Global Mittag-Leffler Stability Research Articles

Finite-time and global Mittag-Leffler stability of fractional-order neural networks with S-type distributed delays

Finite-time and global Mittag-Leffler stability of fractional-order neural networks with S-type distributed delays

Global Asymptotic Stability and Synchronization of Fractional-Order Reaction–Diffusion Fuzzy BAM Neural Networks with Distributed Delays via Hybrid Feedback Controllers

In this paper, the global asymptotic stability and global Mittag–Leffler stability of a class of fractional-order fuzzy bidirectional associative memory (BAM) neural networks with distributed delays is investigated. Necessary conditions are obtained by means of the Lyapunov functional method and inequality techniques. The hybrid feedback controllers are then developed to ensure the global asymptotic synchronization of these neural networks, resulting in two additional synchronization criteria. The derived conditions are applied to check the fractional-order fuzzy BAM neural network’s Mittag–Leffler stability and synchronization. Three examples are given to demonstrate the effectiveness of the achieved results.

New results for dynamical analysis of fractional-order gene regulatory networks with time delay and uncertain parameters

In this paper, a novel class of networks named fractional-order gene regulatory networks with time delay and uncertain parameters (FGRNTDUP) are formulated, and some new results are obtained for the dynamics of FGRNTDUP. Firstly, a general fractional-order differential equality is built to supply a new viewpoint about the study of finite-time stability, stabilization, and synchronization for many fractional systems. Secondly, based on contraction mapping principle, the existence and uniqueness of the equilibrium point are strictly proved for the FGRNTDUP. Meanwhile, by virtue of fractional Lyapunov method and inequality techniques, some sufficient conditions are derived to guarantee the global Mittag–Leffler stability of FGRNTDUP. Moreover, based on a newly developed inequality, some novel criteria are yielded to realize the finite-time synchronization of FGRNTDUP by designing a suitable adaptive controller. Finally, two numerical examples are provided to verify the effectiveness of the obtained results.

Stabilization of a Non-linear Fractional Problem by a Non-Integer Frictional Damping and a Viscoelastic Term

In this paper, we consider a fractional equation of order between one and two which may be looked at as an interpolation between the heat and wave equations. The problem is non-linear as it involves a power-type non-linearity. We investigate the possibilities of stabilizing the system by a lower-order fractional term and/or a memory term involving the Laplacian. We prove a global Mittag–Leffler stability result in case a fractional frictional damping is active and a local Mittag–Leffler stability result when the material is viscoelastic in case of small relaxation functions. Unlike the integer-order problems, additional serious difficulties arise in the present case. These difficulties are highlighted clearly in the introduction. They are mainly due to the memory dependence of the fractional derivatives which is the cause of the invalidity of the product rule in particular. We utilize several properties in fractional calculus. Moreover, we introduce new Lyapunov-type functionals in the context of the multiplier technique.

Mittag-Leffler stability of fractional-order quaternion-valued memristive neural networks with generalized piecewise constant argument

This paper studies the global Mittag-Leffler (M-L) stability problem for fractional-order quaternion-valued memristive neural networks (FQVMNNs) with generalized piecewise constant argument (GPCA). First, a novel lemma is established, which is used to investigate the dynamic behaviors of quaternion-valued memristive neural networks (QVMNNs). Second, by using the theories of differential inclusion, set-valued mapping, and Banach fixed point, several sufficient criteria are derived to ensure the existence and uniqueness (EU) of the solution and equilibrium point for the associated systems. Then, by constructing Lyapunov functions and employing some inequality techniques, a set of criteria are proposed to ensure the global M-L stability of the considered systems. The obtained results in this paper not only extends previous works, but also provides new algebraic criteria with a larger feasible range. Finally, two numerical examples are introduced to illustrate the effectiveness of the obtained results.

The boundedness, global Mittag-Leffler sability and S-asymptotic ω-periodic of fractional-order fuzzy inertial neural networks with delays

The boundedness, global Mittag-Leffler stability (GMLS), and S-asymptotic ω-periodic of fuzzy fractional-order inertial neural networks (FINN) with delays are discussed. Using the properties of Riemann-Liouville fractional-order calculus, variable substitutions and the property of fuzzy functions are adopted to get the boundedness, the GMLS, and the S-asymptotic ω-periodic of the system. Furthermore, a numerical example is given to demonstrate the theorems.

Global Mittag-Leffler stability of Caputo fractional-order fuzzy inertial neural networks with delay

<abstract><p>This paper deals with the global Mittag-Leffler stability (GMLS) of Caputo fractional-order fuzzy inertial neural networks with time delay (CFOFINND). Based on Lyapunov stability theory and global fractional Halanay inequalities, the existence of unique equilibrium point and GMLS of CFOFINND have been established. A numerical example is given to illustrate the effectiveness of our results.</p></abstract>

Global Mittag-Leffler Stability of the Delayed Fractional-Coupled Reaction-Diffusion System on Networks Without Strong Connectedness.

In this article, we mainly consider the existence of solutions and global Mittag-Leffler stability of delayed fractional-order coupled reaction-diffusion neural networks without strong connectedness. Using the Leary-Schauder's fixed point theorem and the Lyapunov method, some criteria for the existence of solutions and global Mittag-Leffler stability are given. Finally, the correctness of the theory is verified by a numerical example.

Global Mittag–Leffler stability and synchronization of discrete-time fractional-order delayed quaternion-valued neural networks

This paper is devoted to investigating discrete-time fractional-order delayed quaternion-valued neural networks (DFDQNNs) by utilizing direct quaternion approach. Firstly, a novel lemma and its two corresponding corollaries have been proposed for estimating nabla fractional difference of the quaternion-valued Lyapunov function. Then, the existence and uniqueness of equilibrium point for DFDQNNs is proved by constructing a new quaternion-valued contraction mapping. In addition, by means of our designed Lyapunov functions and the effective feedback controller as well as neoteric nabla difference inequalities, some sufficient criteria have been obtained to ensure the global Mittag–Leffler stability and Mittag–Leffler synchronization of DFDQNNs, respectively. Finally, some numerical examples are provided to verify the yielded results.

Exponentially Stable Periodic Oscillation and Mittag-Leffler Stabilization for Fractional-Order Impulsive Control Neural Networks With Piecewise Caputo Derivatives.

It is well known that the conventional fractional-order neural networks (FONNs) cannot generate nonconstant periodic oscillation. For this point, this article discusses a class of impulsive FONNs with piecewise Caputo derivatives (IPFONNs). By using the differential inclusion theory, the existence of the Filippov solutions for a discontinuous IPFONNs is investigated. Furthermore, some decision theorems are established for the existence and uniqueness of the (periodic) solution, global exponential stability, and impulsive control global stabilization to IPFONNs. This article achieves four key issues that were not solved in the previously existing literature: 1) the existence of at least one Filippov solution in a discontinuous IPFONN; 2) the existence and uniqueness of periodic oscillation in a nonautonomous IPFONN; 3) global exponential stability of IPFONNs; and 4) impulsive control global Mittag-Leffler stabilization for FONNs.

Global Mittag-Leffler Stability and Global Asymptotic ω-Period for Fractional-Order Cohen–Grossberg Neural Networks with Time-Varying Delays

The dynamic behaviors for fractional-order Cohen–Grossberg neural networks with time-varying delays (FCGNND) are studied in this paper. By introducing the Mittag-Leffler (ML) function, based on properties of fractional calculus, the differential mean-value theorem and Arzela–Ascoli theorem, we give some sufficient theorems to determine the boundedness, global Mittag-Leffler stability (GMLS) and global asymptotical [Formula: see text]-periodicity (GAP) for FCGNND. Finally, a numerical example is given to verify the effectiveness of the theorems.

Global Mittag-Leffler stability for fractional-order quaternion-valued neural networks with piecewise constant arguments and impulses

This paper is devoted to analysing the global Mittag-Leffler stability of fractional-order quaternion-valued neural networks with piecewise constant arguments and impulses. The quaternion direct method is adopted to address the considered model avoiding any decomposition. By utilising the matrix inequality technique and Lyapunov direct method, some sufficient conditions to guarantee the global Mittag-Leffler stability of equilibrium point for the considered model are obtained in the form of quaternion-valued linear matrix inequalities (LMIs). To certify the validity of the derived result, a numerical example is presented.

Lyapunov Approach for Almost Periodicity in Impulsive Gene Regulatory Networks of Fractional Order with Time-Varying Delays

This paper investigates a class of fractional-order delayed impulsive gene regulatory networks (GRNs). The proposed model is an extension of some existing integer-order GRNs using fractional derivatives of Caputo type. The existence and uniqueness of an almost periodic state of the model are investigated and new criteria are established by the Lyapunov functions approach. The effects of time-varying delays and impulsive perturbations at fixed times on the almost periodicity are considered. In addition, sufficient conditions for the global Mittag–Leffler stability of the almost periodic solutions are proposed. To justify our findings a numerical example is also presented.

Global Mittag-Leffler stability and existence of the solution for fractional-order complex-valued NNs with asynchronous time delays.

This paper is dedicated to exploring the global Mittag-Leffler stability of fractional-order complex-valued (CV) neural networks (NNs) with asynchronous time delays, which generates exponential stability of integer-order (IO) CVNNs. Here, asynchronous time delays mean that there are different time delays in different nodes. Two new inequalities concerning the product of two Mittag-Leffler functions and one novel lemma on a fractional derivative of the product of two functions are given with a rigorous theoretical proof. By utilizing three norms, several novel conditions are concluded to guarantee the global Mittag-Leffler stability and the existence and uniqueness of an equilibrium point. Considering the symbols of the matrix elements, the properties of an M-matrix are extended to the general cases, which introduces the excitatory and inhibitory impacts on neurons. Compared with IOCVNNs, exponential stability is the special case of our results, which means that our model and results are general. At last, two numerical experiments are carried out to explain the theoretical analysis.

H∞ Robust Control for Chaotic Motion of the Fractional-Order D-PMSG via the PDC Approach

The stability analysis and controller design problems for the fractional-order (FO) direct-drive permanent magnet synchronous generator (FOD-PMSG) wind turbine with parameter uncertainties and external disturbance are addressed. Takagi–Sugeno (T-S) model is used to approximate nonlinearities, and the parallel distributed compensation (PDC) technique is employed to construct the fuzzy state feedback controller. In order to suppress the external disturbance more effectively, the global Mittag-Leffler stability definition satisfying the H∞ performance index is proposed for the first time. Using the FO Lyapunov direct method, applying the Cauchy matrix inequality (CMI), and combining with the Schur complement lemma, the sufficient conditions of Mittag-Leffler stability meeting the H∞ performance index are given in the form of linear matrix inequalities (LMIs). Simulation results clearly show that the proposed control scheme can make the system get rid of the chaotic state quickly and have strong robustness under parameter uncertainties and external disturbance varying randomly.

Novel methods to global Mittag-Leffler stability of delayed fractional-order quaternion-valued neural networks

In this paper, a type of fractional-order quaternion-valued neural networks (FOQVNNs) with leakage and time-varying delays is established to simulate real-world situations, and the global Mittag-Leffler stability of the system is investigated by using the non-decomposition method. First, to avoid decomposing the system into two complex-valued systems or four real-valued systems, a new sign function for quaternion numbers is introduced based on the ones for real and complex numbers. And two novel lemmas for quaternion-valued sign function and Caputo fractional derivative are established in quaternion domain, which are used to investigate the stability of FOQVNNs. Second, a concise and flexible quaternion-valued state feedback controller is directly designed and a novel 1-norm Lyapunov function composed of the absolute values of real and imaginary parts is established. Then, based on the designed quaternion-valued state feedback controller and the proposed lemmas, some sufficient conditions are given to ensure the global Mittag-Leffler stability of the system. Finally, a numerical simulation is given to verify the theoretical results.

Controlling Wolbachia Transmission and Invasion Dynamics among Aedes Aegypti Population via Impulsive Control Strategy

This work is devoted to analyzing an impulsive control synthesis to maintain the self-sustainability of Wolbachia among Aedes Aegypti mosquitoes. The present paper provides a fractional order Wolbachia invasive model. Through fixed point theory, this work derives the existence and uniqueness results for the proposed model. Also, we performed a global Mittag-Leffler stability analysis via Linear Matrix Inequality theory and Lyapunov theory. As a result of this controller synthesis, the sustainability of Wolbachia is preserved and non-Wolbachia mosquitoes are eradicated. Finally, a numerical simulation is established for the published data to analyze the nature of the proposed Wolbachia invasive model.

Stability of Fractional Order Fuzzy Cellular Neural Networks with Distributed Delays via Hybrid Feedback Controllers

This article studies the global Mittag-Leffler stability of fractional order fuzzy cellular neural networks via hybrid feedback controllers. Based on hybrid feedback control technique Lyapunov approach, and some novel analysis techniques of fractional calculation, some sufficient conditions are obtained to guarantee the Global Mittag-Lefflers stability. Finally, two simulation example are given to illustrate the effectiveness of the proposed method.

A Razumikhin approach to stability and synchronization criteria for fractional order time delayed gene regulatory networks

<abstract> This manuscript is concerned with the stability and synchronization for fractional-order delayed gene regulatory networks (FODGRNs) via Razumikhin approach. First of all, the existence of FODGRNs are established by using homeomorphism theory, 2-norm based on the algebraic method and Cauchy Schwartz inequality. The uniqueness of this work among the existing stability results are, the global Mittag-Leffler stability of FODGRNs is explored based on the fractional-order Lyapunov Razumikhin approach. In the meanwhile, two different controllers such as linear feedback and adaptive feedback control, are designed respectively. With the assistance of fractional Razumikhin theorem and our designed controllers, we have established the global Mittag-Leffler synchronization and adaptive synchronization for addressing master-slave systems. Finally, three numerical cases are given to justify the applicability of our stability and synchronization results. </abstract>

Some Dynamical Behaviors of Fractional-Order Commutative Quaternion-Valued Neural Networks via Direct Method of Lyapunov

Some dynamical behaviors of fractional-order commutative quaternion-valued neural networks (FCQVNNs) are studied in this paper. First, because the commutative quaternion does not satisfy Schwartz triangle inequality, the FCQVNNs are divided into four real-valued neural networks (RVNNs) through quaternion commutative multiplication rules. Furthermore, several types of dynamical behaviors including global Mittag-Leffler stability, the boundedness with bounded disturbances, complete synchronization and quasi-synchronization of FCQVNNs are studied. Simultaneously, several conditions for these dynamical behaviors are driven by fractional-order Lyapunov direct method, some inequality techniques and fractional differential equation theory. At last, the effectiveness and feasibility of the obtained theoretical results are verified by several numerical simulation examples.