Properties Of Mittag-Leffler Function Research Articles

Projective Synchronization of Discrete-Time Variable-Order Fractional Neural Networks With Time-Varying Delays.

This article is committed to studying projective synchronization and complete synchronization (CS) issues for one kind of discrete-time variable-order fractional neural networks (DVFNNs) with time-varying delays. First, two new variable-order fractional (VF) inequalities are built by relying on nabla Laplace transform and some properties of Mittag-Leffler function, which are extensions of constant-order fractional (CF) inequalities. Moreover, the VF Halanay inequality in discrete-time sense is strictly proved. Subsequently, some sufficient projective synchronization and CS criteria are derived by virtue of VF inequalities and hybrid controllers. Finally, we exploit numerical simulation examples to verify the validity of the derived results, and a practical application of the obtained results in image encryption is also discussed.

Quasi-synchronization of fractional-order complex-value neural networks with discontinuous activations

In this paper, under the framework of Filippov solutions, quasi-synchronization issue of fractional-order complex-valued neural networks (FCNNs) with discontinuous activations and uncertainties is investigated. Firstly, using Laplace transform and the property of Mittag-Leffler function, a novel fractional differential inequality is derived. Then combining the newly constructed inequality with delay feedback control scheme, several quasi-synchronization conditions are obtained for the considered FCNNs by means of non-decomposable method. Eventually, a numerical example is provided to substantiate validation of the proposed results.

Stability and adaptive control-based synchronization of delayed uncertain fractional-order gene regulatory networks

This paper is devoted to investigating stability and synchronization issues for delayed uncertain fractional-order gene regulatory networks (DUFOGRNs). First, the existence and uniqueness of the equilibrium point of the DUFOGRNs are explored via constructing contraction map, whereafter, a stability criterion is studied by means of the fractional-order Lyapunov–Razumikhin approach and property of Mittag-Leffler function. Second, a novel fractional-order Lemma is established by using the theory of fractional calculus showing that the derivative of Lyapunov functions in the fractional-order systems (FOSs) satisfy certain form of convergence, which can be able to discuss the problems about finite-time stability and synchronization of FOSs. Third, adaptive control strategies are designed based on the established Lemma, which are different control protocols anent the synchronization of gene regulatory networks proposed in previous work. Furthermore, sufficient synchronization criteria are derived by probing fractional calculus theory and inequality analysis techniques for guaranteeing the complete synchronization (CS) and finite-time synchronization (F-TS) of DUFOGRNs. Finally, three numerical examples are provided to verify the availability of the proposed method.

Stability and controllability results by n–ary aggregation functions in matrix valued fuzzy n–normed spaces

In the present paper, we apply the n–ary aggregation functions on several special functions (Hypergeometric function, generalized exponential function, and Fox's H–function) to define a class of matrix–valued fuzzy controllers which help us to study the Ulam–Hyers stability for a (non) autonomous fractional differential system in the Hilfer sense, through the fixed point theorem, in a matrix valued fuzzy n–normed space. Next, by the properties of Mittag-Leffler functions, the Laplace transform and the non–standard Gronwall inequality, we propose some efficient conditions on the (asymptotic) stability of the governing model, in matrix fuzzy normed spaces.

Containment control for fractional order MASs with nonlinearity and time delay via pull-based event-triggered mechanism

The containment control for fractional order multi-agent systems with nonlinearity and time delay is investigated in this article. Firstly, a distributed control protocol with fixed time delay is proposed for the purpose of achieving containment control for nonlinear fractional order multi-agent systems. Secondly, by using the pull-based event-triggered mechanism, a distributed event-based control law with time delay is further given to accomplish containment control for fractional order multi-agent systems with nonlinearity. In this way, the controller of each agent is merely updated at its own trigger moments. Hence, it can decrease network correspondence congestion and lower computational cost. Thanks to the stability theory of fractional order calculus, matrix theory, and properties of Mittag-Leffler function, the feasibility of the designed control protocols is confirmed. Furthermore, the Zeno behavior is precluded by the strict proof. Ultimately, two simulation examples are used to illustrate the availability of above theoretical analysis respectively.

Synchronization of fractional-order delayed neural networks with reaction–diffusion terms: Distributed delayed impulsive control

In this study, a new fractional Halanay-like impulsive differential inequality, which considers the system delay and impulse delay, is proposed for the first time. And accordingly, the quasi-synchronization problem of delayed fractional-order coupled reaction–diffusion neural networks (CRDNNs) is investigated. It is important to know that there are still relatively few studies on the impulsive synchronization control of fractional-order systems with time delay, and this study is an effective complement to this relevant field. To achieve quasi-synchronization, the distributed delayed impulsive control strategy is applied, which utilizes the local state information and the weighted average delayed information of other available neurons. Then, based on Wirtinger inequality, Lyapunov stability theory, and the properties of Mittag-Leffler function, some quasi-synchronization criteria are derived by solving some linear matrix inequalities (LMIs). Finally, a numerical simulation is provided to demonstrate the effectiveness of theoretical results, and reveal the effects of impulse delay and fractional order on the system synchronization rate.

On Fractional Order Model of Tumor Growth with Cancer Stem Cell

This paper generalizes the integer-order model of the tumour growth into the fractional-order domain, where the long memory dependence of the fractional derivative can be a better fit for the cellular response. This model describes the dynamics of cancer stem cells and non-stem (ordinary) cancer cells using a coupled system of nonlinear integro-differential equations. Our analysis focuses on the existence and boundedness of the solution in correlation with the properties of Mittag-Leffler functions and the fixed point theory elucidating the proof. Some numerical examples with different fractional orders are shown using the finite difference scheme, which is easily implemented and reliably accurate. Finally, numerical simulations are employed to investigate the influence of system parameters on cancer progression and to confirm the evidence of tumour growth paradox in the presence of cancer stem cells.

Synchronization and adaptive control for coupled fractional-order reaction–diffusion neural networks with multiple couplings

In this paper, two kinds of coupled fractional-order reaction–diffusion neural networks (CFORDNNs) with multiple state couplings or spatial diffusion couplings are proposed. By resorting to the Laplace transform and the properties of Mittag-Leffler functions, sufficient synchronization conditions are derived for the concerned network models. In addition, to guarantee the synchronization of these two networks, several appropriate adaptive control schemes are also developed. Ultimately, the validity of the devised adaptive strategies are verified by adopting some numerical examples with simulation results.

Unique positive solutions for boundary value problem of p-Laplacian fractional differential equation with a sign-changed nonlinearity

This paper investigates the existence of a unique positive solution for a class of boundary value problems of p-Laplacian fractional differential equations, where its nonlinearity is signchanged and involves a fractional derivative term, and its boundary involves a nonlinear fractional integral term. By constructing an appropriate auxiliary boundary value problem and applying a generalized fixed point theorem of sum operator and properties of Mittag-Leffler function, some sufficient conditions for the existence of a unique positive solution are presented, and a monotone iterative sequence uniformly converging to the unique solution is constructed. In addition, an example is given to illustrate the main result.

A system of Cauchy fractional differential equations and new properties of Mittag-Leffler functions with matrix argument

Starting from the matrix form of the fractional Cauchy problem, new formulae for the sum of the three-parameter Mittag-Leffler functions are deduced. The derivation is based on the Prabhakar fractional integral operator. A Volterra integral equation relating the solution of the Cauchy problem with that of a perturbed problem is also obtained; from this a condition number measuring the sensitivity of the solution to changes in data can be derived. Several examples are also incorporated to test the bounds.

Impulsive effects on fractional order time delayed gene regulatory networks: Asymptotic stability analysis

This paper address the fractional-order gene regulatory networks (FOGRNs) with both time delays and impulsive effects. Inspired by the integer-order gene regulatory network models, a general class of fractional-order gene regulatory network model is further investigated via fractional-order Lyapunov-Razumikhin stability theory. Firstly, impulsive effects, feedback regulation, and translation time delays are taken well into account and effective analysis techniques are used to reflect the system’s practically dynamic behavior. Secondly, the existence and uniqueness of the equilibrium point of the fractional-order gene regulatory networks are derived based on the Banach contraction mapping principle. Besides, we derived analytically the condition under which the solution of this network is bounded through generalized Gronwall inequality. Thirdly, some novel delay-free sufficient conditions are derived to ensure the global asymptotic stability of the considered model with the help of the fractional-order Lyapunov-Razumikhin method and properties of Mittag-Leffler functions. Finally, an example with numerical simulation is provided to illustrate the validity and effectiveness of the proposed results.

On the Theory of ψ-Hilfer Nonlocal Cauchy Problem

In this paper, we derive the representation formula of the solution for ψ-Hilfer fractional differential equation with constant coefficient in the form of Mittag-Leffler function by using Picard’s successive approximation. Moreover, by using some properties of Mittag-Leffler function and fixed point theorems such as Banach and Schaefer, we introduce new results of some qualitative properties of solution such as existence and uniqueness. The generalized Gronwall inequality lemma is used in analyze Eα -Ulam-Hyers stability. Finally, one example to illustrate the obtained results

Asymptotic stability analysis of Riemann-Liouville fractional stochastic neutral differential equations

The novelty of our paper is to establish results on asymptotic stability of mild solutions in $p$th moment to Riemann-Liouville fractional stochastic neutral differential equations (for short Riemann-Liouville FSNDEs) of order $\alpha \in (\frac{1}{2},1)$ using a Banach's contraction mapping principle. The core point of this paper is to derive the mild solution of FSNDEs involving Riemann-Liouville fractional time-derivative by applying the stochastic version of variation of constants formula. The results are obtained with the help of the theory of fractional differential equations, some properties of Mittag-Leffler functions and asymptotic analysis under the assumption that the corresponding fractional stochastic neutral dynamical system is asymptotically stable.

Existence and stability results of relaxation fractional differential equations with Hilfer--Katugampola fractional derivative.

In this work, we present the existence, uniqueness, and stability result of solution to the nonlinear fractionaldifferential equations involving Hilfer-Katugampola derivative subject to nonlocal fractional integral bound-ary conditions. The reasoning is mainly based upon properties of Mittag-Leffler functions, and fixed-pointmethods such as Banach contraction principle and Krasnoselskii's fixed point theorem. Moreover, the gener-alized Gornwall inequality lemma is used to analyze different types of stability. Finally, one example is givento illustrate our theoretical results.

Stability and Stabilization for a Class of Semilinear Fractional Differential Systems

This paper considers a class of semilinear fractional-order systems with Caputo derivative. New conditions ensuring asymptotic stability and stabilization of fractional systems with the fractional order between 0 and 2 are proposed. The analysis is based on a property of convolution and asymptotic properties of Mittag-Leffler functions. Some numerical examples are provided to illustrate the feasibility and validity of the proposed approach.

Finite-time stability for fractional-order complex-valued neural networks with time delay

This paper explores the finite-time stability of fractional-order complex valued neural networks with time delay. By employing Laplace transform and the properties of Mittag-Leffler function, a lemma of exponent stability is developed to derive the finite-time stability conditions. Further, by using the proposed lemma and the techniques of inequalities, the finite-time stability of fractional-order complex-valued neural networks with time delay is analyzed with and without a controller. In addition, some sufficient conditions are proposed to analyze the finite-time stability of the fractional-order complex-valued neural networks and the setting time for stability is also estimated. Finally, two examples are used to verify the validity and feasibility of the proposed criteria.

Quasi-projective and complete synchronization of fractional-order complex-valued neural networks with time delays

This paper studies quasi-projective synchronization (QPS) and complete synchronization (CS) for a class of fractional-order complex-valued neural networks with time delays by designing suitable controllers. To realize QPS and CS, linear feedback controller and adaptive controller are designed, and a novel fractional-order differential inequality is built by means of Laplace transform and properties of Mittag-Leffler function. By utilizing Lyapunov method, our proposed inequality, fractional-order Razumikhin theorem and some complex analysis techniques, some effective criteria are derived to ensure QPS and CS of the considered networks. Furthermore, the error bound of QPS is obtained. Finally, some numerical results are given to demonstrate the effectiveness of the presented theoretical results.

Robust State Estimation for Fractional-Order Nonlinear Uncertain Systems via Adaptive Event-Triggered Communication Scheme

This paper is concerned with the adaptive event-triggered robust state estimation for a class of fractional-order nonlinear uncertain systems, where the fractional order satisfies 0 <; α <; 1. To save bandwidth resources effectively, an adaptive event-triggered scheme (AETS) is proposed to judge whether the measurement output of the sensor needs to be released to the estimator or not. Firstly, by taking the AETS into consideration, a novel fractional-order estimation error model is established. Then, based on this system model, a sufficient condition that can ensure the robust mean-square stability of the estimation error system has been derived by utilizing the Lyapunov functional approach and some properties of Mittag-Leffler functions. Meanwhile, by applying matrix's singular value decomposition (SVD) technique, the design of the state estimator has been solved in terms of solving the solution of linear matrix inequality (LMI). Finally, one provides a numerical example to verify the feasibility of the proposed method.

Global Synchronization of Coupled Fractional-Order Recurrent Neural Networks.

This paper presents new theoretical results on the global synchronization of coupled fractional-order recurrent neural networks. Under the assumptions that the coupled fractional-order recurrent neural networks are sequentially connected in form of a single spanning tree or multiple spanning trees, two sets of sufficient conditions are derived for ascertaining the global synchronization by using the properties of Mittag-Leffler function and stochastic matrices. Compared with existing works, the results herein are applicable for fractional-order systems, which could be viewed as an extension of integer-order ones. Two numerical examples are presented to illustrate the effectiveness and characteristics of the theoretical results.

Approximate Controllability for Nonlocal Fractional Propagation Systems of Sobolev Type

This paper is concerned with the approximate controllability of a class of fractional propagation systems of Sobolev type with nonlocal conditions in Hilbert spaces. By utilizing the theory of propagation family and techniques of measures of noncompactness, we firstly prove existence of the mild solutions and compactness of solutions set in Banach spaces under weaker conditions, i.e., the compactness condition of propagation family is instead of norm continuous in the sense of uniform operator topology. Secondly, we establish the principle for linear fractional propagation systems is approximately controllable in Hilbert spaces. Then, we present an interesting criteria for approximate controllability of semilinear fractional propagation systems in Hilbert spaces by requiring the corresponding linear fractional propagation systems is approximately controllable and assuming that the resolvent set of the operator pair is compact. Finally, a fractional PDEs model is used to demonstrate that how to check the conditions in the above abstract theorems, In particular, a principle for approximate controllability of linear problem is verified by using the property of Mittag-Leffler function.