Nonlinear Impulsive Systems Research Articles

Existence of a mild solution and approximate controllability for fractional random integro-differential inclusions with non-instantaneous impulses

This paper investigates the existence and approximate controllability (ACA) of fractional neutral-type stochastic differential inclusions (NTSDIs) characterized by non-instantaneous impulses within a separable Hilbert space (HS) framework. Employing the Atangana–Baleanu–Caputo (ABC) derivative, we transform the system into an equivalent fixed-point (FP) problem through an integral operator. Subsequently, the Bohnenblust–Karlin FP theorem is leveraged to establish existence results. By assuming ACA of the corresponding linear system, we derive sufficient conditions for the ACA of the nonlinear stochastic impulsive control system. Our analysis relies on concepts from stochastic analysis, fractional calculus, FP theory, semigroup theory, and the theory of multivalued maps (MVMs). The theoretical findings are illustrated through a concrete example.

Finite time stability of nonlinear impulsive stochastic system and its application to neural networks

In this paper, we employ the Lyapunov theory to generalize the finite time stability (FTS) results from general deterministic impulsive systems to impulsive stochastic time-varying systems, which overcomes inherent challenges. Sufficient conditions for the FTS of the system under stabilizing and destabilizing impulses are established by using the method of average dwell interval (ADT). For FTS of stabilizing impulses, we relax the constraint on the differential operator by allowing it to be indefinite rather than strictly negative or semi-negative definite. Furthermore, the theoretical results are applied to impulsive stochastic neural networks. Finally, two numerical examples are given to validate the reliability and practicability of the obtained results.

Fixed/predefined-time stability and designing of terminal sliding mode control of impulsive dynamical systems

The contribution of the present work is two-fold. Firstly, some results on the predefined and fixed-time stability of non-linear impulsive systems are derived. Impulse-dependent sufficient conditions are derived for predefined-time estimation of settling-time under the influence of stabilising impulses. In contrast, a fixed-time estimate of the settling-time is obtained for destabilising impulses. Secondly, terminal sliding mode (TSM) control is designed in the presence of hybrid (continuous and impulsive) matched disturbances for n-dimensional systems. It shows that the trajectories can reach the sliding surfaces in fixed-time and thus will stay on it after that under the influence of the proposed TSM. Thus, the system states will converge to the origin in a fixed-time. Finally, three examples, which also consist of the stability of Cohen-Grossberg BAM neural networks, are provided to justify the proposed results numerically.

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.

A novel finite-time stability criteria and controller design for nonlinear impulsive systems

The issue of a novel finite-time stability and stabilization design for a class of nonlinear impulsive systems is discussed in this paper. In previous finite-time control designs for impulsive systems, the convergence rate of these control strategies is even less rapid than the exponential convergence rate when the initial state is distant from the origin. To overcome this problem, two sufficient conditions for the novel finite-time stability of impulsive systems are presented, in which the stabilizing impulses and the destabilizing impulses are considered, respectively. Meanwhile, the upper bound of the settling-time is evaluated. In addition, combining Lyapunov functions theory with impulsive control, we design a controller for impulsive dynamical systems to satisfy the novel finite-time stability. Finally, two simulation examples are given, in which the first example demonstrates the superiority of the proposed finite-time stability criterion for nonlinear impulsive systems by comparing it with traditional finite-time stability, and the second example verifies the effectiveness of the presented controller.

Exponential Stability of Time-varying Switched Impulsive Nonlinear Systems Using Comparison Theorem

Exponential Stability of Time-varying Switched Impulsive Nonlinear Systems Using Comparison Theorem

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 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.

Iterative learning based convergence analysis for nonlinear impulsive differential inclusion systems with randomly varying trial lengths

SummaryThis paper studies the finite‐time tracking problem for nonlinear impulsive differential inclusion systems with randomly varying trial lengths. First, we convert the set‐valued mapping in the differential inclusion systems to single‐valued mapping by a Steiner‐type selector. For the tracking problem of random discontinuous output trajectories, this paper defines a piecewise continuous variable by zero‐order holder to correct the tracking error of segmented continuity. Then, we introduce the average operator with forgetting factor to design three novel learning schemes, and establish convergence results by using the mathematical analysis tools such as impulsive Gronwall inequality and ‐norm. Finally, a numerical example verifies the validity of the theoretical results, and we compare the tracking performance of the output trajectories for different forgetting factors.

Stability analysis of nonlinear systems with delayed impulses: a generalised average dwell-time scheme

This paper studies the input-to-state stability (ISS) and uniform stability for impulsive nonlinear systems with delayed impulses, where delays are flexible between two consecutive impulsive instants. Illustrating effects caused by delays, concepts of generalised average impulsive delay and generalised reverse average impulsive delay are introduced, and more applicable in practice. Moreover, sufficient conditions for stability properties are formulated with destabilising and stabilising impulses through a general candidate ISS-Lyapunov function and the generalised conditions. It is shown that the double effects of delays in impulses (i.e. stabilising an initial unstable system, or contrarily, destabilising an initial stable system) coincide with the existing results using the exponential candidate ISS-Lyapunov function. Finally, two examples are provided to demonstrate our developed techniques.

Neural-network-based adaptive control of strict-feedback nonlinear systems with actuator faults: Event-triggered communications strategy

This article considers the neural network-based event-triggered adaptive control problem for a class of strict-feedback nonlinear impulsive systems with actuator faults and external disturbances. It is well known that reducing computational resources and carrier loads is a long-standing task in the field of adaptive control. However, the event-triggered scheme used in traditional research still consumes relatively more resources. To solve the problem, the control scheme in this paper only updates the controller and weight parameters at the event-triggered moments. Dedicated efforts are made to certificate the stability of the system through a new Lyapunov theory, in such a way that the system maintains its stability well even under uncertainties. Finally, two emulation examples are exploited to demonstrate the efficacy and applicability of the proposed approach.

Fixed-Time Stability of Nonlinear Impulsive Systems and its Application to Inertial Neural Networks.

This article is concerned with the fixed-time stability (FTS) problem of nonlinear impulsive systems (NISs). By means of the impulsive control mechanism and Lyapunov functions theory, several sufficient conditions are established to ensure the FTS of general NISs. Meanwhile, some novel impulse-dependent settling-time estimation schemes are developed, which fully considers the influence of stabilizing impulses and destabilizing impulses on the convergence rate of the system states. The proposed schemes establish a quantitative relationship between the upper bound of the settling time and impulse effects. It shows that stabilizing impulses can accelerate the convergence rate of the system states and leads to the upper bound of the settling time being smaller. Conversely, destabilizing impulses can reduce it and make the upper bound of the settling time larger. Then, the theoretical results are applied to delayed inertial neural networks (DINNs), where two kinds of controllers are designed to realize fixed-time synchronization of the considered systems in the impulse sense. Finally, some numerical examples are provided to illustrate the validity of the proposed theoretical results.

Sliding Mode Control for Strict-Feedback Nonlinear Impulsive Systems With Matched Disturbances

Sliding Mode Control for Strict-Feedback Nonlinear Impulsive Systems With Matched Disturbances

Study on the stability and its simulation algorithm of a nonlinear impulsive ABC-fractional coupled system with a Laplacian operator via F-contractive mapping

In this paper, we study the solvability and generalized Ulam–Hyers (UH) stability of a nonlinear Atangana–Baleanu–Caputo (ABC) fractional coupled system with a Laplacian operator and impulses. First, this system becomes a nonimpulsive system by applying an appropriate transformation. Secondly, the existence and uniqueness of the solution are obtained by an F-contractive operator and a fixed-point theorem on metric space. Simultaneously, the generalized UH-stability is established based on nonlinear analysis methods. Thirdly, a novel numerical simulation algorithm is provided. Finally, an example is used to illustrate the correctness and availability of the main results. Our study is a beneficial exploration of the dynamic properties of viscoelastic turbulence problems.

New exploration on approximate controllability of fractional neutral‐type delay stochastic differential inclusions with non‐instantaneous impulse

This paper aims to derive a new set of sufficient conditions for the existence and approximate controllability of neutral‐type fractional stochastic integrodifferential inclusions with infinite delay and non‐instantaneous impulse in a separable Hilbert space using the Atangana–Baleanu Caputo fractional derivative. We investigate the existence of a mild solution for the Atangana–Baleanu Caputo fractional neutral‐type delay integrodifferential stochastic system while taking into account the non‐instantaneous impulses. For this purpose, the Atangana–Baleanu Caputo fractional neutral‐type impulsive delay stochastic system is transferred into an equivalent fixed point problem via an integral operator, and then, the Bohnenblust–Karlin fixed point approach is applied. Further, the approximate controllability results of the proposed nonlinear stochastic impulsive control system are established under the consideration that the corresponding linear system is approximately controllable. The set of sufficient conditions is established by using the concepts of stochastic analysis, fractional calculus, fixed point technique, semigroup theory of bounded linear operators, and the theory of multivalued maps. To illustrate the abstract results, we provide an example at the end of the paper.

Input-to-State Stability of Nonlinear Impulsive Systems with Hybrid Impulses

This note investigates input-to-state stability (ISS) of nonlinear impulsive systems with hybrid impulses. Such systems contain a time-dependent jump map that can generate both stabilizing and destabilizing impulsive jumps. To cope with the two-fold effect of impulses within a single impulse time sequence, we use the concept of average impulsive gain. We provide novel sufficient conditions for ISS in terms of exponential ISS-Lyapunov function equipped with average impulsive gain and appropriate dwell-time condition. Contrary to the existing results we do not impose severe restrictions on impulsive gains, like periodicity or a finite number of different gains. Finally, we demonstrate effectiveness of our results by concluding ISS for a system whose ISS property cannot be verified by any other result available in the literature.

A Concurrent Event-Triggered Approach for Fuzzy Adaptive Control of Nonlinear Strict-Feedback Systems.

If a control input signal constructed by backstepping is used to generate an event-triggered control signal, then not only the real control input is discontinuous, but all the virtual control signals that make up the control signal are also discontinuous and have the same triggering instants with the control input. This fact makes the control design and stability analysis more difficult and complex via backstepping. This article focuses on this problem and works at coming up a new backstepping design procedure of event-triggered fuzzy adaptive control for nonlinear strict feedback systems. A concurrent event-triggered mechanism is proposed to ensure that the real control input and the virtual control signals have the same triggering instants. Then, a systemic backstepping design procedure is developed to construct event-triggered fuzzy adaptive controller. The Lyapunov technique for nonlinear impulsive system is employed to provide the closed-loop stability analysis. It is shown that the derived event-triggered fuzzy adaptive controller ensures that the closed-loop system output well follows the desired tracking trajectory and the other closed-loop signals remain bounded. Eventually, two examples are provided to further verify the applicability and reliability of the presented control scheme.

Existence of weak solutions for stochastic nonlinear impulsive systems

Existence of weak solutions for stochastic nonlinear impulsive systems

Decentralized output-feedback control of triangular large-scale nonlinear impulsive systems with time-varying delays: a gain scaling approach

Decentralized output-feedback control of triangular large-scale nonlinear impulsive systems with time-varying delays: a gain scaling approach

Saturated control for uncertain nonlinear impulsive systems with non-uniformly distributed packet loss

This article aims to provide a unified theoretical framework for studying the dynamic behavior of uncertain impulsive systems involving saturation structure and packet loss phenomenon. The concept named average packet loss interval (APLI) is developed to characterize the distribution of packet loss intervals. In this scenario, the traditional restrictions on the number, the proportion, or the upper bound of packet loss intervals can be significantly relaxed. By modeling the original systems as the asynchronous impulsive switched systems, some sufficient criteria are derived via the set invariant approach, addressing how to integrate the effects of impulsive and external disturbances into input-to-state stability (ISS) performance. It is also shown that the robustness of system performance against packet loss can be guaranteed ideally. Then, an optimization algorithm is designed with the hope of getting the maximum estimate of the domain of initial values. Finally, two examples are discussed to demonstrate the effectiveness of our proposed results.