Scale-Limited Lagrange Stability and Finite-Time Synchronization for Memristive Recurrent Neural Networks on Time Scales

Finite/fixed-time synchronization of inertial memristive neural networks by interval matrix method for secure communication

This paper investigates the finite/fixed-time synchronization problem of delayed inertial memristive neural networks (DIMNNs) using interval matrix-based methods within a unified control framework. By employing set-valued mapping and differential inclusion theory, two distinct methods are applied to handle the switching behavior of memristor parameters: the maximum absolute value method and the interval matrix method. Based on these different approaches, two control strategies are proposed to select appropriate control parameters, enabling the system to achieve finite and fixed-time synchronization, respectively. Additionally, the resulting theoretical criteria differ based on the chosen control strategy, with one expressed in algebraic form and the other in the form of linear matrix inequalities (LMIs). Numerical simulations demonstrate that the interval matrix method outperforms the maximum absolute value method in terms of handling memristor parameter switching, achieving faster finite/fixed-time synchronization. Furthermore, the theoretical results are extended to the field of image encryption, where the response system is utilized for decryption and expanding the keyspace.

Multiple Mismatched Synchronization for Coupled Memristive Neural Networks With Topology-Based Probability Impulsive Mechanism on Time Scales

This article is concerned with the exponential synchronization of coupled memristive neural networks (CMNNs) with multiple mismatched parameters and topology-based probability impulsive mechanism (TPIM) on time scales. To begin with, a novel model is designed by taking into account three types of mismatched parameters, including: 1) mismatched dimensions; 2) mismatched connection weights; and 3) mismatched time-varying delays. Then, the method of auxiliary-state variables is adopted to deal with the novel model, which implies that the presented novel model can not only use any isolated system (regard as a node) in the coupled system to synchronize the states of CMNNs but also can use an external node, that is, not affiliated to the coupled system to synchronize the states of CMNNs. Moreover, the TPIM is first proposed to efficiently schedule information transmission over the network, possibly subject to a series of nonideal factors. The novel control protocol is more robust against these nonideal factors than the traditional impulsive control mechanism. By means of the Lyapunov-Krasovskii functional, robust analysis approach, and some inequality processing techniques, exponential synchronization conditions unifying the continuous-time and discrete-time systems are derived on the framework of time scales. Finally, a numerical example is provided to illustrate the effectiveness of the main results.

Quasisynchronization of Delayed Neural Networks With Discontinuous Activation Functions on Time Scales via Event-Triggered Control

Almost all event-triggered control (ETC) strategies were designed for discrete-time or continuous-time systems. In order to unify these existing theoretical results of ETC and develop ETC strategies for nonlinear systems, whose state variables evolve steadily at one time and change intermittently at another time, this article investigates quasisynchronization of delayed neural networks (NNs) on time scales with discontinuous activation functions via ETC approaches. First, the existence of the Filippov solutions is proved for discontinuous NNs with finite discontinuities. Second, two static event-triggered conditions and two dynamic event-triggered conditions are established to avoid continuous communication between the master-slave systems under algebraic/matrix inequality criteria. Third, under static/dynamic event-triggered conditions, a positive lower bound of event-triggered intervals is demonstrated to be greater than a positive number for each event-based controller, which shows that the Zeno behavior will not occur. Finally, two numerical simulations are carried out to show the effectiveness of the presented theoretical results in this article.

Personalized Teaching Strategy of University Ideology Course Based on Lagrange Neural Network and Big Data Technology

Individualized instruction is a type of educational principle. On the one hand, it necessitates the creation of individualized teaching resources, courses, and methods. Students, on the other hand, require a high level of autonomy and the ability to make personalized plans based on their own cognitive characteristics and needs. The big data (BD) era opens up new possibilities for IE (ideological education) work in universities, but it also poses some challenges. IE development will be greatly aided by recognizing opportunities to meet challenges and optimizing and integrating PL (political lesson) resources. The LNN (Lagrange neural network) model has been established. The simulation results show that the LNN network can converge to the optimal solution quickly and effectively and then reconstruct sparse signals. Individualized college PL instruction using LNN and BD technology helps students communicate more effectively and improves the pertinence, immediacy, and positivity of IE.

Exponential Stability of Impulsive Timescale-Type Nonautonomous Neural Networks With Discrete Time-Varying and Infinite Distributed Delays

Global exponential stability (GES) for impulsive timescale-type nonautonomous neural networks (ITNNNs) with mixed delays is investigated in this article. Discrete time-varying and infinite distributed delays (DTVIDDs) are taken into consideration. First, an improved timescale-type Halanay inequality is proven by timescale theory. Second, several algebraic inequality criteria are demonstrated by constructing impulse-dependent functions and utilizing timescale analytical techniques. Different from the published works, the theoretical results can be applied to GES for ITNNNs and impulsive stabilization design of timescale-type nonautonomous neural networks (TNNNs) with mixed delays. The improved timescale-type Halanay inequality considers time-varying coefficients and DTVIDDs, which improves and extends some existing ones. GES criteria for ITNNNs cover the stability conditions of discrete-time nonautonomous neural networks (NNs) and continuous-time ones, and these theoretical results hold for NNs with discrete-continuous dynamics. The effectiveness of our new theoretical results is verified by two numerical examples in the end.

Fixed-time synchronization of discontinuous competitive neural networks with time-varying delays

In this article, the fixed-time (FXT) synchronization of discontinuous competitive neural networks (CNNs) involving time-varying delays is investigated. Firstly, two kinds of discontinuous FXT control schemes are proposed and two forms of Lyapunov function are constructed based on p-norm and 1-norm to discuss the FXT synchronization of CNNs. By means of nonsmooth analysis and some inequality techniques, some simple criteria are obtained to achieve FXT synchronization and the upper bound of the settling time with less conservativeness is provided. Furthermore, the effect of time scale on FXT synchronization of CNNs is considered. Lastly, some numerical results for an example are provided to demonstrate the derived theoretical results.

Memristor Neural Networks for Linear and Quadratic Programming Problems

This article introduces a new class of memristor neural networks (NNs) for solving, in real-time, quadratic programming (QP) and linear programming (LP) problems. The networks, which are called memristor programming NNs (MPNNs), use a set of filamentary-type memristors with sharp memristance transitions for constraint satisfaction and an additional set of memristors with smooth memristance transitions for memorizing the result of a computation. The nonlinear dynamics and global optimization capabilities of MPNNs for QP and LP problems are thoroughly investigated via a recently introduced technique called the flux-charge analysis method. One main feature of MPNNs is that the processing is performed in the flux-charge domain rather than in the conventional voltage-current domain. This enables exploiting the unconventional features of memristors to obtain advantages over the traditional NNs for QP and LP problems operating in the voltage-current domain. One advantage is that operating in the flux-charge domain allows for reduced power consumption, since in an MPNN, voltages, currents, and, hence, power vanish when the quick analog transient is over. Moreover, an MPNN works in accordance with the fundamental principle of in-memory computing, that is, the nonlinearity of the memristor is used in the dynamic computation, but the same memristor is also used to memorize in a nonvolatile way the result of a computation.

Predefined-Time Stability/Synchronization of Coupled Memristive Neural Networks With Multi-Links and Application in Secure Communication

This paper explores the realization of a predefined-time synchronization problem for coupled memristive neural networks with multi-links (MCMNN) via nonlinear control. Several effective conditions are obtained to achieve the predefined-time synchronization of MCMNN based on the controller and Lyapunov function. Moreover, the settling time can be tunable based on a parameter designed by the controller, which is more flexible than fixed-time synchronization. Then based on the predefined-time stability criterion and the tunable settling time, we propose a secure communication scheme. This scheme can determine security of communication in the aspect of encrypting the plaintext signal with the participation of multi-links topology and coupled form. Meanwhile, the plaintext signals can be recovered well according to the given new predefined-time stability theorem. Finally, numerical simulations are given to verify the effectiveness of the obtained theoretical results and the feasibility of the secure communication scheme.

Finite-/Fixed-Time Synchronization of Delayed Coupled Discontinuous Neural Networks With Unified Control Schemes

In this article, it addresses the problem of finite-/fixed-time synchronization of delayed coupled discontinuous neural networks in the unified framework. To achieve the finite-/fixed-time synchronization and precise estimations of setting time, two novel different kinds of controllers are established, in which one is switching. Then, based on the finite-/fixed-time theorem and Lyapunov function theory, some useful criteria are obtained to select suitable controllers' parameters, which can guarantee error systems converge in the finite time/fixed time with respect to coupled neural networks. Moreover, corresponding estimations of the setting time are also provided. Finally, two numerical examples are introduced to show the effectiveness of the proposed control protocols.

Nonseparation Method-Based Finite/Fixed-Time Synchronization of Fully Complex-Valued Discontinuous Neural Networks

This article mainly focuses on the problem of synchronization in finite and fixed time for fully complex-variable delayed neural networks involving discontinuous activations and time-varying delays without dividing the original complex-variable neural networks into two subsystems in the real domain. To avoid the separation method, a complex-valued sign function is proposed and its properties are established. By means of the introduced sign function, two discontinuous control strategies are developed under the quadratic norm and a new norm based on absolute values of real and imaginary parts. By applying nonsmooth analysis and some novel inequality techniques in the complex field, several synchronization criteria and the estimates of the settling time are derived. In particular, under the new norm framework, a unified control strategy is designed and it is revealed that a parameter value in the controller completely decides the networks are synchronized whether in finite time or in fixed time. Finally, some numerical results for an example are provided to support the established theoretical results.

Synchronization of memristive neural networks with unknown parameters via event-triggered adaptive control

This paper considers the drive-response synchronization of memristive neural networks (MNNs) with unknown parameters, where the unbounded discrete and bounded distributed time-varying delays are involved. Aiming at the unknown parameters of MNNs, the updating law of weight in response system and the gain of adaptive controller are proposed to realize the synchronization of delayed MNNs. In view of the limited communication and bandwidth, the event-triggered mechanism is introduced to adaptive control, which not only decreases the times of controller update and the amount of data sending out but also enables synchronization when parameters of MNNs are unknown. In addition, a relative threshold strategy, which is relative to fixed threshold strategy, is proposed to increase the inter-execution intervals and to improve the control effect. When the parameters of MNNs are known, the algebraic criteria of synchronization are established via event-triggered state feedback control by exploiting inequality techniques and calculus theorems. Finally, one simulation is presented to validate the effectiveness of the proposed results.

Exponential quasi-synchronization of coupled delayed memristive neural networks via intermittent event-triggered control

Firstly, an intermittent event-triggered control (IETC), as a combination of intermittent control and event-triggered control, is proposed. Then, the quasi-synchronization problem of coupled memristive neural networks with time-varying delays (CDMNN) is discussed under this IETC. To include more of the existing work, aperiodic intermittent control and event-triggered control with combined measurement errors are adopted in the IETC. Under the IETC, it is shown that Zeno behavior cannot be exhibited for CDMNN. At the same time, two new differential inequalities are established, and some simple and practical criteria for CDMNN quasi-synchronization and synchronization are obtained by using these inequalities. In the obtained results, synchronization is a spatial case of quasi-synchronization, and the activation functions of DMNN do not need to be bounded. Finally, a numerical example and some simulations are provided to test the results in theoretical analysis.

MultiStep Ahead Forecasting for Hourly PM10 and PM2.5 Based on Two-Stage Decomposition Embedded Sample Entropy and Group Teacher Optimization Algorithm

The randomness, nonstationarity and irregularity of air pollutant data bring difficulties to forecasting. To improve the forecast accuracy, we propose a novel hybrid approach based on two-stage decomposition embedded sample entropy, group teaching optimization algorithm (GTOA), and extreme learning machine (ELM) to forecast the concentration of particulate matter (PM10 and PM2.5). First, the improvement complementary ensemble empirical mode decomposition with adaptive noise (ICEEMDAN) is employed to decompose the concentration data of PM10 and PM2.5 into a set of intrinsic mode functions (IMFs) with different frequencies. In addition, wavelet transform (WT) is utilized to decompose the IMFs with high frequency based on sample entropy values. Then the GTOA algorithm is used to optimize ELM. Furthermore, the GTOA-ELM is utilized to predict all the subseries. The final forecast result is obtained by ensemble of the forecast results of all subseries. To further prove the predictable performance of the hybrid approach on air pollutants, the hourly concentration data of PM2.5 and PM10 are used to make one-step-, two-step- and three-step-ahead predictions. The empirical results demonstrate that the hybrid ICEEMDAN-WT-GTOA-ELM approach has superior forecasting performance and stability over other methods. This novel method also provides an effective and efficient approach to make predictions for nonlinear, nonstationary and irregular data.

Event-Triggered Exponential Synchronization for Complex-Valued Memristive Neural Networks With Time-Varying Delays

This article solves the event-triggered exponential synchronization problem for a class of complex-valued memristive neural networks with time-varying delays. The drive-response complex-valued memristive neural networks are translated into two real-valued memristive neural networks through the method of separating the complex-valued memristive neural networks into real and imaginary parts. In order to reduce the information exchange frequency between the sensor and the controller, a novel event-triggered mechanism with the event-triggering functions is introduced in wireless communication networks. Some sufficient conditions are established to achieve the event-triggered exponential synchronization for drive-response complex-valued memristive neural networks with time-varying delays. In addition, to guarantee that the Zeno behavior cannot occur, a positive lower bound for the interevent times is explicitly derived. Finally, numerical simulations are provided to illustrate the effectiveness and superiority of the obtained theoretical results.

Stabilization of Nonautonomous Recurrent Neural Networks With Bounded and Unbounded Delays on Time Scales

A class of nonautonomous recurrent neural networks (NRNNs) with time-varying delays is considered on time scales. Bounded delays and unbounded delays have been taken into consideration, respectively. First, a new generalized Halanay inequality on time scales is constructed by time-scale theory and some analytical techniques. Based on this inequality, the stabilization of NRNNs with bounded delays is discussed on time scales. The results are also applied to the synchronization of a class of drive-response NRNNs. Furthermore, the stabilization of NRNNs with unbounded delays is investigated. Especially, the stabilization of NRNNs with proportional delays is obtained without any variable transformation. The obtained generalized Halanay inequality on time scales develops and extends some existing ones in the literature. The stabilization criteria for the NRNNs with bounded or unbounded delays cover the results of continuous-time and discrete-time NRNNs and hold the results for the systems that involved on time interval as well. Some examples are given to demonstrate the validity of the results. An application to image encryption and decryption is addressed.

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.

Intermittent Discrete Observation Control for Synchronization of Stochastic Neural Networks

In this paper, to investigate the exponential synchronization of stochastic neural networks, a new periodically intermittent discrete observation control (PIDOC) is first proposed. Different from the existing periodically intermittent control, our control in control time is feedback control based on discrete-time state observations (FCDSOs) instead of a continuous-time one. By employing the Lyapunov method, graph theory, and theory of differential inclusions, the exponential synchronization of stochastic neural networks with a discontinuous right-hand side is realized by PIDOC and some sufficient conditions are presented. Especially, when control width tends to control period, PIDOC will be reduced to a general FCDSO and we give some detailed discussions. Then, we provide some corollaries about synchronization in mean square, asymptotical synchronization in mean square, and exponential synchronization of stochastic neural networks under FCDSO. Finally, some numerical simulations are provided to demonstrate our analytical results.

Prespecified-Time Cluster Synchronization of Complex Networks via a Smooth Control Approach

Most existing finite-/fixed-time synchronization control schemes are nonsmooth or discontinuous, and the settling time is estimated with conservatism. It is due to the utilization of signum function or fraction power state feedback. This brief considers the problem of prespecified-time cluster synchronization of complex networks with a smooth control protocol. The synchronization time is independent of any control parameters or any systems' initial conditions, which is actually uniformly prescribed according to task requirements without any estimations. Moreover, the cluster synchronization can maintain after the specified time, and the smooth control input can always keep uniformly bounded in an infinite time interval as well. Finally, one numerical example is provided to illustrate the effectiveness of the proposed protocol and design method.

Delay-Distribution-Dependent H∞ State Estimation for Discrete-Time Memristive Neural Networks With Mixed Time-Delays and Fading Measurements

This paper addresses the H _∞ state estimation issue for a sort of memristive neural networks in the discrete-time setting under randomly occurring mixed time-delays and fading measurements. The main purpose of the addressed issue is to propose a state estimator design algorithm that ensures the error dynamics of the state estimation to be stochastically stable with a prespecified H _∞ disturbance attenuation index. We put forward certain switching functions to account for the discrete-time yet state-dependent characteristics of the memristive connection weights. By resorting to the robust analysis theory and the Lyapunov-functional analysis theory, we derive some sufficient conditions to guarantee the desired estimation performance. The derived sufficient conditions rely not only on the size of discrete time-delays and the probability distribution law of the distributed time-delays but also on the statistics information of the coefficients of the adopted Rice fading model. Based on the established existence conditions, the gain matrices of the desired estimator are obtained by means of the feasibility of a set of matrix inequalities that can be checked efficiently via available software packages. Finally, the numerical simulation results are provided to show the validity of the main results.

Quasi-Synchronization of Delayed Memristive Neural Networks via Region-Partitioning-Dependent Intermittent Control

This paper aims at investigating the master-slave quasi-synchronization of delayed memristive neural networks (MNNs) by proposing a region-partitioning-dependent intermittent control. The proposed method is described by three partitions of non-negative real region and an auxiliary positive definite function. Whether the control input is imposed on the slave system or not is decided by the dynamical relationships among the three subregions and the auxiliary function. From these ingredients, several succinct criteria with the associated co-design procedure are presented such that the synchronization error converges to a predetermined level. The proposed intermittent control scheme is also applied to the event-triggered control, and an intermittent event-triggered mechanism is devised to investigate the quasi-synchronization of MNNs correspondingly. Such mechanism eliminates the events in rest time, and then it reduces the amount of samplings. Finally, two illustrative examples are presented to verify the effectiveness of our theoretical results.

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

The synchronization problem of multiple/coupled reaction-diffusion neural networks with time-varying delays is investigated. Differing from the existing considerations, state delays among distinct neurons and coupling delays among different subnetworks are included in the proposed model, the assumptions posed on the arisen delays are very weak, time-varying, heterogeneous, even unbounded delays are permitted. To overcome the difficulties from this kind of delay as well as diffusion effects, a comparison-based approach is applied to this model and a series of algebraic criteria are successfully obtained to verify the global asymptotical synchronization. By specifying the existing delays, some M -matrix-based criteria are derived to justify the power-rate synchronization and exponential synchronization. In addition, new criterion on synchronization of general connected neural networks without diffusion effects is also given. Finally, two simulation examples are given to verify the effectiveness of the obtained theoretical results and provide a comparison with the existing criterion.

Global Synchronization of Coupled Fractional-Order Recurrent Neural Networks

This paper presents new theoretical results on the global synchronization of coupled fractional-order recurrent neural networks. Under the assumptions that the coupled fractional-order recurrent neural networks are sequentially connected in form of a single spanning tree or multiple spanning trees, two sets of sufficient conditions are derived for ascertaining the global synchronization by using the properties of Mittag-Leffler function and stochastic matrices. Compared with existing works, the results herein are applicable for fractional-order systems, which could be viewed as an extension of integer-order ones. Two numerical examples are presented to illustrate the effectiveness and characteristics of the theoretical results.

Global Exponential Stability and Synchronization for Discrete-Time Inertial Neural Networks With Time Delays: A Timescale Approach

This paper considers generalized discrete-time inertial neural network (GDINN). By timescale theory, the original network is rewritten as a timescale-type inertial NN. Two different scenarios are considered. In a first scenario, several criteria guaranteeing the global exponential stability for the addressed GDINN are obtained based on the generalized matrix measure concept. In this case, Lyapunov function or functional is not necessary. In a second scenario, some inequality analytical and scaling techniques are used to achieve the global exponential stability for the considered GDINN. The obtained criteria are also applied to the global exponential synchronization of drive-response GDINNs. Several illustrative examples, including applications to the pseudorandom number generator and encrypted image transmission, are given to show the effectiveness of the theoretical 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.

Multiple ψ -Type Stability and Its Robustness for Recurrent Neural Networks With Time-Varying Delays

In this paper, the ψ -type stability and robustness of recurrent neural networks are investigated by using the differential inequality. By utilizing ψ -type functions combined with the inequality techniques, some sufficient conditions ensuring ψ -type stability and robustness are derived for linear neural networks with time-varying delays. Then, by choosing appropriate Lipschitz coefficient in subregion, some algebraic criteria of the multiple ψ -type stability and robust boundedness are established for the delayed neural networks with time-varying delays. For special cases, several criteria are also presented by selecting parameters with easy implementation. The derived results cover both ψ -type mono-stability and multiple ψ -type stability. In addition, these theoretical results contain exponential stability, polynomial stability, and μ -stability, and they also complement and extend some previous results. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed criteria.

Energy-to-Peak State Estimation for Static Neural Networks With Interval Time-Varying Delays

This paper is concerned with energy-to-peak state estimation on static neural networks (SNNs) with interval time-varying delays. The objective is to design suitable delay-dependent state estimators such that the peak value of the estimation error state can be minimized for all disturbances with bounded energy. Note that the Lyapunov-Krasovskii functional (LKF) method plus proper integral inequalities provides a powerful tool in stability analysis and state estimation of delayed NNs. The main contribution of this paper lies in three points: 1) the relationship between two integral inequalities based on orthogonal and nonorthogonal polynomial sequences is disclosed. It is proven that the second-order Bessel-Legendre inequality (BLI), which is based on an orthogonal polynomial sequence, outperforms the second-order integral inequality recently established based on a nonorthogonal polynomial sequence; 2) the LKF method together with the second-order BLI is employed to derive some novel sufficient conditions such that the resulting estimation error system is globally asymptotically stable with desirable energy-to-peak performance, in which two types of time-varying delays are considered, allowing its derivative information is partly known or totally unknown; and 3) a linear-matrix-inequality-based approach is presented to design energy-to-peak state estimators for SNNs with two types of time-varying delays, whose efficiency is demonstrated via two widely studied numerical examples.

Hierarchical Type Stability Criteria for Delayed Neural Networks via Canonical Bessel-Legendre Inequalities

This paper is concerned with global asymptotic stability of delayed neural networks. Notice that a Bessel-Legendre inequality plays a key role in deriving less conservative stability criteria for delayed neural networks. However, this inequality is in the form of Legendre polynomials and the integral interval is fixed on . As a result, the application scope of the Bessel-Legendre inequality is limited. This paper aims to develop the Bessel-Legendre inequality method so that less conservative stability criteria are expected. First, by introducing a canonical orthogonal polynomial sequel, a canonical Bessel-Legendre inequality and its affine version are established, which are not explicitly in the form of Legendre polynomials. Moreover, the integral interval is shifted to a general one . Second, by introducing a proper augmented Lyapunov-Krasovskii functional, which is tailored for the canonical Bessel-Legendre inequality, some sufficient conditions on global asymptotic stability are formulated for neural networks with constant delays and neural networks with time-varying delays, respectively. These conditions are proven to have a hierarchical feature: the higher level of hierarchy, the less conservatism of the stability criterion. Finally, three numerical examples are given to illustrate the efficiency of the proposed stability criteria.