Stability Analysis Of Neural Networks Research Articles

New results on stability analysis for a class of generalized delayed neural networks

This paper addresses the stability analysis of neural networks with a time-varying delay, where the delay is periodically varying bounded function with constrained derivatives. In order to obtain less conservative stability criteria, two novel Lyapunov-Krasovskii functionals (LKFs) with delay-derivative-variation-dependent matrices are constructed. In one LKF, two delay-product terms are double utilized and more free matrices are incorporated. Additionally, a new type of the LKF with delay-derivative-variation-dependent matrices, namely, the neuronal-activation-function-based looped functional, is developed by accounting for the periodic nature of the delay function and the coupling information of the system state and the neural activation function. Then, several stability conditions of delayed neural networks are developed by combining some integral inequalities. Finally, the superiority and validity of developed stability conditions are validated on two benchmark examples.

Stability analysis of delayed neural networks via compound-parameter -based integral inequality

<abstract><p>This paper revisits the issue of stability analysis of neural networks subjected to time-varying delays. A novel approach, termed a compound-matrix-based integral inequality (CPBII), which accounts for delay derivatives using two adjustable parameters, is introduced. By appropriately adjusting these parameters, the CPBII efficiently incorporates coupling information along with delay derivatives within integral inequalities. By using CPBII, a novel stability criterion is established for neural networks with time-varying delays. The effectiveness of this approach is demonstrated through a numerical illustration.</p></abstract>

Formula omitted] stability analysis of neural networks with multiple time-varying delays

This paper studies the problem of Lp stability analysis of neural networks (NNs) with multiple time-varying delays. The delays can be bounded or unbounded. Firstly, a new method based on system solutions is proposed, from which new Lp stability criteria are obtained. On the one hand, the method is convenient to deal with the cases of multiple delays. On the other hand, this method does not need to establish any Lyapunov–Krasovskii functional (LKF), which can greatly reduce the amount of calculations. The obtained Lp stability conditions are simpler, and can be checked by standard software tools. The advantages of the obtained Lp stability conditions are demonstrated with representative numerical examples. It is worth emphasizing that this article is the first time to analyze the Lp stability of NNs with multiple time-varying delays.

A general framework for robust stability analysis of neural networks with discrete time delays

Robust stability of different types of dynamical neural network models including time delay parameters have been extensively studied, and many different sets of sufficient conditions ensuring robust stability of these types of dynamical neural network models have been presented in past decades. In conducting stability analysis of dynamical neural systems, some basic properties of the employed activation functions and the forms of delay terms included in the mathematical representations of dynamical neural networks are of crucial importance in obtaining global stability criteria for dynamical neural systems. Therefore, this research article will examine a class of neural networks expressed by a mathematical model that involves the discrete time delay terms, the Lipschitz activation functions and possesses the intervalized parameter uncertainties. This paper will first present a new and alternative upper bound value of the second norm of the class of interval matrices, which will have an important impact on obtaining the desired results for establishing robust stability of these neural network models. Then, by exploiting wellknown Homeomorphism mapping theory and basic Lyapunov stability theory, we will state a new general framework for determining some novel robust stability conditions for dynamical neural networks possessing discrete time delay terms. This paper will also make a comprehensive review of some previously published robust stability results and show that the existing robust stability results can be easily derived from the results given in this paper.

Improved reciprocally convex inequality for stability analysis of neural networks with time-varying delay

In this paper, the stability of a type of neural network with time-varying delay is studied. Two general (α2,β2) and (α2,β2,γ2)-dependent reciprocally convex inequalities with two and three terms are derived to introduce some quadratic terms into the estimate for the derivative of the Lyapunov–Krasovskii functionals (LKFs). The LKF method is used to develop a stability criterion for time-delay neural networks by using the new inequalities. To demonstrate the improvement of the new criterion, numerical results are provided.

An extended generalized integral inequality based on free matrices and its application to stability analysis of neural networks with time-varying delays

This paper proposes an extended generalized integral inequality based on free matrices (EGIIFM) and applies it to the stability analysis of neural networks with time-varying delays. The EGIIFM estimates an upper bound for a quadratic form of a positive definite matrix with an augmented vector staked not only with the state and its derivative but also with the nonlinear activation function. By reflecting the correlated cross-information among the terms in the augmented vector as free matrices, the EGIIFM provides a tighter upper bound and encompasses various existing single integral inequalities as special cases. In addition, by establishing a new double integral Lyapunov–Krasovskii functional including the correlated cross-information, a less conservative stability criterion is obtained. Through three well-known numerical examples, the effectiveness of the EGIIFM is evaluated.

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.

基于Lyapunov-Krasovskii泛函的时变时滞神经网络稳定性分析

基于Lyapunov-Krasovskii泛函的时变时滞神经网络稳定性分析

Mittag-Leffler synchronization and stability analysis for neural networks in the fractional-order multi-dimension field

This paper proposes the relaxed criteria about the problem of synchronization and stability for the same category fractional-order system of multidimension-valued neural networks (FOSMVNNs). First, we uniformly formulate a new class of FOSMVNNs. Based on Hamilton rules, the researched FOSMVNNs are effectively separated into the four or two fractional-order systems of real-valued neural networks (FOSRVNNs). Moreover, we infer a novel inequality with the quadratic term which distinguishes the new inequality from the existing ones and apply it in the new analysis on the synchronization and stability problem for FOMVNNs. On the basis of the new inequality, some new Lyapunov-Krasovskii functionals (LKFs) and the relaxed criteria can be successfully constructed with quadratic coefficients which can be effectively used for the studied systems mainly via fractional-order Lyapunov method. Finally, we present two numerical simulations to show the possibility and improvement of the obtained relaxed criteria.

Experimental stability analysis of neural networks in classification problems with confidence sets for persistence diagrams

We investigate classification performance of neural networks (NNs) based on topological insight in an attempt to guarantee stability of their inference. NNs which can accurately classify a dataset map it into a hidden space while disentangling intertwined data. NNs sometimes acquire forcible mapping to disentangle the data, and this forcible mapping generates outliers. The mapping around the outliers is unstable because the outputs change drastically. Hence, we define stable NNs to mean that they do not generate outliers. To investigate the possibility of the existence of outliers, we use persistent homology and a method to estimate the confidence set for persistence diagrams. The combined use enables us to test whether the focused geometry is topologically simple, that is, no outliers. In this work, we use the MNIST and CIFAR-10 datasets and investigate the relationship between the classification performance and the topological characteristics with several NNs. Investigation results with the MNIST dataset show that the test accuracy of all the networks is superior, exceeding 98%, even though the transformed dataset is not topologically simple. Results with the CIFAR-10 dataset also show that the possibility of the existence of outliers is shown in the mapping by the accurate convolutional NNs. Therefore, we conclude that the presented investigation is necessary to guarantee that the NNs, in particular deep NNs, do not acquire unstable mapping for forcible classification.

Exponential stability analysis of neural networks with a time‐varying delay via a generalized Lyapunov‐Krasovskii functional method

SummaryAs is known to all that the Lyapunov‐Krasovskii functional (LKF) method plays a significant role in deriving exponential stability criteria of neural networks with a time‐varying delay. However, when the LKF method is adopted, the condition that a functional is required for a neural network with a delay varying in a delay interval is so strong that it may be hard to be satisfied and lead to a conservative criterion. Therefore, a generalized LKF method is proposed by weakening the strong condition in this paper. Then, new exponential stability criteria are derived via applying the proposed method. Finally, the effectiveness of the derived criteria is verified by two numerical examples.

Global Stability Analysis of Neural Networks with Constant Time Delay via Frobenius Norm

This paper deals with the global asymptotic robust stability (GARS) of neural networks (NNs) with constant time delay via Frobenius norm. The Frobenius norm result has been utilized to find a new sufficient condition for the existence, uniqueness, and GARS of equilibrium point of the NNs. Some suitable Lyapunov functional and the slope bounded functions have been employed to find the new sufficient condition for GARS of NNs. Finally, we give some comparative study of numerical examples for explaining the advantageous of the proposed result along with the existing GARS results in terms of network parameters.

Stability analysis of delayed neural network using new delay-product based functionals

This paper concerns with stability analysis of neural networks with time varying delay. Two new delay-product type functionals (DPFs) are developed by introducing new states in the augmented vector of delay-product term. Then using these DPFs, two new Lyapunov-Krasovskii functionals (LKFs) are constructed. Based on these LKFs, two delay-dependent stability criterion are obtained in the form of linear matrix inequalities. The effectiveness of the proposed criterion for delayed neural network is demonstrated by considering two examples.

Global exponential stability analysis of neural networks with a time-varying delay via some state-dependent zero equations

This paper studies the exponential stability problem of neural networks with a time-varying delay. Firstly, an augmented Lyapunov-Krasovskii functional (LKF) containing a single integral state is constructed. Then a generalized free-matrix-based integral inequality and the auxiliary function-based integral inequality combined with an extended reciprocally convex matrix inequality are used to estimate the derivative of the LKF. The novelty of this paper is that some state-dependent zero equations are introduced into before and after bounding the LKF’s derivative so as to increase the freedom and reduce the conservatism. As a result, a less conservative stability criterion is derived in the form of linear matrix inequality, whose superiority is illustrated with three numerical examples.

An Overview of Stability Analysis of Neural Networks with Time-Delays

Stability of neural networks (NNs) with time-delays (TDs) has become a popular research object. The Lyapunov-Krasovskii functional (LKF) method is an extremely effective technique for the stability research of NNs with TDs. Three primary issues, including ideas for constructing a reasonable LKF method, investigation of methods to estimate the LKF's derivative and reviewing approaches for establishing the stability criteria are discussed in details. Finally, some challenges are proposed for the future research directions.

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 of neural networks with time-varying delay using a new augmented Lyapunov–Krasovskii functional

This paper examines the problem of asymptotic stability of continuous neural networks with time-varying delay via a new Lyapunov–Krasovskii functional (LKF). First, a suitable quadratic functional is constructed, which coordinates with the use of the orthogonal-polynomials-based integral inequality. Second, the novel proposed LKF contains more state vectors of neural networks, so that more state information can be exploited adequately. By combining the new proposed LKF and orthogonal-polynomials-based integral inequality, novel delay-dependent stability criteria with less conservatism are established in the form of linear matrix inequalities (LMIs). Finally, two commonly-used numerical examples are provided to show the effectiveness and improvement of the proposed criteria.

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.

Enhanced stability criteria of neural networks with time-varying delays via a generalized free-weighting matrix integral inequality

This paper deals with the problem of delay-dependent stability analysis for neural networks with time-varying delays. First, by constructing an augmented Lyapunov–Krasovskii functional and utilizing a generalized free-weighting matrix integral inequality, an improved stability criterion for the concerned network is derived in terms of linear matrix inequalities. Second, by considering a marginal augmented vector and modifying a Lyapunov–Krasovsii functional, a further enhanced stability criterion is presented. Third, a less conservative stability condition in which a relaxed inequality related to activation functions is added is introduced. Finally, three numerical examples are included to illustrate the advantage and validity of the proposed criteria.

Passivity and stability analysis of neural networks with time-varying delays via extended free-weighting matrices integral inequality

This paper is concerned with the problem of passivity for uncertain neural networks with time-varying delays. First, the recently developed integral inequality called generalized free-matrix-based integral inequality is extended to estimate further tight lower bound of integral terms. By constructing a suitable augmented LKF, an enhanced passivity condition for the concerned network is derived in terms of linear matrix inequalities (LMIs). Here, the integral terms having three states in its quadratic form is estimated by the proposed Lemma. As special cases of main results, for neural networks without uncertainties, passivity and stability conditions are derived. Through three numerical examples, it will be shown that the developed conditions can promote the level of passivity and stability criteria.