Analysis of Fractional-Order Nonlinear Dynamic Systems with General Analytic Kernels: Lyapunov Stability and Inequalities

In this paper, we study the recently proposed fractional-order operators with general analytic kernels. The kernel of these operators is a locally uniformly convergent power series that can be chosen adequately to obtain a family of fractional operators and, in particular, the main existing fractional derivatives. Based on the conditions for the Laplace transform of these operators, in this paper, some new results are obtained—for example, relationships between Riemann–Liouville and Caputo derivatives and inverse operators. Later, employing a representation for the product of two functions, we determine a form of calculating its fractional derivative; this result is essential due to its connection to the fractional derivative of Lyapunov functions. In addition, some other new results are developed, leading to Lyapunov-like theorems and a Lyapunov direct method that serves to prove asymptotic stability in the sense of the operators with general analytic kernels. The FOB-stability concept is introduced, which generalizes the classical Mittag–Leffler stability for a wide class of systems. Some inequalities are established for operators with general analytic kernels, which generalize others in the literature. Finally, some new stability results via convex Lyapunov functions are presented, whose importance lies in avoiding the calculation of fractional derivatives for the stability analysis of dynamical systems. Some illustrative examples are given.

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Stability of Fractional Differential Equations with New Generalized Hattaf Fractional Derivative

Mathematical Problems in Engineering ◽

10.1155/2021/8608447 ◽

2021 ◽

Vol 2021 ◽

pp. 1-7

Author(s):

Khalid Hattaf

Keyword(s):

Stability Analysis ◽

Differential Equations ◽

Fractional Derivative ◽

Fractional Differential Equations ◽

Fractional Derivatives ◽

Direct Method ◽

Lyapunov Direct Method ◽

Science And Engineering ◽

Fractional Differential ◽

The Stability

This paper aims to study the stability of fractional differential equations involving the new generalized Hattaf fractional derivative which includes the most types of fractional derivatives with nonsingular kernels. The stability analysis is obtained by means of the Lyapunov direct method. First, some fundamental results and lemmas are established in order to achieve the goal of this study. Furthermore, the results related to exponential and Mittag–Leffler stability existing in recent studies are extended and generalized. Finally, illustrative examples are presented to show the applicability of our main results in some areas of science and engineering.

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Quadratic Lyapunov Functions for Stability of the Generalized Proportional Fractional Differential Equations with Applications to Neural Networks

Axioms ◽

10.3390/axioms10040322 ◽

2021 ◽

Vol 10 (4) ◽

pp. 322

Author(s):

Ricardo Almeida ◽

Ravi P. Agarwal ◽

Snezhana Hristova ◽

Donal O’Regan

Keyword(s):

Fractional Derivative ◽

Caputo Fractional Derivative ◽

Lyapunov Functions ◽

Fractional Derivatives ◽

Sufficient Conditions ◽

Hopfield Neural Network ◽

Quadratic Lyapunov Functions ◽

Fractional Differential ◽

The Stability ◽

Theoretical Results

A fractional model of the Hopfield neural network is considered in the case of the application of the generalized proportional Caputo fractional derivative. The stability analysis of this model is used to show the reliability of the processed information. An equilibrium is defined, which is generally not a constant (different than the case of ordinary derivatives and Caputo-type fractional derivatives). We define the exponential stability and the Mittag–Leffler stability of the equilibrium. For this, we extend the second method of Lyapunov in the fractional-order case and establish a useful inequality for the generalized proportional Caputo fractional derivative of the quadratic Lyapunov function. Several sufficient conditions are presented to guarantee these types of stability. Finally, two numerical examples are presented to illustrate the effectiveness of our theoretical results.

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Lyapunov functions for fractional order h-difference systems

Filomat ◽

10.2298/fil2104155l ◽

2021 ◽

Vol 35 (4) ◽

pp. 1155-1178

Author(s):

Xiang Liu ◽

Baoguo Jia ◽

Lynn Erbe ◽

Allan Peterson

Keyword(s):

Fractional Order ◽

Quadratic Forms ◽

Lyapunov Functions ◽

Direct Method ◽

Difference Operators ◽

Lyapunov Direct Method ◽

Polynomial Type ◽

Difference Systems ◽

Quadratic Lyapunov Functions ◽

The Stability

This paper presents some new propositions related to the fractional order h-difference operators, for the case of general quadratic forms and for the polynomial type, which allow proving the stability of fractional order h-difference systems, by means of the discrete fractional Lyapunov direct method, using general quadratic Lyapunov functions, and polynomial Lyapunov functions of any positive integer order, respectively. Some examples are given to illustrate these results.

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On the regional controllability of the sub-diffusion process with Caputo fractional derivative

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2016-0065 ◽

2016 ◽

Vol 19 (5) ◽

Cited By ~ 5

Author(s):

Fudong Ge ◽

YangQuan Chen ◽

Chunhai Kou ◽

Igor Podlubny

Keyword(s):

Fractional Derivative ◽

Diffusion Process ◽

Fractional Order ◽

Fractional Derivatives ◽

Minimum Energy ◽

Energy Control ◽

Diffusion Systems ◽

Minimum Energy Control ◽

The Laplace Transform ◽

Regional Controllability

AbstractThis paper is devoted to the investigation of regional controllability of the fractional order sub-diffusion process. We first derive the equivalent integral equations of the abstract sub-diffusion systems with Caputo and Riemann-Liouville fractional derivatives by utilizing the Laplace transform. The new definitions of regional controllability of the system studied are introduced by extending the existence contributions. Then we analyze the regional controllability of the fractional order sub-diffusion system with minimum energy control in two different cases:

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Fractional-Order Colour Image Processing

Mathematics ◽

10.3390/math9050457 ◽

2021 ◽

Vol 9 (5) ◽

pp. 457

Author(s):

Manuel Henriques ◽

Duarte Valério ◽

Paulo Gordo ◽

Rui Melicio

Keyword(s):

Image Processing ◽

Edge Detection ◽

Fractional Derivative ◽

Fractional Order ◽

Fractional Derivatives ◽

Grey Scale ◽

Colour Image ◽

Colour Image Processing ◽

Image Processing Algorithms ◽

Processing Algorithms

Many image processing algorithms make use of derivatives. In such cases, fractional derivatives allow an extra degree of freedom, which can be used to obtain better results in applications such as edge detection. Published literature concentrates on grey-scale images; in this paper, algorithms of six fractional detectors for colour images are implemented, and their performance is illustrated. The algorithms are: Canny, Sobel, Roberts, Laplacian of Gaussian, CRONE, and fractional derivative.

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Vibration Equation of Fractional Order Describing Viscoelasticity and Viscous Inertia

Open Physics ◽

10.1515/phys-2019-0088 ◽

2019 ◽

Vol 17 (1) ◽

pp. 850-856 ◽

Cited By ~ 2

Author(s):

Jun-Sheng Duan ◽

Yun-Yun Xu

Keyword(s):

Fractional Derivative ◽

Fractional Order ◽

Fractional Derivatives ◽

Harmonic Excitation ◽

Steady State Response ◽

Vibration System ◽

Derivative Operator ◽

Vibration Equation ◽

State Response ◽

Frequency Relation

Abstract The steady state response of a fractional order vibration system subject to harmonic excitation was studied by using the fractional derivative operator ${}_{-\infty} D_t^\beta,$where the order β is a real number satisfying 0 ≤ β ≤ 2. We derived that the fractional derivative contributes to the viscoelasticity if 0 < β < 1, while it contributes to the viscous inertia if 1 < β < 2. Thus the fractional derivative can represent the “spring-pot” element and also the “inerterpot” element proposed in the present article. The viscosity contribution coefficient, elasticity contribution coefficient, inertia contribution coefficient, amplitude-frequency relation, phase-frequency relation, and influence of the order are discussed in detail. The results show that fractional derivatives are applicable for characterizing the viscoelasticity and viscous inertia of materials.

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Fractional Lotka-Volterra-Type Cooperation Models: Impulsive Control on Their Stability Behavior

Entropy ◽

10.3390/e22090970 ◽

2020 ◽

Vol 22 (9) ◽

pp. 970 ◽

Cited By ~ 1

Author(s):

Rohisha Tuladhar ◽

Fidel Santamaria ◽

Ivanka Stamova

Keyword(s):

Fractional Order ◽

Fractional Derivatives ◽

Impulsive Control ◽

Control Strategies ◽

Volterra Type ◽

Stability Behavior ◽

Integer Order ◽

Order Case ◽

Cooperation Model ◽

Stability Results

We present a biological fractional n-species delayed cooperation model of Lotka-Volterra type. The considered fractional derivatives are in the Caputo sense. Impulsive control strategies are applied for several stability properties of the states, namely Mittag-Leffler stability, practical stability and stability with respect to sets. The proposed results extend the existing stability results for integer-order n−species delayed Lotka-Volterra cooperation models to the fractional-order case under impulsive control.

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Correction: FRACTIONAL ORDER BARBALAT’S LEMMA AND ITS APPLICATIONS IN THE STABILITY OF FRACTIONAL ORDER NONLINEAR SYSTEMS

Mathematical Modelling and Analysis ◽

10.3846/13926292.2017.1329755 ◽

2017 ◽

Vol 22 (4) ◽

pp. 503-513 ◽

Cited By ~ 5

Author(s):

Fei Wang ◽

Yongqing Yang

Keyword(s):

Nonlinear Systems ◽

Fractional Derivative ◽

Fractional Order ◽

Caputo Fractional Derivative ◽

Barbalat’S Lemma ◽

Liouville Derivative ◽

Liouville Fractional Derivative ◽

The Stability ◽

The Relationship ◽

Theoretical Results

This paper investigates fractional order Barbalat’s lemma and its applications for the stability of fractional order nonlinear systems with Caputo fractional derivative at first. Then, based on the relationship between Caputo fractional derivative and Riemann-Liouville fractional derivative, fractional order Barbalat’s lemma with Riemann-Liouville derivative is derived. Furthermore, according to these results, a set of new formulations of Lyapunov-like lemmas for fractional order nonlinear systems are established. Finally, an example is presented to verify the theoretical results in this paper.

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STABILITY ANALYSIS OF NONLINEAR INTERCONNECTED SYSTEMS VIA T–S FUZZY MODELS

International Journal of Computational Intelligence and Applications ◽

10.1142/s1469026804001033 ◽

2004 ◽

Vol 04 (01) ◽

pp. 41-55 ◽

Cited By ~ 7

Author(s):

WEI-LING CHIANG ◽

CHENG-WU CHEN ◽

FENG-HSIAG HSIAO

Keyword(s):

Stability Analysis ◽

Lyapunov Functions ◽

Direct Method ◽

Interconnected Systems ◽

Linear Matrix ◽

Matrix Inequality ◽

Fuzzy Models ◽

Fuzzy Techniques ◽

The Stability ◽

Takagi Sugeno

This paper is concerned with the stability problem of nonlinear interconnected systems. Each of them consists of a few interconnected subsystems which are approximated by Takagi–Sugeno (T–S) type fuzzy models. In terms of Lyapunov's direct method, a stability criterion is derived to guarantee the asymptotic stability of interconnected systems. It is shown that the stability analysis problems of nonlinear interconnected systems can be reduced to linear matrix inequality (LMI) problems via suitable Lyapunov functions and T–S fuzzy techniques. Finally, numerical examples with simulations are given to demonstrate the validity of the proposed approach.

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The Dirihlet problem for the fractional Poisson’s equation with Caputo derivatives: A finite difference approximation and a numerical solution

Thermal Science ◽

10.2298/tsci110421076b ◽

2012 ◽

Vol 16 (2) ◽

pp. 385-394 ◽

Cited By ~ 3

Author(s):

V.D. Beibalaev ◽

R.P. Meilanov

Keyword(s):

Monte Carlo ◽

Finite Difference ◽

Fractional Derivative ◽

Monte Carlo Methods ◽

Caputo Fractional Derivative ◽

Fractional Derivatives ◽

Finite Difference Approximation ◽

Difference Approximation ◽

Difference Problem ◽

The Stability

A finite difference approximation for the Caputo fractional derivative of the 4-?, 1 < ? ? 2 order has been developed. A difference schemes for solving the Dirihlet?s problem of the Poisson?s equation with fractional derivatives has been applied and solved. Both the stability of difference problem in its right-side part and the convergence have been proved. A numerical example was developed by applying both the Liebman and the Monte-Carlo methods.

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