Generalized Halanay Inequality Research Articles

A note on generalized exponential stability of impulsive stochastic functional differential equations

This brief considers the generalized exponential stability of impulsive stochastic functional differential equations (ISFDEs). Initially, An extension of Halanay inequality is derived. Subsequently, by employing this generalized Halanay inequality, we establish the generalized exponential stability result for the given systems.

Stability analysis for nonautonomous impulsive hybrid stochastic delay systems

In this paper, the problems on the existence and uniqueness, and the input-to-state stability for the global solution of nonlinear nonautonomous impulsive stochastic delay systems with Markovian switching are considered. The existence and uniqueness for the global solution of such systems is studied by using the Lyapunov function, the theory of stochastic analysis, and the function of the impulsive density. In order to overcome the difficulty stemming from the simultaneous presence of the indefinite derivative for the Lyapunov function, the bounded external disturbances, and a nonnegative and bounded time-varying delay, one impulsive generalized Halanay inequality of integral version is established. The input-to-state stability, the integral input-to-state stability, the stochastic input-to-state stability and the eλt(λ>0)-weighted input-to-state stability for the global solution of nonlinear nonautonomous impulsive stochastic delay systems with Markovian switching are investigated, respectively. One example is provided to illustrate the effectiveness of the theoretical results obtained.

Generalized Halanay Inequalities and Asymptotic Behavior of Nonautonomous Neural Networks with Infinite Delays

This paper focuses on the asymptotic behavior of nonautonomous neural networks with delays. We establish criteria for analyzing the asymptotic behavior of nonautonomous recurrent neural networks with delays by means of constructing some new generalized Halanay inequalities. We do not require to constructi any complicated Lyapunov function and our results improve some existing works. Lastly, we provide some illustrative examples to demonstrate the effectiveness of the obtained results.

Asymptotic behavior of some differential inequalities with mixed delays on time scales and their applications

<abstract><p>In this paper, we investigate the asymptotic stability of the trajectories governed by some delay differential inequalities on time scales. Based on time scale theory and the fixed-point theorem, some sufficient conditions are obtained for guaranteeing asymptotic stability. It is interesting that the inequalities studied in this paper include the generalized Halanay inequalities. Due to the fact that dynamic systems on a time scale unify discrete and continuous systems, the results of this paper have wider application value. Furthermore, some numerical examples verify the main results.</p></abstract>

New generalized Halanay inequalities and relative applications to neural networks with variable delays

The asymptotic behavior of solutions for a new class of generalized Halanay inequalities is studied via the fixed point method. This research provides a new approach to the study of the stability of Halanay inequality. To make the application of fixed point method in stability research more flexible and feasible, we introduce corresponding functions to construct an operator according to different characteristics of coefficients. The results obtained in this paper are applied to the stability study of a neural network system, which has high value in application. Moreover, three examples and simulations are given to illustrate the results. The conclusions in this paper greatly improve and generalize the relative results in the current literature.

Projective lag quasi-synchronization of coupled systems with mixed delays and parameter mismatch: a unified theory

This paper investigates the projective lag quasi-synchronization by feedback control of a coupled dynamical system with delays and parameter mismatches on arbitrary time domains. Being formulated on time scales, our results are valid simultaneously for continuous- and discrete-time models as well as for any non-standard time domain. Furthermore, the controller design respects the structure of the equations so that we can characterize the stabilization by a limited controller action. Our proofs rely on the unified matrix-measure theory and the generalized Halanay inequality on time scales. We validate our theoretical results with several simulation examples on various time domains.

Stability of stochastic functional differential equations with impulses

The paper studies the stability of impulsive stochastic functional differential equations. At first, we obtained a new generalization of the Halanay inequality. Then, by applying the generalized Halanay inequality, the exponential stability results of impulsive stochastic functional differential equations are obtained.

Event-Triggered Quantized Quasisynchronization of Uncertain Quaternion-Valued Chaotic Neural Networks With Time-Varying Delay for Image Encryption.

In this article, a class of quaternion-valued master-slave neural networks (NNs) with time-varying delay and parameter uncertainties was first established by conducting the extension from real-valued chaotic NNs to the quaternion field. Then, based on logarithmic quantized output feedback, the quasisynchronization issue of the NNs was investigated via devising a neoteric dynamic event-triggered controller. In virtue of the classical Lyapunov method and a generalized Halanay inequality, not only corresponding synchronization criteria were obtained to realize the quasisynchronization of master-slave NNs but also a precise upper bound was provided. Moreover, Zeno behavior can be eliminated under the presented scheme in this article. The accuracy of the theoretical outcomes was demonstrated by means of Chua's circuit. Ultimately, some experimental results of pragmatic application in image encryption/decryption were exposed to substantiate the feasibility and efficacy of the current algorithm for the proposed quaternion-valued NNs.

Generalized Halanay Inequalities and Relative Application to Time-Delay Dynamical Systems

A class of generalized Halanay inequalities is studied via the Banach fixed point method and comparison principle. The conditions to ensure the boundedness and stability of the zero solution are obtained in this study. This research provides a new approach to the study of the boundedness and stability of Halanay inequality. Numerical examples and simulation results verify the validity and superiority of the conclusions obtained in this study.

Generalized Halanay inequalities for stability and dissipativity of stochastic functional differential equations

Generalized Halanay inequalities for stability and dissipativity of stochastic functional differential equations

Lagrange Stability of Fuzzy Memristive Neural Networks on Time Scales With Discrete Time Varying and Infinite Distributed Delays

The existing results of Lagrange stability for neural networks with distributed time delays are scale-free, which introduces conservativeness naturally. A class of Takagi–Sugeno fuzzy memristive neural networks (FMNNs) on time scales with discrete time-varying and infinite distributed delays is brought in this article. First, a new scale-limited Halanay inequality is demonstrated by timescale theory. Next, on the basis of inequality techniques on time scales, some new scale-limited algebraic criteria and linear matrix inequality criteria of Lagrange stability are obtained by comparison strategy and generalized Halanay inequality. All scale-limited sufficient criteria of Lagrange stability for FMNNs not only apply to continuous-time FMNNs and their discrete-time analogs, but also could deal with the arbitrary combination of them. Finally, two numerical simulations are given to verify the validity of the obtained theoretical results.

Novel Global Asymptotic Stability and Dissipativity Criteria of BAM Neural Networks With Delays

In this article, issues of both stability and dissipativity for a type of bidirectional associative memory (BAM) neural systems with time delays are investigated. By using generalized Halanay inequalities and constructing appropriate Lyapunov functionals, some novelty criteria are obtained for the asymptotic stability for BAM neural systems with time delays. Also, without assuming boundedness and differentiability for activation functions, some new sufficient conditions for proving the dissipativity are established by making use of matrix theory and inner product properties. The received conclusions extend and improve some previously known works on these problems for general BAM neural systems. In the end, numerical simulation examples are made to show the availability of the theoretical conclusions.

Formation Tracking Control of Networked Systems With Time-Varying Delays and Sampling Under Fixed and Markovian Switching Topology

This article studies the formation tracking control (FTC) of nonlinear second-order leader&#x2013;following multiagent systems (LFMASs) with time-varying delays and periodic sampling. The FTC protocols based on position and velocity information of neighbor agents and measurable communication noise in fixed and Markovian switching directed topology are proposed. Some sufficient conditions are obtained to reach formation tracking by using the algebraic graph theory and generalized Halanay inequality. Meanwhile, the upper bound of formation-able time-varying delays <inline-formula><tex-math notation="LaTeX">$\bar{\tau }$</tex-math></inline-formula> is obtained. Results show that the LFMASs can achieve formation tracking in mean-square exponential stability as well as almost sure exponential stability under the proposed control algorithms. Finally, the availability of the theoretical results is verified by numerical simulation.

Global dissipativity of non-autonomous BAM neural networks with mixed time-varying delays and discontinuous activations

Abstract In this paper, we investigate the dissipativity of a class of BAM neural networks with both time-varying and distributed delays, as well as discontinuous activations. First, the concept of the Filippov solution is extended to functional differential equations with discontinuous right-hand sides via functional differential inclusions. Then, by constructing Lyapunov functional and employing a generalized Halanay inequality, several sufficient easy-to-test conditions are successfully obtained to guarantee the global dissipativity of the Filippov solution of the considered system. The derived results extend and improve some previous publications on conventional BAM neural networks. Meanwhile, the estimations of the positive invariant and globally attractive set are given. Finally, numerical simulations are provided to demonstrate the effectiveness of our proposed 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.

Global dissipativity and exponential synchronization of mixed time-varying delays neural networks with discontinuous activations

Abstract In this paper, the matters of dissipativity and synchronization for non-autonomous Hopfield neural networks with discontinuous activations are investigated. Firstly, under the framework of extending Filippov differential inclusion theory, several effective new criteria are derived. The global dissipativity of Filippov solution to neural networks is proved by using generalized Halanay inequality and matrix measure method. Secondly, the global exponential synchronization of the addressed network drive system and the response system is realized by utilizing inequality and some analysis techniques and designing the discontinuous state feedback controller. Finally, several numerical examples are given to verify the validity of the theoretical results.

Global dissipativity and finite-time synchronization of mixed time-varying delayed memristor-based neural networks with discontinuous activations

In this paper, the matters of dissipativity and finite time synchronization for memristor-based neural networks (MNNs) with mixed time-varying discontinuities are investigated. Firstly, under the framework of extending Filippov differential inclusion theory, several effective new criteria are derived. Then, the global dissipativity of Filippov solution to neural networks is proved by using generalized Halanay inequality and matrix measure method. Secondly, some novel sufficient conditions are introduced to guarantee the finite-time synchronization of the drive-response MNNs based on a simple Lyapunov function and two different feedback controllers. Finally, several numerical examples are given to verify the validity of the theoretical results.

Global Exponential Stability Criteria for Proportional Delay High-Order Neural Networks: A Hyper-Exponential Stability Technique

A new method is presented for stability analysis of proportional delay high-order neural networks. The network model is first transformed into a system with a constant time delay and unbounded time-varying coefficients, and then it is proven that the former is globally exponentially stable if and only if the laster is globally hyper-exponentially stable. The global hyper-exponential stability criteria of the laster are investigated by employing the generalized Halanay inequality and constructing a novel Lyapunov function that can avoid the computation of upper-right derivative. From which, the global exponential stability criteria of the former are derived. To illustrate the advantages of this proposed method, numerical simulation examples are given. Compared with the existing results, the contributions of this paper lie in: (i) An Lyapunov function different from ones in literature is constructed; (ii) The derived global exponential stability criteria possess simple forms, which are easy to verify; and (iii) The concept of hyper-exponential stability is proposed. The proposed method is also available to multi-proportional delay neural networks.

Generalized Halanay Inequalities with Applications to Generalized Exponential Stability and Boundedness of Time-Delay Systems

In this paper, a class of generalized Halanay inequalities is investigated. Under some assumptions, the generalized exponential stability and boundedness are discussed by means of integral inequalities. We do not require the boundedness of the time-varying delays and coefficients. In addition, our results improve some previous works. At last, three examples and simulations are given to illustrate the effectiveness of the main results.

Synchronization of memristive neural networks with leakage delay and parameters mismatch via event-triggered control

In this paper, we investigate the synchronization problem on delayed memristive neural networks (MNNs) with leakage delay and parameters mismatch via event-triggered control. We divide MNNs with parameters mismatch into two categories for discussion. One is state-dependent and can achieve synchronization by designing a suitable controller. A novel Lyapunov functional is constructed to analyze the synchronization problem. Moreover, the triggering conditions are independent from the delay boundaries and can be static or dynamic. Another category of parameters mismatch is structure-dependent and can only achieve quasi-synchronization by appropriate controller. By using matrix measure method and generalized Halanay inequality, a quasi-synchronization criterion is established. The controllers in this paper are discrete state-dependent and can be updated under the event-based triggering condition, which is more simpler than the previous results. In the end of our paper, two illustrative examples are given to support our results.