Global Mittag-Leffler Stability Research Articles

Global Mittag–Leffler Stabilization of Fractional-Order BAM Neural Networks with Linear State Feedback Controllers

In this paper, the global Mittag–Leffler stabilization of fractional-order BAM neural networks is investigated. First, a new lemma is proposed by using basic inequality to broaden the selection of Lyapunov function. Second, linear state feedback control strategies are designed to induce the stability of fractional-order BAM neural networks. Third, based on constructed Lyapunov function, generalized Gronwall-like inequality, and control strategies, several sufficient conditions for the global Mittag–Leffler stabilization of fractional-order BAM neural networks are established. Finally, a numerical simulation is given to demonstrate the effectiveness of our theoretical results.

Stability of discrete‐time fractional‐order time‐delayed neural networks in complex field

Dynamics of discrete‐time neural networks have not been well documented yet in fractional‐order cases, which is the first time documented in this manuscript. This manuscript is mainly considered on the stability criterion of discrete‐time fractional‐order complex‐valued neural networks with time delays. When the fractional‐order β holds 1 < β < 2, sufficient criteria based on a discrete version of generalized Gronwall inequality and rising function property are established for ensuring the finite stability of addressing fractional‐order discrete‐time‐delayed complex‐valued neural networks (FODCVNNs). In the meanwhile, when the fractional‐order β holds 0 < β < 1, a global Mittag–Leffler stability criterion of a class of FODCVNNs is demonstrated with two classes of neuron activation function by means of two different new inequalities, fractional‐order discrete‐time Lyapunov method, discrete version Laplace transforms as well as a discrete version of Mittag–Leffler function. Finally, computer simulations of two numerical examples are illustrated to the correctness and effectiveness of the presented stability results.

New Results on Stability for a Class of Fractional-Order Static Neural Networks

This paper investigates the stability of a class of fractional-order static neural networks. Two new Lyapunov functions with proper integral terms are constructed. These integrals with variable upper limit are convex functions. Based on the fractional-order Lyapunov direct method and some inequality skills, several novel stability sufficient conditions which ensure the global Mittag–Leffler stability of fractional-order projection neural networks (FPNNs) are presented in the forms of linear matrix inequalities (LMIs). Two LMI-based Mittag–Leffler stability criteria with less conservativeness are given for a special kind of FPNNs. Finally, the effectiveness of the proposed method is demonstrated via four numerical examples.

Global Finite-time Stability for Fractional-order Neural Networks

This paper is concerned with the global Mittag-Leffler stability (GMLS) and global finite-time stability (GFTS) for fractional Hopfield neural networks (FHNNs) with Holder neuron activation functions subject to nonlinear growth. Firstly, four functions possessing convexity are proposed, which can guarantee that four formulas with respect to the fractional derivative are established. Correspondingly, a novel principle of convergence in finite-time for FHNNs is developed based on the proposed formulas. In addition, by applying the Brouwer topological degree theory and inequality analysis techniques, the proof of the existence and uniqueness of equilibrium point is addressed. Subsequently, by means of the Lur’e-type Postnikov Lyapunov functional approach, and the presented principle of convergence in finite-time, the GMLS and GFTS conditions are achieved in terms of linear matrix inequalities (LMIs). Moreover, the upper bound of the setting time for the GFTS is calculated accurately. Finally, three numerical examples are given to verify the validity of the theoretical results.

Global Mittag–Leffler Stability and Stabilization Analysis of Fractional-Order Quaternion-Valued Memristive Neural Networks

This paper studies the global Mittag–Leffler stability and stabilization analysis of fractional-order quaternion-valued memristive neural networks (FOQVMNNs). The state feedback stabilizing control law is designed in order to stabilize the considered problem. Based on the non-commutativity of quaternion multiplication, the original fractional-order quaternion-valued systems is divided into four fractional-order real-valued systems. By using the method of Lyapunov fractional-order derivative, fractional-order differential inclusions, set-valued maps, several global Mittag–Leffler stability and stabilization conditions of considered FOQVMNNs are established. Two numerical examples are provided to illustrate the usefulness of our analytical results.

Global Mittag-Leffler stability for fractional-order coupled systems on network without strong connectedness

This study investigates the global Mittag-Leffler stability (MLS) problem of the equilibrium point for a new fractional-order coupled system (FOCS) on a network without strong connectedness. In particular, an integer-order coupled system is extended into the FOCS on a complex network without strong connectedness. Based on the theory of asymptotically autonomous systems and graph theory, sufficient conditions are derived to ensure the existence, uniqueness, and global MLS of the solutions of this FOCS on a network. Finally, a numerical example is provided to demonstrate the validity and potential of the proposed method for studying the MLS of FOCSs.

Robust state estimation for fractional-order delayed BAM neural networks via LMI approach

This paper is concerned with the robust state estimation problem for a class of delayed fractional-order bidirectional associative memory (FOBAM) neural networks (NNs) with norm-bounded uncertainties. The objective of the study is to construct an efficient estimator in such a way that the behavior of the corresponding error state is stable in the Mittag-Leffler sense. Distinct to the previous studies, the state estimation problem of FOBAM NNs is investigated through fractional-order Lyapunov direct method. The sufficient conditions that ensure the global Mittag-Leffler stability of the error system are derived as a set of solvable linear matrix inequalities (LMIs) without uncertainties. The proposed stability conditions are further extended to the FOBAM NNs by considering the effect of uncertainties. In order to validate the effectiveness of the proposed theoretical results, two numerical examples have been illustrated.

Mittag-Leffler stability analysis of multiple equilibrium points in impulsive fractional-order quaternion-valued neural networks

In this study, we investigate the problem of multiple Mittag-Leffler stability analysis for fractional-order quaternion-valued neural networks (QVNNs) with impulses. Using the geometrical properties of activation functions and the Lipschitz condition, the existence of the equilibrium points is analyzed. In addition, the global Mittag-Leffler stability of multiple equilibrium points for the impulsive fractional-order QVNNs is investigated by employing the Lyapunov direct method. Finally, simulation is performed to illustrate the effectiveness and validity of the main results obtained.

Global Mittag-Leffler stability and synchronization of discrete-time fractional-order complex-valued neural networks with time delay

Without decomposing complex-valued systems into real-valued systems, this paper investigates existence, uniqueness, global Mittag-Leffler stability and global Mittag-Leffler synchronization of discrete-time fractional-order complex-valued neural networks (FCVNNs) with time delay. Inspired by Lyapunov’s direct method on continuous-time systems, a class of discrete-time FCVNNs is further discussed by employing the fractional-order extension of Lyapunov’s direct method. Firstly, by means of contraction mapping theory and Cauchy’s inequality, a sufficient condition is presented to ascertain the existence and uniqueness of the equilibrium point for discrete-time FCVNNs. Then, based on the theory of discrete fractional calculus, discrete Laplace transform, the theory of complex functions and discrete Mittag-Leffler functions, a sufficient condition is established for global Mittag-Leffler stability of the proposed networks. Additionally, by applying the Lyapunov’s direct method and designing a effective control scheme, the sufficient criterion is derived to ensure the global Mittag-Leffler synchronization of discrete-time FCVNNs. Finally, two numerical examples are also presented to manifest the feasibility and validity of the obtained results.

Global Mittag-Leffler stability analysis of impulsive fractional-order complex-valued BAM neural networks with time varying delays

This paper investigates the stability of impulsive fractional-order complex-valued BAM neural networks with time varying delays. As the extension of fractional-order real-valued BAM neural networks, fractional-order complex-valued BAM neural networks have complex-valued states, synaptic weights, and the activation functions. Two different kinds of activation functions are considered, along with popular bounded and Lipschitz-kind activation functions. By using Lyapunov function and Homomorphic mapping theorem, sufficient conditions for the existence of unique equilibrium and global asymptotic stability of complex-valued systems are derived. In derivation we separated nonlinear complex-valued activation functions into real and imaginary parts. Moreover, Mittag-Leffler stability for BAM neural networks(BAMNNs) have been proposed when the nonlinear complex activation functions are bounded. Simulation results are presented to prove the efficiency of the obtained methods.

Global Mittag-Leffler Stability of Fractional Hopfield Neural Networks with δ-Inverse Hölder Neuron Activations

In this paper, the global Mittag-Leffler stability of fractional Hopfield neural networks (FHNNs) with $$\delta $$-inverse holder neuron activation functions are considered. By applying the Brouwer topological degree theory and inequality analysis techniques, the proof of the existence and uniqueness of equilibrium point is addressed. By constructing the Lure’s Postnikov-type Lyapunov functions, the global Mittag-Leffler stability conditions are achieved in terms of linear matrix inequalities (LMIs). Finally, three numerical examples are given to verify the validity of the theoretical results.

LMI conditions for some dynamical behaviors of fractional-order quaternion-valued neural networks

This paper addresses the issue of three dynamical behaviors including global Mittag-Leffler stability, robust stability and projection synchronization for fractional-order quaternion-valued neural networks (FQVNNs). Some linear matrix inequality conditions for these dynamical behaviors of FQVNNs are given by Lyapunov stability theory, quaternion matrix theory, Homeomorphic mapping theory and fractional differential equation theory. Furthermore, these obtained sufficient conditions for stability and synchronization are superior to those in existing literature. Finally, three examples are given to illustrate the effectiveness of the theoretical results.

Global Stabilization of Fractional-Order Memristor-Based Neural Networks With Time Delay.

This paper addresses the global stabilization of fractional-order memristor-based neural networks (FMNNs) with time delay. The voltage threshold type memristor model is considered, and the FMNNs are represented by fractional-order differential equations with discontinuous right-hand sides. Then, the problem is addressed based on fractional-order differential inclusions and set-valued maps, together with the aid of Lyapunov functions and the comparison principle. Two types of control laws (delayed state feedback control and coupling state feedback control) are designed. Accordingly, two types of stabilization criteria [algebraic form and linear matrix inequality (LMI) form] are established. There are two groups of adjustable parameters included in the delayed state feedback control, which can be selected flexibly to achieve the desired global asymptotic stabilization or global Mittag-Leffler stabilization. Since the existing LMI-based stability analysis techniques for fractional-order systems are not applicable to delayed fractional-order nonlinear systems, a fractional-order differential inequality is established to overcome this difficulty. Based on the coupling state feedback control, some LMI stabilization criteria are developed for the first time with the help of the newly established fractional-order differential inequality. The obtained LMI results provide new insights into the research of delayed fractional-order nonlinear systems. Finally, three numerical examples are presented to illustrate the effectiveness of the proposed theoretical results.

Stability analysis of memristor-based time-delay fractional-order neural networks

Abstract In this paper, a class of memristor-based time-delay fractional-order neural networks has been investigated. Through constructing Lyapunov functional and applying the Gronwall’s integral inequality, new criteria to guarantee the global Mittag–Leffler stability have been proposed for the addressed neural networks. At the same time, by employing contraction mapping principle, sufficient conditions for ensuring the existence and uniqueness of the equilibrium point are also presented. Numerical examples are given to illustrate the effectivity and feasibility of our results.

Global Mittag–Leffler stabilization of fractional-order complex-valued memristive neural networks

This paper presents the theoretical results about global Mittag–Leffler stabilization for a class of fractional-order complex-valued memristive neural networks with the designed two types of control rules. As the extension of fractional-order real-valued memristive neural networks, fractional-order complex-valued memristive neural networks have complex-valued states, synaptic weights, and the activation functions. By utilizing the set-valued maps, a generalized fractional derivative inequality as well as fractional-order differential inclusions, several stabilization criteria for global Mittag–Leffler stabilization of fractional-order complex-valued memristive neural networks are established. A numerical example is provided here to illustrate our theoretical results.

Global Mittag-Leffler stability and synchronization analysis of fractional-order quaternion-valued neural networks with linear threshold neurons

This paper talks about the stability and synchronization problems of fractional-order quaternion-valued neural networks (FQVNNs) with linear threshold neurons. On account of the non-commutativity of quaternion multiplication resulting from Hamilton rules, the FQVNN models are separated into four real-valued neural network (RVNN) models. Consequently, the dynamic analysis of FQVNNs can be realized by investigating the real-valued ones. Based on the method of M-matrix, the existence and uniqueness of the equilibrium point of the FQVNNs are obtained without detailed proof. Afterwards, several sufficient criteria ensuring the global Mittag-Leffler stability for the unique equilibrium point of the FQVNNs are derived by applying the Lyapunov direct method, the theory of fractional differential equation, the theory of matrix eigenvalue, and some inequality techniques. In the meanwhile, global Mittag-Leffler synchronization for the drive–response models of the addressed FQVNNs are investigated explicitly. Finally, simulation examples are designed to verify the feasibility and availability of the theoretical results.

Lur’e Postnikov Lyapunov functional technique to global Mittag-Leffler stability of fractional-order neural networks with piecewise constant argument

Abstract In this paper, the global Mittag-Leffler stability issue of fractional-order neural networks (FNNs) with piecewise constant argument is investigated. Firstly, a new inequality with respect to the fractional derivative of integer-order variable upper limit integral is proposed, which not only is favorable to the construction of Lyapunov function but also enriches the fractional order calculus theory. Secondly, based on topological degree theory, the existence and uniqueness of equilibrium point is certified. In addition, under some suitable assumptions, by applying Picard successive approximation technique, the proof of the existence and uniqueness of solution with the initial value is given. Moreover, by introducing integral term into Lyapunov functional, Lur’e Postnikov type Lyapunov functional is constructed, and a sufficient condition is addressed in terms of linear matrix inequalities (LMIs) to guarantee that the considered FNNs are Mittag-Leffler stable. Finally, an example is given to demonstrate the validity of the obtained results.

Stability analysis of fractional-order neural networks: An LMI approach

Abstract This paper analyzes the stability of fractional-order neural networks (FNNs) without and with delay by the fractional Lyapunov direct method and the fractional Razumikhin-type theorem, respectively. S-procedure is applied to handle the nonlinear constraints to obtain a wider parameter selection of the systems. For FNNs without delay, the improved less conservative conditions of the existence and uniqueness of the equilibrium point and the global Mittag-Leffler stability are all derived in the form of linear matrix inequalities (LMIs). Moreover, an LMI-based uniform stability condition of FNNs with time delay is established, which simplifies and extends some previous work. Finally, the validity of the presented results is indicated by some numerical examples.

Adaptive type‐2 fuzzy system for synchronisation and stabilisation of chaotic non‐linear fractional order systems

This study proposes an adaptive interval type-2 Takagi–Sugeno–Kang (IT2 TSK) fuzzy system with a supervisory mode to control and stabilise a certain class of non-linear fractional order systems. In this study, a fractional order adaptation law is derived which adjusts the free parameters and bounds them by utilising a projection algorithm. The global Mittag–Leffler stability of the closed-loop system is proved in the sense that all the involved signals are uniformly bounded. Moreover, if the non-linear system tends to be unstable, a supervisory controller starts cooperating with the adaptive IT2 TSK fuzzy controller to guarantee the stability of the closed-loop system. In addition, a new inference mechanism for the adaptive IT2 TSK fuzzy system is introduced for which the antecedent part is chosen as a type-2 fuzzy set and the consequent parameters are represented as interval sets. According to the practical nature of the proposed inference equation, it would be applicable in online and real-time applications. Numerical simulations show the validity and effectiveness of the introduced control strategy for stabilisation and control of a general class of non-linear fractional order systems perturbed by disturbance and uncertainty.

Global Mittag-Leffler Synchronization of Fractional-Order Neural Networks Via Impulsive Control

This paper aims at analyzing the impulsive synchronization of fractional-order neural works. Firstly, in view of control theory, by constructing a suitable impulsive response system with the designed controller, the synchronization error system between the drive system and the corresponding response system is given. Afterwards, based on the theory of impulsive differential equation, the theory of fractional differential equation, Lyapunov direct method, and inequality techniques, some effective sufficient criteria are established to guarantee the global Mittag-Leffler stability for the synchronization error system. Finally, several simulation examples are designed to demonstrate the effectiveness and feasibility of the obtained results.