Exponential Stability and Stabilization of Delayed Memristive Neural Networks Based on Quadratic Convex Combination Method

$ \mathcal{L}_{2}-\mathcal{L}_{\infty} $ control for memristive NNs with non-necessarily differentiable time-varying delay

This paper investigates $ \mathcal{L}_{2}-\mathcal{L}_{\infty} $ control for memristive neural networks (MNNs) with a non-necessarily differentiable time-varying delay. The objective is to design an output-feedback controller to ensure the $ \mathcal{L}_{2}-\mathcal{L}_{\infty} $ stability of the considered MNN. A criterion on the $ \mathcal{L}_{2}-\mathcal{L}_{\infty} $ stability is proposed using a Lyapunov functional, the Bessel-Legendre inequality, and the convex combination inequality. Then, a linear matrix inequalities-based design scheme for the required output-feedback controller is developed by decoupling nonlinear terms. Finally, two examples are presented to verify the proposed $ \mathcal{L}_{2}-\mathcal{L}_{\infty} $ stability criterion and design method.

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.

Passivity and Dissipativity of Fractional-Order Quaternion-Valued Fuzzy Memristive Neural Networks: Nonlinear Scalarization Approach

In this article, the problem of passivity and dissipativity analysis is investigated for a class of fractional-order quaternion-valued fuzzy memristive neural networks. Based on the famous nonlinear scalarizing function, a nonlinear scalarization method is developed, which can be employed to compare the "size" of two different quaternions. In this way, the convex closure proposed by the quaternion-valued connection weights is meaningful. By constructing proper Lyapunov functional, several improved passivity criteria and dissipativity conclusions are established, which can be checked efficiently by utilizing some standard mathematical calculations. Finally, the obtained results are validated by simulation examples.

Producing Stable Periodic Solutions of Switched Impulsive Delayed Neural Networks Using a Matrix-Based Cubic Convex Combination Approach

This article is dedicated to designing a novel periodic impulsive control strategy for producing globally exponentially stable periodic solutions for switched neural networks with discrete and finite distributed time-varying delays. First, tunable parameters and cubic convex combination approach are proposed to study the globally exponential convergence of switched neural networks. Second, a sufficient criterion for the existence, uniqueness, and globally exponential stability of a periodic solution is demonstrated by using contraction mapping theorem and the impulse-delay-dependent Lyapunov-Krasovskii functional method. It is worth emphasizing that the addressed Lyapunov-Krasovskii functional covers both triple integral terms and novel quadruple integral terms, which makes the conservatism of the above criteria decrease. Even if the original neural network models are unstable or the impulsive effects are strong, the addressed neural network model can produce a globally exponentially stable periodic solution. These results here, which include boundedness, globally uniformly exponential convergence, and globally exponentially stability of the periodic solution, generalize and improve the earlier publications. Finally, two numerical examples and their computer simulations are given to show the effectiveness of theoretical results.

Global Exponential Stability of Memristive Neural Networks With Mixed Time-Varying Delays

This article investigates the Lagrange exponential stability and the Lyapunov exponential stability of memristive neural networks with discrete and distributed time-varying delays (DMNNs). By means of inequality techniques, theories of the M-matrix, and the comparison strategy, the Lagrange exponential stability of the underlying DMNNs is considered in the sense of Filippov, and the globally exponentially attractive set is estimated through employing the M-matrix and external input. Especially, when the external input is not concerned, the Lyapunov exponential stability of the corresponding DMNNs is developed immediately in the form of an M-matrix, which contains some published outcomes as special cases. Furthermore, by constructing an M-matrix-based differential system, the Lyapunov exponential stability of the DMNNs is studied, which is less conservative than some existing ones. Finally, three simulation examples are carried out to examine the validness of the theories.

Anti-Synchronization of Delayed State-Based Switched Inertial Neural Networks

In this article, global anti-synchronization control for a class of state-based switched inertial neural networks (SBSINNs) with time-varying delays is considered. Based on the hybrid control strategies and Lyapunov stability theory, several criteria are obtained to ensure global anti-synchronization of the underlying SBSINNs. Furthermore, we consider the global asymptotic anti-synchronization directly from the SBSINNs themselves with a nonreduced-order method. Finally, a numerical simulation is given to illustrate the effectiveness of the 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.

Exponential State Estimation for Stochastically Disturbed Discrete-Time Memristive Neural Networks: Multiobjective Approach

The state estimation of the discrete-time memristive model is studied in this article. By applying the stochastic analysis technique, sufficient formulas are established to ensure the exponentially mean-square stability of the error model. Moreover, the derived control gain matrix can be calculated via the linear matrix inequality (LMI). It should be mentioned that, by extending the derived conclusion to a multiobjective optimization problem, the maximum bound of the active function and the minimum bound of the disturbance attenuation are derived. The corresponding simulation figures are provided in the end.

New Varying-Parameter ZNN Models With Finite-Time Convergence and Noise Suppression for Time-Varying Matrix Moore-Penrose Inversion

This article aims to solve the Moore-Penrose inverse of time-varying full-rank matrices in the presence of various noises in real time. For this purpose, two varying-parameter zeroing neural networks (VPZNNs) are proposed. Specifically, VPZNN-R and VPZNN-L models, which are based on a new design formula, are designed to solve the right and left Moore-Penrose inversion problems of time-varying full-rank matrices, respectively. The two VPZNN models are activated by two novel varying-parameter nonlinear activation functions. Detailed theoretical derivations are presented to show the desired finite-time convergence and outstanding robustness of the proposed VPZNN models under various kinds of noises. In addition, existing neural models, such as the original ZNN (OZNN) and the integration-enhanced ZNN (IEZNN), are compared with the VPZNN models. Simulation observations verify the advantages of the VPZNN models over the OZNN and IEZNN models in terms of convergence and robustness. The potential of the VPZNN models for robotic applications is then illustrated by an example of robot path tracking.

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.

New Criteria on Global Stabilization of Delayed Memristive Neural Networks With Inertial Item

In this paper, we are concerned with global stabilization for a kind of delayed memristive neural network with an inertial term. By building a new Lyapunov functional and designing a feedback controller, we obtain some new results on global stabilization of the addressed delayed memristive inertial neural networks (MINNs). An adaptive control strategy is also designed to realize the global stabilization. Compared with the reduced-order method used in the existing literature, we consider the stabilization directly from the MINNs themselves without a reduced-order method. In addition, the new results proposed here are shown as algebraic criteria, which are easy to test. At last, some simulations are given to show the validity of the derived criteria.

An improved result on synchronization control for memristive neural networks with inertial terms and reaction-diffusion items

This paper investigates the synchronization issue of the memristive neural networks (MNNs) with inertial terms and reaction-diffusion items. In order to smoothly derive the controller gains and obtain an excellent control effect, the desired controller that contains a discontinuous function is proposed. Moreover, by constructing a novel Lyapunov-Krasovskii functional and combining the inequality techniques, several sufficient conditions in terms of algebraic inequalities are obtained to guarantee the synchronization of the proposed drive and response systems. Finally, three numerical simulations are exploited to support the acquired theoretical results.

Recurrent Neural Networks With External Addressable Long-Term and Working Memory for Learning Long-Term Dependences

Learning long-term dependences (LTDs) with recurrent neural networks (RNNs) is challenging due to their limited internal memories. In this paper, we propose a new external memory architecture for RNNs called an external addressable long-term and working memory (EALWM)-augmented RNN. This architecture has two distinct advantages over existing neural external memory architectures, namely the division of the external memory into two parts-long-term memory and working memory-with both addressable and the capability to learn LTDs without suffering from vanishing gradients with necessary assumptions. The experimental results on algorithm learning, language modeling, and question answering demonstrate that the proposed neural memory architecture is promising for practical applications.

A Switched Operation Approach to Sampled-Data Control Stabilization of Fuzzy Memristive Neural Networks With Time-Varying Delay

This paper investigates the issue of sampled-data stabilization for Takagi-Sugeno fuzzy memristive neural networks (FMNNs) with time-varying delay. First, the concerned FMNNs are transformed into the tractable fuzzy NNs based on the excitatory and inhibitory of memristive synaptic weights using a new convex combination technique. Meanwhile, a switched fuzzy sampled-data controller is employed for the first time to tackle stability problems related to FMNNs. Then, the novel stabilization criteria of the FMNNs are established using the fuzzy membership functions (FMFs)-dependent Lyapunov-Krasovskii functional. This sufficiently utilizes information from not only the delayed state and the actual sampling pattern but also the FMFs. Two simulation examples are presented to demonstrate the feasibility and validity of the proposed 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.

Effects of State-Dependent Impulses on Robust Exponential Stability of Quaternion-Valued Neural Networks Under Parametric Uncertainty

This paper addresses the state-dependent impulsive effects on robust exponential stability of quaternion-valued neural networks (QVNNs) with parametric uncertainties. In view of the noncommutativity of quaternion multiplication, we have to separate the concerned quaternion-valued models into four real-valued parts. Then, several assumptions ensuring every solution of the separated state-dependent impulsive neural networks intersects each of the discontinuous surface exactly once are proposed. In the meantime, by applying the B -equivalent method, the addressed state-dependent impulsive models are reduced to fixed-time ones, and the latter can be regarded as the comparative systems of the former. For the subsequent analysis, we proposed a novel norm inequality of block matrix, which can be utilized to analyze the same stability properties of the separated state-dependent impulsive models and the reduced ones efficaciously. Afterward, several sufficient conditions are well presented to guarantee the robust exponential stability of the origin of the considered models; it is worth mentioning that two cases of addressed models are analyzed concretely, that is, models with exponential stable continuous subsystems and destabilizing impulses, and models with unstable continuous subsystems and stabilizing impulses. In addition, an application case corresponding to the stability problem of models with unstable continuous subsystems and stabilizing impulses for state-dependent impulse control to robust exponential synchronization of QVNNs is considered summarily. Finally, some numerical examples are proffered to illustrate the effectiveness and correctness of the obtained 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.

Exponential Stabilization of Fuzzy Memristive Neural Networks With Hybrid Unbounded Time-Varying Delays

This paper is concerned with exponential stabilization for a class of Takagi-Sugeno fuzzy memristive neural networks (FMNNs) with unbounded discrete and distributed time-varying delays. Under the framework of Filippov solutions, algebraic criteria are established to guarantee exponential stabilization of the addressed FMNNs with hybrid unbounded time delays via designing a fuzzy state feedback controller by exploiting inequality techniques, calculus theorems, and theories of fuzzy sets. The obtained results in this paper enhance and generalize some existing ones. Meanwhile, a general theoretical framework is proposed to investigate the dynamical behaviors of various neural networks with mixed infinite time delays. Finally, two simulation examples are performed to illustrate the validity of the derived outcomes.

Stability Analysis for Memristor-Based Complex-Valued Neural Networks with Time Delays

In this paper, the problem of stability analysis for memristor-based complex-valued neural networks (MCVNNs) with time-varying delays is investigated extensively. This paper focuses on the exponential stability of the MCVNNs with time-varying delays. By means of the Brouwer's fixed-point theorem and M-matrix, the existence, uniqueness, and exponential stability of the equilibrium point for MCVNNs are studied, and several sufficient conditions are obtained. In particular, these results can be applied to general MCVNNs whether the activation functions could be explicitly described by dividing into two parts of the real parts and imaginary parts or not. Two numerical simulation examples are provided to illustrate the effectiveness of the theoretical results.

Rejecting Chaotic Disturbances Using a Super-Exponential-Zeroing Neurodynamic Approach for Synchronization of Chaotic Sensor Systems

Due to the existence of time-varying chaotic disturbances in complex applications, the chaotic synchronization of sensor systems becomes a tough issue in industry electronics fields. To accelerate the synchronization process of chaotic sensor systems, this paper proposes a super-exponential-zeroing neurodynamic (SEZN) approach and its associated controller. Unlike the conventional zeroing neurodynamic (CZN) approach with exponential convergence property, the controller designed by the proposed SEZN approach inherently possesses the advantage of super-exponential convergence property, which makes the synchronization process faster and more accurate. Theoretical analyses on the stability and convergence advantages in terms of both faster convergence speed and lower error bound within the task duration are rigorously presented. Moreover, three synchronization examples substantiate the validity of the SEZN approach and the related controller for synchronization of chaotic sensor systems. Comparisons with other approaches such as the CZN approach, show the convergence superiority of the proposed SEZN approach. Finally, extensive tests further investigate the impact on convergence performance by choosing different values of design parameter and initial state.

A Flexible Terminal Approach to Sampled-Data Exponentially Synchronization of Markovian Neural Networks With Time-Varying Delayed Signals

This paper investigates the problem of sampled-data (SD) exponentially synchronization for a class of Markovian neural networks with time-varying delayed signals. Based on the tunable parameter and convex combination computational method, a new approach named flexible terminal approach is proposed to reduce the conservatism of delay-dependent synchronization criteria. The SD subject to stochastic sampling period is introduced to exhibit the general phenomena of reality. Novel exponential synchronization criterion are derived by utilizing uniform Lyapunov-Krasovskii functional and suitable integral inequality. Finally, numerical examples are provided to show the usefulness and advantages of the proposed design procedure.

Event-Triggered State Estimation for Delayed Stochastic Memristive Neural Networks With Missing Measurements: The Discrete Time Case

In this paper, the event-triggered state estimation problem is investigated for a class of discrete-time stochastic memristive neural networks (DSMNNs) with time-varying delays and missing measurements. The DSMNN is subject to both the additive deterministic disturbances and the multiplicative stochastic noises. The missing measurements are governed by a sequence of random variables obeying the Bernoulli distribution. For the purpose of energy saving, an event-triggered communication scheme is used for DSMNNs to determine whether the measurement output is transmitted to the estimator or not. The problem addressed is to design an event-triggered estimator such that the dynamics of the estimation error is exponentially mean-square stable and the prespecified disturbance rejection attenuation level is also guaranteed. By utilizing a Lyapunov-Krasovskii functional and stochastic analysis techniques, sufficient conditions are derived to guarantee the existence of the desired estimator, and then, the estimator gains are characterized in terms of the solution to certain matrix inequalities. Finally, a numerical example is used to demonstrate the usefulness of the proposed event-triggered state estimation scheme.

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.

Robust Estimation for Neural Networks With Randomly Occurring Distributed Delays and Markovian Jump Coupling

This paper studies the issue of robust state estimation for coupled neural networks with parameter uncertainty and randomly occurring distributed delays, where the polytopic model is employed to describe the parameter uncertainty. A set of Bernoulli processes with different stochastic properties are introduced to model the randomly occurrences of the distributed delays. Novel state estimators based on the local coupling structure are proposed to make full use of the coupling information. The augmented estimation error system is obtained based on the Kronecker product. A new Lyapunov function, which depends both on the polytopic uncertainty and the coupling information, is introduced to reduce the conservatism. Sufficient conditions, which guarantee the stochastic stability and the performance of the augmented estimation error system, are established. Then, the estimator gains are further obtained on the basis of these conditions. Finally, a numerical example is used to prove the effectiveness of the results.

Dissipativity Analysis for Stochastic Memristive Neural Networks With Time-Varying Delays: A Discrete-Time Case

In this paper, the dissipativity problem of discrete-time memristive neural networks (DMNNs) with time-varying delays and stochastic perturbation is investigated. A class of logical switched functions are put forward to reflect the memristor-based switched property of connection weights, and the DMNNs are then recast into a tractable model. Based on the tractable model, the robust analysis method and Refined Jensen-based inequalities are applied to establish some sufficient conditions that ensure the of DMNNs. Two numerical examples are presented to illustrate the effectiveness of the obtained results.

Stability of Recurrent Neural Networks With Time-Varying Delay via Flexible Terminal Method

This brief is concerned with the stability criteria for recurrent neural networks with time-varying delay. First, based on convex combination technique, a delay interval with fixed terminals is changed into the one with flexible terminals, which is called flexible terminal method (FTM). Second, based on the FTM, a novel Lyapunov-Krasovskii functional is constructed, in which the integral interval associated with delayed variables is not fixed. Thus, the FTM can achieve the same effect as that of delay-partitioning method, while their implementary ways are different. Guided by FTM, Wirtinger-based integral inequality and free-weight matrix method are employed to develop several stability criteria, respectively. Finally, the feasibility and the effectiveness of the proposed results are tested by two numerical examples.

Stability Analysis of Discrete-Time Neural Networks With Time-Varying Delay via an Extended Reciprocally Convex Matrix Inequality

This paper is concerned with the stability analysis of discrete-time neural networks with a time-varying delay. Assessment of the effect of time delays on system stability requires suitable delay-dependent stability criteria. This paper aims to develop new stability criteria for reduction of conservatism without much increase of computational burden. An extended reciprocally convex matrix inequality is developed to replace the popular reciprocally convex combination lemma (RCCL). It has potential to reduce the conservatism of the RCCL-based criteria without introducing any extra decision variable due to its advantage of reduced estimation gap using the same decision variables. Moreover, a delay-product-type term is introduced for the first time into the Lyapunov function candidate such that a delay-variation-dependent stability criterion with the bounds of delay change rate is established. Finally, the advantages of the proposed criteria are demonstrated through two numerical examples.