Impulsive control for passivity and exponential synchronization of coupled neural networks with multiple weights

This paper investigates the passivity and synchronization problems for two classes of multiple weighted coupled neural networks (MWCNNs) with or without time delays. Firstly, by utilizing an impulsive control strategy and some inequality techniques, several passivity criteria for MWCNNs with diverse dimensions of output and input are established. Then, based on the Lyapunov functional, some sufficient conditions to ensure the synchronization of MWCNNs via impulsive control are derived. In addition, combined with the comparison principle and the impulsive delay differential inequality, the global exponential synchronization of MWCNNs with time-varying delays is considered under impulsive control. Finally, two numerical examples illustrate the effectiveness of the obtained results.     

Synchronization of dynamical networks with nonlinearly coupling function under hybrid pinning impulsive controllers

In this paper, the globally exponential synchronization problem of dynamical networks with nonlinearly coupling function is considered. Hybrid pinning control strategies are established to force the states of the network to follow some objective trajectory. The impulsive pinning controllers are used to control a fringe of nodes at the impulsive instants, while during the impulsive instants, pinning state-feedback controllers are designed to achieve the control objective. Finally, the validity of the developed techniques is evidenced by an illustrative example.     

Graph theory-based finite-time synchronization of fractional-order complex dynamical networks

This paper considers the finite-time synchronization problem for a class of fractional-order complex dynamical networks (FOCDNs). By utilizing the properties of fractional calculus and fractional-order comparison principle, we propose a new lemma. Base on the new lemma, some analysis techniques and algebraic graph theory method, some novel criteria are given to ensure finite-time synchronization of FOCDNs, and the upper bound of the setting time for synchronization is estimated. At last, numerical simulations are given to verify the effectiveness of the obtained results.     

Exponential synchronization for coupled complex networks with time-varying delays and stochastic perturbations via impulsive control

This paper is concerned with the problem of exponential synchronization of coupled complex networks with time-varying delays and stochastic perturbations (CCNTDSP). Different from previous works, both the internal time-varying delay and the coupling time-varying delay are taken into account in the network model. Meanwhile, an impulsive controller is designed to realize exponential synchronization in mean square of CCNTDSP. Combining the Lyapunov method with Kirchhoff’s Matrix Tree Theorem, some sufficient criteria are obtained to guarantee exponential synchronization in mean square of CCNTDSP. Furthermore, we apply the theoretical results to study exponential synchronization of stochastic coupled oscillators with the internal time-varying delay and the coupling time-varying delay. And a synchronization criterion is also obtained. Finally, two numerical examples are given to demonstrate the effectiveness and feasibility of our theoretical results and the superiority of impulsive control.     

Synchronization of linear dynamical networks under stochastic impulsive coupling protocols

This paper investigates a stochastic impulsive coupling protocol for synchronization of linear dynamical networks based on discrete-time sampled-data. The convergence of the networks under the proposed protocol is discussed, and some sufficient conditions are showed to guarantee almost sure exponential synchronization. Moreover, this coupling protocol with a pinning control scheme is developed to lead the state of all nodes to almost sure exponentially converge to a virtual synchronization target. It is shown that the almost sure exponential synchronization can be achieved by only interacting based on the stochastic feedback information at discrete-time instants. Some numerical examples are finally provided to present the effectiveness of the proposed stochastic coupling protocols.     

Impulsive effects on weak projective synchronization of parameter mismatched stochastic memristive neural networks

This paper analyses the weak projective synchronization (WPS) of the parameter mismatched memristive neural networks (MNNs) with stochastic disturbance and time delays via impulsive control. Complete synchronization cannot achieve because of the projective factor and mismatched parameters. Therefore, the WPS of practical MNNs under impulsive control strategy is studied. The augmented systems are built to utilize more information of the system and reduce the constraint conditions. Meanwhile, two types of comparison principles are used owing to the impulsive controller with and without time delays. Then, sufficient criteria for the exponential convergence of systems are obtained under the positive and negative effects of impulses. Finally, the validity of the theoretical results is verified by simulations of different conditions.     

Boundary control for synchronization of fractional-order complex spatiotemporal networks based on PDEs with multiple delays and its application in image encryption

This paper addresses the synchronization problem of fractional-order complex spatiotemporal networks (CSNs) based on partial differential equations with delays via boundary control. First, fractional-order CSNs with time-invariant and time-varying delays are studied separately due to the widespread existence of time delays in complex networks. Moreover, two boundary controllers are proposed to solve the synchronization issue of fractional-order CSNs, in which nodes communicate with each other only on the spatial boundary. Furthermore, according to the fractional-order inequality, the synchronization criteria of fractional-order CNSs with multiple delays are obtained. Finally, the numerical simulations are given to verify the feasibility of the presented results. A case provides the application of CSNs in image encryption.     

Lag quasi-synchronization of incommensurate fractional-order memristor-based neural networks with nonidentical characteristics via quantized control: A vector fractional Halanay inequality approach

This work realizes lag quasi-synchronization of incommensurate fractional-order memristor-based neural networks (FMNNs) with nonidentical characteristics via quantized control. The motivations behind this research work are threefold: (1) quantized controllers, which generate discrete control signals, can be more easily realized in computers than non-quantized controllers, and can consume smaller communication capacity; (2) incommensurate orders in a single FMNN and nonidentical characteristics in drive-response FMNNs are inescapable due to the differences among the circuit elements used to implement FMNNs; (3) convergence analysis of delayed incommensurate fractional-order nonlinear systems, which is the basis for the derivation of synchronization criterion, has not been handled perfectly. As an effective tool for convergence analysis of delayed incommensurate fractional-order nonlinear systems, especially for estimation of ultimate state bound, a vector fractional Halanay inequality is established at first. Then, a quantized synchronization controller, in which the dead-zone is introduced into some logarithmic quantizers to avoid chattering phenomenon, is designed. By means of vector Lyapunov function together with the newly derived vector fractional Halanay inequality, the synchronization criterion is proved theoretically. Lastly, numerical simulations supplementarily illustrate the correctness of the synchronization criterion. In contrast with the hypotheses in the relevant literature, the hypotheses in this paper are weaker.     

On designing decentralized impulsive controllers for synchronization of complex dynamical networks with nonidentical nodes and coupling delays

This paper investigates the problem of designing decentralized impulsive controllers for synchronization of a class of complex dynamical networks (CDNs) about some prescribed goal function. The CDNs are allowed to possess nonidentical nodes and coupling delays. Two cases of time-varying coupling delays are considered: the case where the coupling delays are uniformly bounded, and the case where the derivatives of the coupling delays are not greater than 1. The synchronization analysis for the first case is performed by applying a time-varying Lyapunov function based method combined with Razumikhin-type technique, while the synchronization analysis for the second case is conducted based on a time-varying Lyapunov functional based method. For each case, by utilizing a convex combination technique, the resulting synchronization criterion is formulated as the feasibility problem of a set of linear matrix inequalities (LMIs). Then, sufficient conditions on the existence of a decentralized impulsive controller are presented by employing these newly obtained synchronization criteria. The local impulse gain matrices can be designed by solving a set of LMIs. Finally, two representative examples are given to illustrate the correctness of the theoretical results.     

Finite-time inter-layer projective synchronization of Caputo fractional-order two-layer networks by sliding mode control

Finite-time inter-layer projective synchronization (FIPS) of Caputo fractional-order two-layer networks (FTN) based on sliding mode control (SMC) technique is investigated in this article. Firstly, in order to realize the FIPS of FTN, a fractional-order integral sliding mode surface (SMS) is established. Then, through the theory of SMC, we design a sliding mode controller (SMCr) to ensure the appearance of sliding mode motion. According to the fractional Lyapunov direct method, the trajectories of the system are driven to the proposed SMS, and some novel sufficient conditions for FIPS of FTN are derived. Furthermore, as two special cases of FIPS, finite-time inter-layer synchronization and finite-time inter-layer anti-synchronization for the FTN are studied. Finally, this paper takes the fractional-order chaotic Lü’s system and the fractional-order chaotic Chen’s system as the isolated node of the first layer system and the second layer system, respectively. And the numerical simulations are given to demonstrate the feasibility and validity of the proposed theoretical results.     

Impulsive effects on the stability and stabilization of positive systems with delays

This study addresses the exponential stability and positive stabilization problems of impulsive positive systems (IPSs) with time delay. Specially, three types of impulses, namely, disturbance, “neutral”, and stabilizing impulses, are considered. For each type of impulsive effect, the exponential stability criterion is established utilizing the Lyapunov–Razumikhin techniques. Moreover, on the basis of the obtained stability results, the state-feedback controller design problem is investigated to positively stabilize the IPSs with time delay under different types of impulsive effects. Finally, numerical examples are provided to illustrate the effectiveness of the theoretical results.     

Event-triggered synchronization control of complex networks with adaptive coupling strength

The event-triggered synchronization control problem is concerned for a class of complex networks with nonlinearly coupling function and adaptive coupling strength. Given a state-based event-trigger mechanism and the threshold, an event-triggered control method is introduced to make complex networks achieve exponential synchronization. By combining the Lyapunov stability theory and the knowledge of graph theory, a sufficient condition is established such that complex networks can achieve exponential synchronization. Then, the feasibility of the event-triggered control is analyzed. Moreover, the second-order Kuramoto oscillators is taken into account. And the event-triggered control strategy is used to make the oscillators achieve exponential synchronization. Meanwhile, two simulation results about the second-order Kuramoto oscillators are given to show the effectiveness of results.     

Synchronization of coupled heterogeneous complex networks

By only designing the internal coupling, quasi synchronization of heterogeneous complex networks coupled by N nonidentical Duffing-type oscillators without any external controller is investigated in this paper. To achieve quasi synchronization, the average of states of all nodes is designed as the virtual target. Heterogeneous complex networks with two kinds of nonlinear node dynamics are analyzed firstly. Some sufficient conditions on quasi synchronization are obtained without designing any external controller. Quasi synchronization means that the states of all nonidentical nodes will keep a bounded error with the virtual target. Then the heterogeneous complex network with impulsive coupling which means the network only has coupling at some discrete impulsive instants, is further discussed. Some sufficient conditions on heterogeneous complex network with impulsive coupling are derived. Based on these results, heterogeneous complex network can still reach quasi synchronization even if its nodes are only coupled at discrete impulsive instants. Finally, two examples are provided to verify the theoretical results.     

Synchronization of stochastic discrete-time complex networks with partial mixed impulsive effects

In this paper, the synchronization problem is studied for a class of stochastic discrete-time complex networks with partial mixed impulsive effects. The involving impulsive effects, called partial mixed impulses, can be regarded as local and time-varying impulses, which means that impulses are not only injected into a fraction of nodes in networks but also contain synchronizing and desynchronizing impulses at the same time. In order to handle this case, several mathematical techniques are proposed to tackle mixed impulsive effects in discrete-time dynamical systems. Based on the variation of parameters formula, several sufficient criteria are derived to ensure that synchronization of the addressed networks can be achieved in mean square. The obtained criteria not only rely on the strengths of mixed impulses and the impulsive intervals, but also can reduce conservativeness. Finally, a numerical example is presented to show the effectiveness of our results for neural networks.     

Delay-dependent cluster synchronization of time-varying complex dynamical networks with noise via delayed pinning impulsive control

This paper investigates the problem of cluster synchronization of complex dynamical networks with noise and time-varying delays by using a delayed pinning impulsive control scheme. Different from the traditional impulsive control schemes without the effects of input delays, it designs a pinning impulsive control scheme to successfully address the aforementioned problem subject to impulsive input delays. By employing a time-dependent Lyapunov function and the mathematical induction, some novel criteria are established to guarantee the cluster synchronization of the noisy complex networks, revealing the closed relationship between the synchronization performance and the related factors, including the impulsive input delays, the number of the pinned nodes, the frequency and strength of the impulsive control, and the noisy perturbations. Some numerical examples and computer simulations are presented to illustrate the effectiveness of the theoretical results.     

Finite-time and fixed-time synchronization of discontinuous complex networks: A unified control framework design

This paper is concerned with the finite-time and fixed-time synchronization of complex networks with discontinuous nodes dynamics. Firstly, under the framework of Filippov solution, a new theorem of finite-time and fixed-time stability is established for nonlinear systems with discontinuous right-hand sides by using mainly reduction to absurdity. Furthermore, for a class of discontinuous complex networks, a general control law is firstly designed. Under the unified control framework and the same conditions, the considered networks are ensured to achieve finite-time or fixed-time synchronization by only adjusting the value of a key control parameter. Based on the similar discussion, a unified control strategy is also provided to realize respectively asymptotical, exponential and finite-time synchronization of the addressed networks. Finally, the derived theoretical results are supported by an example with numerical simulations.     

Pinning impulsive stabilization for BAM reaction-diffusion neural networks with mixed delays

The exponential stabilization of BAM reaction-diffusion neural networks with mixed delays is discussed in this article. At first, a general pinning impulsive controller is introduced, in which the control functions are nonlinear and the pinning neurons are determined by reordering the state error. Next, based on the designed control protocol and the Lyapunov–Krasovskii functional approach, some novel and useful criteria, which depend on the diffusion coefficients and controlling parameters, are established to guarantee the global exponential stabilization of the considered neural networks. Finally, the effectiveness of the proposed control strategy is shown by two numerical examples.     

Synchronization criteria for coupled stochastic neural networks with time-varying delays and leakage delay

This paper proposes new delay-dependent synchronization criteria for coupled stochastic neural networks with time-varying delays and leakage delay. By constructing a suitable Lyapunov–Krasovskii's functional and utilizing Finsler's lemma, novel synchronization criteria for the networks are established in terms of linear matrix inequalities (LMIs) which can be easily solved by using the LMI toolbox in MATLAB. Three numerical examples are given to illustrate the effectiveness of the proposed methods.     

Pinning and impulsive synchronization control of complex dynamical networks with non-derivative and derivative coupling

In this paper, the problem of pinning and impulsive synchronization between two complex dynamical networks with non-derivative and derivative coupling is investigated. A hybrid controller, which contains a pinning controller and a pinning impulsive controller, is proposed simultaneously. Based on the Lyapunov stability theory and mathematical analysis technique, some novel criteria of synchronization are derived, which can guarantee that the response network asymptotically synchronizes to the drive network by combining pinning control and pinning impulsive control. Moreover, the restrictions about non-derivative coupling matrix, impulsive intervals and the number of pinned nodes are removed. Numerical examples are presented finally to illustrate the effectiveness of the theoretical results.