Markov jump process with the coexistence of positive and negative interaction weights. Specifically, a positive weight indicates an alliance relationship between two nodes and a negative one shows an adversary relationship. This article aims to design a bipartite synchronization controller for the aforementioned networks with the switching topology such that a prescribed H∞ bipartite synchronization is satisfied. Then, some sufficient criteria to ensure the stochastic stability of bipartite synchronization error systems are established in view of an appropriate Lyapunov function. Finally, two simulation examples are presented to verify the validity of the proposed bipartite synchronization control method.

Mean-square finite-time synchronization of stochastic competitive neural networks with infinite time-varying delays and reaction–diffusion terms

This paper focuses on the mean-square finite-time synchronization (MFTS) of stochastic competitive neural networks with infinite time-varying discrete delays and reaction–diffusion terms (IRSCNNs). Different from other articles, this paper presents a new approach, which does not use finite-time stability theorem but uses integral inequality, Gronwall-type inequality and comparison strategy to study MFTS. In addition, two control schemes are designed: a feedback control scheme and a new adaptive control strategy, which can achieve MFTS of IRSCNNs. Finally, the correctness of the results is verified by examples.