Mittag-Leffler Stability Research Articles

Mittag-Leffler Stability and Synchronization of Multi-delayed Fractional Neural Networks via Halanay Inequality

Mittag-Leffler Stability and Synchronization of Multi-delayed Fractional Neural Networks via Halanay Inequality

Designing a switching law for Mittag-Leffler stability in nonlinear singular fractional-order systems and its applications in synchronization

In this work, the stability and synchronization issue of switched singular continuous-time fractional-order systems with nonlinear perturbation are examined. Using the fixed-point principle and S-procedure lemma, a sufficient condition for the existence and uniqueness of the solution to the switched singular fractional-order system is first stated. Next, using the Lyapunov functional method in combination with some techniques related to singular systems and fractional calculus, a switching rule for regularity, impulse-free, and Mittag-Leffler stability is developed based on the formation of a partition of the stability state regions in convex cones. For synchronizing switched fractional singular dynamical systems, we propose a state feedback controller that ensures regularity, impulse-free, and Mittag-Leffler stable in the error closed-loop system. Finally, the ease of use and computational convenience of our proposed methods are illustrated by two numerical examples and a practical example about DC motor controlling an inverted pendulum accompanied by simulation results.

Mittag-Leffler stability and stabilization of some classes of time-varying fractional systems

UDC 517.9 We consider some classes of time-varying fractional systems and study the problem of stabilization for these systems with norm-bounded controls. We use time-varying Lyapunov functions to analyze the Mittag-Leffler stability of these systems. A numerical example is given to illustrate the efficiency of the obtained result.

Ulam–Hyers–Rassias Mittag-Leffler stability of ϖ–fractional partial differential equations

This paper offers a comprehensive analysis of solution representations for ϖ-fractional partial differential equations, specifically focusing on the linear case of the Darboux problem. We exhibit a representation of the solutions for the Darboux problem of ϖ-fractional partial differential equations in the linear case in the space of continuous functions. Through the application of the generalized Gronwall inequality, we establish the Ulam–Hyers–Rassias Mittag–Leffler stability in the space of continuous functions. Three numerical examples are presented to show the effectiveness and the applicability of our results.

Robustness and dynamical features of fractional difference spacecraft model with Mittag–Leffler stability

Abstract Spacecraft models that mimic the Planck satellite’s behaviour have produced information on cosmic microwave background radiation, assisting physicists in their understanding of the composition and expansion of the universe. For achieving the intended formation, a framework for a discrete fractional difference spacecraft model is constructed by the use of a discrete nabla operator of variable order containing the Mittag–Leffler kernel. The efficacy of the suggested framework is evaluated employing a numerical simulation of the concerning dynamic systems of motion while taking into account multiple considerations such as exterior disruptions, parameterized variations, time-varying feedback delays, and actuator defects. The implementation of the Banach fixed-point approach provides sufficient requirements for the presence of the solution as well as a distinctive feature for such mechanisms Furthermore, the consistent stability is examined. With the aid of discrete nabla operators, we monitor the qualitative behavioural patterns of spacecraft systems to provide justification for structure’s chaos. We acquire the fixed points of the proposed trajectory. At each fixed point, we calculate the eigenvalue of the spacecraft system’s Jacobian matrix and check for zones of instability. The outcomes exhibit a wide range of multifaceted behaviours resulting from the interaction with various fractional orders in the offered system. To maintain stability and synchronize the system, nonlinear controllers are additionally provided. The study highlights the technique’s vulnerability to fractional-order factors, resulting in exclusive, changing trends and equilibrium frameworks. Because of its diverse and convoluted behaviour, the spacecraft chaotic model is an intriguing and crucial subject for research.

Mittag-Leffler stability and application of delayed fractional-order competitive neural networks

In the article, the Mittag-Leffler stability and application of delayed fractional-order competitive neural networks (FOCNNs) are developed. By virtue of the operator pair, the conditions of the coexistence of equilibrium points (EPs) are discussed and analyzed for delayed FOCNNs, in which the derived conditions of coexistence improve the existing results. In particular, these conditions are simplified in FOCNNs with stepped activations. Furthermore, the Mittag-Leffler stability of delayed FOCNNs is established by using the principle of comparison, which enriches the methodologies of fractional-order neural networks. The results on the obtained stability can be used to design the horizontal line detection of images, which improves the practicability of image detection results. Two simulations are displayed to validate the superiority of the obtained results.

Stability of a fractional heat equation with memory

Of concern is a fractional differential problem of order between zero and one. The model generalizes an existing well-known problem in heat conduction theory with memory. First, we justify the replacement of the first order derivative by a fractional one. Then, we establish a Mittag-Leffler stability result for a class of heat flux relaxation functions. We will combine the energy method with some properties from fractional calculus.

Mittag-Leffler stability and Lyapunov stability for a problem arising in porous media

Mittag-Leffler stability and Lyapunov stability for a problem arising in porous media

New Results on the Ulam–Hyers–Mittag–Leffler Stability of Caputo Fractional-Order Delay Differential Equations

The author considers a nonlinear Caputo fractional-order delay differential equation (CFrDDE) with multiple variable delays. First, we study the existence and uniqueness of the solutions of the CFrDDE with multiple variable delays. Second, we obtain two new results on the Ulam–Hyers–Mittag–Leffler (UHML) stability of the same equation in a closed interval using the Picard operator, Chebyshev norm, Bielecki norm and the Banach contraction principle. Finally, we present three examples to show the applications of our results. Although there is an extensive literature on the Lyapunov, Ulam and Mittag–Leffler stability of fractional differential equations (FrDEs) with and without delays, to the best of our knowledge, there are very few works on the UHML stability of FrDEs containing a delay. Thereby, considering a CFrDDE containing multiple variable delays and obtaining new results on the existence and uniqueness of the solutions and UHML stability of this kind of CFrDDE are the important aims of this work.

Impulsive Control of Variable Fractional-Order Multi-Agent Systems

The main goal of the paper is to present and study models of multi-agent systems for which the dynamics of the agents are described by a Caputo fractional derivative of variable order and a kernel that depends on an increasing function. Also, the order of the fractional derivative changes at update times. We study a case for which the exchanged information between agents occurs only at initially given update times. Two types of linear variable-order Caputo fractional models are studied. We consider both multi-agent systems without a leader and multi-agent systems with a leader. In the case of multi-agent systems without a leader, two types of models are studied. The main difference between the models is the fractional derivative describing the dynamics of agents. In the first one, a Caputo fractional derivative with respect to another function and with a continuous variable order is applied. In the second one, the applied fractional derivative changes its constant order at each update time. Mittag–Leffler stability via impulsive control is defined, and sufficient conditions are obtained. In the case of the presence of a leader in the multi-agent system, the dynamic of the agents is described by a Caputo fractional derivative with respect to an increasing function and with a constant order that changes at each update time. The leader-following consensus via impulsive control is defined, and sufficient conditions are derived. The theoretical results are illustrated with examples. We show with an example the leader’s influence on the consensus.

Impulsive Control Discrete Fractional Neural Networks in Product Form Design: Practical Mittag-Leffler Stability Criteria

The planning, regulation and effectiveness of the product design process depend on various characteristics. Recently, bio-inspired collective intelligence approaches have been applied in this process in order to create more appealing product forms and optimize the design process. In fact, the use of neural network models in product form design analysis is a complex process, in which the type of network has to be determined, as well as the structure of the network layers and the neurons in them; the connection coefficients, inputs and outputs have to be explored; and the data have to be collected. In this paper, an impulsive discrete fractional neural network modeling approach is introduced for product design analysis. The proposed model extends and complements several existing integer-order neural network models to the generalized impulsive discrete fractional-order setting, which is a more flexible mechanism to study product form design. Since control and stability methods are fundamental in the construction and practical significance of a neural network model, appropriate impulsive controllers are designed, and practical Mittag-Leffler stability criteria are proposed. The Lyapunov function strategy is applied in providing the stability criteria and their efficiency is demonstrated via examples and a discussion. The established examples also illustrate the role of impulsive controllers in stabilizing the behavior of the neuronal states. The proposed modeling approach and the stability results are applicable to numerous industrial design tasks in which multi-agent systems are implemented.

Coexistence and locally exponential stability of multiple equilibrium points for fractional-order impulsive control Cohen–Grossberg neural networks

Different from the existing multiple Mittag-Leffler stability or multiple asymptotic stability, the multiple exponential stability, which has explicit and faster convergence rate, is investigated in this paper for fractional-order impulsive control Cohen–Grossberg neural networks. First, by using the definition of Dirac delta function, the fractional-order control Cohen–Grossberg neural networks are translated into fractional-order impulsive neural networks, in which pulse effect relies on the fractional order of the addressed system. Then, based on maximum norm, 1−norm and general q−norm (q=2n), a series of novel criteria are obtained respectively to ensure that such n−neuron neural networks can have ∏i=1n(2Li+1) total equilibrium points and ∏i=1n(Li+1) locally exponentially stable equilibrium points, by utilizing the known fixed point theorem, the method of average impulsive interval, the theory of fractional-order differential equations, and the method of Lyapunov function. This paper’s investigation reveals the effects of impulsive function, impulsive interval, and fractional order on the dynamic behaviors. Finally, theoretical results are shown to be effective by four illustrative examples.

Exponential stability of fractional‐order uncertain systems with asynchronous switching and impulses

AbstractDifferent from the Mittag–Leffler stability or asymptotic stability, the exponential stability issue, which provides faster and explicit convergence rate, is studied in this paper for fractional‐order uncertain systems with asynchronous switching and impulses, where the impulsive functions rely on not only switching modes but also impulsive time. Instead of using the inequality , by utilizing the theory of fractional‐order differential equations, the methods of Lyapunov function and mathematical induction, some novel and less‐conservative stability criteria are developed, respectively, for the case of switched stable subsystems or switched unstable subsystems. The obtained results build a tradeoff between impulsive function, impulsive interval, average dwell time and fractional order. In addition, our results with are also novel in contrast with the ones of integer‐order switched impulsive systems. Finally, five numerical examples are given to show the effectiveness of theoretical results.

Commutator of the Caputo fractional derivative and the shift operator and applications

In this paper, we study the commutator of the Caputo fractional derivative and the shift operator. Consequently, we sharpen the recent (Diethelm and Tuan, 2022) on the separation of solutions to fractional equations. As a byproduct, we also obtain the sub and super multiplicative laws of Mittag-Leffler functions, which play an essential role in our approach to fractional Halanay type inequalities. Furthermore, by utilizing the estimate of the commutator, we provide an upper bound with an exact decaying rate for the separation of solutions to delay fractional equations under a flexible condition. This upper bound can be seen as a generalization of previous fractional Halanay type inequalities with bounded delay, which improves both (Wang and Zou, 2019) and (Huong et al., 2023). Finally, we gain practical sufficient criteria for the (strict) Mittag-Leffler stability and finite-time attractivity for a class of delay fractional equations. Some concrete examples are examined to illustrate these results.

Mittag-Leffler stability analysis of a class of homogeneous fractional systems

Mittag-Leffler stability analysis of a class of homogeneous fractional systems

Practical Mittag-Leffler stability of quasi-one-sided Lipschitz fractional order systems

Practical Mittag-Leffler stability of quasi-one-sided Lipschitz fractional order systems

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.

Global Stability of Fractional Nonlinear Differential Systems With State-Dependent Delayed Impulses.

The fractional-order (FO) nonlinear differential system with state-dependent (SD) delayed impulses (DI) is considered in this brief. The considered impulses are related to the delayed state of the system and the delays are SD. A novel lemma for the monotonicity of the solution of Caputo's FO derivative equation is given. By means of linear matrix inequality (LMI) and several comparative arguments, criteria of uniform stability, uniform asymptotical stability, and Mittag-Leffler stability are obtained. Compared with other works on integer-order (IO) impulsive delayed systems with SD delays or fixed delays, how to impose constraints on parameters and impulses is explored, without imposing the boundedness on the state delays. Two examples are implemented to examine the practicality and sharpness of our theoretical analysis.

Multiple Mittag-Leffler Stability of Almost Periodic Solutions for Fractional-Order Delayed Neural Networks: Distributed Optimization Approach.

This article proposes new theoretical results on the multiple Mittag-Leffler stability of almost periodic solutions (APOs) for fractional-order delayed neural networks (FDNNs) with nonlinear and nonmonotonic activation functions. Profited from the superior geometrical construction of activation function, the considered FDNNs have multiple APOs with local Mittag-Leffler stability under given algebraic inequality conditions. To solve the algebraic inequality conditions, especially in high-dimensional cases, a distributed optimization (DOP) model and a corresponding neurodynamic solving approach are employed. The conclusions in this article generalize the multiple stability of integer-or fractional-order NNs. Besides, the consideration of the DOP approach can ameliorate the excessive consumption of computational resources when utilizing the LMI toolbox to deal with high-dimensional complex NNs. Finally, a simulation example is presented to confirm the accuracy of the theoretical conclusions obtained, and an experimental example of associative memories is shown.

Boundary Disturbance Rejection Control for Fractional-order Multi-agent Systems with Reaction-diffusion

This paper focuses on boundary disturbance rejection control (BDRC) for fractional-order multi-agent systems (FMASs) with reaction diffusion terms. Firstly, based on the output information, the corresponding observer is designed to obtain information about all the agents and the unknown external perturbations. Then, a novel distributed boundary control protocol is designed based on the observed values. Thus, the convergence analysis of the closed-loop system is accomplished through the Mittag-Leffler stability theory and the linear matrix inequality condition. Finally, a simulation example is used to verify the correctness of the results.