# Asymptotical Stability Of Riemann–Liouville Fractional Nonlinear Systems

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### On asymptotic properties of solutions to fractional di

the notion of Mittag-Le er stability which is suitable for systems of fractional-order. Next, we use this notion to describe the asymptotical behavior of solutions to FDEs by two approaches: Lyapunov s rst method and Lyapunov s second method. Finally, we give a discus-sion on the relation between Lipschitz condition, stability and speed

### Adaptive Fuzzy Control for Uncertain Fractional-Order

on fractional Lyapunov stability theorem, an adaptive fuzzy controller is designed, and the asymptotical stability can be guaranteed. Finally, simulation studies are given to indicate

### Dynamical analysis of the Irving Mullineux oscillator

Fractional systems get increasing attention in science and engineering problems, see, for example [5,15]. Remember that stability is a crucial property of systems, see for example , for the stability of traditional systems of integer order and [14,20], for fractional system s. One of the traditional systems of integer order is the

### A New Fractional Order Observer Design for Fractional Order

world physical systems are better characterized by fractional order differential equations. For the fractional order systems, there are many theories and criterions regarding the controllability, observability and stability, including the linear and nonlinear systems [9-15]. Besides, the design of different controllers for fractional order

### A SEPARATION PRINCIPLE FOR THE STABILISATION OF A CLASS OF

Keywords and phrases: Riemann Liouville fractional system, nonlinear time delay system, observer design, asymptotical stability, Lyapunov functional. 1.Introduction

### Robust Output Feedback Control for Fractional Order Nonlinear

a class of commensurate fractional order nonlinear systems with uncertain constant parameters. However, for fractional order nonlinear systems with time-varying delays, there is none related work. Motivated by the mentioned situation, we devote to solve the stabilization problem of fractional order nonlinear systems with time-varying delays.

### Global Asymptotical Stability Analysis for Fractional Neural

Abstract: In this paper, the global asymptotical stability of Riemann-Liouville fractional-order neural networks with time-varying delays is studied. By combining the Lyapunov functional function and LMI approach, some sufﬁcient criteria that guarantee the global asymptotical stability of such

### abstract - Pennsylvania State University

approaches to the generalization of the notion of diﬀerentiation to fractional or-ders e.g. Riemann-Liouville, Grnwald-Letnikov, Caputo and generalized functions approach . Riemann-Liouville fractional derivative is mostly used by mathe-maticians but this approach is not suitable for real world physical problems since

### Research Article Stability and Bifurcation of Two Kinds of

Two kinds of three-dimensional fractional Lotka-Volterra systems are discussed. For one system, the asymptotic stability of the equilibria is analyzed by providing some su cient conditions.

### Stability Analysis of Conformable Fractional Systems

. of Iranian Journal of Numerical Analysis and Optimization Vol. 7, No. 1, (2017), pp 13-32 DOI: 10.22067/ijnao.v7i1.46917 Stability Analysis of Conformable

### Adaptive Impulsive Synchronization for a Class of Delay

method of delay fractional-order chaotic systems is proposed in Section 3, based on the theory of Lyapunov stability and impulsive differential equations. Finally, conclusions are addressed in Section 4. 2. Preliminaries of Fractional Derivative The theory of the fractional order calculus is the arbitrary

### STABILITY AND F-ASYMPTOTIC STABILITY ANALYSIS OF NONLINEAR

STABILITY AND F-ASYMPTOTIC STABILITY ANALYSIS OF NONLINEAR FRACTIONAL ORDER SYSTEMS WITH RIEMANN-LIOUVILLE DERIVATIVE JAVAD ALIDOUSTI1, ALI BAYATI ESHKAFTAKI2, AND REZA KHOSHSIAR GHAZIANI3 1,2,3Department of Applied Mathematics Shahrekord University P.O. Box 115, Shahrekord, IRAN

### Projective Lag Synchronization of Fractional Order Chaotic

Abstract Projective lag synchronization of fractional order chaotic systems with unknown parameters is investigated. It is shown that the slave system can be synchronized with the past states of the driver up to a scaling matrix. According to the stability theorem of linear fractional order systems, a nonlinear controller

### Generalized dichotomous linear part for a class of

mathematics, but also in dynamical systems, control systems, physical sciences and engineering to construct the mathematical modeling. Fractional di erential equations concerning the Caputo derivative or the Riemann-Liouville fractional operators have been organized in di erent classes of fractional di erential equations. In general,

### Robust Synchronization of Fractional-Order Uncertain Chaotic

Jul 05, 2019 time when the systems were affected by uncertainties and external noises in . He also discussed the ﬁnite time control problem for a class of uncertain fractional-order nonlinear systems with model uncertainties and external disturbances via the fractional Lyapunov stability theory in .

### Asymptotical Stability of Nonlinear Fractional Differential

In this paper, we further study the stability of nonlinear fractional diﬀerential systems with Caputo derivative by utilizing a Lyapunov-like function. Taking into account the re-lation between asymptotical stability and generalized Mittag-Leﬄer stability, we are able to weaken the conditions assumed for the Lyapunov-like function.

### New results on stability and stabilization of a class of

Nonlinear Dyn (2014) 75:633 641 DOI 10.1007/s11071-013-1091-5 ORIGINAL PAPER New results on stability and stabilization of a class of nonlinear fractional-order systems

### Research Article Stability Analysis of Fractional-Order

cient conditions ensuring asymptotical stability of fractional-order nonlinear system with delay are proposed rstly. And the application of Riemann-Liouville fractional-order systems is extended by the fractional comparison principle and the Caputo fractional-order systems. Numerical simulations of an example demonstrate the universality and the

### arXiv:1608.00760v2 [math.DS] 8 Nov 2016

fractional-order dynamical systems, and hence, the analysis of fractional-order dynamical systems is a very important ﬁeld of research. For example, it has been shown (Kaslik and Sivasundaram, 2012a) that the fractional-order derivative (of Caputo, Grunwald-Letnikov or Riemann-Liouville type)

### Stability with respect to part of the variables of nonlinear

In , Yan Li et al. presented the Mittag-Leﬄer stability of fractional order non-linear dynamic systems. Furthermore, stability analysis of Hilfer fractional diﬀer-ential systems is shown in . On the other hand, in , the authors described the asymptotical stability of nonlinear fractional diﬀerential system with Caputo derivative.

### Stability of Linear Multiple Different Order Caputo

FDEs with Riemann-Liouville derivative and the same fractional order α, where 0 1< <α After that, many researchers have been investigated stability of linear and nonlinear FDEs with 0 1< <α -. W. H. Deng et al. [19, 20] studied the stability and asymptotical stability for linear (linear time delay) fractional system with

### Stability analysis of linear fractional differential system

410 Nonlinear Dyn (2007) 48:409 416 Recently, Chen and Moore  studied stability of 1-dimensional fractional systems with retard time. Similar to  and , in this paper, we introduce multiple time delays to the fractional differential equa-tions. Then we study the (asymptotical) stability of such systems.

### Asymptotical stability analysis of conformable fractional systems

formable time-fractional schrödinger model , con-formablefractionalBiswas Milovicmodel.Thesta-bility of the differential system is also attracted for researchers, that is because the stable system is very important in our life. Recently, stability problems of nonlinear fractional systems have been extensively

### Research Article Sliding Mode Control of the Fractional-Order

promising topic. On one hand, more and more fractional nonlinear systems which exhibit chaos have been discov-ered, and their chaotic behaviors have been studied with numerical simulations, such as the fractional-order Chua circuit [ ], the fractional-order Van der Pol oscillator [ ], the fractional-order Lorenz system [ , ], the fractional-

### DESIGN OF UNKNOWN INPUT FRACTIONAL ORDER OBSERVERS FOR

Fig. 2. Stability region of linear fractional-order systems with order 1 <α<2. The two drawings in Figs. 1 and 2 illustrate the stability regions of linear fractional-order systems with a fractional order belonging to 0 <α<1 and 1 ≤ α<2, respectively. Note that the conditions presented in Lemma 2 are equivalent to those given by Sabatier

### IEEE/CAA JOURNAL OF AUTOMATICA SINICA 1 Stability analysis of

The stability analysis of nonlinear fractional differential systems is much more difﬁcult and only a few available. For example, Li et al. investigated the Mittag-Lefﬂer stability of fractional order nonlinear dynamic systems  and proposed Lyapunov direct method to check stability of fractional order nonlinear dynamic systems .

### Introduction f - umexpert.um.edu.my

the Lyapunov s methods; Deng  has derived suﬃcient conditions for the local asymptotical stability of nonlinear fractional diﬀerential equations, and Li et all  has studied the stability of fractional-order nonlinear dynamic systems using the Lyapunov direct method and generalized Mittag-Leﬄer stability.

### Uncorrected Proof

linear and nonlinear fractional differential equations. In , stability analysis of fractional-order nonlinear systems with delay is studied. The authors proposed the deﬁnition of Mittag-Lefﬂer stability of the time-delay system and in-troduced the fractional Lyapunov direct method by using the properties of Mittag-Lefﬂer

### Fractional-Order Adaptive Fault Estimation for a Class of

nonlinear systems in both integral-order and fractional-order cases. Using indirect Lyapunov method and LMI techniques, Boroujeni et al.  and Lan et al.  investigated the observer design for fractional-order nonlinear systems. The advantage of an adaptive fault diagnosis observer is that the state vector estimation and actuator fault

### On the robustness of linear and non-linear fractional-order

cient and necessary condition for asymptotical stability of fractional-order systems with fractional-order α belonging to 0 <α<1 is presented. In Section4, uncertain non-linear fractional-order systems are presented. In Sections 5 and 6, the important issues related to linear and non-linear fractional-order systems are studied, respectively.

### Asymptotical stability of RiemannￃﾢￂﾀￂﾓLiouville fractional

investigate fractional nonlinear systems and singular systems by extending the inequality to Riemann Liouville derivatives, several criteria on delay-independent stability are obtained. In , Li et al. study asymptotical stability of fractional neutral systems, the presented criteria are expressed in terms of matrix * Corresponding author.

### VWHPV - Institute of Physics

The stability of impulsive incommensurate fractional order chaotic systems with Caputo derivative Runzi Luo and Haipeng Su-A new contribution for the impulsive synchronization of fractional-order discrete-time chaotic systems Ouerdia Megherbi et al-Impulsive synchronization of fractional Takagi-Sugeno fuzzy complex networks Weiyuan Ma et al-

### Fault Tolerant Control Using Fractional-order Terminal

asymptotical stability in the presence of actuator failures . The Sliding Mode Control (SMC) which belongs to the class of a nonlinear controller has the capability to deal efficiently nonlinear systems with uncertainties, bounded external load disturbances, and small sensitivity to the time varying parameters. For finite-time stability,

### www.researchgate.net

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### Research Article Stability of a Class of Fractional-Order

as linear systems. erefore, stability of nonlinear system is of great signi cance, and it also has important value in application. In [ ], the stability of fractional nonlinear time-delay systems for Caputo s derivative are investigated, and two theorems for Mittag-Le er stability of the fractional order nonlinear time-delay systems are proved.

### arXiv:1901.05588v1 [math.OC] 17 Jan 2019

fractional order adaptive backstepping controller design, the bound of disturbance is estimated, and saturation is compensated by the virtual signal of an auxiliary system as well. In spite of the existence of nonsmooth nonlinearities, the output is guaranteed to track the reference signal asymptotically on the basis of our proposed method.

### Fractional-order sliding mode controller for the two-link

Hence, recently, the fractional-order sliding mode controller has been investigated and applied into many system controls such as: single-link flexible manipulator , antilock braking systems [20, 21], speed control system for permanent magnet synchronous motor . As a result, the systems with the fractional-order sliding