Stability for neural networks with time-varying delays via some new approaches

Relaxed stability criteria of delayed neural networks using delay-parameters-dependent slack matrices

This note aims to reduce the conservatism of stability criteria for neural networks with time-varying delay. To this goal, on the one hand, we construct an augmented Lyapunov-Krasovskii functional (LKF), incorporating some delay-product terms that capture more information about neural states. On the other hand, when dealing with the derivative of the LKF, we introduce several parameter-dependent slack matrices into an affine integral inequality, zero equations, and the S-procedure. As a result, more relaxed stability criteria are obtained by employing the so-called Lyapunov-Krasovskii Theorem. Two numerical examples show that the proposed stability criteria are of less conservatism compared with some existing methods.

Stability Analysis of Recurrent Neural Networks With Time-Varying Delay by Flexible Terminal Interpolation Method

This brief studies the stability problem of recurrent neural networks with time-varying delay. Based on one tunable parameter α , a flexible terminal interpolation method is proposed to change the interval with fixed terminals as 2k+1-3 ones with flexible terminals. Associated with the flexible subintervals, a novel Lyapunov-Krasovskii functional with more delay information is constructed. In order to estimate the Lyapunov-Krasovskii functional, a quadratic reciprocally convex inequality is proposed, which covers some existing ones as its special cases. Based on these ingredients, a new stability criterion is derived in the form of linear matrix inequalities. A comprehensive comparison of results is given to illustrate the newly proposed stability criterion.

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.

Hierarchical Passivity Criterion for Delayed Neural Networks via A General Delay-Product-Type Lyapunov-Krasovskii Functional

This article is concerned with passivity analysis of neural networks with a time-varying delay. Several techniques in the domain are improved to establish the new passivity criterion with less conservatism. First, a Lyapunov-Krasovskii functional (LKF) is constructed with two general delay-product-type terms which contain any chosen degree of polynomials in time-varying delay. Second, a general convexity lemma without conservatism is developed to address the positive-definiteness of the LKF and the negative-definiteness of its time-derivative. Then, with these improved results, a hierarchical passivity criterion of less conservatism is obtained for neural networks with a time-varying delay, whose size and conservatism vary with the maximal degree of the time-varying delay polynomial in the LKF. It is shown that the conservatism of the passivity criterion does not always reduce as the degree of the time-varying delay polynomial increases. Finally, a numerical example is given to illustrate the proposed criterion and benchmark against the existing results.

State Observer Design of Coupled Genetic Regulatory Networks With Reaction-Diffusion Terms via Time-Space Sampled-Data Communications

In this paper, state observation of coupled reaction-diffusion genetic regulatory networks (GRNs) with time-varying delays is investigated under Dirichlet boundary conditions. First, the above GRNs are constructed to model gene regulatory properties, where the feedback regulation function of the GRNs is assumed to exhibit the Hill form and a novel method to deal with it is introduced. Then a time-space sampled-data state observer is designed for the mentioned networks and new criteria are established by utilizing the Lyapunov stability theory and the inequality techniques of Halanay et al. Finally, the validity of the theoretical results is proved by numerical examples.

Improved Stability Criteria for Discrete-Time Delayed Neural Networks via Novel Lyapunov-Krasovskii Functionals

This article investigates the stability problem for discrete-time neural networks with a time-varying delay by focusing on developing new Lyapunov-Krasovskii (L-K) functionals. A novel L-K functional is deliberately tailored from two aspects: 1) the quadratic term and 2) the single-summation term. When the variation of the discrete-time delay is further considered, the constant matrix involved in the quadratic term is extended to be a delay-dependent one. All these innovations make a contribution to a quadratic function with respect to the delay from the forward differences of L-K functionals. Consequently, tractable stability criteria are derived that are shown to be more relaxed than existing results via numerical examples.

Improved Stability Criteria for Delayed Neural Networks via a Relaxed Delay-Product-Type Lapunov–Krasovskii Functional

In this paper, the asymptotic stability problem of neural networks with time-varying delays is investigated. First, a new sufficient and necessary condition on a general polynomial inequality was developed. Then, a novel augmented Lyapunov–Krasovskii functional (LKF) was constructed, which efficiently introduces some new terms related to the previous information of neuron activation function. Furthermore, based on the suitable LKF and the stated negative condition of the general polynomial, two criteria with less conservatism were derived in the form of linear matrix inequalities. Finally, two numerical examples were carried out to confirm the superiority of the proposed criteria, and a larger allowable upper bound of delays was achieved.

Dissipativity-Based Synchronization for Switched Discrete-Time-Delayed Neural Networks With Combined Switching Paradigm

The present study concerns the dissipativity-based synchronization problem for the discrete-time switched neural networks with time-varying delay. Different from some existing research depending on the arbitrary and time-dependent switching mechanisms, all subsystems of the investigated delayed neural networks are permitted to be nondissipative. For reducing the switching frequency, the combined switching paradigm constituted by the time-dependent and state-dependent switching strategies is then constructed. In light of the proposed dwell-time-dependent storage functional, sufficient conditions with less conservativeness are formulated, under which the resultant synchronization error system is strictly (~X,~Y,~Z) - ϑ -dissipative on the basis of the combined switching mechanism or the joint action of the switching mechanism and time-varying control input. Finally, the applicability and superiority of the theoretical results are adequately substantiated with the synchronization issue of two discrete-time switched Hopfield neural networks with time-varying delay, and the relationship among the performance index, time delay, and minimum dwell time is also revealed.

Stability Analysis for Delayed Neural Networks via a Novel Negative-Definiteness Determination Method

The stability of neural networks with a time-varying delay is studied in this article. First, a relaxed Lyapunov-Krasovskii functional (LKF) is presented, in which the positive-definiteness requirement of the augmented quadratic term and the delay-product-type terms are set free, and two double integral states are augmented into the single integral terms at the same time. Second, a new negative-definiteness determination method is put forward for quadratic functions by utilizing Taylor's formula and the interval-decomposition approach. This method encompasses the previous negative-definiteness determination approaches and has less conservatism. Finally, the proposed LKF and the negative-definiteness determination method are applied to the stability analysis of neural networks with a time-varying delay, whose advantages are shown by two numerical examples.

Some Novel Results on Stability Analysis of Generalized Neural Networks With Time-Varying Delays via Augmented Approach

This article proposes three new methods to enlarge the feasible region for guaranteeing stability for generalized neural networks having time-varying delays based on the Lyapunov method. First, two new zero equalities in which three states are augmented are proposed and inserted into the results of the time derivative of the constructed Lyapunov-Krasovskii functionals for the first time. Second, inspired by the Wirtinger-based integral inequality, new Lyapunov-Krasovskii functionals are introduced. Finally, by utilizing the relationship among the augmented vectors and from the original equation, newly augmented zero equalities are established and Finsler's lemma are applied. Through three numerical examples, it is verified that the proposed methods can contribute to enhance the allowable region of maximum delay bounds.

Improved Stability Criteria for Delayed Neural Networks Using a Quadratic Function Negative-Definiteness Approach

This brief is concerned with the stability of a neural network with a time-varying delay using the quadratic function negative-definiteness approach reported recently. A more general reciprocally convex combination inequality is taken to introduce some quadratic terms into the time derivative of a Lyapunov-Krasovskii (L-K) functional. As a result, the time derivative of the L-K functional is estimated by a novel quadratic function on the time-varying delay. Moreover, a simple way is introduced to calculate the coefficients of a quadratic function, which avoids tedious works by hand as done in some studies. The L-K functional approach is applied to derive a hierarchical type stability criterion for the delayed neural networks, which is of less conservatism in comparison with some existing results through two well-studied numerical examples.

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.

Space-Dividing-Based Cluster Synchronization of Reaction-Diffusion Genetic Regulatory Networks via Intermittent Control

In this paper, we focus on the cluster synchronization of reaction-diffusion genetic regulatory networks (RDGRNs) with time-varying delays, where the state of the system is not only time-dependent but also spatially-dependent due to the presence of the reaction-diffusion terms. First, we construct an intermittent space-dividing controller that effectively combines the two control strategies, making it more cost-effective. Furthermore, based on the activation function division approach, we propose a regulation function division method that can improve the delay upper bound of RDGRNs; meanwhile, the cluster synchronization criteria of RDGRNs under the proposed controller are derived based on the Lyapunov theory and Halanay's et al. inequality techniques. Finally, the criteria's effectiveness is demonstrated by numerical examples of the system in one- and two-dimensional space.

H∞ State Estimation for Neural Networks With General Activation Function and Mixed Time-Varying Delays

This article deals with H_∞ state estimation of neural networks with mixed delays. In order to make full use of delay information, novel delay-product Lyapunov-Krasovskii functional (LKF) by using parameterized delay interval is first constructed. Then, generalized free-weighting-matrix integral inequality is used to estimate the derivative of LKF to reduce the conservatism. Also, a more general activation function is further applied by combining with parameterized delay interval in order to obtain a more accurate estimator model. Finally, sufficient conditions are derived to confirm that the estimation error system is asymptotically stable with a prescribed H_∞ performance. Numerical examples are simulated to show the benefits of our proposed method.

Relaxed Stability Criteria for Neural Networks With Time-Varying Delay Using Extended Secondary Delay Partitioning and Equivalent Reciprocal Convex Combination Techniques

This article investigates global asymptotic stability for neural networks (NNs) with time-varying delay, which is differentiable and uniformly bounded, and the delay derivative exists and is upper-bounded. First, we propose the extended secondary delay partitioning technique to construct the novel Lyapunov-Krasovskii functional, where both single-integral and double-integral state variables are considered, while the single-integral ones are only solved by the traditional secondary delay partitioning. Second, a novel free-weight matrix equality (FWME) is presented to resolve the reciprocal convex combination problem equivalently and directly without Schur complement, which eliminates the need of positive definite matrices, and is less conservative and restrictive compared with various improved reciprocal convex inequalities. Furthermore, by the present extended secondary delay partitioning, equivalent reciprocal convex combination technique, and Bessel-Legendre inequality, two different relaxed sufficient conditions ensuring global asymptotic stability for NNs are obtained, for time-varying delays, respectively, with unknown and known lower bounds of the delay derivative. Finally, two examples are given to illustrate the superiority and effectiveness of the presented method.

Hidden Markov Model-Based Nonfragile State Estimation of Switched Neural Network With Probabilistic Quantized Outputs

This paper focuses on the state estimator design problem for a switched neural network (SNN) with probabilistic quantized outputs, where the switching process is governed by a sojourn probability. It is assumed that both packet dropouts and signal quantization exist in communication channels. Asynchronous estimator and quantification function are described by two different hidden Markov model between the SNNs and its estimator. To deal with the small uncertain of estimators in a random way, a probabilistic nonfragile state estimator is introduced, where uncertain information is described by the interval type of gain variation. A sufficient condition on mean square stable of the estimation error system is obtained and then the desired estimator is designed. Finally, a simulation result is provided to verify the effectiveness of the proposed design method.

Passivity Analysis of Delayed Neural Networks Based on Lyapunov-Krasovskii Functionals With Delay-Dependent Matrices

This paper is concerned with passivity of a class of delayed neural networks. In order to derive less conservative passivity criteria, two Lyapunov-Krasovskii functionals (LKFs) with delay-dependent matrices are introduced by taking into consideration a second-order Bessel-Legendre inequality. In one LKF, the system state vector is coupled with those vectors inherited from the second-order Bessel-Legendre inequality through delay-dependent matrices, while no such coupling of them exists in the other LKF. These two LKFs are referred to as the coupled LKF and the noncoupled LKF, respectively. A number of delay-dependent passivity criteria are derived by employing a convex approach and a nonconvex approach to deal with the square of the time-varying delay appearing in the derivative of the LKF. Through numerical simulation, it is found that: 1) the coupled LKF is more beneficial than the noncoupled LKF for reducing the conservatism of the obtained passivity criteria and 2) the passivity criteria using the convex approach can deliver larger delay upper bounds than those using the nonconvex approach.

Stability Analysis for Delayed Neural Networks With an Improved General Free-Matrix-Based Integral Inequality

This paper revisits the problem of stability analysis for neural networks with a time-varying delay. An improved general free-matrix-based (FMB) integral inequality is proposed with an undetermined number m . Compared with the conventional FMB ones, the improved inequality involves a much smaller number of free matrix variables. In particular, the improved FMB integral inequality is expressed in a concrete form for any value of m . By employing the new inequality with a properly constructed Lyapunov-Krasovskii functional, a new stability condition is derived for neural networks with a time-varying delay. Two commonly used numerical examples are given to show strong competitiveness of the proposed approach in both the conservatism and computation burdens.

Stability Analysis for Neural Networks With Time-Varying Delay via Improved Techniques

This paper is concerned with the stability problem for neural networks with a time-varying delay. First, an improved generalized free-weighting-matrix integral inequality is proposed, which encompasses the conventional one as a special case. Second, an improved Lyapunov-Krasovskii functional is constructed that contains two complement triple-integral functionals. Third, based on the improved techniques, a new stability condition is derived for neural networks with a time-varying delay. Finally, two widely used numerical examples are given to demonstrate that the proposed stability condition is very competitive in both conservatism and complexity.

New Results on Stability Analysis for Delayed Markovian Generalized Neural Networks With Partly Unknown Transition Rates

The stability of delayed Markovian generalized neural networks is studied where the transition rates of the modes are partly unknown. The partly unknown transition rates generalize the traditional works that are with all known transition rates. Then, a Lyapunov-Krasovskii functional (LKF) with a delay-product-type (DPT) term is constructed. The DPT term is not only simple but also fully utilizes the information of time delay. Based on the new DPT LKF, stability criteria are presented, which are with lower computational complexity and less conservative. In the end, the validity and superiorities of the analytical results are verified by several examples.

Stability Analysis for Delayed Neural Networks via Improved Auxiliary Polynomial-Based Functions

This brief is concerned with stability analysis for delayed neural networks (DNNs). By establishing polynomials and introducing slack variables reasonably, some improved delay-product type of auxiliary polynomial-based functions (APFs) is developed to exploit additional degrees of freedom and more information on extra states. Then, by constructing Lyapunov-Krasovskii functional using APFs and integrals of quadratic forms with high order scalar functions, a novel stability criterion is derived for DNNs, in which the benefits of the improved inequalities are fully integrated and the information on delay and its derivative is well reflected. By virtue of the advantages of APFs, more desirable performance is achieved through the proposed approach, which is demonstrated by the numerical examples.

A New Approach to Stochastic Stability of Markovian Neural Networks With Generalized Transition Rates

This paper investigates the stability problem of Markovian neural networks (MNNs) with time delay. First, to reflect more realistic behaviors, more generalized transition rates are considered for MNNs, where all transition rates of some jumping modes are completely unknown. Second, a new approach, namely time-delay-dependent-matrix (TDDM) approach, is proposed for the first time. The TDDM approach is associated with both time delay and its time derivative. Thus, the TDDM approach can fully capture the information of time delay and would play a key role in deriving less conservative results. Third, based on the TDDM approach and applying Wirtinger's inequality and improved reciprocally convex inequality, stability criteria are derived. In comparison with some existing results, our results are not only less conservative but also involve lower calculation complexity. Finally, numerical examples are provided to show the effectiveness and advantages of the proposed results.

Admissible Delay Upper Bounds for Global Asymptotic Stability of Neural Networks With Time-Varying Delays

This paper is concerned with global asymptotic stability of a neural network with a time-varying delay, where the delay function is differentiable uniformly bounded with delay-derivative bounded from above. First, a general reciprocally convex inequality is presented by introducing some slack vectors with flexible dimensions. This inequality provides a tighter bound in the form of a convex combination than some existing ones. Second, by constructing proper Lyapunov-Krasovskii functional, global asymptotic stability of the neural network is analyzed for two types of the time-varying delays depending on whether or not the lower bound of the delay derivative is known. Third, noticing that sufficient conditions on stability from estimation on the derivative of some Lyapunov-Krasovskii functional are affine both on the delay function and its derivative, allowable delay sets can be refined to produce less conservative stability criteria for the neural network under study. Finally, two numerical examples are given to substantiate the effectiveness of the proposed method.