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

Fixed-time periodic stabilization of discontinuous reaction-diffusion Cohen-Grossberg neural networks

This paper aims to study the fixed-time stabilization of a class of delayed discontinuous reaction-diffusion Cohen-Grossberg neural networks. Firstly, by providing some relaxed conditions containing indefinite functions and based on inequality techniques, a new fixed-time stability lemma is given, which can improve the traditional ones. Secondly, based on state-dependent switching laws, the periodic wave solution of the formulated networks is transformed into the periodic solution of ordinary differential system. By utilizing differential inclusions theory and coincidence theorem, the existence of periodic solutions is obtained. Thirdly, based on the new fixed-time stability lemma, the periodic solutions are stabilized at zero in a fixed-time, which is a new topic on reaction-diffusion networks. Moreover, the established criteria are all delay-dependent, which are less conservative than the previous delay-independent ones for ensuring the stabilization of delayed reaction-diffusion networks. Finally, two examples give numerical explanations of the proposed results and highlight the influence of delays.

Adaptive event-triggered extended dissipative synchronization of delayed reaction-diffusion neural networks under deception attacks

Under spatially averaged measurements (SAMs) and deception attacks, this article mainly studies the problem of extended dissipativity output synchronization of delayed reaction-diffusion neural networks via an adaptive event-triggered sampled-data (AETSD) control strategy. Compared with the existing ETSD control methods with constant thresholds, our scheme can be adaptively adjusted according to the current sampling and latest transmitted signals and is realized based on limited sensors and actuators. Firstly, an AETSD control scheme is proposed to save the limited transmission channel. Secondly, some synchronization criteria under SAMs and deception attacks are established by utilizing Lyapunov-Krasovskii functional and inequality techniques. Then, by solving linear matrix inequalities (LMIs), we obtain the desired AETSD controller, which can satisfy the specified level of extended dissipativity behaviors. Lastly, one numerical example is given to demonstrate the validity of the proposed method.

Tree-structured neural networks: Spatiotemporal dynamics and optimal control

How the network topology drives the response dynamic is a basic question that has not yet been fully answered in neural networks. Elucidating the internal relation between topological structures and dynamics is instrumental in our understanding of brain function. Recent studies have revealed that the ring structure and star structure have a great influence on the dynamical behavior of neural networks. In order to further explore the role of topological structures in the response dynamic, we construct a new tree structure that differs from the ring structure and star structure of traditional neural networks. Considering the diffusion effect, we propose a diffusion neural network model with binary tree structure and multiple delays. How to design control strategies to optimize brain function has also been an open question. Thus, we put forward a novel full-dimensional nonlinear state feedback control strategy to optimize relevant neurodynamics. Some conditions about the local stability and Hopf bifurcation are obtained, and it is proved that the Turing instability does not occur. Moreover, for the formation of the spatially homogeneous periodic solution, some diffusion conditions are also fused together. Finally, several numerical examples are carried out to illustrate the results' correctness. Meanwhile, some comparative experiments are rendered to reveal the effectiveness of the proposed control strategy.

On Pinning Linear and Adaptive Synchronization of Multiple Fractional-Order Neural Networks With Unbounded Time-Varying Delays

In this article, the synchronization of multiple fractional-order neural networks with unbounded time-varying delays (FNNUDs) is investigated. By introducing a pinning linear control, sufficient conditions are provided for achieving the synchronization of multiple FNNUDs via an extended Halanay inequality. Moreover, a new effective adaptive control which applies to the fractional differential equations with unbounded time-varying delays is designed, under which sufficient criteria are presented to ensure the synchronization of multiple FNNUDs. The introduced control in this article is also workable in traditional integer-order neural networks. Finally, the validity of obtained results is demonstrated by a numerical example.

New Criteria on Stability of Dynamic Memristor Delayed Cellular Neural Networks

Dynamic memristor (DM)-cellular neural networks (CNNs), which replace a linear resistor with flux-controlled memristor in the architecture of each cell of traditional CNNs, have attracted researchers' attention. Compared with common neural networks, the DM-CNNs have an outstanding merit: when a steady state is reached, all voltages, currents, and power consumption of DM-CNNs disappeared, in the meantime, the memristor can store the computation results by serving as nonvolatile memories. The previous study on stability of DM-CNNs rarely considered time delay, while delay is quite common and highly impacts the stability of the system. Thus, taking the time delay effect into consideration, we extend the original system to DM-D(delay)CNNs model. By using the Lyapunov method and the matrix theory, some new sufficient conditions for the global asymptotic stability and global exponential stability with a known convergence rate of DM-DCNNs are obtained. These criteria generalized some known conclusions and are easily verified. Moreover, we find DM-DCNNs have 3ⁿ equilibrium points (EPs) and 2ⁿ of them are locally asymptotically stable. These results are obtained via a given constitutive relation of memristor and the appropriate division of state space. Combine with these theoretical results, the applications of DM-DCNNs can be extended to other fields, such as associative memory, and its advantage can be used in a better way. Finally, numerical simulations are offered to illustrate the effectiveness of our theoretical 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.

Exponential Synchronization of Memristor-Based Competitive Neural Networks With Reaction-Diffusions and Infinite Distributed Delays

Taking into account the infinite distributed delays and reaction-diffusions, this article investigates the global exponential synchronization problem of a class of memristor-based competitive neural networks (MCNNs) with different time scales. Based on the Lyapunov-Krasovskii functional and inequality approach, an adaptive control approach is proposed to ensure the exponential synchronization of the addressed drive-response networks. The closed-loop system is a discontinuous and delayed partial differential system in a cascade form, involving the spatial diffusion, the infinite distributed delays, the parametric adaptive law, the state-dependent switching parameters, and the variable structure controllers. By combining the theories of nonsmooth analysis, partial differential equation (PDE) and adaptive control, we present a new analytical method for rigorously deriving the synchronization of the states of the complex system. The derived m-norm (m ≥ 2)-based synchronization criteria are easily verified and the theoretical results are easily extended to memristor-based neural networks (NNs) without different time scales and reaction-diffusions. Finally, numerical simulations are presented to verify the effectiveness of the theoretical results.

Intralayer Synchronization of Multiplex Dynamical Networks via Pinning Impulsive Control

These days, the synchronization of multiplex networks is an emerging and important research topic. Grounded framework and theory about synchronization and control on multiplex networks are yet to come. This article studies the intralayer synchronization on a multiplex network (i.e., a set of networks connected through interlayer edges), via the pinning impulsive control method. The topologies of different layers are independent of each other, and the individual dynamics of nodes in different layers are different as well. Supra-Laplacian matrices are adopted to represent the topological structures of multiplex networks. Two cases are considered according to impulsive sequences of multiplex networks: 1) pinning controllers are applied to all the layers simultaneously at the instants of a common impulse sequence and 2) pinning controllers are applied to each layer at the instants of distinct impulse sequences. Using the Lyapunov stability theory and the impulsive control theory, several intralayer synchronization criteria for multiplex networks are obtained, in terms of the supra-Laplacian matrix of network topology, self-dynamics of nodes, impulsive intervals, and the pinning control effect. Furthermore, the algorithms for implementing pinning schemes at every impulsive instant are proposed to support the obtained criteria. Finally, numerical examples are presented to demonstrate the effectiveness and correctness of the proposed schemes.

Reachable Set Estimation for Markovian Jump Neutral-Type Neural Networks With Time-Varying Delays

The reachable set estimation problem for a class of Markovian jump neutral-type neural networks (MJNTNNs) with bounded disturbances and time-varying delays is tackled in this article. With the aid of the delay partitioning method, a novel stochastic Lyapunov-Krasovskii functional containing triple integral terms is constructed in mode-dependent augmented form. To begin with, transition probabilities of the concerned Markovian jump neural networks (NNs) are considered to be completely known. By employing the integral inequality approach and reciprocally convex combination method, it is proved that all state trajectories which start from the origin by bounded inputs can be constrained by an ellipsoid-like set if a group of linear matrix inequalities (LMIs) is feasible. Then, the free-connection weighting matrix technique is utilized to handle the case of partially known transition probabilities. As byproducts, some sufficient conditions are also obtained to guarantee the stochastic stability of the concerned NNs. The validity of the theoretical analysis is confirmed by numerical simulations.

Guaranteed Cost Finite-Time Control of Uncertain Coupled Neural Networks

This article investigates a robust guaranteed cost finite-time control for coupled neural networks with parametric uncertainties. The parameter uncertainties are assumed to be time-varying norm bounded, which appears on the system state and input matrices. The robust guaranteed cost control laws presented in this article include both continuous feedback controllers and intermittent feedback controllers, which were rarely found in the literature. The proposed guaranteed cost finite-time control is designed in terms of a set of linear-matrix inequalities (LMIs) to steer the coupled neural networks to achieve finite-time synchronization with an upper bound of a guaranteed cost function. Furthermore, open-loop optimization problems are formulated to minimize the upper bound of the quadratic cost function and convergence time, it can obtain the optimal guaranteed cost periodically intermittent and continuous feedback control parameters. Finally, the proposed guaranteed cost periodically intermittent and continuous feedback control schemes are verified by simulations.

Quantized Sampled-Data Synchronization of Delayed Reaction-Diffusion Neural Networks Under Spatially Point Measurements

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.

Gain-Scheduled Finite-Time Synchronization for Reaction-Diffusion Memristive Neural Networks Subject to Inconsistent Markov Chains

An innovative class of drive-response systems that are composed of Markovian reaction-diffusion memristive neural networks, where the drive and response systems follow inconsistent Markov chains, is proposed in this article. For this kind of nonlinear parameter-varying systems, a suitable gain-scheduled controller that involves a mode and memristor-dependent item is designed, so that the error system is bounded within a finite-time interval. Moreover, by constructing a novel Lyapunov-Krasovskii functional and employing the canonical Bessel-Legendre inequality and free-weighting matrix method, the conservatism of the finite-time synchronization criterion can be greatly reduced. Finally, two numerical examples are provided to illustrate the feasibility and practicability of the obtained results.

Synchronization of recurrent neural networks with unbounded delays and time-varying coefficients via generalized differential inequalities

In this paper, we revisit the drive-response synchronization of a class of recurrent neural networks with unbounded delays and time-varying coefficients, contrary to usual in the literature about time-varying neural networks, the signs of self-feedback coefficients are permitted to be indefinite or the time-varying coefficients can be unbounded. A generalized scalar delay differential inequality considering indefinite self-feedback coefficient and unbounded delay simultaneously is established, which covers the existing result with bounded delay, the applicabilities of the sufficient conditions are discussed. Some novel criteria for network synchronization are then derived by constructing different candidate functions. These results have been improved in some aspects compared with the existing ones. Differential inequality in vector form is also derived to obtain a more refined synchronization criterion which removes some strong assumptions. Three examples are presented to verify the effectiveness and show the superiorities of our theoretical results.

Synchronization of Nonidentical Neural Networks With Unknown Parameters and Diffusion Effects via Robust Adaptive Control Techniques

This paper considers the self-synchronization and tracking synchronization issues for a class of nonidentically coupled neural networks model with unknown parameters and diffusion effects. Using the special structure of neural networks with global Lipschitz activation function, nonidentical terms are treated as external disturbances, which can then be compensated via robust adaptive control techniques. For the case where no common reference trajectory is given in advance, a distributed adaptive controller is proposed to drive the synchronization error to an adjustable bounded area. For the case where a reference trajectory is predesigned, two distributed adaptive controllers are proposed, respectively, to address the tracking synchronization problem with bounded and unbounded reference trajectories, different decomposition methods are given to extract the heterogeneous characteristics. To avoid the appearance of global information, such as the spectrum of the coupling matrix, corresponding adaptive designs on coupling strengths are also provided for both cases. Moreover, the upper bounds of the final synchronization errors can be gradually adjusted according to the parameters of the adaptive designs. Finally, numerical examples are given to test the effectiveness of the control algorithms.

Output Synchronization of Complex Dynamical Networks With Multiple Output or Output Derivative Couplings

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.

Global Stabilization of Fuzzy Memristor-Based Reaction-Diffusion Neural Networks

This article investigates the global stabilization problem of Takagi-Sugeno fuzzy memristor-based neural networks with reaction-diffusion terms and distributed time-varying delays. By using the Green formula and proposing fuzzy feedback controllers, several algebraic criteria dependent on the diffusion coefficients are established to guarantee the global exponential stability of the addressed networks. Moreover, a simpler stability criterion is obtained by designing an adaptive fuzzy controller. The results derived in this article are generalized and include some existing ones as special cases. Finally, the validity of the theoretical results is verified by two examples.

Set Stabilization of Probabilistic Boolean Control Networks: A Sampled-Data Control Approach

This article investigates the set stabilization of probabilistic Boolean control networks (PBCNs) under sampled-data (SD) state-feedback control within finite and infinite time, respectively. First, the algorithms are, respectively, proposed to find the sampled point set and the largest sampled point control invariant set (SPCIS) of PBCNs by SD state-feedback control. Based on this, a necessary and sufficient criterion is proposed for the global set stabilization of PBCNs by SD state-feedback control within finite time. Moreover, the time-optimal SD state-feedback controller is designed. It is interesting that if the sampled period (SP) is changed, the time of global set stabilization of PBCNs may also change or even the PBCNs cannot achieve set stabilization. Second, a criterion for the global set stabilization of PBCNs by SD state-feedback control within infinite time is obtained. Furthermore, all possible SD state-feedback controllers are obtained by using all the complete families of reachable sets. Finally, three examples are presented to illustrate the effectiveness of the obtained results.

Lagrange Stability and Finite-Time Stabilization of Fuzzy Memristive Neural Networks With Hybrid Time-Varying Delays

This paper focuses on Lagrange exponential stability and finite-time stabilization of Takagi-Sugeno (T-S) fuzzy memristive neural networks with discrete and distributed time-varying delays (DFMNNs). By resorting to theories of differential inclusions and the comparison strategy, an algebraic condition is developed to confirm Lagrange exponential stability of the underlying DFMNNs in Filippov's sense, and the exponentially attractive set is estimated. When external input is not considered, global exponential stability of DFMNNs is derived directly, which includes some existing ones as special cases. Furthermore, finite-time stabilization of the addressed DFMNNs is analyzed by exploiting inequality techniques and the comparison approach via designing a nonlinear state feedback controller. The boundedness assumption of activation functions is removed herein. Finally, two simulations are presented to demonstrate the validness of the outcomes, and an application is performed in pseudorandom number generation.

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.

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.