Time-varying Impulses Research Articles

(Integral)ISS of Switched and Time-Varying Impulsive Systems Based on Global State Weak Linearization

It is shown that impulsive systems of nonlinear, time-varying, and/or switched form that allow a stable global state weak linearization are jointly input-to-state stable (ISS) under small inputs and integral ISS. The system is said to allow a global state weak linearization if its flow and jump equations can be written as a (time-varying, switched) linear part plus a (nonlinear) perturbation satisfying a bound of affine form on the state. This bound reduces to a linear form under zero input but does not force the system to be linear under zero input. The given results generalize and extend previously existing ones in many directions: 1) no (dwell-time or other) constraints are placed on the impulse time sequence; 2) the system needs not be linear under zero input; 3) existence of a (common) Lyapunov function is not required; and 4) the perturbation bound needs not be linear on the input.

Input-to-State Stability Criteria of Discrete-Time Time-Varying Impulsive Switched Delayed Systems With Applications to Multi-Agent Systems

This paper considers weakly and strongly input-to-state stability (ISS) of a discrete-time time-varying delayed system with synchronous impulses and switches. By employing the uniformly asymptotically stable function and mode-dependent dwell time conditions, ISS criteria are derived using the Krasovskii and Razumikhin techniques. The time differences of the Krasovskii functionals or the Razumikhin functions can be mode-dependent and time-varying, which release constraints of some existing results. As subsequent results, we apply the derived ISS criteria to the multi-agent systems (MASs). Finally, two examples considering the quasi-consensus of MASs are given to verify the established results.

Variable-time impulsive control for bipartite synchronization of coupled complex networks with signed graphs

This paper investigates the bipartite synchronization of coupled complex networks (CCNs) with signed graphs via variable-time impulsive (VTI) control. Different from the traditional complex networks, the communication topology between nodes of the complex networks can be either collaborative or antagonistic. In addition, the appearance of impulses is always dependent on each node in the network instead of appearing at fixed instants. Furthermore, by utilizing the B-equivalent technique, the VTI network models can be changed as fixed-time impulsive CCNs. The fixed-time impulsively controlled CCNs can be regarded as the comparison systems of the signed CCNs with variable-time impulses. By 1-norm analytical techniques, several assumptions are derived to guarantee every solution of the coupled error nodes intersect each discontinuous impulsive surface exactly once. Some sufficient conditions with theoretical demonstration are presented to guarantee the bipartite synchronization under mathematical induction. Simulation results are carried out to show the effectiveness of the obtained results.

Uniform Input-To-State Stability for Switched and Time-Varying Impulsive Systems

We provide a Lyapunov-function-based method for establishing different types of uniform input-to-state stability (ISS) for time-varying impulsive systems. The method generalizes to impulsive systems with inputs the well-established philosophy of assessing the stability of a system by reducing the problem to that of the stability of a scalar system given by the evolution of the Lyapunov function on the system trajectories. This reduction is performed in such a way so that the resulting scalar system has no inputs. Novel sufficient conditions for ISS are provided, which generalize existing results for time-invariant and time-varying, switched and nonswitched, impulsive and nonimpulsive systems in several directions.

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.

Unified Stability Criteria of Random Nonlinear Time-Varying Impulsive Switched Systems

This paper investigates the problem of noise-to-state practical stability in mean (NSpS-M) (which is a natural generalization of noise-to-state stability in mean) and the problem of almost sure global asymptotic stability (GAS a.s.) for a class of random nonlinear time-varying impulsive switched systems. By using the notions of average impulsive switched interval and Poisson process, unified sufficient stability criteria on NSpS-M and GAS a.s. are derived. Two remarkable distinctions from the existing results lie in that: (1) stabilizing, inactive and destabilizing impulses are simultaneously considered; (2) the coefficient of the derivative of a Lyapunov function is allowed to be a time-varying function which can be both positive and negative and may even be unbounded. As an accompaniment, a less conservative unified criterion on NSpS-M for a special case is also presented by taking into account the stabilization role of the gain constant of the time-varying coefficient. Two examples are provided to illustrate the effectiveness of our derived criteria.

Converging-Input Convergent-State and Related Properties of Time-Varying Impulsive Systems

Very recently, it has been shown that the standard notion of stability for impulsive systems, whereby the state is ensured to approach the equilibrium only as continuous time elapses, is too weak to allow for any meaningful type of robustness in a time-varying impulsive system setting. By strengthening the notion of stability so that convergence to the equilibrium occurs not only as time elapses but also as the number of jumps increases, some facts that are well-established for time-invariant nonimpulsive systems can be recovered for impulsive systems. In this context, our contribution is to provide novel results consisting in rather mild conditions under which stability under zero input implies stability under inputs that converge to zero in some appropriate sense.

Exponential stability of inertial BAM neural network with time-varying impulses and mixed time-varying delays via matrix measure approach

This article is concerned with the effects of time-varying impulses on exponential stability to a unique equilibrium point of inertial Bidirectional Associative Memories (BAM) neural network with mixed time-varying delays. A suitable variable transformation is chosen to transform the original system into a system of first order differential equations. The concept of homeomorphism has been implemented to find a distributed delay-dependent sufficient condition which assures that the system has a unique equilibrium point. In order to study the impulsive effects on stability problems, a time-varying impulses, including stabilizing and destabilizing impulses, are considered with the transformed system. Based on the matrix measure approach and an extended impulsive differential inequality for a time-varying delayed system, we have derived sufficient criteria in matrix measure form which ensure the exponential stability of the system towards an equilibrium point for two classes of activation functions. Further, different convergence rates of the system’s trajectory have been discussed for the cases of time-varying stabilizing and destabilizing impulses using the concept of an average impulsive interval. Finally, the efficiency of the theoretical results has been illustrated by providing two numerical examples.

Existence and finite-time stability of solutions for a class of nonlinear fractional differential equations with time-varying delays and non-instantaneous impulses

In this paper, we mainly consider the existence and finite-time stability of solutions for a kind of ψ-Hilfer fractional differential equations involving time-varying delays and non-instantaneous impulses. By Schauder’s fixed point theorem, the contraction mapping principle and the Lagrange mean-value theorem, we present new constructive results as regards existence and uniqueness of solutions. In addition, under some new criteria and by applying the generalized Gronwall inequality, we deduce that the solutions of the addressed equation have finite-time stability. Some results in the literature can be generalized and improved. As an application, three typical examples are delineated to demonstrate the effectiveness of our theoretical results.

On Exponential Stability of Neural Networks with Proportional Delays and Periodic Distribution Impulsive Effects

This paper is concerned with the problem of generalized exponential stability of impulsive neural networks with a proportional delay. More specifically, the considered network models are subject to both time-varying impulses, whose strengths are in a type of periodic distributions, and a special kind of unbounded time-varying delays called proportional delays. Based on the comparison principle, a unified delay-independent stability criterion is first derived. As an application of the derived stability conditions, the problem of designing a local state feedback control law with bounded controller gains is addressed. Finally, three examples with numerical simulations are given to demonstrate the effectiveness and advantages of the results.

Dynamics of a Nonautonomous Plant Disease Model with General Nonlinear Incidence Rate and Time-Varying Impulse

In this paper, for controlling the spread of plant diseases, a nonautonomous SEIS (Susceptible → Exposed → Infectious → Susceptible) epidemic model with a general nonlinear incidence rate and time-varying impulsive control strategy is proposed and investigated. This novel model could result in an objective criterion on how to control plant disease transmission by replanting of healthy plants and removal of infected plants. Using the method of small amplitude perturbation, the sufficient conditions under which guarantee the globally attractive of the disease-free periodic solution and the permanence of the disease are obtained, that is, the disease dies out if R12>1.

Comparison System Method for a class of Stochastic Systems with Variable-time Impulses

This paper studies the stability of stochastic impulsive systems with variable-time impulses. We consider the case that the trajectory of the stochastic system intersects each surface of discontinuity exactly once. Then we shall show that under the well-selected conditions the variable-time impulsive systems can be reduced to the fixed-time impulsive systems with impulse time window. By using comparison method, we obtain two theorems to determine the different impulsive time windows for stable and unstable stochastic dynamical systems, respectively. The effectiveness of the theoretical results are illustrated by two numerical examples.

Comparison principle for difference equations with variable-time impulses

In this paper, a class of difference equations with variable-time impulses is considered. By applying comparison principle, we shall show that difference equations with variable-time impulse can be reduced to the corresponding difference equations with fixed-time impulses under well-selected conditions. Meanwhile, the fixed-time impulsive systems can be regarded as the comparison system of the difference equations with variable-time impulses. Furthermore, we use a series of sufficient criteria to illustrate the same stability properties between variable-time impulsive difference equations and the fixed-time ones. We then establish several sufficient conditions guaranteeing the global exponential stability of variable-time impulsive difference equations by comparison principle. As an application, global exponential stability of discrete-time neural networks with variable-time impulses is discussed. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed results.

Effects of variable-time impulses on global exponential stability of Cohen–Grossberg neural networks

We investigate the global exponential stability of Cohen–Grossberg neural networks (CGNNs) with variable moments of impulses using B-equivalence method. Under certain conditions, we show that each solution of the considered system intersects each surface of discontinuity exactly once, and that the variable-time impulsive systems can be reduced to the fixed-time impulsive ones. The obtained results imply that impulsive CGNN will remain stability property of continuous subsystem even if the impulses are of somewhat destabilizing, and that stabilizing impulses can stabilize the unstable continuous subsystem at its equilibrium points. Moreover, two stability criteria for the considered CGNN by use of proposed comparison system are obtained. Finally, the theoretical results are illustrated by two examples.

Stability of nonlinear systems with variable-time impulses: B-equivalence method

This paper addresses the stability problem of nonlinear systems with variable-time impulses. By B-equivalence method, we shall show that under the well-selected conditions each solution of the considered systems will intersect each surface of discontinuity exactly once, and that the considered systems can be reduced to the fixed-time impulsive ones, which can be regarded as the comparison systems of the considered variable-time impulsive systems. Based on the stability theory of fixed-time impulsive systems, we propose a set of stability criteria for the variable-time impulsive systems. The theoretical results are illustrated by impulsive stabilization of Chua circuit.

Stability of Variable-Time Impulsive Systems with Delays via Generalized Razumikhin Technique and Application to Impulsive Neural Networks

This paper investigates the stability of variable-time impulsive systems with time delays. A novel stability result is obtained via the generalized Razumikhin technique. By viewing fixed-time impulsive systems as degenerative variable-time impulsive systems, a new stability result for fixed-time impulsive systems is also proposed. In order to demonstrate the effectiveness of the generalized Razumikhin technique, we give some comparison results with the existing stability criteria which are derived from the classical Razumikhin technique. Stability of impulsive neural networks and synchronization of complex networks are employed to indicate the effectiveness and the practicality of the theoretical results.

Periodicity and stability for variable-time impulsive neural networks

The paper considers a general neural networks model with variable-time impulses. It is shown that each solution of the system intersects with every discontinuous surface exactly once via several new well-proposed assumptions. Moreover, based on the comparison principle, this paper shows that neural networks with variable-time impulse can be reduced to the corresponding neural network with fixed-time impulses under well-selected conditions. Meanwhile, the fixed-time impulsive systems can be regarded as the comparison system of the variable-time impulsive neural networks. Furthermore, a series of sufficient criteria are derived to ensure the existence and global exponential stability of periodic solution of variable-time impulsive neural networks, and to illustrate the same stability properties between variable-time impulsive neural networks and the fixed-time ones. The new criteria are established by applying Schaefer’s fixed point theorem combined with the use of inequality technique. Finally, a numerical example is presented to show the effectiveness of the proposed results.

Synchronization of stochastic discrete-time complex networks with partial mixed impulsive effects

In this paper, the synchronization problem is studied for a class of stochastic discrete-time complex networks with partial mixed impulsive effects. The involving impulsive effects, called partial mixed impulses, can be regarded as local and time-varying impulses, which means that impulses are not only injected into a fraction of nodes in networks but also contain synchronizing and desynchronizing impulses at the same time. In order to handle this case, several mathematical techniques are proposed to tackle mixed impulsive effects in discrete-time dynamical systems. Based on the variation of parameters formula, several sufficient criteria are derived to ensure that synchronization of the addressed networks can be achieved in mean square. The obtained criteria not only rely on the strengths of mixed impulses and the impulsive intervals, but also can reduce conservativeness. Finally, a numerical example is presented to show the effectiveness of our results for neural networks.

Stability analysis of nonlinear fractional-order systems with variable-time impulses

This paper aims at analyzing the stability analysis for a class of variable-time impulsive fractional-order nonlinear systems. Based on the theory of fractional calculus, the theory of impulsive differential equation, inequality techniques, and the B-equivalence method, the variable-time jump operator of the considered system can be updated as a fixed-time substitution, and the fractional-order system with the latter operator can be regarded as the comparison system of the original system. In addition, both graphic illustration and theoretical explanation are presented. Finally, two numerical examples are shown to demonstrate the validity and feasibility of the obtained results.

Exponential Stability of Non-Autonomous Neural Networks with Heterogeneous Time-Varying Delays and Destabilizing Impulses

In this paper, the problem of global exponential stability analysis of a class of non-autonomous neural networks with heterogeneous delays and time-varying impulses is considered. Based on the comparison principle, explicit conditions are derived in terms of testable matrix inequalities ensuring that the system is globally exponentially stable under destabilizing impulsive effects. Numerical examples are given to demonstrate the effectiveness of the obtained results.