Jianquan Lu

Adaptive synchronization of neural networks with or without time-varying delay.

In this paper, based on the invariant principle of functional differential equations, a simple, analytical, and rigorous adaptive feedback scheme is proposed for the synchronization of almost all kinds of coupled identical neural networks with time-varying delay, which can be chaotic, periodic, etc. We do not assume that the concrete values of the connection weight matrix and the delayed connection weight matrix are known. We show that two coupled identical neural networks with or without time-varying delay can achieve synchronization by enhancing the coupling strength dynamically. The update gain of coupling strength can be properly chosen to adjust the speed of achieving synchronization. Also, it is quite robust against the effect of noise and simple to implement in practice. In addition, numerical simulations are given to show the effectiveness of the proposed synchronization method.

Synchronization Control for Nonlinear Stochastic Dynamical Networks: Pinning Impulsive Strategy

In this paper, a new control strategy is proposed for the synchronization of stochastic dynamical networks with nonlinear coupling. Pinning state feedback controllers have been proved to be effective for synchronization control of state-coupled dynamical networks. We will show that pinning impulsive controllers are also effective for synchronization control of the above mentioned dynamical networks. Some generic mean square stability criteria are derived in terms of algebraic conditions, which guarantee that the whole state-coupled dynamical network can be forced to some desired trajectory by placing impulsive controllers on a small fraction of nodes. An effective method is given to select the nodes which should be controlled at each impulsive constants. The proportion of the controlled nodes guaranteeing the stability is explicitly obtained, and the synchronization region is also derived and clearly plotted. Numerical simulations are exploited to demonstrate the effectiveness of the pinning impulsive strategy proposed in this paper.

Exponential Synchronization of Linearly Coupled Neural Networks With Impulsive Disturbances

This brief investigates globally exponential synchronization for linearly coupled neural networks (NNs) with time-varying delay and impulsive disturbances. Since the impulsive effects discussed in this brief are regarded as disturbances, the impulses should not happen too frequently. The concept of average impulsive interval is used to formalize this phenomenon. By referring to an impulsive delay differential inequality, we investigate the globally exponential synchronization of linearly coupled NNs with impulsive disturbances. The derived sufficient condition is closely related with the time delay, impulse strengths, average impulsive interval, and coupling structure of the systems. The obtained criterion is given in terms of an algebraic inequality which is easy to be verified, and hence our result is valid for large-scale systems. The results extend and improve upon earlier work. As a numerical example, a small-world network composing of impulsive coupled chaotic delayed NN nodes is given to illustrate our theoretical result.

Stochastic Synchronization of Complex Networks With Nonidentical Nodes Via Hybrid Adaptive and Impulsive Control

In this paper, the global exponential synchronization of delayed complex dynamical networks with nonidentical nodes and stochastic perturbations is studied. By combining adaptive control and impulsive control schemes, the considered network can be synchronized onto any given goal dynamics. The adaptive control is discontinuous and can overcome the unknown difference between dynamical nodes and goal system. As for the impulsive control, the concept named “average impulsive interval” with “elasticity number” of impulsive sequence is utilized to get less conservative synchronization criterion. Numerical simulations are given to show the effectiveness and less conservativeness of the theoretical results.

Pinning Stabilization of Linearly Coupled Stochastic Neural Networks via Minimum Number of Controllers

Pinning stabilization problem of linearly coupled stochastic neural networks (LCSNNs) is studied in this paper. A minimum number of controllers are used to force the LCSNNs to the desired equilibrium point by fully utilizing the structure of the network. In order to pinning control the LCSNNs to a certain desired state, only one controller is required for strongly connected network topology, and m controllers, which will be shown to be the minimum number, are needed for LCSNNs with m -reducible coupling matrix. The isolate node of the LCSNNs can be stable, periodic, or even chaotic. The coupling Laplacian matrix of the LCSNNs can be symmetric irreducible, asymmetric irreducible, or m-reducible, which means that the network topology can be strongly connected, weakly connected, or even unconnected. There is no constraint on the network topology. Some criteria are derived to judge whether the LCSNNs can be controlled in mean square by using designed controllers. The given criteria are expressed in terms of strict linear matrix inequalities, which can be easily checked by resorting to recently developed algorithm. Moreover, numerical examples including small-world and scale-free networks are also given to demonstrate that our theoretical results are valid and efficient for large systems.

Globally Exponential Synchronization and Synchronizability for General Dynamical Networks

The globally exponential synchronization problem for general dynamical networks is considered in this paper. One quantity will be distilled from the coupling matrix to characterize the synchronizability of the corresponding dynamical networks. The calculation of such a quantity is very convenient even for large-scale networks. The network topology is assumed to be directed and weakly connected, which implies that the coupling configuration matrix can be asymmetric, weighted, or reducible. This assumption is more consistent with the realistic network in practice than the constraint of symmetry and irreducibility. By using the Lyapunov functional method and the Kronecker product techniques, some criteria are obtained to guarantee the globally exponential synchronization of general dynamical networks. In addition, numerical examples, including small-world and scale-free networks, are given to demonstrate the theoretical results. It will be shown that our criteria are available for large-scale dynamical networks.

Synchronization of Randomly Coupled Neural Networks With Markovian Jumping and Time-Delay

This paper studies synchronization in an array of coupled neural networks with Markovian jumping and random coupling strength. The array of neural networks are coupled in a random fashion which is governed by Bernoulli random variable and each node has an interval time-varying delay. By designing a novel Lyapunov functional, using some inequalities and the properties of random variables, several delay-dependent synchronization criteria are derived for the coupled networks of continuous-time version. Discrete-time analogues of the continuous-time networks are also formulated and studied. Some new lemmas are developed to obtain less conservative synchronization criteria of both continuous-time model and its discrete-time analogues. Numerical examples of both continuous-time system and its discrete-time analogues are finally given to demonstrate the effectiveness of the theoretical results.

Synchronization in an array of output-coupled Boolean networks with time delay.

This brief presents an analytical study of synchronization in an array of coupled deterministic Boolean networks (BNs) with time delay. Two kinds of models are considered. In one model, the outputs contain time delay, while in another one, the outputs do not. One restriction in this brief is that the state delay and output delay are restricted to be equal. By referring to the algebraic representations of logical dynamics and using the techniques of semitensor product of matrices, some necessary and sufficient conditions are derived for the synchronization of delay-coupled BNs. Examples including a practical epigenetic example are given for illustration.

PINNING IMPULSIVE STABILIZATION OF NONLINEAR DYNAMICAL NETWORKS WITH TIME-VARYING DELAY

In this paper, a new impulsive control strategy, namely pinning impulsive control, is proposed for the stabilization problem of nonlinear dynamical networks with time-varying delay. In this strategy, only a small fraction of nodes is impulsively controlled to globally exponentially stabilize the whole dynamical network. By employing the Lyapunov method combined with the mathematical analysis approach as well as the comparison principle for impulsive systems, some criteria are obtained to guarantee the success of the global exponential stabilization process. The obtained criteria are closely related to the proportion of the controlled nodes, the impulsive strength, the impulsive interval and the time-delay. Numerical examples are given to demonstrate the effectiveness of the designed pinning impulsive controllers.

A Unified Approach to Practical Consensus with Quantized Data and Time Delay

In this paper, we study the consensus problem of multi-agent networks subject to communication constrains. Undirected and weighted network is considered here. Two types of communication constrains are discussed in this paper: i) each agent can only exchange quantized data with its neighbors and ii) each agent can only obtain the delayed information from its neighbors. The main contribution of this paper is to provide a precise mathematical treatment for the continuous multi-agent network with quantization and time delay. The existence of a global solution to the resulting system is firstly proved in the Filippov sense and then we prove that the solution converges to a practical consensus set asymptotically. Here, practical consensus means that the final consensus values are bounded within an interval, but not a value. Further, an explicit relationship among time delay, quantization parameter and the practical consensus set are theoretically presented. Numerical examples are finally given to demonstrate the effectiveness of the obtained theoretical results.