Unbounded Time-varying Delays Research Articles

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.

Stability of positive coupled differential-difference equations with unbounded time-varying delays

This paper considers the stability problem of a class of positive coupled differential-difference equations with unbounded time-varying delays. A new method, which is based on upper bounding of the state vector by a decreasing function, is presented to analyze the stability of the system. Different from the existing methods, our method does not use the usual Lyapunov–Krasovskii functional method or the comparison method based on positive systems with constant delays. A new criterion is derived which ensures asymptotic stability of the system with unbounded time-varying delays. A numerical example with simulation results is given to illustrate the stability criterion.

Global Dissipativity of Inertial Neural Networks with Proportional Delay via New Generalized Halanay Inequalities

This article is devoted to the global dissipativity of inertial neural networks with proportional delay. A novel generalized Halanay inequality which involves proportional delay is established. By constructing a new generalized Halanay inequality, several new explicit delay-independent conditions are derived in terms of linear matrix inequalities to ensure the global dissipativity of the considered system. Moreover, a new differential delay inequality which involves unbounded time-varying delay is considered. Due to the proportional delay is one type of unbounded time-varying delays, new analysis techniques can effectively avoid the difficulties caused by proportional delay by applying a new differential delay inequality. Especially, several novel delay-dependent sufficient conditions are obtained to guarantee the global dissipativity of the considered system. Finally, two simulations examples are provided to illustrate the validity of the proposed theoretical analysis.

Coexistence and local $$\varvec{\mu }$$μ-stability of multiple equilibrium points for complex-valued Cohen–Grossberg neural networks with unbounded time-varying delays

This paper concentrates on the multistability problem of the complex-valued Cohen–Grossberg neural networks (CVCGNNs) with two classes of nondecreasing activation functions and unbounded time-varying delays. By virtue of the properties of the activation functions and the fixed point theorem, sufficient conditions are established to ascertain that the n-dimensional CVCGNNs can have \(9^n\) equilibrium points, \(4^n\) of which are locally \(\mu \)-stable. Some criteria that guarantee the multiple exponential stability, multiple power stability, multiple log-stability, multiple log–log-stability are also derived. The obtained results generalize some previous works in the literature. Finally, two examples with simulations are presented to clarify the validity of the theoretical results.

Stability Analysis of Monotone Systems via Max-Separable Lyapunov Functions

We analyze stability properties of monotone nonlinear systems via max-separable Lyapunov functions, motivated by the following observations: first, recent results have shown that asymptotic stability of a monotone nonlinear system implies the existence of a max-separable Lyapunov function on a compact set; second, for monotone linear systems, asymptotic stability implies the stronger properties of D-stability and insensitivity to time delays. This paper establishes that for monotone nonlinear systems, equivalence holds between asymptotic stability, the existence of a max-separable Lyapunov function, D-stability, and insensitivity to bounded and unbounded time-varying delays. In particular, a new and general notion of D-stability for monotone nonlinear systems is discussed, and a set of necessary and sufficient conditions for delay-independent stability are derived. Examples show how the results extend the state of the art.

Multiple μ-stability analysis for memristor-based complex-valued neural networks with nonmonotonic piecewise nonlinear activation functions and unbounded time-varying delays

Abstract This paper concentrates on the coexistence and dynamical behaviors of multiple equilibrium points for a class of memristor-based complex-valued neural networks (MCVNNs) with non-monotonic piecewise nonlinear activation functions and unbounded time-varying delays. Based on the geometrical configuration of activation functions, by utilizing the Intermediate Value Theorem and other analytical tools, some novel algebraic criteria are proposed to guarantee the coexistence of 25n equilibrium points in which 9n equilibrium points are locally μ-stable for such MCVNNs. As a direct application of these results, some criteria that assure the multiple exponential stability, multiple power stability, multiple log-stability and multiple log–log-stability are established. The proposed results show that the complex-valued neural networks introduced in this paper can have greater storage capacity than the real-valued ones. Finally, numerical example is presented to demonstrate the validity and feasibility of the obtained theoretical findings.

Impulsive control of unstable neural networks with unbounded time-varying delays

This paper considers the impulsive control of unstable neural networks with unbounded time-varying delays, where the time delays to be addressed include the unbounded discrete time-varying delay and unbounded distributed time-varying delay. By employing impulsive control theory and some analysis techniques, several sufficient conditions ensuring μ-stability, including uniform stability, (global) asymptotical stability, and (global) exponential stability, are derived. It is shown that an unstable delay neural network, especially for the case of unbounded time-varying delays, can be stabilized and has μ-stability via proper impulsive control strategies. Three numerical examples and their simulations are presented to demonstrate the effectiveness of the control strategy.

Razumikhin-type theorems for time-delay systems with Persistent impulses

The main results of the paper are generalizations of the Razumikhin classical theorems for stability analysis of impulsive time-delay systems. By employing Lyapunov–Razumikhin method and impulsive control theory, several Razumikhin-type theorems for uniform stability and global exponential stability are obtained. The significance and novelty of the results lie in that the criteria for stability of impulsive time-delay systems admit the existence of persistent impulses and unbounded time-varying delays. Two examples are provided to show the effectiveness of the proposed results.

On the Decay Rate of Discrete-Time Linear Delay Systems With Cone Invariance

This technical note is concerned with the decay rate constrained stability analysis for linear delay systems that possess cone-invariant property. In order to capture the decay rate of such systems, we introduce a nondecreasing positive function whose reciprocal represents the decay rate. Under mild assumptions on the growth rate of this function, an explicit condition is given to ensure that a cone-preserving linear system with unbounded time-varying delays is asymptotically stable with a given decay rate. As typical cases, necessary and sufficient conditions are given to characterize the decay rate when the delay is restricted by a linear, sublinear or logarithmic growth rate. Finally, some numerical examples are given to illustrate the effectiveness of the theoretical results.

Multistability of Recurrent Neural Networks With Nonmonotonic Activation Functions and Unbounded Time-Varying Delays

This paper is concerned with the coexistence of multiple equilibrium points and dynamical behaviors of recurrent neural networks with nonmonotonic activation functions and unbounded time-varying delays. Based on a state space partition by using the geometrical properties of the activation functions, it is revealed that an -neuron neural network can exhibit equilibrium points with . In particular, several sufficient criteria are proposed to ascertain the asymptotical stability of equilibrium points for recurrent neural networks. These theoretical results cover both monostability and multistability. Furthermore, the attraction basins of asymptotically stable equilibrium points are estimated. It is shown that the attraction basins of the stable equilibrium points can be larger than their originally partitioned subsets. Finally, the results are illustrated by using the simulation results of four examples.

Coexistence and local [formula omitted]-stability of multiple equilibrium points for memristive neural networks with nonmonotonic piecewise linear activation functions and unbounded time-varying delays

In this paper, the coexistence and dynamical behaviors of multiple equilibrium points are discussed for a class of memristive neural networks (MNNs) with unbounded time-varying delays and nonmonotonic piecewise linear activation functions. By means of the fixed point theorem, nonsmooth analysis theory and rigorous mathematical analysis, it is proven that under some conditions, such n-neuron MNNs can have 5n equilibrium points located in ℜn, and 3n of them are locally μ-stable. As a direct application, some criteria are also obtained on the multiple exponential stability, multiple power stability, multiple log-stability and multiple log–log-stability. All these results reveal that the addressed neural networks with activation functions introduced in this paper can generate greater storage capacity than the ones with Mexican-hat-type activation function. Numerical simulations are presented to substantiate the theoretical results.

Effects of bounded and unbounded leakage time-varying delays in memristor-based recurrent neural networks with different memductance functions

This paper is concerned with the problem of μ-stability analysis of memristor-based recurrent neural networks with the effects of bounded and unbounded leakage time-varying delays. A new idea of μ-stability analysis of neural networks is given first, then by means of the linear matrix inequality (LMI) approach, stability criteria are presented. Two types of memductance functions is used to derive the proposed stability results. Obviously, the memristive neural network with different memductance functions is a state-dependent switched system or a state-dependent continuous system, which is the generalization of those for conventional artificial neural networks. Under the framework of Filippov solutions, μ-stability analysis of solutions for functional differential inclusions and memristor-based neural networks can be guaranteed by constructing suitable Lyapunov–Krasovskii functionals, suitable inequalities and LMIs. The dynamic analysis in this paper utilizes the theory of set-valued maps and functional differential equations with discontinuous right-hand sides of Filippov type. Leakage time-varying delay is considered in this paper to be bounded, unbounded and differentiable. Taking into account of the information of the neuron activation functions and unbounded time-varying delays, several improved results have been obtained in terms of LMIs and these are tested by MATLAB LMI toolbox. The model based on the memristor widens the relevance scope for the design of neural networks, and the new effective results also enrich the toolbox for the qualitative analysis of neural networks. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed theoretical results.

Global μ-stability criteria for quaternion-valued neural networks with unbounded time-varying delays

In this paper, we first propose quaternion-valued neural networks (QVNNs) with unbounded time-varying delays. Some sufficient conditions on the global μ-stability in the form of both complex-valued and real-valued linear matrix inequalities (LMIs) are provided by solving two difficulties. One is decomposing the QVNN into two complex-valued systems with the plural decomposition method of quaternion, which can reduce the complexity of calculations by avoiding the non-commutativity of quaternion multiplication. The other is choosing the appropriate Lyapunov–Krasovskii functional in the form of Hermitian matrices, which is a big challenge. Finally, two numerical examples are provided to verify the effectiveness of the obtained results.

$\mu $ -Stability of Nonlinear Positive Systems With Unbounded Time-Varying Delays.

The stability of the zero solution plays an important role in the investigation of positive systems. In this brief, we discuss the μ -stability of positive nonlinear systems with unbounded time-varying delays. The system is modeled by the continuous-time ordinary differential equation. Under some assumptions on the nonlinear functions, such as homogeneous, cooperative, and nondecreasing, we propose a novel transform, by which the nonlinear system reduces to a new system. Thus, we analyze its dynamics, which can simplify the nonlinear homogenous functions with respect to the arbitrary dilation map to those with respect to the standard dilation map. We finally get some new criteria for the global μ -stability taking the degree into consideration. A numerical example is given to demonstrate the validity of obtained results.

Decentralized risk-sensitive design for large-scale stochastic interconnected systems with time-varying delays

In this paper, the problem of risk-sensitivity for a class of large-scale interconnected stochastic nonlinear systems with unbounded time-varying delays is investigated. The design procedure is dedicated to designing a decentralized controller by using only local measurements in each subsystems, and the designed controller can guarantee any desired achievable level of long-term average cost for a given risk-sensitivity parameter μ. Under some suitable conditions, we prove the globally asymptotic stability in probability by applying the design procedure. Finally, two examples are presented to illustrate our results.

Global μ-stability of complex-valued delayed neural networks with leakage delay

In this paper, the global μ-stability is investigated for the complex-valued neural networks with leakage time delay and unbounded time-varying delays. The activation functions considered are no longer required to be derivable. By constructing appropriate Lyapunov–Krasovskii functionals and taking advantage of the free weighting matrix method, several sufficient and delay-dependent criteria are established to ensure the global μ-stability of the complex-valued neural networks, which can be easily checked by the Matlab Toolbox. Finally, three numerical examples are provided to illustrate the effectiveness of our theoretical results.

Delay-dependent passivity analysis of impulsive neural networks with time-varying delays

This paper investigates delay-dependent passivity for a class of impulsive neural networks with bounded or unbounded time-varying delays. By applying Lyapunov–Krasovskii functional and matrix inequality approach, some new delay-dependent passivity conditions are proposed in terms of the full use of the conditions of neuron activation functions and involved time-varying delays. These passivity conditions are presented in accordance with matrix inequalities, which can be easily verified via standard numerical software. Meanwhile, the results derived here include the existing relative results on the passivity for neural networks without impulse effects as special cases and can also be extended to other neural networks with more complex impulse disturbance. Finally, two numerical examples with simulations are given to demonstrate the effectiveness of the proposed criteria.

Power-rate synchronization of coupled genetic oscillators with unbounded time-varying delay.

In this paper, a new synchronization problem for the collective dynamics among genetic oscillators with unbounded time-varying delay is investigated. The dynamical system under consideration consists of an array of linearly coupled identical genetic oscillators with each oscillators having unbounded time-delays. A new concept called power-rate synchronization, which is different from both the asymptotical synchronization and the exponential synchronization, is put forward to facilitate handling the unbounded time-varying delays. By using a combination of the Lyapunov functional method, matrix inequality techniques and properties of Kronecker product, we derive several sufficient conditions that ensure the coupled genetic oscillators to be power-rate synchronized. The criteria obtained in this paper are in the form of matrix inequalities. Illustrative example is presented to show the effectiveness of the obtained results.

Further analysis of global [formula omitted]-stability of complex-valued neural networks with unbounded time-varying delays

In this paper, we consider the problem of global μ-stability for complex-valued neural networks (CVNNs) with unbounded time-varying delays and it has been widely investigated. Under mild conditions, some new sufficient conditions for global μ-stability of considered CVNNs are derived. Moreover, some new sufficient conditions are obtained to ensure the global μ-stability of CVNNs in the form of complex-valued LMIs as well as real-valued LMIs by using an appropriate Lyapunov–Krasovskii functional and linear matrix inequalities (LMIs). Both of complex-valued LMIs as well as real-valued LMIs are easily solved by using standard numerical algorithms. Finally, two numerical examples are presented to demonstrate the effectiveness and usefulness of our theoretical results.

<inline-formula> <tex-math notation="TeX">$\ell_{\infty}/L_{\infty} $</tex-math></inline-formula>-Gain Analysis for Positive Linear Systems With Unbounded Time-Varying Delays

This technical note addresses the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> -gain (respectively, L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> -gain) analysis problem for discrete-time (respectively, continuous-time) positive linear systems with unbounded time-varying delays. For the discrete-time case, by virtue of the monotonicity of an auxiliary system and the corresponding delay-free system, it is shown that the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> -gain of discrete-time positive systems with unbounded delays is insensitive of the magnitude of delays. For the continuous-time case, we first construct a sampled-data positive system and analyze its monotonicity and asymptotic behavior. Then, by comparing the delayed positive system with the proposed sampled-data positive system and the corresponding delay-free system, it is further proved that the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> -gain of continuous-time positive systems with unbounded delays is also independent of the delays and fully determined by the system matrices. The application of the results is illustrated by a sampled-data system with unbounded sampling intervals.