Unbounded Time-varying Delays Research Articles

Multistability of recurrent neural networks with general periodic activation functions and unbounded time-varying delays

This paper investigates the multistability of recurrent neural networks (RNNs) with unbounded time-varying delays whose activation functions are general periodic functions. The activation function can be linear, nonlinear, or have multiple corner points, as long as it satisfies Lipschitz continuous condition. According to the characteristics of the parameters of the RNNs and the state space division method, the number of equilibrium points (EPs) of the RNNs is split into three categories, which can be unique, finite, or countable infinite. Some sufficient conditions for determining the number of EPs are presented, the criterion of asymptotically stable EPs is deduced, and the attraction basins of stable EPs are estimated. Furthermore, the multistability results in this paper are an extension of some previous results. The theoretical results are validated using three numerical simulation examples.

Finite-Time Stabilization of Inertial Memristive Neural Networks via Nonreduced Order Method.

This article investigates the finite-time stabilization problem of inertial memristive neural networks (IMNNs) with bounded and unbounded time-varying delays, respectively. To simplify the theoretical derivation, the nonreduced order method is utilized for constructing appropriate comparison functions and designing a discontinuous state feedback controller. Then, based on the controller, the state of IMNNs can directly converge to 0 in finite time. Several criteria for finite-time stabilization of IMNNs are obtained and the setting time is estimated. Compared with previous studies, the requirement of differentiability of time delay is eliminated. Finally, numerical examples illustrate the usefulness of the analysis results in this article.

Unified analysis on multistablity of fraction-order multidimensional-valued memristive neural networks

This article provides a unified analysis of the multistability of fraction-order multidimensional-valued memristive neural networks (FOMVMNNs) with unbounded time-varying delays. Firstly, based on the knowledge of fractional differentiation and memristors, a unified model is established. This model is a unified form of real-valued, complex-valued, and quaternion-valued systems. Then, based on a unified method, the number of equilibrium points for FOMVMNNs is discussed. The sufficient conditions for determining the number of equilibrium points have been obtained. By using 1-norm to construct Lyapunov functions, the unified criteria for multistability of FOMVMNNs are obtained, these criteria are less conservative and easier to verify. Moreover, the attraction basins of the stable equilibrium points are estimated. Finally, two numerical simulation examples are provided to verify the correctness of the results.

Observer-based resilient dissipativity control for discrete-time memristor-based neural networks with unbounded or bounded time-varying delays

This work focuses on the issue of observer-based resilient dissipativity control of discrete-time memristor-based neural networks (DTMBNNs) with unbounded or bounded time-varying delays. Firstly, the Luenberger observer is designed, and additionally based on the observed states, the observer-based resilient controller is proposed. An augmented system is presented by considering both the error system and the DTMBNNs with the controller. Secondly, a novel sufficient extended exponential dissipativity condition is obtained for the augmented system with unbounded time-varying delays by proposing a system solutions-based estimation approach. This method is based on system solutions and without constructing any Lyapunov–Krasovskii functionals (LKF), thereby reducing the complexity of theoretical derivation and computational workload. In addition, an algorithm is proposed to solve the nonlinear inequalities in the sufficient condition. Thirdly, the sufficient extended exponential dissipativity condition for the augmented system with bounded time-varying delays is also obtained. Finally, the effectiveness of the theoretical results is illustrated through two simulation examples.

Adaptive Drive-Response Synchronization of Timescale-Type Neural Networks With Unbounded Time-Varying Delays.

In recent years, adaptive drive-response synchronization (DRS) of two continuous-time delayed neural networks (NNs) has been investigated extensively. For two timescale-type NNs (TNNs), how to develop adaptive synchronization control schemes and demonstrate rigorously is still an open problem. This article concentrates on adaptive control design for synchronization of TNNs with unbounded time-varying delays. First, timescale-type Barbalat lemma and novel timescale-type inequality techniques are first proposed, which provides us practical methods to investigate timescale-type nonlinear systems. Second, using timescale-type calculus, novel timescale-type inequality, and timescale-type Barbalat lemma, we demonstrate that global asymptotic synchronization can be achieved via adaptive control under algebraic and matrix inequality criteria even if the time-varying delays are unbounded and nondifferentiable. Adaptive DRS is discussed for TNNs, which implies our control schemes are suitable for continuous-time NNs, their discrete-time counterparts, and any combination of them. Finally, numerical examples on TNNs and timescale-type chaotic Ikeda-like oscillator with unbounded time-varying delays are carried out to verify the adaptive control schemes.

$\mu$-Stability of Positive Homogeneous Differential-difference Equations with Unbounded Time-varying Delays

$\mu$-Stability of Positive Homogeneous Differential-difference Equations with Unbounded Time-varying Delays

Finite-Time Stabilization of Semi-Markov Reaction-Diffusion Memristive NNs With Unbounded Time-Varying Delays

Finite-Time Stabilization of Semi-Markov Reaction-Diffusion Memristive NNs With Unbounded Time-Varying Delays

Predefined-time synchronization of incommensurate fractional-order competitive neural networks with time-varying delays

This paper focuses on the predefined-time synchronization (PTS) of incommensurate fractional-order competitive neural networks (IFCNNs) with bounded or unbounded time-varying delays in the sense of Caputo fractional derivative (CFD). To this end, based on the piecewise Lyapunov function, a novel predefined-time stability theorem (PTST) is presented, where the setting of an adjustable time parameter in the deduced results makes it to be more versatile and more flexible, which refines former studies and can infer a series of new results. Besides, a double-layers predefined-time controller with two different fractional-order integrals is proposed, where the predefined time (PT) is predefined during the control design and can ensure that before a known time. Using this new PTST and some inequality techniques, an effective criteria for ensuring the PTS for two IFCNNs is derived in terms of algebraic inequalities. The PT is set to any positive parameter in the controllers and wholly independent of the initial values. Finally, examples are given to verify the proposed results.

Stability and [formula omitted]-gain of positive fractional-order coupled differential-difference systems with unbounded time-varying delays

This work aims to research the positivity, stability, and L∞-gain of incommensurate fractional-order coupled differential-difference systems (FOCDDSs) with unbounded time-varying delays. By virtual of the Banach’s fixed point theorem, the existence and uniqueness of the solution of FOCDDs is given. A necessary and sufficient criterion ensuring the positivity of delayed FOCDDSs is put forward. Then, by analyzing the monotonicity of system trajectories and constructing sample data systems, a necessary and sufficient criterion realizing the stability of positive FOCDDSs is obtained. Different from related works, there is no need to require the Lyapunov–Krasovskii function or the comparison between the counterpart with constant time delays by the proposed approach. To obtain the L∞-gain of FOCDDSs, two auxiliary systems are developed to calculate the limit of the state trajectory. Eventually, a numerical experiment is performed to verify the rationality of the results.

Anti-synchronization of inertial neural networks with quaternion-valued and unbounded delays: Non-reduction and non-separation approach

This article investigates the anti-synchronization (AS) problem for quaternion-valued inertial neural networks (QVINNs) with unbounded time delays. Using two different types of control strategies (feedback and adaptive controllers) and Lyapunov theory, various conditions are established to guarantee the AS of QVINNs. Most of the results available for QVINNs are based on the variable substitution approach, which reduces the order of the original second-order system into the first-order system. The quaternion-valued networks can be separated into four equivalent real-valued neural networks (RVNNs), which have made the process more complicated. However, in the present study, the authors deal with the non-reduction order method and non-separation approach for QVINNs, making the analysis approach more concise and easier to deal with inertial neural networks (INNs). Here, the QVINNs have unbounded time-varying delays since neurons’ actions are related to their previous states. Hence the approach adopted here is more realistic and easy to deal with the QVINNs with unbounded time delays, which has not yet been considered by any researcher. Finally, two numerical examples are provided to demonstrate the efficiency and effectiveness of the proposed approach. In the first example, the present theoretical results have been validated, whereas the second example contains the application of quaternion-valued neural networks (QVNNs) related to associative memory to demonstrate their capacity to restore true color image patterns accurately.

Observer-based [formula omitted] control of memristor-based neural networks with unbounded time-varying delays

This work is devoted to developing observer-based H∞ control of memristor-based neural networks with unbounded time-varying delays. A suitable observer is first designed, and then the controller is implemented based on the estimated states. Taking into account the dynamic equation of the MNN and that of the observer error, an augmented closed-loop system is given. By proposing a system solutions-based estimation method, sufficient conditions are obtained to guarantee that the augmented system is globally exponentially stable and satisfies a prescribed H∞ performance level. This approach requires neither model transformation nor the construction of Lyapunov–Krasovskii functionals. In addition, the obtained sufficient conditions contain only a few scalar inequalities, which can be easily addressed by MATLAB. Finally, illustrative simulations are given to test the validity of the theoretical results.

On Pinning Linear and Adaptive Synchronization of Multiple Fractional-Order Neural Networks With Unbounded Time-Varying Delays.

In this article, the synchronization of multiple fractional-order neural networks with unbounded time-varying delays (FNNUDs) is investigated. By introducing a pinning linear control, sufficient conditions are provided for achieving the synchronization of multiple FNNUDs via an extended Halanay inequality. Moreover, a new effective adaptive control which applies to the fractional differential equations with unbounded time-varying delays is designed, under which sufficient criteria are presented to ensure the synchronization of multiple FNNUDs. The introduced control in this article is also workable in traditional integer-order neural networks. Finally, the validity of obtained results is demonstrated by a numerical example.

Bounded real lemmas and exponential [formula omitted] control for memristor-based neural networks with unbounded time-varying delays

This paper focuses on developing a bounded real lemma (BRL) and designing a state-feedback controller which guarantees a prescribed H∞ performance level for a class of memristor-based neural networks (MNNs) with unbounded time-varying delays. Firstly, a BRL for MNNs is presented by taking a new approach based on system solutions. This approach requires neither transformation of the model nor construction of Lyapunov–Krasovskii functionals, thereby reducing computational effort and complexity. In addition, the obtained BRL contains only a few simple inequalities, which can be easily solved by using MATLAB. Secondly, the condition for the existence of exponential H∞ controller is given based on the obtained BRL. Finally, two simulation examples demonstrate the validity of the theoretical results.

Global synchronization of complex-valued neural networks with unbounded time-varying delays

This paper investigates global synchronization of complex-valued neural networks (CVNNs) with unbounded time-varying delays. By applying analytical method and inequality techniques, an algebraic criterion is established to ensure global synchronization of the CVNNs via a devised feedback controller, which generalizes some existing outcomes. Finally, two numerical simulations and one application in image encryption are provided to verify the effectiveness of the theoretical results.

Multipleμ‐stability analysis of time‐varying delayed quaternion‐valued neural networks

This article addresses the multiple ‐stability analysis of ‐dimensional quaternion‐valued neural networks (QVNNs) with unbounded time‐varying delays (UTVD) and two general classes of activation functions (AFs). Firstly, the QVNNs are decomposed into four equivalent real‐valued systems, and based on the geometrical configuration of the AFs, the state space is divided into disjoint regions. Considering the properties of AFs, several sufficient conditions are derived to ensure the coexistence of equilibria, out of which are locally ‐stable. Moreover, some sufficient conditions for multiple exponential stability, multiple power stability, and multiple log stability are also derived in this article. Finally, two numerical examples are presented. The first example validates the effectiveness of the proposed theoretical results while the second example illustrates the application of QVNNs on the associative memory, which shows that QVNNs have the ability to reliably retrieve true‐color image patterns.

Impulsive Stabilization of Nonautonomous Timescale-Type Neural Networks With Constant and Unbounded Time-Varying Delays

This article focuses on the impulsive stabilization of nonautonomous timescale-type neural networks (NTNNs) with constant and unbounded time-varying delays. By choosing two discontinuous piecewise linear functions and employing the convex combination method, several algebraic criteria are demonstrated to achieve globally asymptotic stabilization of NTNNs with constant and unbounded time-varying delays. First, the asymptotic stability theorem for timescale-type systems is constructed by utilizing the timescale theory. Based on this asymptotic stability theorem, an impulsive control scheme is designed to guarantee global asymptotic stabilization of NTNNs with constant delay. Second, we propose several impulsive control schemes to achieve globally asymptotic stabilization of NTNNs with unbounded delay by virtue of the comparison strategy. Globally asymptotic stabilization criteria for NTNNs include the theoretical results of continuous-time neural networks (NNs), their discrete-time forms and NNs on continuous-discrete hybrid time scales. Especially, impulsive control for globally asymptotic stabilization of NTNNs with proportional delay is demonstrated without any variable transformation. Finally, the effectiveness of our proposed theoretical results is verified by four numerical examples.

Bounded real lemmas for inertial neural networks with unbounded mixed delays and state-dependent switching

This paper mainly studies the bounded real lemma for inertial neural networks with unbounded time-varying transmission delays, unbounded distribution delays and state-dependent switching. Bounded real lemmas of inertial neural networks under consideration are presented by proposing a parameterizing approach based on the system solutions. The advantage of this approach is that it neither decomposes the model into two first-order differential equations nor constructs any Lyapunov–Krasovskii functional, thus reducing computational effort and complexity. Furthermore, the obtained sufficient condition contains only a few simple linear scalar inequalities, which can be easily solved by using MATLAB. Finally, a numerical example and its numerical simulation are used to demonstrate the validity of the theoretical results.

Positivity and Stability Analysis of T-S Fuzzy Descriptor Systems With Bounded and Unbounded Time-Varying Delays.

This work focuses on the stability analysis of positive Takagi-Sugeno (T-S) fuzzy descriptor systems with time-varying delays. An equivalent augmented system is constructed to investigate the positivity and stability of T-S fuzzy descriptor time-delay systems. By using this transformed system, a necessary and sufficient positivity condition is first derived for systems, which can be verified by linear programming (LP). Then, based on the positivity of T-S fuzzy descriptor systems, a sufficient condition is put forward for the asymptotic stability of systems with bounded and unbounded time-varying delays. Finally, several examples are provided to show the effectiveness of the obtained results.

Global exponential p-norm stability of BAM neural networks with unbounded time-varying delays: A method based on the representation of solutions

This paper studies the global exponential [Formula: see text]-norm stability of bidirectional associative memory (BAM) neural networks with unbounded time-varying delays. A novel method based on the representation of solutions is put forward to deduce a global exponential [Formula: see text]-norm stability criterion. This method does not need to set up any Lyapunov–Krasovskii functionals (LKF), which can greatly reduce a large amount of computations and is simpler than the existing methods. In the end, representative numerical examples are given to illustrate the availability of the method.

Multistability and Stabilization of Fractional-Order Competitive Neural Networks With Unbounded Time-Varying Delays

This article investigates the multistability and stabilization of fractional-order competitive neural networks (FOCNNs) with unbounded time-varying delays. By utilizing the monotone operator, several sufficient conditions of the coexistence of equilibrium points (EPs) are obtained for FOCNNs with concave-convex activation functions. And then, the multiple μ -stability of delayed FOCNNs is derived by the analytical method. Meanwhile, several comparisons with existing work are shown, which implies that the derived results cover the inverse-power stability and Mittag-Leffler stability as special cases. Moreover, the criteria on the stabilization of FOCNNs with uncertainty are established by designing a controller. Compared with the results of fractional-order neural networks, the obtained results in this article enrich and improve the previous results. Finally, three numerical examples are provided to show the effectiveness of the presented results.