Average Impulsive Interval Research Articles

Fractional exponential stability of nonlinear conformable fractional-order delayed systems with delayed impulses and its application

This article studies the fractional exponential stability (FES) of nonlinear conformable fractional-order delayed systems (CFODSs) and the fractional exponential synchronization of conformable fractional-order delayed inertial neural networks (CFODINNs), both with delayed impulses (DIs). Under the conformable fractional-order derivative framework, a novel Halanay inequality is established by generalizing the average impulsive interval (AII), which is a key basis for further investigations. Additionally, based on the proposed inequality, the requirement of the magnitude relationship among the system delay, the impulsive intervals, and the impulsive delays is removed. Moreover, using the improved average impulsive delay (AID), a relaxed FES criterion for nonlinear CFODSs with DIs is derived. Furthermore, as an application of the obtained theoretical results, the fractional exponential synchronization of CFODINNs with DIs is studied. Finally, simulations are given to illustrate the validity of our results, where the positive effect of impulsive delays is verified.

Finite-time hybrid impulsive formation tracking control of multi-agent systems via aperiodic intermittent communication

This article studies the problem of formation tracking control in multi-agent systems, achieved in finite time, under challenging conditions such as strong nonlinearity, aperiodic intermittent communication, and time-delay effects, all within a hybrid impulsive framework. The impulses are categorized as either stabilizing control impulses or disruptive impulses. Furthermore, by integrating Lyapunov-based stability theory, graph theory, and the linear matrix inequality (LMI) method, new stability criteria are established. These criteria ensure finite-time intermittent formation tracking while considering weak Lyapunov inequality conditions, intermittent communication rates, and time-varying gain strengths. Additionally, the approach manages an indefinite number of impulsive moments and adjusts the control domain’s width based on the average impulsive interval and state-dependent control width. Numerical simulations are provided to validate the applicability and effectiveness of the proposed formation tracking control protocols.

Input-to-state hybrid impulsive formation stabilization for multi-agent systems with impulse delays

This paper addresses the input-to-state formation stabilization problem of nonlinear multi-agent systems within a hybrid impulsive framework, considering delay-dependent impulses, strong nonlinearity, and deception attack signals. By leveraging Lyapunov functionals, impulsive comparison theory, average impulsive interval methods, and graph theory, we develop novel criteria for possessing locally input-to-state and integral input-to-state formation stabilization across different impulse sequence classes. These criteria are expressed in terms of continuous/impulsive feedback gains, time delay size, nonlinearity strength, uniform upper bound of impulsive interval, and length of average impulsive interval. Notably, the design of control impulses benefit the destabilizing continuous dynamics in the formation stabilization process. To demonstrate the effectiveness and validity of the analytical results, we provide numerical simulation examples involving various types of external attack signals.

Stability analysis for neutral stochastic time-varying systems with delayed impulses

This article investigates the stability of stochastic neutral delayed systems (SNDS) with delayed impulses, which includes the stability in input-to-state exponentially stable in pth moment (p-ISES), integral ISES in pth moment (p-iISES) and eλt-weighted ISES in pth moment (eλt-p-ISES). Compared with the existing works, we allow the neutral term, delayed impulses, time-varying coefficients in the diffusion condition, bounded time-varying delay (BTVD) to be in a stochastic system, which arise the difficulty. By utilizing the techniques of Lyapunov-Krasovskii (L-K) function, generalized delay integral inequality and average impulsive interval (AII), some desired results are obtained by surmounting the underlying difficulty. Finally, an example is given to show the validity of our work.

New fixed-time/preassigned-time stability results of impulsive systems and its application to synchronization of delayed octonion-valued neural networks

In this paper, in order to obtain more accurate settling time estimation and overcome the complexity of the theoretical analysis caused by using decomposition method, the problem of fixed-time (F-T) and preassigned-time (P-T) synchronization of delayed octonion-valued neural networks with impulsive effects is explored by utilizing direct octonion approach. First, a new fixed-time stability theorem for impulsive system is presented by using the comparison principle, average impulsive interval and Bernoulli differential equation theory. Second, to solve the problems of non associative and non commutative laws of octonion, some novel lemmas are introduced, which are the basis of the direct octonion approach. Third, under the improved F-T stability result, a new P-T stability theorem is derived and the settling time is preassigned according to realistic need and is irrelevant with any initial values and any parameters. Then, based on newly established theorems and developed controller strategies, some sufficient conditions are deduced to guarantee F-T and P-T synchronization of the discussed systems. Finally, a numerical simulation is provided to verify the effectiveness of the theoretical results.

Almost sure exponential stability and stochastic stabilization of discrete-time stochastic systems with impulses

This paper considers the almost sure exponential stability and stochastic stabilization problems of discrete-time stochastic systems (DTSSs) with impulsive effects, where the average impulsive interval is taken into account. By using the average impulsive interval approach and the strong law of large numbers, we not only establish the criteria for almost sure exponential stability of general nonlinear discrete-time impulsive stochastic systems (DTISSs) but also design an exact method of a stochastic perturbation to stabilize a given unstable impulsive discrete-time systems. Adopting the average impulsive interval approach and the strong law of large numbers, we established a criterion for almost sure exponential stability of general nonlinear DTISSs. Furthermore, a method of stochastic perturbation has been developed to stabilize an unstable impulsive discrete-time system. Finally, two simulation examples demonstrate the effectiveness of the derived results.

Adaptive finite-time synchronization of quaternion-valued inertial neural networks with mixed delays under quantized event-triggered impulsive strategy

By incorporating inertial terms, quaternions, and time-varying discrete and distributed delays into traditional neural networks (NNs), this paper establishes a category of drive-response quaternion-valued inertial neural networks (QVINNs) with mixed delays. The issue of adaptive finite-time synchronization (FTS) of the QVINNs with mixed delays is explored. To do this, a fresh and low-cost adaptive controller is devised by combining event-triggered impulsive control (ETIC) with quantized control to greatly lower the amount of communication and economic costs. In virtue of the variable substitution, the Lyapunov method, the comparison principle, the inequality technique, and the concept of average impulsive interval (AII), the novel criteria that the QVINNs can accomplish FTS are derived. Meanwhile, it is possible to determine that Zeno-behavior and chattering are absent, and the settling time is predicted. Ultimately, by performing the numerical instance, the efficiency and accuracy of the theoretical results are confirmed.

Input-to-state stability of impulsive switched systems under mode-dependent event-triggered impulsive control

This paper aims to investigate input-to-state stability (ISS) of nonlinear impulsive switched systems under mode-dependent event-triggered impulsive control (MDETIC) method. A novel mode-dependent event-triggered mechanism (MDETM) is devised, where triggering impulsive instants are determined by some predesigned event conditions. Different from existing MDETM, there are two types of impulsive behavior involved, including the impulsive behavior of system itself and event-triggered one. The novel MDETM not only reduces communication cost and energy consumption but also excludes “Zeno behavior.” Then, the ISS criteria of the closed-loop system are established based on MDETIC strategy, admissible edge-dependent average dwell time (AED-ADT), and admissible edge-dependent average impulsive interval (AED-AII) methods. Finally, two examples are given to verify the effectiveness of the main results.

Input-to-state stability of discrete-time impulsive switched delayed systems with delay-dependent impulse under two asynchronous cases

This paper studies a class of discrete-time impulsive switched delayed systems with delay-dependent impulses, aiming to solving the input-to-state stability (ISS) problem in the case of asynchronous switching signal between subsystem and controller and asynchrony of switching signals and impulsive signals. Among them, the delay effect is handled by a new Lyapunov-Krasovskii function divided into a delay-dependent part and a delay-independent part. In contrast to admissible edge-dependent average dwell time (AED-ADT) methods that address switching signals, we propose admissible edge-dependent average impulsive interval (AED-AII) to deal with delayed impulses, thereby establishing the relationship between these methods and the decay rate. Meanwhile, a stabilizing controller for discrete impulsive switched linear delayed systems is proposed and the minimum average dwell time and controller gain are obtained. Compared with the previous work, the parameters of Lyapunov functional relation depend on the directed edges; the operation of subsystem is divided into asynchronous time and synchronous time, which are less conservative; the combination of AED-ADT and AED-AII can be applied to a wider range of impulsive switched systems. Finally, two examples are given to show the effectiveness of the results.

Fixed‐time stability of nonlinear impulsive systems: Sufficient criteria and necessary condition

AbstractThis article explores the criteria for fixed‐time stability (FxTS) within nonlinear impulsive dynamical systems, encompassing both stabilizing and destabilizing impulses. Initially, employing the Lyapunov method, the article delineates a criterion for FxTS, particularly when impulsive sequences are delineated by an average impulsive interval. Following this, the article innovatively applies two integral inequalities to introduce an optimized criterion for uniform impulsive sequences, evidencing its lesser conservativeness relative to the initial criterion. Moreover, this article derives a necessary condition for dynamical systems subjected to destabilizing impulses. Intriguingly, the investigation reveals the existence of a minimal threshold for the maximum impulsive interval, below which the system cannot achieve FxTS. This result elucidates why existing literature uniformly imposes a specific condition on the impulsive interval for destabilizing impulses. Finally, the efficacy of the proposed theory is underscored through numerical examples.

Stability of impulsive and switched stochastic delay systems via the event-triggered impulsive mechanism

Under the event-triggered impulsive mechanism, this work introduces the pth moment (integral) input-to-state stability (p-(i)ISS) for a general group of impulsive and switched stochastic delay systems. Using the means of mode-dependent average dwell time and mode-dependent average impulsive interval, some sufficient conditions of p-ISS and p-iISS are supplied. By incorporating the input of subsystems and impulses into the triggering functions, two distinct types of event-triggered impulsive constraint is proposed, which is defined as mode-dependent event-triggered impulsive mechanism with input (MDETIMI). Different from the traditional event-triggered impulsive control, two MDETIMI strategy are individually discussed, which one is the supremum of K∞ function of subsystem input and the superposition of impulsive input, and the other is the integral of K∞ function of subsystem input and the superposition of impulsive input. Moreover, it is unveiled that even with the indeterminate derivatives of the Lyapunov functions, the Zeno phenomenon can effectively be avoided by using MDETIMI strategy. Finally, an example shows the effectiveness and practicability of the results.

Impulsive synchronization control for dynamic networks subject to double deception attacks

This paper investigates the problem of secure synchronization for coupled dynamic networks subject to two types of deception attacks. Deception attacks are considered in both the controller-to-actuator and sensor-to-controller communication channels and modeled as Bernoulli distributions. The same and different attack probabilities of each node in dynamic networks are considered, respectively. A novel secure impulsive controller is introduced, mitigating malicious information from deception attacks and ensuring bounded synchronization. The average impulsive interval (AII) method is employed to relax the constraints on the upper and lower bounds of consecutive impulses, which is not adequately addressed by the previous secure impulsive control methods. Additionally, the paper establishes an upper bound for synchronization errors and proposes a method to design the AII based on attack probabilities. Finally, two numerical examples are provided to show the validity of our control approach.

Coexistence and locally exponential stability of multiple equilibrium points for fractional-order impulsive control Cohen–Grossberg neural networks

Different from the existing multiple Mittag-Leffler stability or multiple asymptotic stability, the multiple exponential stability, which has explicit and faster convergence rate, is investigated in this paper for fractional-order impulsive control Cohen–Grossberg neural networks. First, by using the definition of Dirac delta function, the fractional-order control Cohen–Grossberg neural networks are translated into fractional-order impulsive neural networks, in which pulse effect relies on the fractional order of the addressed system. Then, based on maximum norm, 1−norm and general q−norm (q=2n), a series of novel criteria are obtained respectively to ensure that such n−neuron neural networks can have ∏i=1n(2Li+1) total equilibrium points and ∏i=1n(Li+1) locally exponentially stable equilibrium points, by utilizing the known fixed point theorem, the method of average impulsive interval, the theory of fractional-order differential equations, and the method of Lyapunov function. This paper’s investigation reveals the effects of impulsive function, impulsive interval, and fractional order on the dynamic behaviors. Finally, theoretical results are shown to be effective by four illustrative examples.

Hybrid impulses for almost sure quasi-synchronization of stochastic complex networks: an indefinite Lyapunov function method

This article studies quasi-synchronization of stochastic complex networks under hybrid impulses. With parameter mismatches, different from the previous quasi-synchronization results in the sense of mean square on stochastic complex networks, almost sure quasi-synchronization is investigated, in which noises have a positive effect on the synchronization. By utilizing the Lyapunov method, stochastic analysis theory, average impulsive interval, and average impulsive gain, we formulate the almost sure quasi-synchronization criteria when the average impulsive interval z 0 satisfies z 0 < + ∞ and z 0 = + ∞ , respectively, where the synchronizing impulses and the desynchronizing impulses are simultaneously considered. It is worth noticing that in our criteria of quasi-synchronization, the mutual restraints among the average impulsive interval, average impulsive gain, and noise intensity are given. Moreover, for the criteria in this article, the piecewise continuous scalar functions cannot only replace the constant coefficient of the upper bound estimation for the diffusion operator of a Lyapunov function in available literature but also even be unbounded, which has wider applications than some existing works. Our corollaries regarding complete synchronization and analyses of the synchronization for stochastic complex networks with aperiodic intermittent noises provide sufficient conditions in closed form. Finally, the theoretical results are applied to a coupled chaotic Lorentz system, and several numerical examples are presented to demonstrate the advantages of the theoretical results.

Parameter variational based adaptive bipartite formation control of second-order multi-agent systems with antagonistic interactions

This article mainly concentrates on researching the leader-following bipartite formation of second-order nonlinear multi-agent systems (MASs) with antagonistic interactions and time-varying delay. A novel distributed feedback control and impulsive control strategy is proposed for realizing the bipartite formation. By establishing an error system, the bipartite formation issue of the MASs is transformed into the stability issue of the error systems. Sufficient conditions for achieving the bipartite formation of second-order nonlinear MASs are obtained by jointly employing the extended parameters variation formula, the comparison principle and the definition of average impulsive interval. In addition, in order to obtain the optimal control gains and control effects, an adaptive control strategy is taken into account by designing the adaptive updating laws. Ultimately, the effectiveness of the theoretical results is demonstrated by two numerical simulations.

Exponential stability of impulsive random delayed nonlinear systems with average-delay impulses

In this paper, we investigate the pth moment exponential stability of impulsive random delayed nonlinear systems (RDNS) with average-delay impulses. Utilizing the average impulsive interval, average impulsive delay, and the Lyapunov method, we present several criteria for pth exponential stability in these systems under a novel condition, diverging from previous literature. Additionally, we explore the effects of delayed impulses, analyzing both destabilizing and stabilizing impulses. Our findings reveal the dual effects of time delay in impulses, which can potentially destabilize a stable system or stabilize an unstable one.

Exponential stability of delayed discrete‐time systems with averaged delayed impulses

AbstractThis article investigates the exponential stability of nonlinear discrete‐time systems with time‐varying state delay and delayed impulses, where the delays in impulses are not fixed. Specifically, the study is separated into two cases: (1) stability of delayed discrete‐time systems with destabilizing delayed impulses, where the time delays in impulses can be flexible and even larger than the length of the impulsive interval; (2) stability of delayed discrete‐time systems with mixed delayed impulses (means stabilizing and destabilizing delayed impulses exist simultaneously), where the time delays in impulses are unfixed between two adjacent impulsive instants. First, the concept of average impulsive delay (AID) is extended to discrete‐time systems with delayed impulses. Then, some Lyapunov‐based exponential stability criteria are provided for delayed discrete‐time systems with delayed impulses, where the impulses satisfy the average impulsive interval (AII) condition and the delays in impulses satisfy the AID condition. For the delayed discrete‐time systems with mixed delayed impulses, the exponential stability criterion is provided by limiting the geometric average of the impulses' strengths. The obtained results can be used to study the delayed discrete‐time systems with some large impulses which may even be unbounded. It's also shown that the delays in impulses can have a stabilizing effect on the stability of delayed discrete‐time systems. Some numerical examples are also provided to illustrate the effectiveness and advantage of the theoretical results.

Finite-time stability for impulsive stochastic nonlinear system with uncertain parameter

This paper considered the finite-time mean square stability for impulsive stochastic nonlinear system(ISNS) with uncertain parameter. By selecting the modal dependent Lyapunov functional, using average impulsive interval approach and combining Linear Matrix Inequality (LMI) method, the correlation stability criterion of ISNS was obtained. A gain matrix of the feedback controller is obtained on the basis of this sufficient condition. Finally, the LMI toolbox in Matlab was used for data simulation, and the relevant state trajectory demonstrates the availability of the proposed theory.

Novel LMI-Based Boundary Stabilization of Stochastic Delayed Reaction-Diffusion Cohen–Grossberg BAM Neural Networks with Impulsive Effects

The stabilization problem of stochastic delayed reaction-diffusion Cohen–Grossberg BAM neural networks (SDRDCGBAMNNs) with impulsive effects and boundary control is studied in this paper. By using suitable boundary controllers, Lyapunov–Krasovskii functional, linear matrix inequalities and average impulsive interval method, new sufficient criteria are found to ensure that the SDRDCGBAMNNs achieve boundary stabilization in finite-time. Based on these criteria, the effects of impulsive and boundary controllers on finite-time stability are analyzed. Numerical simulations are performed to demonstrate the feasibility of the theoretical results.

Non-fragile exponential synchronisation of stochastic neural networks via aperiodic intermittent impulsive control

This study aims to address the non-fragile exponential synchronisation problem of stochastic neural networks (SNNs). To cut down unnecessary control costs, a novel aperiodic intermittent-based impulsive control (APIIC) is designed in this investigation. Besides, the randomly occurring gain fluctuation (ROGF) is considered in APIIC, which satisfies certain Bernoulli distributed white noise sequences. By exploiting the Lyapunov approach and the average dwell-time technique, some sufficient criteria are derived in terms of linear matrix inequalities, which ensure that APIIC can achieve exponential synchronisation of SNNs with and without ROGF. More intriguingly, a technical definition of aperiodic window-based average impulsive interval is developed to cut back the conservativeness of these results. At last, the effectiveness of our explored results is confirmed by several numerical examples.