Infinite Distributed Delays Research Articles

Positive periodic solutions of Lotka-Volterra-like impulsive functional differential equations with infinite distributed time delays on time scales

This paper is concerned with the Lotka-Volterra-like functional differential equations with impulses and infinite distributed time delays on time scales. By applying a fixed point theorem of strict-set contractions, some sufficient conditions have been established to guarantee the existence of periodic solutions. As applications, the existence of positive periodic solutions is given for some common Lotka-Volterra systems on time scales.

Comparison principle for impulsive functional differential equations with infinite delays and applications

We introduce the Razumikhin technique to comparison principle and establish some comparison results for impulsive functional differential equations (IFDEs) with infinite delays, where the infinite delays may be infinite time-varying delays or infinite distributed delays. The idea is, under the help of Razumikhin technique, to reduce the study of IFDEs with infinite delays to the study of scalar impulsive differential equations (IDEs) in which the solutions are easy to deal with. Based on the comparison principle, we study the qualitative properties of IFDEs with infinite delays , which include stability, asymptotic stability, exponential stability, practical stability, boundedness, etc. It should be mentioned that the developed results in this paper can be applied to IFDEs with not only infinite delays but also persistent impulsive perturbations. Moreover, even for the special cases of non-impulsive effects or/and finite delays, the criteria prove to be simpler and less conservative than some existing results. Finally, two examples are given to illustrate the effectiveness and advantages of the proposed results.

A sampled-data approach to distributed H∞ resilient state estimation for a class of nonlinear time-delay systems over sensor networks

This paper deals with the distributed sampled-data H∞ state estimation problem for a class of continuous-time nonlinear systems with infinite-distributed delays. To cater for possible implementation errors, the estimator gain is allowed to have certain bounded parameter variations. A sensor network is deployed to acquire the plant output by collaborating with their neighbors according to a given network topology. The individually sampled sensor measurement is transmitted to the corresponding estimator through a digital communication channel. By utilizing the input delay approach, the effect of the sampling intervals is transformed into an equivalent bounded time-varying delay. A set of sampled-data distributed estimators is designed for the addressed nonlinear systems in order to meet the following three performance requirements: (1) the asymptotic convergence of the estimation error dynamics; (2) the H∞ disturbance attenuation/rejection behavior against the exogenous disturbances; and (3) the resilience against possible gain variations. A Lyapunov functional approach is put forward to obtain the existence conditions for the desired estimators which are then parameterized in light of the feasibility of some matrix inequalities. An illustrative numerical example is given to demonstrate the usefulness of the proposed estimator design algorithm.

Reliable Control of Fuzzy Systems With Quantization and Switched Actuator Failures

This paper is concerned with the problem of reliable switched controller design for a class of discrete-time Takagi–Sugeno fuzzy systems with randomly occurring infinite-distributed delays and quantization as well as actuator failures. A random Bernoulli process is used to describe the stochastic infinite-distributed delays. Due to limited communication capacity, the control signal is quantized before being transmitted to the actuator by the logarithmic quantizer. We apply the switching mechanisms to categorize the stochastic behavior of actuator faults. Based on the parallel distributed compensation, the switched feedback controller is designed. By the fuzzy-basis-dependent Lyapunov functional approach, sufficient conditions are obtained to ensure that the resulting closed-loop system is exponentially stable in the mean-square sense with a given $ {l}_{ {2}}$ - $ {l}_{ {\infty }}$ performance index. Then, a numerical example is presented to demonstrate the effectiveness of the proposed new design method.

On passivity and robust passivity for discrete-time stochastic neural networks with randomly occurring mixed time delays

In this paper, the passivity analysis problem is investigated for a class of discrete-time stochastic neural networks (DSNNs) with randomly occurring mixed time delays (ROMDs). The mixed delays comprise time-varying discrete delays, infinite-distributed delays as well as finite-distributed delays. A set of Bernoulli-distributed white sequences is used to account for the random nature of the occurrence of the mixed time delays. In addition, stochastic disturbances are taken into consideration to describe the state-dependent noises caused possibly by electronic devices and hardware implementation of neural networks. By using a combination of Lyapunov-Krasovskii functional, free-weighting matrix approach and stochastic analysis technique, we establish sufficient conditions guaranteeing the passivity performance of the underlying DSNNs. Furthermore, a delay-dependent robust passivity criterion is presented to deal with the parameter uncertainties in the DSNNs with ROMDs. A simulation example is provided to verify the effectiveness of the proposed approach.

Finite time boundedness of neutral high-order Hopfield neural networks with time delay in the leakage term and mixed time delays

This article deals with the finite time boundedness (FTB) and FTB-stabilization problem for a general class of neutral high-order Hopfield neural networks (NHOHNNs) with time delay in the leakage term and mixed time delays. The mixed time delays consist of both discrete time-varying delays and infinite distributed delays. By using the topological degree theory, sufficient conditions are established to prove the existence of equilibrium points. Then, the Lyapunov–Krasovskii functional (LKF) method is used to prove sufficient conditions for the FTB. These conditions are in the form of linear matrix inequalities (LMIs) and can be numerically checked. Furthermore, a state feedback control is constructed to solve the FTB-stabilization problem. Finally, some numerical examples are presented to show the effectiveness of our main results.

Modelling and analysis of global resurgence of mumps: A multi-group epidemic model with asymptomatic infection, general vaccinated and exposed distributions

Based on the possible mechanism of the recent global resurgence of mumps, a novel multi-group SVEIAR epidemic model with infinite distributed delays of vaccination and latency, asymptomatic infection and nonlinear incidence is formulated, which is also equivalent to the multi-group version with ages of vaccination and latency. Here, it is assumed that the vaccine confers complete protection against infection. By constructing Lyapunov functionals, it is shown that the disease will die out if the vaccinated reproduction number Rv≤1 and the disease becomes endemic if Rv>1. Moreover, the threshold dynamics of the multi-group SVEIAR epidemic model of ordinary differential equations (ODEs) with imperfect vaccination is also established. Our main conclusions generalize and improve some existing results. In the end, it is found that the possible reasons for occurrence of backward bifurcation in epidemic models with vaccination may include partial immunity, standard incidence by comparing our results obtained with the existing literatures.

Finite-time stability analysis for neutral-type neural networks with hybrid time-varying delays without using Lyapunov method

In this paper, we study the finite-time stability problem of neutral-type neural networks with various hybrid time-varying delays. These delays include general time-varying delays, finite distributed time-varying delays, neutral-type time-varying delays and infinite distributed time-varying delays. Based on the definition of finite-time stability instead of Lyapunov function method, with the aid of inequality techniques, some simple and novel sufficient conditions are derived to guarantee the finite-time stability of our proposed network model. The deduction process is simple and easy to understand. Finally, three simulation examples are given to show the effectiveness of our main results.

H∞ state estimation for artificial neural networks over redundant channels

In this paper, a new design problem of the H∞ state estimator is developed for a kind of artificial neural networks (ANNs), where both infinite distributed delays and redundant channels are happening. These adopted redundant channels can effectively improve the reliability of networked systems from the viewpoint of engineering. Two series of stochastic variables satisfying Bernoulli distribution, are introduced to govern the infinite distributed delays and schedule the redundant channels. By utilizing both stochastic analysis and Lyapunov functional approach, we obtain a lot of sufficient conditions to ensure the desired H∞ performance, while the mean-square stability is also satisfied for this investigated state estimation issues of ANNs. The needed estimator gains are designed making use of the matrix inequalities’ solution. In final, a simulation is showed to demonstrate the effectiveness and usefulness of the developed state estimator in this paper.

Threshold dynamics of a nonlinear multi‐group epidemic model with two infinite distributed delays

The global stability of equilibria is investigated for a nonlinear multi‐group epidemic model with latency and relapses described by two distributed delays. The results show that the global dynamics are completely determined by the basic reproduction number under certain reasonable conditions on the nonlinear incidence rate. Moreover, compared with the results in Michael Y. Li and Zhisheng Shuai, Journal Differential Equations 248 (2010) 1–20, it is found that the two distributed delays have no impact on the global behaviour of the model. Our study improves and extends some known results in recent literature. Copyright © 2016 John Wiley & Sons, Ltd.

Synchronization of coupled reaction‐diffusion neural networks with mixed delays

The reaction‐diffusion neural network is often described by semilinear diffusion partial differential equation (PDE). This article focuses on the asymptotical synchronization and synchronization for coupled reaction‐diffusion neural networks with mixed delays (that is, discrete and infinite distributed delays) and Dirichlet boundary condition. First, using the Lyapunov–Krasoviskii functional scheme, the sufficient condition is obtained for the asymptotical synchronization of coupled semilinear diffusion PDEs with mixed time‐delays and this condition is represented by linear matrix inequalities (LMIs), which is easy to be solved. Then the robust synchronization is considered in temporal‐spatial domain for the coupled semilinear diffusion PDEs with mixed delays and external disturbances. In terms of the technique of completing squares, the sufficient condition is obtained for the robust synchronization. Finally, a numerical example of coupled semilinear diffusion PDEs with mixed time‐delays is given to illustrate the correctness of the obtained results. © 2016 Wiley Periodicals, Inc. Complexity 21: 42–53, 2016

Mean-square stability analysis of discrete-time stochastic Markov jump recurrent neural networks with mixed delays

In this paper, the problem of mean-square stability analysis for discrete-time stochastic Markov jump recurrent neural networks with time-varying mixed delays is considered. The Markov jumping transition probabilities are assumed completely unknown but piecewise homogeneous, and the mixed time delays under consideration comprise both time-varying discrete delay and infinite distributed delay. In the framework of the delay partitioning approach, the informations of the delay distribution probability are fully considered. With a novel Lyapunov functional, a sufficient delay-dependent condition is established, which is characterized in terms of Linear Matrix Inequalities (LMIs). Finally, a numerical example is given to demonstrate the effectiveness of the obtained results.

Exponential p-convergence analysis for stochastic BAM neural networks with time-varying and infinite distributed delays

This paper is concerned with the problem of global exponential p-convergence for stochastic BAM neural networks with time-varying and infinite distributed delays. By constructing a new delay differential-integral inequality and a novel L-operator differential-integral inequality, and coupling with stochastic analysis techniques, some delay-dependent sufficient conditions are derived to guarantee exponential p-convergence and the state variables of the discussed stochastic BAM neural networks are globally exponentially convergent to a ball in the state space with a pre-specified convergence rate. Meanwhile, the exponential p-convergent balls are also estimated. Here, the existence and the uniqueness of the equilibrium point needs not to be considered. Finally, two examples with numerical simulations are given to illustrate the effectiveness of the theoretical results.

Global asymptotic stability of nonautonomous Cohen–Grossberg neural network models with infinite delays

For a general Cohen–Grossberg neural network model with potentially unbounded time-varying coefficients and infinite distributed delays, we give sufficient conditions for its global asymptotic stability. The model studied is general enough to include, as subclass, the most of famous neural network models such as Cohen–Grossberg, Hopfield, and bidirectional associative memory. Contrary to usual in the literature, in the proofs we do not use Lyapunov functionals. As illustrated, the results are applied to several concrete models studied in the literature and a comparison of results shows that our results give new global stability criteria for several neural network models and improve some earlier publications.

Global attractivity and extinction for Lotka–Volterra systems with infinite delay and feedback controls

The paper deals with a multiple species Lotka–Volterra model with infinite distributed delays and feedback controls, for which we assume a weak form of diagonal dominance of the instantaneous negative intra-specific terms over the infinite delay effect in both the population variables and controls. General sufficient conditions for the existence and attractivity of a saturated equilibrium are established. When the saturated equilibrium is on the boundary of , sharper criteria for the extinction of all or part of the populations are given. While the literature usually treats the case of competitive systems only, here no restrictions on the signs of the intra- and inter-specific delayed terms are imposed. Moreover, our technique does not require the construction of Lyapunov functionals.

A class of delayed viral infection models with general incidence rate and adaptive immune response

To model the role of the adaptive immune response in viral infection, take account the time needed for infected cells to produce new virions after viral entry and the time necessary for the newly produced virions to become mature and infectious, we propose three generalized models that describe the interactions between virus, host cells, antibodies and cytotoxic T lymphocytes (CTL) cells. The incidence rate of the infection into three models is given by a general function which includes many special cases and depends on the uninfected cells, infected cells and virus. Our main results show that the global dynamics of the three models under certain hypotheses on the incidence function is fully determined by five threshold parameters called the reproduction numbers for viral infection, for antibody immune response, for CTL immune response, for antibody immune competition and for CTL immune competition. Several viral infection models with discrete, finite and infinite distributed delays existing in other previous studies are extended and generalized. Furthermore, we give some biological findings of our analytical results.

Spreading speeds in a lattice differential equation with distributed delay

This paper studies the spreading speed for a lattice differential equation with infinite distributed delay and we find that the spreading speed coincides with the minimal wave speed of traveling waves. Here the model has been proposed to describe a single species living in a 1D patch environment with infinite number of patches connected locally by diffusion.

Non‐fragile H ∞ control with randomly occurring gain variations, distributed delays and channel fadings

This study is concerned with the non-fragile H ∞ control problem for a class of discrete-time systems subject to randomly occurring gain variations (ROGVs), channel fadings and infinite-distributed delays. A new stochastic phenomenon (ROGVs), which is governed by a sequence of random variables with a certain probabilistic distribution, is put forward to better reflect the reality of the randomly occurring fluctuation of controller gains implemented in networked environments. A modified stochastic Rice fading model is then exploited to account for both channel fadings and random time-delays in a unified representation. The channel coefficients are a set of mutually independent random variables which abide by any (not necessarily Gaussian) probability density function on [0, 1]. Attention is focused on the analysis and design of a non-fragile H ∞ output-feedback controller such that the closed-loop control system is stochastically stable with a prescribed H ∞ performance. Through intensive stochastic analysis, sufficient conditions are established for the desired stochastic stability and H ∞ disturbance attenuation, and the addressed non-fragile control problem is then recast as a convex optimisation problem solvable via the semi-definite programme method. An example is finally provided to demonstrate the effectiveness of the proposed design method.

Reliable H∞ state estimation for 2-D discrete systems with infinite distributed delays and incomplete observations

In this note, we consider a novel reliable estimation problem for a class of two-dimensional (2-D) discrete linear systems with incomplete observations. The time delays existing in 2-D systems are assumed to be infinitely distributed in the discrete-time domain. Factors leading to incomplete observations include delayed and missing measurements, sensor quantization and saturations. A uniform mathematical model is employed to describe the phenomenon of incomplete observations. Our attention is focused on the design of a linear state estimator for the discussed 2-D systems, such that for all admissible sensor failures, possible infinite distributed time delays and incomplete observations, the error dynamics is mean-square asymptotically stable, and the steady-state error of the estimation is not more than a prescribed level . The flexibility of the proposed approach is confirmed by a numerical simulation.

Global robust exponential synchronization of BAM recurrent FNNs with infinite distributed delays and diffusion terms on time scales

In this article, the global robust exponential synchronization of reaction-diffusion BAM recurrent fuzzy neural networks (FNNs) with infinite distributed delays on time scales is investigated. Applied Lyapunov functional and inequality skills, some sufficient criteria are established to guarantee the global robust exponential synchronization of reaction-diffusion BAM recurrent FNNs with infinite distributed delays on time scales. One example is given to illustrate the effectiveness of our results.