B-equivalence Method Research Articles

Stability analysis of state-dependent impulsive systems via a new two-sided looped functional

In this paper, the asymptotic stability of state-dependent impulsive systems is addressed by virtue of looped-functional approach. Firstly, based on B-equivalence method, a fixed-time impulsive comparison system is derived, which is B-equivalent with the original state-dependent impulsive system. Then the asymptotic stability criteria of state-dependent impulsive system are obtained by employing a new two-sided looped functional to analyze the comparison system with fixed-time impulses. The theoretical results are applied to Takagi-Sugeno (T-S) fuzzy impulsive control. The effectiveness of the proposed results is verified by two numerical examples.

Global exponential stability of high-order Hopfield neural networks with state-dependent impulses

In this paper, we discuss the stability of high-order Hopfield neural networks with state-dependent impulses. Under some necessary assumptions, that every solution of the considered system intersects each impulsive surface exactly once is proved. Meanwhile, by using B-equivalence method, the considered system can be simplified to a system with fixed-time impulses. Moreover, some sufficient criteria are derived to ensure the stability between high-order Hopfield neural networks with state-dependent impulses and the corresponding system with fixed-time impulses. The main results show that the stability of high-order Hopfield neural networks with state-dependent impulses maintains no matter the stable continuous subsystems with unstabilizing impulses or the unstable continuous subsystems with stabilizing impulses. Finally, some numerical examples are given to illustrate the effectiveness of our results.

Exponential consensus of discrete-time non-linear multi-agent systems via relative state-dependent impulsive protocols

In this paper, we discuss the exponential consensus problem of discrete-time multi-agent systems with non-linear dynamics via relative state-dependent impulsive protocols. Impulsive protocols of which the impulsive instants are dependent on the weighted relative states of any two agents are introduced for general discrete-time multi-agent systems. The analysis of such impulsive protocols is transformed into an investigation on reduced fixed-time impulsive protocols by constructing a map, which is achieved mainly by a derived B-equivalence method in discrete-time domain. Our main results indicate that the exponential consensus of the multi-agent systems via relative state-dependent impulsive protocols can be achieved if the reduced systems via fixed-time impulsive protocols can achieve exponential consensus, which need to satisfy suitable sufficient conditions. Numerical simulations are presented to support the theoretical results.

State‐dependent impulsive synchronisation of complex dynamical networks with non‐linear coupling

This study investigates the synchronisation problem for complex dynamical networks with non-linear coupling via state-dependent impulsive control. In contrast to most existing publications, impulses occurring are not at fixed instants but depend on the states of each node. Based on the differential mean value theorem and inequality techniques, several sufficient conditions with theoretical demonstration ensuring every solution of the error nodes intersect each surface of the discontinuity exactly once are derived. Moreover, by applying the B-equivalence method, the error system with state-dependent impulses can be reduced to the fixed-time impulsive ones, which can be analysed via Lyapunov stability theory and mathematical induction. Finally, two numerical examples are carried out to demonstrate the performance of the obtained 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.

Impacts of state-dependent impulses on the stability of switching Cohen-Grossberg neural networks

This paper investigates the impacts of state-dependent impulses on the stability of switching Cohen-Grossberg neural networks (CGNN) by means of B-equivalence method. Under certain conditions, the state-dependent impulsive switching systems can be reduced to the fixed-time ones. Furthermore, a stability criterion for the considered CGNN using the proposed comparison system is established. Finally, two numerical examples are provided to illustrate the efficiency of the theoretical results.

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 Analysis of TS Fuzzy System with State-Dependent Impulses

This paper investigates the stability of TS fuzzy system via state-dependent impulsive control. Based on the Lyapunov stability theory, comparison theorem, and inequality techniques, the sufficient conditions with theoretical demonstration ensuring every solution of concerned models intersect each surface of the discontinuity exactly once are derived. Moreover, by applying B-equivalence method, the state-dependent impulsive TS fuzzy system can be reduced to the fixed-time impulsive ones, which can be analyzed via comparison method and mathematical induction. The control approach guarantees that the proposed state-dependent impulsive TS fuzzy system converge to zero. Finally, two numerical examples are carried out to demonstrate the effectiveness of the obtained results.

Hybrid impulsive and switching Hopfield neural networks with state-dependent impulses

We discuss the global stability of switching Hopfield neural networks (HNN) with state-dependent impulses using B-equivalence method. Under certain conditions, we show that the state-dependent impulsive switching systems can be reduced to the fixed-time ones, and that the global stability of corresponding comparison system implies the same stability of the considered system. On this basis, a novel stability criterion for the considered HNN is established. Finally, two numerical examples are given to demonstrate the effectiveness of our results.

A Memristor-Based Lorenz Circuit and Its Stabilization via Variable-Time Impulsive Control

In this paper, an off-the-shelf memristor emulator of the quadratic memristor is developed and applied to a Lorenz circuit. The impulsive stabilization of the chaotic system by variable moments of impulses is investigated. By B-equivalence method, the considered variable-time impulsive system can be reduced to the fixed-time impulsive one. The simulations of the chaotic behavior verify the effectiveness of the emulator-based memristive system. The stability simulations support the validity of the method.

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.

Synchronization of coupled memristive chaotic circuits via state-dependent impulsive control

This paper investigates the synchronization of two memristive chaotic circuits via state-dependent impulsive control. Different from most existing publications, impulses occurring is not at fixed instants but depends on the states of systems. Furthermore, the state variables of the driving system (driving system which does not involve the impulses) are transmitted to the response system, and then the state variables of response system are subjected to jumps at the state-dependent impulsive instants, and ultimately to achieve synchronization. Based on the Lyapunov stability theory, impulsive differential equation, and inequality techniques, the sufficient conditions with theoretical demonstration ensuring every solution of error system intersects each surface of the discontinuity exactly once are derived. Then, by applying B-equivalence method, the error system with state-dependent impulses can be reduced to the case of fixed-time impulses. Finally, the numerical simulations are carried out to demonstrate the effectiveness of the obtained results.

Mittag–Leffler stability analysis on variable-time impulsive fractional-order neural networks

Mittag–Leffler stability analysis of fractional-order neural networks with variable-time impulses is addressed in this paper. Several well-proposed conditions with theoretical demonstration ensuring that every solution of concerned models intersects each surface of the discontinuities exactly once are provided. Meanwhile, by applying B-equivalence method, the reduced fractional-order neural networks with fixed-time impulsive effects can be regarded as the comparison systems of the investigated original network models. Furthermore, a series of sufficient criteria illustrating the same stability properties between both variable-time impulsive fractional-order neural networks and the fixed-time alternative, and guaranteeing the stability of the considered models are presented. Finally, two simulation examples are presented to demonstrate the efficiency and feasibility of the achieved results.

State-dependent impulsive Cohen–Grossberg neural networks with time-varying delays

In this paper, a more general class of state-dependent impulsive Cohen–Grossberg neural networks having variable coefficients with time-varying delays is addressed. By means of B-equivalence method, we reduce this state-dependent impulsive neural networks system to a fix time impulsive neural networks system. Sufficient conditions for existence and global exponential stability of the equilibrium point as well as periodic solution are obtained by employing a suitable Lyapunov function, the Banach fixed point theorem and the Halanay-type impulsive differential inequality technique. Finally, two examples with numerical simulations to show the effectiveness of our theoretical results are illustrated.

Periodic solution for state-dependent impulsive shunting inhibitory CNNs with time-varying delays

In this paper, we consider existence and global exponential stability of periodic solution for state-dependent impulsive shunting inhibitory cellular neural networks with time-varying delays. By means of B-equivalence method, we reduce these state-dependent impulsive neural networks system to an equivalent fix time impulsive neural networks system. Further, by using Mawhin’s continuation theorem of coincide degree theory and employing a suitable Lyapunov function some new sufficient conditions for existence and global exponential stability of periodic solution are obtained. Previous results are improved and extended. Finally, we give an illustrative example with numerical simulations to demonstrate the effectiveness of our theoretical results.

Global robust asymptotic stability of variable-time impulsive BAM neural networks

In this paper, the global robust asymptotic stability of the equilibrium point for a more general class of bidirectional associative memory (BAM) neural networks with variable time of impulses is addressed. Unlike most existing studies, the case of non-fix time impulses is focused on in the present study. By means of B-equivalence method, which was introduced in Akhmet (2003, 2005, 2009, 2010), Akhmet and Perestyuk (1990) and Akhmet and Turan (2009), we reduce these networks to a fix time impulsive neural networks system. Sufficient conditions ensuring the existence, uniqueness and global robust asymptotic stability of the equilibrium point are obtained by employing an appropriate Lyapunov function and linear matrix inequality (LMI). Finally, we give one illustrative example to show the effectiveness of the theoretical results.