Fractional Comparison Principle Research Articles

Asymptotic Boundedness of Conformable Fractional Delay Differential Systems

This article focuses on the problem of asymptotic boundedness for non-autonomous conformable fractional delay differential systems. Firstly, by utilizing the definition of conformable fractional derivative and fractional comparison principle, we propose non-autonomous conformable fractional Halanay inequalities. And then, based on the Halanay inequalities, sufficient conditions for the asymptotic boundedness are constructed for the systems. Finally, a numerical example shows the applicability of the proposed technique.

Robust synchronization of multi-weighted fractional order complex dynamical networks under nonlinear coupling via non-fragile control with leakage and constant delays

In this article, we examine the impact of leakage delays on robust synchronization for fractional order multi-weighted complex dynamical networks(MFCDN) under non-linear coupling via non-fragile control. By employing the fractional order comparison principle, suitable Lyapunov method, and some fractional order inequality techniques, we ensured the robust asymptotical synchronization for MFCDN. In addition to common findings, we have done some specific research in order to get reliable synchronization for multi-weighted complex dynamical network(MCDN) without leakage delay. Additionally, our findings gained are applicable to single weighted FCDN and integer order complex dynamical networks, regardless of whether they have a single weight or many weights. Our suggested approach is shown to be more effective and practical in this article by providing a numerical simulation.

Fractional-Order Impulsive Delayed Reaction-Diffusion Gene Regulatory Networks: Almost Periodic Solutions

The paper is oriented on the existence of almost periodic solutions of factional-order impulsive delayed reaction-diffusion gene regulatory networks. Caputo type fractional-order derivatives and impulsive disturbances at not fixed instants of time are considered. New almost periodic and perfect Mittag–Leffler stability criteria are proposed. Lyapunov’s like impulsive functions, the properties of the fractional derivatives and comparison principle are the main tools in the investigation. Illustrative examples are also presented to demonstrate the proposed criteria. Our results contribute to the development of qualitative the theory of fractional-order gene regulatory networks.

On stability and event trigger control of fractional neural networks by fractional non-autonomous Halanay inequalities

This paper studies the stability and control of fractional neural networks by Halanay inequality technique. Based on the fractional comparison principle and supremum and infimum principle, a novel fractional non-autonomous Halanay inequality is developed. The fractional non-autonomous Halanay inequality is in a form of integral, which considers the global nature of the system parameters and reduces estimation error. By combining the Halanay inequality with a maximum auxiliary function, an asymptotically stable discriminant condition for fractional Hopfield time-delay neural networks is established in an algebraic form. Moreover, event trigger control for fractional neural networks is studied. Low network bandwidth costs and high control efficiency are guaranteed by a Mittag-Leffler type event-triggered mechanism. Then, a discriminant condition on the event trigger control for fractional neural networks is established. The advantages of the proposed methods are demonstrated by three numerical examples.

Delay-coupled fractional order complex Cohen-Grossberg neural networks under parameter uncertainty: Synchronization stability criteria

<abstract><p>This paper inspects the issues of synchronization stability and robust synchronization stability for fractional order coupled complex interconnected Cohen-Grossberg neural networks under linear coupling delays. For investigation of synchronization stability results, the comparison theorem for multiple delayed fractional order linear system is derived at first. Then, by means of given fractional comparison principle, some inequality methods, Kronecker product technique and classical Lyapunov-functional, several asymptotical synchronization stability criteria are addressed in the voice of linear matrix inequality (LMI) for the proposed model. Moreover, when parameter uncertainty exists, we also the investigate on the robust synchronization stability criteria for complex structure on linear coupling delayed Cohen-Grossberg type neural networks. At last, the validity of the proposed analytical results are performed by two computer simulations.</p></abstract>

Mittag–Leffler stability of nabla discrete fractional-order dynamic systems

In the present study, the definition of discrete Mittag–Leffler stability is derived to characterize convergence rule of the pseudostates for nabla discrete fractional-order dynamic systems. Applying the Lyapunov stability theory, some new criteria are proposed to determine asymptotic stability of the zero equilibrium. In addition, by applying fractional comparison principle, the results are extended from Caputo discrete fractional-order systems to Riemann–Liouville systems. Moreover, a useful inequality is proposed to further improve the availability of the presented methods. Finally, some meticulously designed simulations are provided to verify the correctness and practicability of the elaborated stability notions.

Stability analysis for generalized fractional differential systems and applications

Abstract This paper shows solicitude for the stability analysis issue of a class of generalized fractional differential systems via fractional comparison principle and Lyapunov direct method. With the concept of (Generalized) Mittag-Leffler (M-L) stability given, we first establish a new framework to consider the global M-L stability of generalized fractional differential systems with or without time-delay, in which new stability criterion is achieved by some novel differential inequalities satisfied by the fractional derivative of Lyapunov functions. Second, through the employment of less conservative comparison principle for the generalized fractional system, a sufficient theorem for the (Generalized) M-L stability is derived. Third, a generalized fractional memristor-based impulsive neural network is investigated to illustrate the proposed stability theory, this model is more general which includes a memristor-based Caputo fractional neural network or common Hopfield neural network as special cases. In the end, two simple numerical examples are listed to affirm the theoretical findings.

A GENERAL COMPARISON PRINCIPLE FOR CAPUTO FRACTIONAL-ORDER ORDINARY DIFFERENTIAL EQUATIONS

In this paper, we work on a general comparison principle for Caputo fractional-order ordinary differential equations. A full result on maximal solutions to Caputo fractional-order systems is given by using continuation of solutions and a newly proven formula of Caputo fractional derivatives. Based on this result and the formula, we prove a general fractional comparison principle under very weak conditions, in which only the Caputo fractional derivative is involved. This work makes up deficiencies of existing results.

Tempered Mittag–Leffler Stability of Tempered Fractional Dynamical Systems

Due to finite lifespan of the particles or boundedness of the physical space, tempered fractional calculus seems to be a more reasonable physical choice. Stability is a central issue for the tempered fractional system. This paper focuses on the tempered Mittag–Leffler stability for tempered fractional systems, being much different from the ones for pure fractional case. Some new lemmas for tempered fractional Caputo or Riemann–Liouville derivatives are established. Besides, tempered fractional comparison principle and extended Lyapunov direct method are used to construct stability for tempered fractional system. Finally, two examples are presented to illustrate the effectiveness of theoretical results.

A new Lyapunov stability analysis of fractional-order systems with nonsingular kernel derivative

This study introduces a new and promising stability approach for Caputo-Fabrizio (CF)-fractional-order system. A new fractional comparison principle for this nonsingular kernel fractional derivative is proposed. Next, a key inequality is suggested to analysis the Lyapunov-based stability of assumed systems. Afterwards, class-K functions are established to analysis of fractional Lyapunov direct method. At last, an explanatory example is given to validate the proposed idea. This new and novel approach can be expanded to the other types of nonsingular kernel derivatives due to a simple and effective idea beyond the proposed procedure.

Sliding mode control for a class of variable-order fractional chaotic systems

This paper concerns the stability analysis and controller design for a class of variable order fractional chaotic systems. By use of Fractional Comparison Principle, the Mittag–Leffler stability criterion is proposed and proved for variable order fractional systems. For the unperturbed system, two different kinds of sliding mode controllers are proposed and the stability of the controlled systems is proved based on the proposed stability criterion. The first control law is designed on the constructing of a variable order fractional integral sliding mode surface and results in a free of chattering signal. While, the second one introduces a variable order fractional derivative sliding mode surface, which is also adapted for the variable order fractional system with uncertainty and external disturbance. And the sign · function in the switching control law is transferred into the fractional derivative of the control input such that it avoids the undesirable chattering. In addition, for the systems with uncertainty and external disturbance, an adaptive sliding mode control law is designed for the variable order fractional chaotic system. And the unknown bounds of the uncertainty and external disturbance are estimated by the variable order fractional derivative adaptive laws. Based on the fractional order Barbalat’s Lemma which is extended from the integer order Barbalat’s Lemma, the asymptotical stability is proved for the controlled uncertain system. At last, numerical simulations are presented to verify the validity and efficiency of the proposed fractional order controllers.

Multi-weighted Complex Structure on Fractional Order Coupled Neural Networks with Linear Coupling Delay: A Robust Synchronization Problem

This sequel is concerned with the analysis of robust synchronization for a multi-weighted complex structure on fractional-order coupled neural networks (MWCFCNNs) with linear coupling delays via state feedback controller. Firstly, by means of fractional order comparison principle, suitable Lyapunov method, Kronecker product technique, some famous inequality techniques about fractional order calculus and the basis of interval parameter method, two improved robust asymptotical synchronization analysis, both algebraic method and LMI method, respectively are established via state feedback controller. Secondly, when the parameter uncertainties are ignored, several synchronization criterion are also given to ensure the global asymptotical synchronization of considered MWCFCNNs. Moreover, two type of special cases for global asymptotical synchronization MWCFCNNs with and without linear coupling delays, respectively are investigated. Ultimately, the accuracy and feasibility of obtained synchronization criteria are supported by the given two numerical computer simulations.

Finite time impulsive synchronization of fractional order memristive BAM neural networks

This paper is concerned with the finite time impulsive synchronization problem of fractional order memristive BAM neural networks (MBAMNNs) with switching jumps mismatch. By utilizing the properties of fractional calculus and comparison principle, a novel fractional order result about finite time impulsive stability is obtained. In the light of the double layer structure of fractional order MBAMNNs, two impulsive controllers are designed for the response fractional order MBAMNNs. By combing properties of Gamma functions and some analysis techniques of impulsive interval, several sufficient conditions are given which ensure finite time synchronization of fractional order MBAMNNs, and the upper bound of the setting time for synchronization is estimated, which is related to the fractional order of the system. Finally, numerical simulation is provided to demonstrate the effectiveness of the obtained results.

Impulsive Delayed Lasota–Wazewska Fractional Models: Global Stability of Integral Manifolds

In this paper we deal with the problems of existence, boundedness and global stability of integral manifolds for impulsive Lasota–Wazewska equations of fractional order with time-varying delays and variable impulsive perturbations. The main results are obtained by employing the fractional Lyapunov method and comparison principle for impulsive fractional differential equations. With this research we generalize and improve some existing results on fractional-order models of cell production systems. These models and applied technique can be used in the investigation of integral manifolds in a wide range of biological and chemical processes.

Dissipativity Analysis of Memristor-Based Fractional-Order Hybrid BAM Neural Networks with Time Delays

Abstract A new class of memristor-based time-delay fractional-order hybrid BAM neural networks has been put forward. The contraction mapping principle has been adopted to verify the existence and uniqueness of the equilibrium point of the addressed neural networks. By virtue of fractional Halanay inequality and fractional comparison principle, not only the dissipativity has been analyzed, but also a globally attractive set of the new model has been formulated clearly. Numerical simulation is presented to illustrate the feasibility and validity of our theoretical results.

Existence of the solution and stability for a class of variable fractional order differential systems

Abstract In this paper, the existence results of the solution and stability are focused for the variable fractional order differential equation. In view of the definitions of three kinds of Caputo variable fractional order operator, the sufficient condition of the solution existence for the variable fractional order differential system is obtained by use of Arzela–Ascoli theorem. Moreover, some criterions of the Mittag–Leffler stability and asymptotical stability are proposed for the variable fractional order differential system according to the Fractional Comparison Principle.

Stability analysis of nonlinear Hadamard fractional differential system

The stability of the zero solution of a class of nonlinear Hadamard type fractional differential system is investigated by utilizing a new fractional comparison principle. The novelty of this paper is based on some new fractional differential inequalities along the given nonlinear Hadamard fractional differential system. A comparison principle employing the new fractional differential inequality for scalar Hadamard fractional differential system is presented. Based on the new comparison principle, some sufficient conditions for the (generalized) stability and the (generalized) Mittag-Leffler stability are given.

Generalized Mittag–Leffler Stability of Hilfer Fractional Order Nonlinear Dynamic System

This article studies the generalized Mittag–Leffler stability of Hilfer fractional nonautonomous system by using the Lyapunov direct method. A new Hilfer type fractional comparison principle is also proved. The novelty of this article is the fractional Lyapunov direct method combined with the Hilfer type fractional comparison principle. Finally, our main results are explained by some examples.

Distributed consensus tracking of unknown nonlinear chaotic delayed fractional-order multi-agent systems with external disturbances based on ABC algorithm

In this paper, the distributed consensus tracking of unknown nonlinear chaotic delayed fractional-order multi-agent systems (FOMASs) with external disturbances is investigated, where the fractional orders and system parameters are unknown. Firstly, in order to identify the unknown parameters of the delayed nonlinear FOMASs, an efficient artificial bee colony algorithm (ABC)-based parameters estimation approach is put forward. Secondly, based on the estimated parameters, by using fractional derivative inequality and comparison principle of linear fractional equation with delay, a discontinuous distributed control protocol is proposed to make the tracking errors converge to zero asymptotically. Thirdly, to overcome the undesirable chattering phenomenon caused by the discontinuous controller, a continuous distributed control algorithm is further designed and uniformly ultimately bounded (UUB) tracking errors can be obtained and reduced as small as desired. Finally, numerical simulations are given to test the effectiveness of the proposed parameters estimation scheme and the deigned control algorithms.

Fractional comparison method and asymptotic stability results for multivariable fractional order systems

This paper investigates the asymptotic stability of equilibrium points of multivariable fractional order systems where its state equations contain different fractional orders which lie between 0 and 1. First, a fractional comparison principle is presented. Then, some elementary comparison results for linear fractional order systems of different fractional orders are developed. Based on continuously differentiable quadratic function and comparison results, new sufficient conditions are established for the asymptotic stability of linear and nonlinear multivariable fractional order systems. Finally, a few illustrative examples are presented.