Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability

Stability of fractional-order nonlinear dynamic systems is studied using Lyapunov direct method with the introductions of Mittag–Leffler stability and generalized Mittag–Leffler stability notions. With the definitions of Mittag–Leffler stability and generalized Mittag–Leffler stability proposed, the decaying speed of the Lyapunov function can be more generally characterized which include the exponential stability and power-law stability as special cases. Finally, four worked out examples are provided to illustrate the concepts.     

Global Synchronization in the Finite Time for Variable-Order Fractional Neural Networks with Discontinuous Activations

In this paper, we focus the global synchronization in the finite time for variable-order fractional neural networks with discontinuous activation functions. Global Mittag–Leffler synchronization and synchronization in the finite time. Firstly, the order α(t) of the fractional derivative of Caputo is changed with time, the α(t) is designed and improved, which plays an important role in the synchronization analysis. Secondly, the fractional Lyapunov method and the Mittag–Leffler function are applied, the linear matrix inequalities (LMI) are used to guarantee the conditions for satisfying the finite time synchronization. With this method finite-time synchronization and time estimation can be achieved simultaneously. Finally, the effectiveness of the method is verified by two examples.     

An efficient authentication with key agreement procedure using Mittag–Leffler–Chebyshev summation chaotic map under the multi-server architecture

The recent technological advancement and rapid development of computer networks have increased the popularity of remote password authentication protocols. Toward this end, the emphasis has shifted to protocols that apply to smart cards-empowered multi-server environments. In order to defend against the replay attack, these protocols usually depend on the nonce or timestamp. In this paper, an efficient Mittag–Leffler–Chebyshev Summation Chaotic Map (MLCSCM)-enabled multi-server authentication protocol with the key agreement is proposed and generalized to address this peculiarity in multi-server-oriented applications. The security proof and efficiency analysis of the presented MLCSCM authenticated key agreement protocol is rigorously derived and validated. Compared to the recently published literature, the proposed protocol presents high efficiency with unique features, and it is highly resistant to sophisticated attacks and achieves perfect forward secrecy.

     

Mittag‐Leffler stabilization for an unstable time‐fractional anomalous diffusion equation with boundary control matched disturbance

This paper addresses the Mittag‐Leffler stabilization for an unstable time‐fractional anomalous diffusion equation with boundary control subject to the control matched disturbance. The active disturbance rejection control (ADRC) approach is adopted for developing the control law. A state‐feedback scheme is designed to estimate the disturbance by constructing two auxiliary systems: One is to separate the disturbance from the original system to a Mittag‐Leffler stable system and the other is to estimate the disturbance finally. The proposed control law compensates the disturbance using its estimation and stabilizes system asymptotically. The closed‐loop system is shown to be Mittag‐Leffler stable and the constructed auxiliary systems in the closed loop are proved to be bounded. This is the first time for ADRC to be applied to a system described by the fractional partial differential system without using the high gain.     

Global Practical Mittag Leffler Stabilization by Output Feedback for a Class Of Nonlinear Fractional‐Order Systems

In this paper, the problem of a global practical Mittag Leffler feedback stabilization for a class of nonlinear fractional order systems by means of observer is described. The linear matrix inequality approach is used to guarantee the practical stability of the proposed feedback fractional order system. An illustrative example is given to show the applicability of the results.     

Regional output feedback stabilization of semilinear time‐fractional diffusion systems in a parallelepipedon with control constraints

This article is concerned with the regional output feedback stabilization problems for semilinear time‐fractional diffusion systems in a 1≤n?dimensional parallelepipedon with control inequality constraints. For this, the spectrum decomposition method is used to derive a finite‐dimensional fractional ordinary differential equation (ODE) system that captures the dominant dynamics of the considered system. With this ODE system, we propose a finite‐dimensional fractional compensator to guarantee that the constrained closed‐loop semilinear systems are Mittag‐Leffler stable in some subregions of their evolution domains. An example is finally included to illustrate our results.     

Pseudo-State Estimation for Fractional Order Neural Networks

Neural Processing Letters - This paper studies the pseudo-state estimation of fractional order neural networks based on the event-triggered mechanism. First, a novel Mittag–Leffler type event...     

Boundary Mittag–Leffler stabilization of a class of time fractional-order nonlinear reaction–diffusion systems

Boundary control design of a class of time fractional-order nonlinear reaction–diffusion systems (FNRDSs) is considered in three cases: domain-averaged measurement, collocated boundary point measurement, and anti-collocated boundary point measurement. For domain-averaged measurement and collocated boundary point measurement, boundary controllers are designed directly based on the measurements, respectively. For the anti-collocated boundary point measurement, to overcome the difficulty that the measurement cannot be used for the controller design directly, an observer is constructed firstly, and then, an observer-based boundary controller is designed. Sufficient conditions for Mittag–Leffler (M-L) stability of the closed-loop system are all provided in terms of linear matrix inequalities (LMIs). Numerical simulation results are provided to illustrate the feasibility and effectiveness of the proposed methods.     

Impulsively fractional-order time-delayed gene regulatory networks: Existence and asymptotic stability

This paper presents asymptotic stability criteria for fractional-order gene regulatory networks (FOGRNs) with impulses, time delays, and two numerical cases to illustrate the applicability of the results. The established system's boundedness, existence, and uniqueness are discussed using the Mittag–Leffler function, homeomorphism theory, and Cauchy–Schwartz inequality. The delay-independent asymptotic stability criteria for FOGRNs are derived using algebraic and LMI methods, famous inequality techniques, and Lyapunov stability theory.     

Fractional neural observer design for a class of nonlinear fractional chaotic systems

In this paper, a novel observer structure for nonlinear fractional-order systems is presented to estimate the states of fractional-order nonlinear chaotic system with unknown dynamical model. A new fractional error back-propagation learning algorithm is derived to adapt weights of the artificial neural network, by taking advantage of the Lyapunov stability strategy of fractional-order systems which is called Miattag–Leffler stability. The main contribution is the extension of neural observer for fractional dynamics in a way that satisfies Miattag–Leffler conditions. Observer design procedure guarantees the convergence of observer error to the neighborhood of zero. Furthermore, the robustness of the proposed estimator against uncertainties and external disturbances are the main benefits of the proposed method. Simulation results show the effectiveness and capabilities of the proposed observer.

     

Lyapunov stability analysis for nonlinear nabla tempered fractional order systems

Due to the restriction of practical systems in time or space, tempered fractional calculus becomes more reasonable than the traditional fractional calculus. It is known that stability analysis is a crucial issue for control systems. This paper concerns the stability analysis issue of nabla tempered fractional order systems for the first time. The (discrete time) tempered Mittag–Leffler stability is defined firstly and then a stability criterion is derived via Lyapunov method. Besides, boundedness and attractiveness are also investigated.     

Control and Synchronization Of A Class Of Uncertain Fractional Order Chaotic Systems Via Adaptive Backstepping Control

This paper presents the stabilization and synchronization problem of a class of fractional order chaotic systems with unknown parameters. A systematic step by step approach is explained to derive control results using an adaptive backstepping strategy. The analytically obtained control structure, derived by blending a systematic backstepping procedure with Mittag‐Leffler stability results, helps in obtaining the stability of a strict feedback‐like class of uncertain fractional order chaotic systems. The results are further extended to achieve synchronization of these systems in master–slave configuration. Thereafter, the methodology has been applied to two example systems, that is, chaotic Chua's circuit and Genesio‐Tesi system, which belong to addressed class, in order to show the application of results. Numerical simulation given at the end confirms the efficacy of the scheme presented here.     

Local radial basis function–finite-difference method to simulate some models in the nonlinear wave phenomena: regularized long-wave and extended Fisher–Kolmogorov equations

In this investigation, we concentrate on solving the regularized long-wave (RLW) and extended Fisher–Kolmogorov (EFK) equations in one-, two-, and three-dimensional cases by a local meshless method called radial basis function (RBF)–finite-difference (FD) method. This method at each stencil approximates differential operators such as finite-difference method. In each stencil, it is necessary to solve a small-sized linear system with conditionally positive definite coefficient matrix. This method is relatively efficient and has low computational cost. How to choose the shape parameter is a fundamental subject in this method, since it has a palpable effect on coefficient matrix. We will employ the optimal shape parameter which results from algorithm of Sarra (Appl Math Comput 218:9853–9865, 2012). Then, we compare the approximate solutions acquired by RBF–FD method with theoretical solution and also with results obtained from other methods. The numerical results show that the RBF–FD method is suitable and robust for solving the RLW and EFK equations.

     

Finite‐time stability of fractional order impulsive switched systems

This paper considers the finite‐time stability of fractional order impulsive switched systems. First, by using the fractional order Lyapunov function, Mittag–Leffler function, and Gronwall–Bellman lemma, two sufficient conditions are given to verify the finite‐time stability of fractional order nonlinear systems. Then, the concept of finite‐time stability is extended to fractional order impulsive switched systems. A sufficient condition is given to verify the finite‐time stability of fractional order impulsive switched systems by combining the method of average dwell time with fractional order Lyapunov function. Finally, two numerical examples are provided to illustrate the theoretical results. Copyright © 2014 John Wiley & Sons, Ltd.     

Laguerre Functions and Their Applications to Tempered Fractional Differential Equations on Infinite Intervals

Tempered fractional diffusion equations (TFDEs) involving tempered fractional derivatives on the whole space were first introduced in Sabzikar et al. (J Comput Phys 293:14–28, 2015), but only the finite-difference approximation to a truncated problem on a finite interval was proposed therein. In this paper, we rigorously show the well-posedness of the models in Sabzikar et al. (2015), and tackle them directly in infinite domains by using generalized Laguerre functions (GLFs) as basis functions. We define a family of GLFs and derive some useful formulas of tempered fractional integrals/derivatives. Moreover, we establish the related GLF-approximation results. In addition, we provide ample numerical evidences to demonstrate the efficiency and “tempered” effect of the underlying solutions of TFDEs.     

Mittag–Leffler Stability and Global Asymptotically $$\omega $$-Periodicity of Fractional-Order BAM Neural Networks with Time-Varying Delays

This article studies the Mittag–Leffler stability and global asymptotical \(\omega \)-periodicity for a class of fractional-order bidirectional associative memory (BAM) neural networks with time-varying delays by using Laplace transform, stability theory of fractional systems and some integration technique. Firstly, some sufficient conditions are given to ensure the boundedness and global Mittaag-Leffler stability of fractional-order BAM neural networks with time-varying delays. Next, S-asymptotical \(\omega \)-periodicity and global asymptotical \(\omega \)-periodicity of fractional-order BAM neural networks with time-varying delays are also explored. Finally, some numerical examples and simulation are performed to show the effectiveness of theoretical results.     

Jacobi–Gauss–Lobatto collocation approach for non-singular variable-order time fractional generalized Kuramoto–Sivashinsky equation

This paper introduces the non-singular variable-order (VO) time fractional version of the generalized Kuramoto–Sivashinsky (GKS) equation with the aid of fractional differentiation in the Caputo–Fabrizio sense. The Jacobi–Gauss–Lobatto collocation technique is developed for solving this equation. More precisely, the derivative matrix of the classical Jacobi polynomials and the VO fractional derivative matrix of the shifted Jacobi polynomials (which is obtained in this study) together with the collocation technique are used to transform the solution of problem into the solution of an algebraic system of equations. Numerical simulations for several test problems have been shown to accredit the established algorithm.

     

Fractional fault-tolerant dynamical controller for a class of commensurate-order fractional systems

A fault-tolerant control scheme is proposed for a class of commensurate-order fractional nonlinear systems that consists of two fractional-order observers (hybrid scheme). The diagnosis of the faults is performed by means of a model-free fractional proportional integral reduced-order observer that uses the fractional algebraic observability property. A fractional dynamical controller obtained in a natural way from the dynamics of a fractional high-gain observer is designed, which is constructed from a fractional generalised observability canonical form; the controller performs output tracking, thus eliminating the effects of the faults. A stability analysis on the overall system demonstrates that the origin is Mittag–Leffler stable. The proposed methodology is assessed by means of simulations on the fractional models of the Van der Pol oscillator and a DC motor.     

Numerical solution of nonlinear delay differential equations of fractional variable-order using a novel shifted Jacobi operational matrix

This paper presents the generalized nonlinear delay differential equations of fractional variable-order. In this article, a novel shifted Jacobi operational matrix technique is introduced for solving a class of multi-terms variable-order fractional delay differential equations via reducing the main problem to an algebraic system of equations that can be solved numerically. The suggested technique is successfully developed for the aforementioned problem. Comprehensive numerical experiments are presented to demonstrate the efficiency, generality, accuracy of proposed scheme and the flexibility of this method. The numerical results compared it with other existing methods such as fractional Adams method (FAM), new predictor–corrector method (NPCM), a new approach, Adams–Bashforth–Moulton algorithm and L1 predictor–corrector method (L1-PCM). Comparing the results of these methods as well as comparing the current method (NSJOM) with the exact solution, indicating the efficiency and validity of this method. Note that the procedure is easy to implement and this technique will be considered as a generalization of many numerical schemes. Furthermore, the error and its bound are estimated.

     

Convergence of method of lines approximations to partial differential equations

Many existing numerical schemes for evolutionary problems in partial differential equations (PDEs) can be viewed as method of lines (MOL) schemes. This paper treats the convergence of one-step MOL schemes. Our main purpose is to set up a general framework for a convergence analysis applicable to nonlinear problems. The stability materials for this framework are taken from the field of nonlinear stiff ODEs. In this connection, important concepts are the logarithmic matrix norm and C-stability. A nonlinear parabolic equation and the cubic Schrödinger equation are used for illustrating the ideas.