Fractional-order Differential Inequality Research Articles

Exponential Quasi-Synchronization of Fractional-Order Fuzzy Cellular Neural Networks via Impulsive Control

This paper investigates the exponential quasi-synchronization of fractional-order fuzzy cellular neural networks with parameters mismatch via impulsive control. Firstly, under the framework of the generalized Caputo fractional-order derivative, a new fractional-order impulsive differential inequality is established. Secondly, based on this fractional-order impulsive differential inequality, a general criterion for the quasi-synchronization of fractional-order systems is obtained. Then, specific to the fractional-order fuzzy cellular neural network model in this paper, the criteria and error estimation of the exponential quasi-synchronization of fractional-order fuzzy cellular neural networks can be obtained. Finally, two numerical examples are given to illustrate the effectiveness of the obtained results.

Finite-time stability of fractional-order quaternion-valued memristive neural networks with time delay

This study addresses the finite-time stability (FTS) problem of fractional-order quaternion-valued memristive neural networks (FQMNNs) with time delay. Firstly, by leveraging set-valued mapping, differential inclusion, and contracting mapping principle, the sufficient condition is provided to assure the existence and uniqueness of the equilibrium point. Secondly, given the fractional order differential inequality and Gronwall inequality, a finite-time stability (FTS) criterion is obtained for the delayed FQMNNs with the fractional order between 0 and 1. Additionally, a sufficient condition is proposed to guarantee the FTS of FQMNNs with the fractional order between 1 and 2. Compared with existing results, this study not only extends the fractional order between 0 and 2, but also incorporates quaternions, which significantly enhances the FTS theory. Finally, two numerical examples demonstrate the practicality of the obtained results.

Mittag–Leffler stability and stabilization of delayed fractional-order memristive neural networks based on a new Razumikhin-type theorem

The Mittag–Leffler stability and stabilization of delayed fractional-order memristive neural networks(DFMNNs) are investigated in this paper. First, two new fractional Halanay inequalities are established by solving two fractional-order non-autonomous differential inequalities. Next, by using the proposed fractional Halanay inequalities, a novel Razumikhin-type theorem for Mittag–Leffler stability of delayed fractional-order systems is presented, which is an extension of the so-called Razumikhin theorem for integer-order delayed differential systems. Applying the Razumikhin-type theorem to the DFMNNs, several Mittag–Leffler stability and stabilization criteria are obtained. Finally, the validity of the proposed results is shown by two numerical examples.

New Approach to Quasi-Synchronization of Fractional-Order Delayed Neural Networks

This article investigates quasi-synchronization for a class of fractional-order delayed neural networks. By utilizing the properties of the Laplace transform, the Caputo derivative, and the Mittag–Leffler function, a new fractional-order differential inequality is introduced. Furthermore, an adaptive controller is designed, resulting in the derivation of an effective criterion to ensure the aforementioned synchronization. Finally, a numerical illustration is provided to demonstrate the validity of the presented theoretical findings.

Predefined-time synchronization of fractional-order memristive competitive neural networks with time-varying delays

This article focuses on the predefined-time synchronization (PTS) of fractional-order memristive competitive neural networks (FMCNNs) with time-varying delays. According to the two-layer structural characteristics of CNNs, two kinds of distinctive discontinuous bilayer predefined-time control schemes with the fractional integrals are proposed: one is the double controllers based on piecewise Lyapunov function and the other is a controller with Lyapunov function and exponential function. Using the predefined-time stability theorems and applying fractional-order differential inequalities and other inequality techniques, some effective criteria are obtained to assure the PTS of two FMCNNs in terms of algebraic inequalities, which are very succinct and avert complicated calculations. Besides, the predefined time (PT) is set to an arbitrary positive parameter in these controllers and is entirely irrelevant to the initial values. Finally, two concrete examples are given to verify the theoretical results.

Anti-synchronization of fractional-order complex-valued neural networks with a leakage delay and time-varying delays

In this paper, the anti-synchronization for fractional-order complex-valued neural networks (FCVNNs) with a leakage delay and time-varying delays is investigated. The solution of a fractional-order differential inequality is considered. By using the Laplace transform and the inverse Laplace transform, a new inequality is proved, which provides an important tool for studying the anti-synchronization of the FCVNNs with delays. Two different types of controllers are designed. Based on this new inequality and some inequality techniques, several sufficient conditions are obtained to ensure the anti-synchronization of the FCVNNs with a leakage term and time-varying delays. Finally, two numerical examples are presented to illustrate the validity and feasibility of the main results.

Finite-time synchronization of fractional-order delayed memristive fuzzy neural networks

This paper deals with the finite-time synchronization (FTS) for a class of fractional-order memristive fuzzy neural networks (FOMFNNs) with leakage and transmission delays. Two types of discontinuous control schemes with leakage and transmission delays are designed: one is the state feedback control and the other is the fractional-order adaptive control. Based on differential inclusion theory and fractional-order differential inequalities, several new delay-independent algebraic conditions are derived to guarantee the FTS of drive-response FOMFNNs with leakage and transmission delays. Moreover, the estimation of the upper bound of the settling time for the FTS is given as well. Ultimately, two numerical examples with simulations are furnished to demonstrate the feasibility and availability of the theoretical results.

Adaptive finite-time synchronization of fractional-order delayed fuzzy cellular neural networks

The adaptive finite-time synchronization problems are investigated for a class of fractional-order delayed fuzzy cellular neural networks (FODFCNNs). First, with the aid of the properties of the Mittag-Leffler function and the Laplace transform with the parameter, some novel fractional-order differential inequalities are established to provide new perspective to explore finite-time synchronization of fractional-order systems. Second, based on these inequalities and the designed adaptive controller, some novel sufficient conditions are given to guarantee the adaptive finite-time synchronization of FODFCNNs. Finally, the effectiveness of the results is demonstrated by a numerical example.

PROJECTION SYNCHRONIZATION OF FUNCTIONAL FRACTIONAL-ORDER NEURAL NETWORKS WITH VARIABLE COEFFICIENTS

In this paper, the projection synchronization problem of functional fractional-order neural networks with variable coefficients and Caputo derivatives is studied. Firstly, a simple global projection synchronization scheme is designed according to the open-loop and adaptive feedback control. Secondly, by constructing a suitable Lyapunov function and utilizing the properties of delayed fractional-order differential inequalities, some criteria for the global projective synchronization of the variable coefficient functional neural networks with Caputo derivatives are obtained. Finally, a numerical example with many numerical simulations is employed to demonstrate the correctness and validity of the proposed method in this paper.

Switching-Jumps-Dependent Quasi-Synchronization Criteria for Fractional-Order Memrisive Neural Networks

This paper investigates the switching-jumps-dependent quasi-synchronization issue for fractional-order memristive neural networks (FMNNs). First, a simplied linear feedback controller is applied. Then, in terms of several fractional order differential inequalities and two kinds of Lyapunov functions, two quasi-synchronization criteria expressed by linear matrix inequality (LMI)-based form and algebraic form are established, respectively. Meanwhile, the co-designed scheme for error bound and control gain is established. Compared with the previous quasi-synchronization results, a strong assumption that the system states must be bounded is removed. Finally, some simulation examples are carried out to display the feasibility and validity of the proposed analysis methods.

Quasiconsensus of fractional-order heterogeneous multiagent systems under event-triggered impulsive control method

This paper investigates the quasiconsensus problem of fractional-order heterogeneous multiagent systems, the distributed impulsive control protocol is designed for the multiagent system. In contrast to some existing results, the impulsive moments are determined by preset events, i.e., the event-triggered mechanism is used. Based on the fractional-order Lyapunov stability theory and fractional-order differential inequality, the quasiconsensus criteria are derived; furthermore, the prescribed error bound is given. Then, Zeno behavior for the considered event-triggered control method is excluded. Finally, numerical examples are given to shown the effectiveness of the proposed method.

Synchronization analysis of fractional-order inertial-type neural networks with time delays

This paper is dedicated to the global Mittag-Leffler synchronization (GMLS) of fractional-order inertial-type neural networks (FOITNNs) with time delays. To begin with, based on the semigroup property of the Caputo fractional derivative (CFD), an appropriate variable substitution is chosen to transform the original fractional-order inertial system into a traditional fractional-order system. Secondly, two types of discontinuous control schemes with delays and only a control input are proposed: one is the state feedback control and the other is the fractional-order adaptive control. On the basis of Lyapunov stability theory and fractional-order differential inequalities, some new sufficient criteria for the GMLS of two FOITNNs are established. Furthermore, the control gains here can be selected more widely, which makes the results more applicative and less conservative. Finally, two numerical examples validate the efficacy of the obtained results.

Finite-time synchronization for fractional-order quaternion-valued coupled neural networks with saturated impulse

In this paper, the issue of finite-time synchronization (FTS) of fractional-order quaternion-valued coupled neural networks (FOQVCNNs) with saturated impulse was discussed. The impulsive effect was firstly applied to the FOQVCNNs, which was subjected to saturation. Therefore, the constructed FOQVNNs was more suitable for realistic life and had a wider range of applications. The synchronization criteria were extended to a less conservative form. In addition, FTS was considered with the purpose of having a faster convergence rate. Based on the fractional-order differential inequality, polytopic representation approach, Lyapunov function, and the concept of FTS, some synchronization criteria were obtained to accomplish the synchronization for the FOQVCNNs, which were expressed as some inequalities. Then, the settling time was bounded, which could be accurately predicted with ease. Lastly, the theoretical outcome was proved to be valid by a numerical example.

Robust Stability of Fractional Order Memristive BAM Neural Networks with Mixed and Additive Time Varying Delays

This paper is concerned with the problem of the robust stability of fractional-order memristive bidirectional associative memory (BAM) neural networks. Based on Lyapunov theory, fractional-order differential inequalities and linear matrix inequalities (LMI) are applied to obtain a robust asymptotical stability. Finally, numerical examples are presented.

Adaptive Synchronization of Fractional-Order Complex-Valued Neural Networks With Time-Varying Delays

In this paper, the adaptive synchronization of fractional-order complex-valued neural networks with time-varying delays(FOCVNNTDs) is investigated. First, two novel fractional-order differential inequalities with time delays are established, which can be seen as an extension of Halanay inequality. Besides, complete synchronization and quasi-projective synchronization of FOCVNNTDs are investigated based on the two novel inequalities using a novel adaptive controller. In addition, instead of separating the complex-valued neural networks into two real-valued networks, a non-decomposition method is adopted to study the adaptive synchronization of FOCVNNTDs, which avoids the difficulty and complexity of theoretical analysis. Finally, some numerical simulation examples are provided to demonstrate the validity of our theory.

Estimating the ultimate bounds and synchronization of fractional-order plasma chaotic systems

• The global Mittag–Leffler ultimate bounds and synchronization for fractional order plasma chaotic systems are studied. • Some new ellipsoid and cylindrical domains as well as a rotatory paraboloid of the bounds are derived via generalized Lyapunov function theory. • Linear feedback control tactics with two inputs or one input as well as fractional order adaptive control with one input are proposed. • The boundaries and feedback gain constants are easier to be calculated and the controller structures are simpler. This paper focuses on the Mittag–Leffler ultimate bounds (MLUBs) and synchronization of fractional-order plasma chaotic systems (FOPCSs). For one thing, by choosing some appropriate generalized Lyapunov functions and combining fractional-order differential inequalities, some new estimates on the 3D ellipsoid and cylindrical domains as well as a rotatory paraboloid of the bounds for the FOPCS are acquired according to system parameters, which perfect the previous studies and may infer a few new estimates. For another, linear feedback control tactics with two inputs or one input as well as fractional-order adaptive control with one input and a single variable are presented to realize the synchronization of two FOPCSs. Several new sufficient conditions for synchronization are analytically obtained by using inequality methods. The boundaries and feedback gain constants are easier to be calculated and the controller structures are simpler. Finally, numerical simulations are shown to confirm the correctness of the presented synchronization schemes.

Intermittent control for finite-time synchronization of fractional-order complex networks

This paper is concerned with the finite-time synchronization problem for fractional-order complex dynamical networks (FCDNs) with intermittent control. Using the definition of Caputo’s fractional derivative and the properties of Beta function, the Caputo fractional-order derivative of the power function is evaluated. A general fractional-order intermittent differential inequality is obtained with fewer additional constraints. Then, the criteria are established for the finite-time convergence of FCDNs under intermittent feedback control, intermittent adaptive control and intermittent pinning control indicate that the setting time is related to order of FCDNs and initial conditions. Finally, these theoretical results are illustrated by numerical examples.

Finite-Time Synchronization of Fractional-Order Complex-Variable Dynamic Networks

In this paper, without dividing complex-variable networks into two subsystems with real values, the finite-time synchronization is considered for complex-valued dynamical networks with fractional order by means of the theory of complex-variable functions. First of all, as a generalization of the real-valued sign function, the sign functions of complex-valued numbers and complex-valued vectors are introduced and some formulas about them are established. Under the sign function framework, two complex-valued control strategies are designed based on two different norms of complex numbers. Some synchronization criteria are derived and the settling times of synchronization are effectively estimated by developing fractional-order finite-time differential inequalities and utilizing the theory of complex-variable functions. The established theoretical results are demonstrated and the effect of the fractional order of the network model on the finite-time synchronization is revealed finally by providing some numerical simulations.

Quasi-synchronization of fractional-order heterogeneous dynamical networks via aperiodic intermittent pinning control

This paper focuses on the quasi-synchronization problem for fractional-order heterogeneous dynamical networks via aperiodic intermittent pinning control. First, based on the properties of the Mittag–Leffler function, a new fractional-order differential inequality is established. By utilizing the new inequality and Lyapunov function method, a general sufficient condition is then derived to ensure the addressed dynamical networks can achieve global quasi-synchronization through pinning part of the network nodes with simple aperiodic intermittent controllers, which is followed by some easily-verified quasi-synchronization criteria. In addition, the exponential convergence rate and the error bound of the quasi-synchronization are also estimated, respectively. Moreover, a detailed algorithm about how to design suitable aperiodic intermittent pinning controllers is provided. Finally, a numerical example is presented to verify the validity of theoretical analysis.

Exponential quasi-synchronization of conformable fractional-order complex dynamical networks

Abstract This article concerns the global exponential quasi-synchronization issue of conformable fractional-order complex dynamical networks (FCDNs) by periodically intermittent pinning control. We first establish two conformable fractional-order differential inequality, which are useful to deal with the global exponential quasi-synchronization of conformable FCDNs, and then by the obtained inequalities, the Lyapunov approach as well as the periodically intermittent pinning controller, related sufficient conditions are obtained to ensure the global exponential quasi-synchronization of the addressed networks. Finally, numerical simulations are provided to show the effectiveness of the theoretical results.