This article, respectively, tackles two types of lag consensus problems for a class of nonlinear multiagent systems (MASs) suffering external disturbances. Based on a nodes-based adaptive event-triggered control (ETC) protocol, the lag consensus can be realized for MASs by utilizing Barbalat’s lemma and inequality techniques. By employing the nodes-based adaptive ETC method, lag$H_{\infty } $consensus is also addressed for MASs. Furthermore, we also prove the facts that MASs with and without external disturbances under the nodes-based adaptive ETC schemes don’t exhibit Zeno behavior. In addition, two edge-based adaptive ETC strategies are also developed to deal with two kinds of lag consensus problems of MASs. Finally, the effectiveness of the devised nodes- and edges-based adaptive ETC protocols is demonstrated via two numerical examples.

This paper considers a type of high-order nonlinear multiagent systems (MASs), and the time-varying optimal lag formation control problem for such MAS is addressed. To eliminate the influences of the unknown system dynamics and external disturbance on single agent, a neural network (NN) identifier and disturbance observer based compensation controller is developed. Then, an optimal control law to ensure the practical time-varying lag formation for the MAS is introduced by exploiting the sliding-mode control (SMC) method and reinforcement learning (RL) technique. Finally, a numerical example is given to validate the effectiveness of the proposed optimal control law.

Based on PD and PI control methods, the lag output consensus problem for second-order nonlinear multiagent systems (MASs) without and with external disturbances is addressed, which generalizes the existing lag consensus results for MASs. Firstly, a lag output consensus condition is derived for the MAS by designing a proper PD control scheme. Furthermore, a PI control strategy is developed to ensure that the MAS realizes the lag output consensus. In addition, two lag${\mathcal {H}}_{\infty } $output consensus criteria are also established on the basis of PD and PI control schemes for the MAS with external disturbances. Lastly, two examples are utilized to verify the correctness and effectiveness of the devised control schemes. Note to Practitioners—Recently, virtual leader-based approach has been widely utilized to address the output consensus control problem for various MASs. Practically, each agent receiving its own output information may be later than the virtual leader. Therefore, it is more reasonable and meaningful to discuss the lag output consensus for MASs. Furthermore, considering that PID control is very simple and easy to implement, some PD and PI control strategies are developed to tackle the lag output consensus for MASs. In addition, since external disturbances are ubiquitous and may destroy the lag output consensus, the lag${\mathcal {H}}_{\infty } $output consensus for MASs is also investigated. Multiple single-link robot systems experiments verify that the approaches given in this paper are feasible and valid.

This paper studies the state observer design of a spatial two-dimensional (2-D) linear diffusion process described by a linear parabolic partial differential equation (PDE) under mobile sensors. Firstly, we analyze the well-posedness of the PDE system and give the structure form of the state observer with mobile sensors. Subsequently, according to the number of mobile sensors, the 2-D space domain is divided into multiple sub-domains, and the mobile sensors are guided by the projection operator method, which can guarantee that the mobile sensors can only move in their respective 2-D sub-domains. Then, in the light of Lyapunov theory, Poincaré-Wirtinger inequality and Barbalat lemma, we propose an observation-plus-guidance design method to ensure the asymptotic stability of the state estimation error system. In the designed mobile strategy, the actual guidance of mobile sensors is essentially a physical synthesis of two direction guidance laws, where two dimensional guidance laws are designed separately. Moreover, the existence condition of the observer is given by linear matrix inequalities. At last, a numerical example is provided to demonstrate the efficacy of the proposed design scheme. Note to Practitioners—For the actual physical temporal-space process, the spatial 2-D case makes more sense. With the increase of space dimension, the difficulty of system design increases sharply. The observer design methodology and mobile sensor guidance approach developed for spatial 1-D systems are not directly extendable to 2-D systems. However, there have also been few reports on the observer design of 2-D physical temporal-space process using mobile sensors, and it is still a challenging problem. In this article, this study addresses the problem of designing state observer for 2-D linear parabolic PDE systems utilizing mobile sensors. To prevent collisions among mobile sensors, the 2-D spatial domain is partitioned into several subregions, and the projection operator approach is employed to develop the motion strategy. Then, a Lyapunov-based observation-plus-guidance design method is provided to achieve the desired design goals. At last, the finite difference method is used to verify the design method.

This article deals with the bipartite synchronization in coupled fractional-order reaction-diffusion neural networks (CFRNN) with multiple state or multiple spatial diffusion couplings. A bipartite synchronization condition is derived for the multiple state CFRNN (MSCFRNN) with the help of the Lyapunov functional combined with inequality techniques, and an adaptive state-feedback control scheme is also devised to ensure the bipartite synchronization of the MSCFRNN. Similarly, several sufficient conditions are also given to guarantee the bipartite synchronization in the multiple spatial diffusion CFRNN (MSDCFRNN). Finally, the effectiveness of the devised control strategies is demonstrated through two numerical examples.

Multistate coupled coupled neural networks (MSCCNN) and multiderivative coupled coupled neural networks (MDCCNN) are introduced in this article, and the lag outer synchronization for these two networks are tackled. First, a lag outer synchronization criterion for MSCCNN is derived using a node-based adaptive event-triggered control scheme, and the fact that the Zeno behavior does not exist is also proved. Moreover, the edge-based adaptive event-triggered control method is also utilized to address the lag outer synchronization for MSCCNN, and the existence of Zeno behavior is ruled out. In addition, two lag outer synchronization criteria for MDCCNN are given on the basis of the node- and edge-based adaptive event-triggered control strategies, and the nonexistence of Zeno behavior is also established. Finally, two examples are provided to demonstrate the feasibility of the proposed control schemes.

This article is concerned with passivity and synchronization for fuzzy coupled reaction-diffusion neural networks (FCRDNNs) with multistate couplings or multiple spatial-diffusion couplings. First, by utilizing an adaptive state feedback controller, several passivity criteria for the FCRDNNs with multistate couplings are obtained. Moreover, a sufficient condition to guarantee the synchronization for the multistate coupled FCRDNNs is also given on the basis of the devised adaptive state feedback control scheme. Additionally, the problems of passivity and synchronization are also addressed for the FCRDNNs with multiple spatial-diffusion couplings by employing the adaptive control technique and Lyapunov functional method. Finally, two numerical examples are presented to validate the effectiveness of the devised adaptive control schemes.

In this article, two kinds of dynamic event-triggered lag bipartite consensus (LBC) control problems for a linear multiagent system (MAS) with structurally balanced signed directed graph are tackled, that is, the cases with known agent state and with unknown agent state. For the former, based on an additional internal dynamic variable and the parametric Lyapunov equation (PLE), a dynamic event-triggered state-feedback control (DETSFC) protocol is devised, which not only does not exhibit the Zeno behavior but also can guarantee that the MAS realizes the LBC. Similarly, we further utilize the state observer and the dynamic event-triggered control (DETC) method to address the LBC for the MAS with unknown agent state. Finally, the double-integrator MAS and the linearized reduced-order aircraft are used to verify the advantages of the obtained theoretical results.

Owing to the immaturity of current artificial intelligence techniques, practical multiagent systems (MASs) often require supervision and intervention from humans. However, it is unrealistic for a human to monitor the entire MAS and provide appropriate input in some circumstances. A viable approach is to allow a human to control an agent as the leader which in turn influences the other autonomous followers. To this end, a critical issue is how to learn human behavior to improve the autonomy of followers for collaborating with human effectively, since the autonomous followers do not have prior knowledge of human behavior. In this article, the human leading behavior learning problem is studied for a class of human-in-the-loop (HiTL) MASs that are not fully connected. A linear quadratic differential game framework is applied to formulate the collaborative control problem in the HiTL MAS where the human behavior is represented as a cost function whose weighting matrix is unknown to the followers. In the HiTL MAS, we select a follower that has strong computing power called follower 1 to learn the human behavior via an online adaptive inverse differential game (IDG) approach. Based on concurrent learning (CL) technique, an adaptive law is developed for follower 1 to determine the human feedback matrix online, and at the same time the interaction strategies for the autonomous followers are also calculated by follower 1 in case of the constrained communication topology. Subsequently, the weighting matrix in the human cost function is recovered by addressing a linear matrix inequality (LMI) optimization problem. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed method.

In this article, we investigate the lag consensus and lag$ \mathcal{H}_{\infty}$consensus problems for second-order nonlinear multiagent systems (MASs) by utilizing the proportional-derivative (PD) and proportional-integral (PI) control methods. On the one hand, a criterion is developed for ensuring the lag consensus of the MAS by choosing an appropriate PD control protocol. Moreover, a PI controller is also provided to guarantee that the MAS can achieve lag consensus. On the other hand, several lag$\mathcal{H}_{\infty}$consensus criteria are also given for the case in which external disturbances appear in the MAS; these criteria are developed by exploiting the PD and PI control strategies. Finally, the devised control schemes and the developed criteria are verified by employing two numerical examples.

In this article, the distributed formation control problem for a class of human-in-the-loop (HiTL) multiagent systems (MASs) is studied. A hidden Markov jump MAS is employed to model the HiTL MAS, which integrates the human models, the MAS model, and their interactions. The HiTL MAS investigated in this article is composed of two parts: a leader without human in the control loop and a group of followers in which each follower is simultaneously controlled by a human operator and an automation. For each follower, a hidden Markov model is used for modeling the human behaviors in consideration of the random nature of human internal state (HIS) reasoning and the uncertainty from HIS observation. By means of a stochastic Lyapunov function, a necessary and sufficient condition is first developed in terms of the linear matrix inequalities (LMIs) to ensure the formation of the HiTL MAS in the mean-square sense. Then, an LMI approach to the human-assistance control design is proposed for the automations in the followers to guarantee the mean-square formation of the HiTL MAS. Finally, simulation results are presented to verify the effectiveness of the proposed methods.

This article investigates the finite-time synchronization (FTS) and$H_{\infty }$synchronization for two types of coupled neural networks (CNNs), that is, the cases with multistate couplings and with multiderivative couplings. By designing appropriate state feedback controllers and parameter adjustment strategies, some FTS and finite-time$H_{\infty }$synchronization criteria for CNNs with multistate couplings are derived. In addition, we further consider the FTS and finite-time$H_{\infty }$synchronization problems for CNNs with multiderivative couplings by utilizing state feedback control approach and selecting suitable parameter adjustment schemes. Finally, two simulation examples are given to demonstrate the effectiveness of the proposed criteria.

This article focuses on finite-time passivity (FTP) and finite-time synchronization (FTS) for complex dynamical networks with multiple state/derivative couplings based on the proportional-derivative (PD) control method. Several criteria of FTP for complex dynamical networks with multiple state couplings (CDNMSCs) are formulated by utilizing the PD controller and constructing an appropriate Lyapunov function. Furthermore, FTP is further used to investigate the FTS in CDNMSCs under the PD controller. In addition, the FTP and FTS for complex dynamical networks with multiple derivative couplings (CDNMDCs) are also studied by exploiting the PD control method and some inequality techniques. Finally, two numerical examples are worked out to demonstrate the validity of the presented PD controllers.

This article deals with the fully distributed pull-based event-triggered bipartite fixed-time output consensus problem of heterogeneous linear multiagent systems (HLMASs) with an active leader, whose information can be merely accessed by a small fraction of followers. First, a class of fully distributed fixed-time observers is proposed for each follower to estimate the leader's system matrices, position, and control input under the signed communication topology, respectively. Then, based on the estimations of leader's system matrices, two adaptive algorithms are given to solve the regulator equations. Furthermore, the fully distributed fixed-time observer-based controllers associated with state feedback and output feedback are, respectively, proposed by employing the pull-based event-triggered mechanism (ETM) where each agent merely updates controller at its own triggering instants. Correspondingly, some sufficient criteria and the rigorous proofs are provided to ensure the implementation of bipartite output consensus in fixed time by using the Lyapunov stability theory and fixed-time stability theory. Moreover, the strictly positive lower bounds of intervals between two adjacent event-triggered times are derived, which means the Zeno behavior is ruled out. Finally, numerical simulations are performed to demonstrate the theoretical analysis.

This article studies the boundary fuzzy control problem for nonlinear parabolic partial differential equation (PDE) systems under spatially noncollocated mobile sensors. In a real setup, sensors and actuators can never be placed at the same location, and the noncollocated setting may be beneficial in some application scenarios. The control design is very difficult due to the noncollocated mobile observation, which can be solved by an observer-based technique. At first, a Takagi–Sugeno fuzzy PDE model is devoted to accurately representing the nonlinear parabolic PDE system. Next, we present a state estimation scheme including fuzzy Luenberger-type PDE state observer plus mobile sensor guidance. Then, an observer-based boundary fuzzy controller is posed to render the resulting closed-loop system exponentially stable, and the exponential decay rate is increased by the designed mobile sensor guidance laws. At last, two examples verify the proposed method.

This article investigates the stabilization of stochastic highly non-linear coupled systems (SHNCSs) with multiple time delays by using periodically intermittent control (PIC). It is worth noting that coefficients in SHNCSs dissatisfy the linear growth condition, which weakens the previous stability conditions. In addition, PIC and multiple time delays are first introduced into the study of highly nonlinear systems, which leads to the existing methods being inapplicable to investigate the stability of SHNCSs with multiple time delays. Therefore, a novel Halanay-type differential inequality is established, which can be employed to deal with highly nonlinear systems with PIC. Based on the Lyapunov method, the graph theory, and the novel differential inequality, SHNCSs with multiple time delays are first studied, and stability criteria are presented. Next, the theoretical results can be applied to modified FitzHugh-Nagumo models. At last, a numerical example is presented to show the effectiveness of our results.

This article mainly delves into the finite-time passivity (FTP) for coupled fractional-order neural networks with multistate couplings (CFNNMSCs) or coupled fractional-order neural networks with multiderivative couplings (CFNNMDCs). Distinguishing from the traditional FTP definitions, several concepts of FTP for fractional-order systems are given. On one hand, we present several sufficient conditions to ensure the FTP for CFNNMSCs by artfully designing a state-feedback controller and an adaptive state-feedback controller. On the other hand, by utilizing some inequality techniques, two sets of FTP criteria for CFNNMDCs are also established on the basis of the state-feedback and adaptive state-feedback controllers. Finally, numerical examples are used to demonstrate the validity of the derived FTP criteria.

This article presents several new α -passivity and α -finite-time passivity ( α -FTP) concepts for the fractional-order systems with different input and output dimensions, which are distinct from the concepts for integer-order systems and extend the existing passivity and FTP definitions to some extent. On one hand, we not only develop some sufficient conditions for ensuring the α -passivity of the multi-weighted fractional-order complex dynamical networks (MWFOCDNs) with fixed and adaptive couplings, but also discuss the synchronization for the MWFOCDNs based on the α -output-strict passivity ( α -OSP). On the other hand, the α -FTP for the MWFOCDNs with fixed and adaptive couplings are also studied on the basis of the designed state feedback controller, and the relationship between finite-time synchronization (FTS) and α -FTP for the MWFOCDNs is also illustrated. Finally, two numerical examples with simulation results are used to demonstrate the validity of the obtained criteria.

In this article, we, respectively, take the synchronization into consideration for directed and undirected complex networks (CNs) with multiple state or delayed state couplings subject to recoverable attacks. By selecting appropriate Lyapunov functional, employing inequality techniques, and adopting the designed state-feedback controller, two criteria of the synchronization are established for the directed CN with multiple state couplings (CNMSCs). Moreover, the synchronization of CNMSCs is also discussed for the case that the network topology is undirected. In addition, two types of CNs with multiple delayed state couplings are also proposed, and several criteria of synchronization are formulated for these networks. Finally, two examples are given to verify the correctness of the derived synchronization criteria.

This article studies dissipativity analysis of fractional-order quaternion-valued neural networks (FOQVNNs) with time delays. Two specific activation functions are considered along with common bounded and activation functions of Lipschitz-kind. Since quaternion multiplication is not commutative, we must divide the model, which is evaluated by quaternion, into four elements that are real-valued elements. On the basis of the construction of novel Lyapunov functional, and applying fractional-calculus theory, new criteria for the test of the global dissipativity and exponential stability of FOQVNNs model are established. FOQVNNs have also been suggested to provide global dissipativity and exponential stability, whereas nonlinear complex activation functions are constrained by the usage of linear matrix inequality methods, which utilize quaternion matrices and positive quaternion definite matrices. Finally, the effectiveness and superiority of the proposed approach is validated through numerical examples.

In this paper, the output synchronization problem for complex dynamical networks (CDNs) with multiple output or output derivative couplings is discussed in detail. Under the help of Lyapunov functional and inequality techniques, an output synchronization criterion is presented for CDNs with multiple output couplings (CDNMOCs). To ensure the output synchronization of CDNMOCs, an adaptive control scheme is also devised. Similarly, we also take into account the adaptive output synchronization and output synchronization of CDNs with multiple output derivative couplings. At last, several numerical examples are designed to testify the effectiveness of the proposed results.

In this article, two kinds of complex dynamical networks (CDNs) with state and derivative coupling are investigated, respectively. First, some important concepts about finite-time passivity (FTP), finite-time output strict passivity, and finite-time input strict passivity are introduced. By making use of state-feedback controllers and adaptive state-feedback controllers, several sufficient conditions are given to guarantee the FTP of these two network models. On the other hand, based on the obtained FTP results, some finite-time synchronization criteria for the CDNs with state and derivative coupling are gained. Finally, two simulation examples are proposed to verify the availability of the derived results.

This article investigates the finite-time output synchronization and$H_{\infty }$output synchronization problems for coupled neural networks with multiple output couplings (CNNMOC), respectively. By choosing appropriate state feedback controllers, several finite-time output synchronization and$H_{\infty }$output synchronization criteria are proposed for the CNNMOC. Moreover, a coupling-weight adjustment scheme is also developed to guarantee the finite-time output synchronization and$H_{\infty }$output synchronization of CNNMOC. Finally, two numerical examples are given to verify the effectiveness of the presented criteria.

This article considers the synchronization problem of delayed reaction-diffusion neural networks via quantized sampled-data (SD) control under spatially point measurements (SPMs), where distributed and discrete delays are considered. The synchronization scheme, which takes into account the communication limitations of quantization and variable sampling, is based on SPMs and only available in a finite number of fixed spatial points. By utilizing inequality techniques and Lyapunov-Krasovskii functional, some synchronization criteria via a quantized SD controller under SPMs are established and presented by linear matrix inequalities, which can ensure the exponential stability of the synchronization error system containing the drive and response dynamics. Finally, two numerical examples are offered to support the proposed quantized SD synchronization method.

This article focuses on the finite-time output synchronization problem for undirected and directed coupled neural networks with output coupling (CNNOC). Based on the designed state feedback controllers and some inequality techniques, we present several finite-time output synchronization criteria for these network models. In addition, two kinds of coupling-weight adjustment strategies are also developed to guarantee the finite-time output synchronization of undirected and directed CNNOC. Finally, two numerical examples are also provided to demonstrate the effectiveness of the theoretical results.

In this paper, two kinds of multiweighted and adaptive state coupled complex networks (CNs) with or without coupling delays are presented. First, we develop the appropriate state feedback controller and adaptive law for the sake of guaranteeing that the proposed network models without coupling delays can be finite-timely synchronized and H∞synchronized. Furthermore, for the multiweighted CNs with coupling delays and adaptive state couplings, some finite-time synchronization and H∞synchronization criteria are presented by choosing the appropriate adaptive law and controllers. Eventually, we give two numerical simulations to verify the validity of the theoretical results.

In this paper, the finite-time passivity (FTP) problem for two classes of coupled neural networks (CNNs) with adaptive coupling weights is discussed. By selecting appropriate adaptive laws and controllers, several FTP conditions are given for CNNs with undirected and directed topologies. Furthermore, some finite-time synchronization conditions are also established by employing the FTP of the CNNs. At last, two numeral examples are used to check the correctness of the obtained criteria.

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.

The output synchronization and H∞output synchronization problems for multiweighted complex network are discussed in this paper. First, we analyze the output synchronization of multiweighted complex network by exploiting Lyapunov functional and Barbalat's lemma. In addition, some nodes- and edges-based pinning control strategies are developed to ensure the output synchronization of multiweighted complex network. Similarly, the H∞output synchronization problem of multiweighted complex network is also discussed. Finally, two numerical examples are presented to verify the correctness of the obtained results.

In this paper, two multiple weighted coupled reaction-diffusion neural networks (CRDNNs) with and without coupling delays are introduced. On the one hand, some finite-time passivity (FTP) concepts are proposed for the spatially and temporally system with different dimensions of output and input. By choosing appropriate Lyapunov functionals and controllers, several sufficient conditions are presented to ensure the FTP of these CRDNNs. On the other hand, the finite-time synchronization (FTS) problem is also discussed for the multiple weighted CRDNNs with and without coupling delays, respectively. Finally, two numeral examples with simulation results are provided to verify the effectiveness of the obtained FTP and FTS criteria.

This paper presents a complex network model consisting of N uncertain reaction-diffusion neural networks with multiple time delays. We analyze the passivity and synchronization of the proposed network model and derive several passivity and synchronization criteria based on some inequality techniques. In addition, by considering the difficulty in achieving passivity (synchronization) in such a network, an adaptive control scheme is also developed to ensure that the proposed network achieves passivity (synchronization). Finally, we design two numerical examples to verify the effectiveness of the derived passivity and synchronization criteria.

In this paper, we introduce two complex dynamical networks with multiweights, which have several different sorts of weights between two nodes. By means of Lyapunov functional method and pinning control technique, some sufficient conditions are derived to ensure the synchronization for proposed network models. Moreover, some adaptive strategies are given to acquire suitable coupling strengths and feedback gains. By exploiting these designed adaptive laws, several general criteria for network synchronization are established. Finally, two numerical examples are also provided to show the validity of the theoretical results.

This paper considers a complex dynamical network model, in which the input and output vectors have different dimensions. We, respectively, investigate the passivity and the relationship between output strict passivity and output synchronization of the complex dynamical network with fixed and adaptive coupling strength. First, two new passivity definitions are proposed, which generalize some existing concepts of passivity. By constructing appropriate Lyapunov functional, some sufficient conditions ensuring the passivity, input strict passivity and output strict passivity are derived for the complex dynamical network with fixed coupling strength. In addition, we also reveal the relationship between output strict passivity and output synchronization of the complex dynamical network with fixed coupling strength. By employing the relationship between output strict passivity and output synchronization, a sufficient condition for output synchronization of the complex dynamical network with fixed coupling strength is established. Then, we extend these results to the case when the coupling strength is adaptively adjusted. Finally, two examples with numerical simulations are provided to demonstrate the effectiveness of the proposed criteria.

This research focuses on the problem of output synchronization in undirected and directed complex dynamical networks, respectively, by applying Barbalat's lemma. First, to ensure the output synchronization, several sufficient criteria are established for these network models based on some mathematical techniques, such as the Lyapunov functional method and matrix theory. Furthermore, some adaptive schemes to adjust the coupling weights among network nodes are developed to achieve the output synchronization. By applying the designed adaptive laws, several criteria for output synchronization are deduced for the network models. In addition, a design procedure of the adaptive law is shown. Finally, two simulation examples are used to show the effectiveness of the previous results.

A complex dynamical network consisting of N identical neural networks with reaction-diffusion terms is considered in this paper. First, several passivity definitions for the systems with different dimensions of input and output are given. By utilizing some inequality techniques, several criteria are presented, ensuring the passivity of the complex dynamical network under the designed adaptive law. Then, we discuss the relationship between the synchronization and output strict passivity of the proposed network model. Furthermore, these results are extended to the case when the topological structure of the network is undirected. Finally, two examples with numerical simulations are provided to illustrate the correctness and effectiveness of the proposed results.

Two coupled reaction-diffusion neural networks (CRDNNs) with different dimensions of input and output are considered in this paper. The only difference between them is whether time-varying delay is incorporated in the mathematical model of network. We respectively analyze dissipativity and passivity of these CRDNNs. First, for the systems with different dimensions of input and output vectors, two new passivity definitions are proposed. Then, by exploiting some inequality techniques, several dissipativity and passivity criteria for these CRDNNs are established. Furthermore, we analyze stability of passive CRDNNs. Finally, two examples with simulation results are presented to verify the effectiveness of the proposed criteria.

Two types of coupled neural networks with reaction-diffusion terms are considered in this paper. In the first one, the nodes are coupled through their states. In the second one, the nodes are coupled through the spatial diffusion terms. For the former, utilizing Lyapunov functional method and pinning control technique, we obtain some sufficient conditions to guarantee that network can realize synchronization. In addition, considering that the theoretical coupling strength required for synchronization may be much larger than the needed value, we propose an adaptive strategy to adjust the coupling strength for achieving a suitable value. For the latter, we establish a criterion for synchronization using the designed pinning controllers. It is found that the coupled reaction-diffusion neural networks with state coupling under the given linear feedback pinning controllers can realize synchronization when the coupling strength is very large, which is contrary to the coupled reaction-diffusion neural networks with spatial diffusion coupling. Moreover, a general criterion for ensuring network synchronization is derived by pinning a small fraction of nodes with adaptive feedback controllers. Finally, two examples with numerical simulations are provided to demonstrate the effectiveness of the theoretical results.

This paper proposes a directed complex dynamical network consisting of N linearly and diffusively coupled identical reaction-diffusion neural networks. Based on the Lyapunov functional method and the pinning control technique, some sufficient conditions are obtained to guarantee the synchronization of the proposed network model. In addition, an adaptive strategy is proposed to obtain appropriate coupling strength for achieving network synchronization. Furthermore, the pinning adaptive synchronization problem is also investigated in this paper, and a general criterion for ensuring network synchronization is established. Finally, a numerical example is provided to illustrate the effectiveness of the proposed criteria.

In this paper, we study a general array model of coupled reaction-diffusion neural networks (NNs) with adaptive coupling. In order to ensure the passivity of the coupled reaction-diffusion neural networks, some adaptive strategies to tune the coupling strengths among network nodes are designed. By utilizing some inequality techniques and the designed adaptive laws, several sufficient conditions ensuring passivity are obtained. In addition, we reveal the relationship between passivity and synchronization of the coupled reaction-diffusion NNs. Based on the obtained passivity results and the relationship between passivity and synchronization, a global synchronization criterion is established. Finally, numerical simulations are presented to illustrate the correctness and effectiveness of the proposed results.

In this paper, we propose a general array model of coupled reaction-diffusion neural networks with hybrid coupling, which is composed of spatial diffusion coupling and state coupling. By utilizing the Lyapunov functional method combined with the inequality techniques, a sufficient condition is given to ensure that the proposed network model is synchronized. In addition, when the external disturbances appear in the network, a criterion is obtained to guarantee the H∞ synchronization of the network. Moreover, some adaptive strategies to tune the coupling strengths among network nodes are designed for reaching synchronization and H∞ synchronization. Some criteria for synchronization and H∞ synchronization are derived by using the designed adaptive laws. Numerical simulations are presented finally to demonstrate the effectiveness of the obtained theoretical results.

In this paper, two types of linearly coupled neural networks with reaction-diffusion terms are proposed. We respectively investigate the adaptive synchronization of these two types of complex network models. With local information of node dynamics, some novel adaptive strategies to tune the coupling strengths among network nodes are designed. By constructing appropriate Lyapunov functionals and using inequality techniques, several sufficient conditions are given for reaching synchronization by using the designed adaptive laws. Finally, two examples with numerical simulations are provided to demonstrate the effectiveness of the theoretical results.

This paper is concerned with the passivity and stability problems of reaction-diffusion neural networks (RDNNs) in which the input and output variables are varied with the time and space variables. By utilizing the Lyapunov functional method combined with the inequality techniques, some sufficient conditions ensuring the passivity and global exponential stability are derived. Furthermore, when the parameter uncertainties appear in RDNNs, several criteria for robust passivity and robust global exponential stability are also presented. Finally, a numerical example is provided to illustrate the effectiveness of the proposed criteria.