Halanay Inequality Research Articles

Distributed Event-Triggered Output Synchronization of Complex-Valued Memristive Reaction-Diffusion Complex Networks with Spatial Sampled-Data

This study addresses the problem of output quasisynchronization for coupled complex-valued memristive reaction-diffusion complex networks via the distributed event-triggered control scheme. First, by using the separate method, set value mapping, and intermediate value theorem, the complex-valued memristive reaction-diffusion complex networks can be transferred into two semi-uncertain real-valued reaction-diffusion complex networks. Second, a distributed output piecewise event-triggered control (OPETC) scheme with spatial sampled-data is first proposed including a spatial sampling event-triggered generator and spatiotemporal sampling state feedback controller. Furthermore, this scheme can effectively save the measurement resources and lower the update rate of controllers in spatial and time domain. Third, the synchronization analysis is considered by utilizing an appropriate Lyapunov function, the Halanay inequality, and the improved Wirtinger inequality. Subsequently, several output event-triggered quasisynchronization criteria are derived. The relations among event trigger conditions, spatial sampling interval, convergence rate, and control gain are given by rigorous mathematical derivation. Finally, multiple simulations are compared to substantiate the validation of the OPETC scheme.

Synchronization analysis for discrete fractional-order complex-valued neural networks with time delays

In this paper, we do not separate the complex-valued neural networks into two real-valued systems, the quasi-projective synchronization and complete synchronization of fractional-order complex-valued neural networks with time delays are investigated. First, the generalized discrete fractional Halanay inequality with bounded time delays is established. Further, based on the generalized discrete fractional Halanay inequality and Lyapunov functional method, several novel quasi-projective synchronization and complete synchronization conditions of fractional-order complex-valued neural networks with time delays are derived. Finally, several examples are presented to illustrated the results.

Global Exponential Stability and Synchronization for Novel Complex-Valued Neural Networks With Proportional Delays and Inhibitory Factors.

In this article, complex-valued neural networks (CVNNs) with proportional delays and inhibitory factors are proposed. First, the global exponential stability of the model addressed is investigated by employing the Halanay inequality technique and the matrix measure method. Some criteria are derived to guarantee the global exponential stability of CVNNs with proportional delays and inhibitory factors. The obtained criteria are applicable not only to systems with proportional delays but also to systems with arbitrary delays. Here, the Lyapunov functions are not constructed. Compared with the Lyapunov method, the matrix measure method makes the obtained criteria more concise, and the Halanay inequality makes the analytical procedure more compact. Furthermore, the global exponential synchronization of two neural-network models with proportional delays and inhibitory factors is also studied. By designing a feedback controller and giving some limitation conditions, the drive system and the response system realize global exponential synchronization. Finally, numerical simulation examples are provided to validate the effectiveness of the theoretical results obtained.

Sampled-Data State Estimation of Reaction Diffusion Genetic Regulatory Networks via Space-Dividing Approaches.

A novel state estimator is designed for genetic regulatory networks with reaction-diffusion terms in this study. First, the diffusion space (where mRNA and protein exist) is divided into several parts and only a point, a line, or a plane, etc., is measured in every subspace to reduce the measurement cost effectively. Then, samplers and network-induced time delay are considered to meet the network transmission requirement. A new criterion to ensure that the estimation error converges to zero is established by using the Lyapunov functional combined with Wirtinger's inequality, reciprocally convex approach, and Halanay's inequality; furthermore, the estimator's parameters are derived by solving linear matrix inequalities. Finally, two simulation examples (including one-dimensional and two-dimensional spaces) are presented to demonstrate the developed scheme's applicability.

Time-Varying Halanay Inequalities With Application to Stability and Control of Delayed Stochastic Systems

In this article, the Halanay inequality is generalized into the time-varying version under simple and natural conditions, without constant to bound the time-varying coefficients. The solution of the underlying inequality is estimated by virtue of the feasible function pair associated with the coefficients and delays. Corollaries are given for typical special cases. Some approaches are illustrated for the searching of the initial feasible pairs, and an iterative procedure is proposed for the optimization of the feasible pairs. The usage and superiority of the obtained result are exhibited by numerical examples with simulations. As a base, the comparison principle for the functional differential inequalities and equations with arbitrary time-varying delays is studied first. As an application of the main result, the stability and control of the stochastic systems with time-varying coefficients and delays are investigated, with the main concern to the time variance. An asymptotic stability theorem and a criterion are established, and an interesting control strategy with unbounded time delay, namely the pantograph delay sampled data based feedback, is proposed. The stability criterion and the control strategy are illustrated and verified with numerical experiments.

Synchronization analysis for delayed spatio-temporal neural networks with fractional-order

In this paper, the problem of synchronization for a class of fractional-order delayed spatio-temporal neural networks with Dirichlet boundary conditions is investigated. Firstly, based on the L’Hopital law, a Caputo fractional partial differential inequality with p-norm is established to improve the previous results. Besides, a generalized fractional-order Halanay inequality is introduced by rigorous theoretical deduction. Additionally, to realize synchronization, two different control schemes are designed for the networks, respectively, and several criteria are derived based on the developed fractional inequalities. Finally, numerical examples and simulations are provided to verify the correctness of the results.

Distributed Observers With Time-Varying Delays

The distributed estimation problem is solved for continuous-time observer nodes that obtain real-time measurements but communicate with their neighbors over a communication network. To this end, the digital communication between the observer nodes is modeled by the time-delay approach where variable sampling intervals, transmission delays, and packet dropouts are taken into account. An LMI for the design of the observer gains is derived using Halanay's inequality, the feasibility of which guarantees exponential stability with a selected convergence rate up to a maximum total delay. A comparison of the maximal delay on a numerical example shows the advantage of a distributed observer over a centralized one.

The synchronization and stability analysis of delayed fuzzy Cohen-Grossberg neural networks via nonlinear measure method

ABSTRACT This paper examines the problem of master-slave synchronization for a class of fuzzy Cohen-Grossberg neural networks (FCGNNs) subject to fuzzy effects and time-delays (time-varying and distributed). Some sufficient and new conditions are given in order to establish the exponential lag synchronization for the considered model. Also, the existence, the uniqueness, and exponential stability of the equilibrium point are investigated, based on the nonlinear measure method and Halanay inequality. Finally, two examples with numerical simulations are given to show the effectiveness of the derived results.

Asymptotic behavior of a BAM neural network with delays of distributed type

In this paper, we examine a Bidirectional Associative Memory neural network model with distributed delays. Using a result due to Cid [J. Math. Anal. Appl.281(2003) 264–275], we were able to prove an exponential stability result in the case when the standard Lipschitz continuity condition is violated. Indeed, we deal with activation functions which may not be Lipschitz continuous. Therefore, the standard Halanay inequality is not applicable. We will use a nonlinear version of this inequality. At the end, the obtained differential inequality which should imply the exponential stability appears ‘state dependent’. That is the usual constant depends in this case on the state itself. This adds some difficulties which we overcome by a suitable argument.

A Note on Sampled-Data Synchronization of Memristor Networks Subject to Actuator Failures and Two Different Activations

This brief is concerned with the drive-response synchronization problem of memristor networks subject to actuator failures and two different activations via sampled-data (SD) control strategy. By using δ-matrix measure method and Halanay inequality technique, the sufficient conditions for global exponential synchronization of driving-response memristor systems are obtained. Unlike the existing works, the common Lyapunov functionals are not employed, but two simple and general stability criteria are derived. Particularly, the established criteria are easier to demonstrate and execute in practice than the relevant results. And in order to verify the validity and superiority of the proposed theoretical results, a numerical simulation is finally provided.

Takagi–Sugeno fuzzy‐model‐based event‐triggered point control for semilinear partial differential equation systems using collocated pointwise measurements

AbstractThis paper addresses event‐triggered point control design for semilinear partial differential equation systems by using pointwise measurements. First, the nonlinear terms existing in the plant are approximated by Takagi–Sugeno fuzzy model, based on which a novel fuzzy controller is constructed. The main novelties of the control strategy are summarized as follows: (1) The one‐dimensional space is divided into L parts, and only a few points need to measure. (2) Event‐triggered communication scheme is introduced to save the limited transmission channel. (3) Pointwise control is considered to reduce the control cost via utilizing limited actuators. Second, the conditions that guarantee the stability of the closed‐loop system are established by using Lyapunov direct method and some inequality techniques including Wirtinger's and Halanay's inequalities, and the controller gains can be obtained via solving linear matrix inequalities. Third, three simulation examples are provided to verify the effectiveness and superiority of the proposed approach.

Dissipativity of Fractional Navier–Stokes Equations with Variable Delay

We use classical Galerkin approximations, the generalized Aubin–Lions Lemma as well as the Bellman–Gronwall Lemma to study the asymptotical behavior of a two-dimensional fractional Navier–Stokes equation with variable delay. By modifying the fractional Halanay inequality and the comparison principle, we investigate the dissipativity of the corresponding system, namely, we obtain the existence of global absorbing set. Besides, some available results are improved in this work. The existence of a global attracting set is still an open problem.

Effects of heterogeneous impulses on synchronization of complex-valued neural networks with mixed time-varying delays

This article investigates a new type of impulsive effect, namely the heterogeneous impulsive effect on the synchronization of complex-valued neural networks (CVNNs) with mixed time-varying delays. Heterogeneous impulsive effect is a unified framework of non-identical and time-varying impulses, i.e., the impulsive effect to state of each neuron is not only non-identical to each other but also distinct at different impulsive times. Here heterogeneous impulsive control is designed, which is an extension of hybrid impulsive control. The problem is studied with the following procedures. Firstly, the master–slave CVNNs are explicitly separated into two real-valued neural networks (RVNNs). Secondly, the two RVNNs are transformed into impulsive error dynamical system using the heterogeneous impulsive controller. Thirdly, the concepts of average impulsive interval (AII) and average impulsive gain (AIG) are implemented to obtain the synchronization of delayed CVNNs under heterogeneous impulsive effects. By employing a matrix measure method and extended impulsive Halanay inequality, sufficient criteria are derived to guarantee exponential synchronization of the given delayed CVNNs. The dependency of convergence rates on AIIs and AIGs is explicitly discussed. Finally, two examples are provided to show the efficiency of the proposed theoretical results.

Projective Synchronization Analysis of Fractional-Order Neural Networks With Mixed Time Delays.

In this article, we analyze the projective synchronization of fractional-order neural networks with mixed time delays. By introducing an extended Halanay inequality that is applicable for the case of fractional differential equations with arbitrary initial time and multiple types of delays, sufficient criteria are deduced for ensuring the projective synchronization of fractional-order neural networks with both discrete time-varying delays and distributed delays. Furthermore, sufficient criteria are presented for ensuring the projective synchronization in the Mittag-Leffler sense if there is no delay in fractional-order neural networks. The results derived herein include complete synchronization, anti-synchronization, and stabilization of fractional-order neural networks as particular cases. Moreover, the testable criteria in this article are a meaningful extension of projective synchronization of neural networks with mixed time delays from integer-order to fractional-order ones. A numerical simulation with four cases is provided to verify the validity of the obtained results.

Pinning synchronization of fractional-order memristor-based neural networks with multiple time-varying delays via static or dynamic coupling

This paper investigates global asymptotical synchronization between fractional-order memristor-based neural networks (FMNNs) with multiple time-varying delays (MTDs) by pinning control. Two classes of coupling manners, static manner and dynamic manner, are introduced into the pinning controller respectively. For the case of static coupling, to make the controller exclude fraction, 1-norm Lyapunov function and fractional Halanay inequality in MTDs case are utilized for synthesis of controller and convergence analysis of synchronization error. For the case of dynamic coupling, a fractional differential inequality is proved and discussed in an elaborate way, and then global asymptotical synchronization is analyzed by means of Lyapunov-like function and the newly-proved inequality. Lastly, numerical simulations are carried out to show the practicability of the pinning controllers and the feasibility of the obtained synchronization criteria.

Synchronization of Complex Networks With Nondifferentiable Time-Varying Delay.

In this article, we investigate the synchronization of complex networks with general time-varying delay, especially with nondifferentiable delay. In the literature, the time-varying delay is usually assumed to be differentiable. This assumption is strict and not easy to verify in engineering. Until now, the synchronization of networks with nondifferentiable delay through adaptive control remains a challenging problem. By analyzing the boundedness of the adaptive control gain and extending the well-known Halanay inequality, we solve this problem and establish several synchronization criteria for networks under the centralized adaptive control and networks under the decentralized adaptive control. Particularly, the boundedness of the centralized adaptive control gain is theoretically proved. Numerical simulations are provided to verify the theoretical results.

Pinning synchronization of nonlinear hybrid‐coupled complex networks with double time delays via aperiodically intermittent adaptive control

SummaryIn this article, the synchronization problem for nonlinear hybrid‐coupled complex networks with both coupling and internal self‐feedback delays is studied. An aperiodically intermittent adaptive control is employed for a fraction of nodes to pin the whole network to the synchronization state. Some sufficient conditions for realizing global synchronization are derived based on Halanay inequality and Lyapunov stability theory. Finally, a simulation example verifies the effectiveness of the proposed control method.

Halanay Inequality on Time Scales with Unbounded Coefficients and Its Applications

In the present paper, we obtain a Halanay inequality on time scales with unbounded coefficient for a dynamic problem, which extends a result of Wen et al. (J. Math. Anal. Appl., 347 (2008), 169–178.) to the inequality of integral type on time scales. Moreover, we list two dynamic problems to which the Halanay inequality obtained above can be applied and prove the zero solution of two delay dynamic problems are asymptotically stable. Moreover, it is worth mentioning that the Halanay inequality obtained in the present paper is more precise than the results in [3, 14, 17].

Synchronization Analysis for Stochastic Inertial Memristor-Based Neural Networks with Linear Coupling

This paper concerns the synchronization problem for a class of stochastic memristive neural networks with inertial term, linear coupling, and time-varying delay. Based on the interval parametric uncertainty theory, the stochastic inertial memristor-based neural networks (IMNNs for short) with linear coupling are transformed to a stochastic interval parametric uncertain system. Furthermore, by applying the Lyapunov stability theorem, the stochastic analysis approach, and the Halanay inequality, some sufficient conditions are obtained to realize synchronization in mean square. The established criteria show that stochastic perturbation is designed to ensure that the coupled IMNNs can be synchronized better by changing the state coefficients of stochastic perturbation. Finally, an illustrative example is presented to demonstrate the efficiency of the theoretical results.

Polynomial dissipativity of proportional delayed BAM neural networks

This paper pays close attention to the global polynomial dissipativity (GPD) for proportional delayed BAM neural networks (PDBAMNNs). The global exponential dissipativity (GED) and the global dissipativity (GD) are also talked about. Under the help of novel Lyapunov functionals and a generalized Halanay inequality, a set of dissipative criteria for such systems are led out, together with the global polynomial attracting set (GPAS) and the global attracting set (GAS). Further, the relationship among GPD, GED and GD is unveiled. Finally, a proposed theoretical condition is validated through a simulation experiment.