Stability analysis of nonlinear Hadamard fractional differential system☆
Introduction
In recent years, fractional calculus is a topic of growing interest based on the superiority of integrals and derivatives of complex order and the ability to model certain physical systems in a more adequate and precise fashion than integer order alternative. There are many applications in different fields such as electrical circuit, cosmology, control theory, biomedical engineering, economics, etc. In terms of applied mathematics to study many problems from several diverse disciplines of engineering and technical sciences, the fractional calculus is a powerful tool. For details, we refer the reader to the works in [1], [2], [3], [4], [5], [6], [7]. While the most common ones are the Riemann–Liouville and Caputo fractional operators, recently, there has been an increasing interest in the development of Hadamard fractional operators. Details and properties of the Hadamard fractional derivative and integral can be found in book [4] and papers [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20].
Recently, fractional calculus in the control theory is widely seen. Fractional-order controller is playing a very vital role in almost every field of control subject. Stability is one of the important characteristics of the control problem. It is also an essential condition for any control problem. The initial work about stability of fractional order systems can be dated back to Matignon [21]. It has achieved great strides [22], [23], [24], [25], [26], [27], [28], [29]. For its latest developments, readers of interest could refer to [30], [31], [32], [33], [34], [35], [36], [37], [38]. So far, there are several approaches to the study of the stability of fractional differential systems, one of which is the fractional comparison principle approach. The main difficulty is to establish a fractional comparison principle. To overcome this difficulty, we developed several fractional differential inequalities, which play a crucial role in this paper.
In this paper, the stability of the zero solution of nonlinear Caputo-type Hadamard fractional system is investigated. We establish a Hadamard type fractional differential inequality. Comparison principle using this new fractional differential inequality and scalar Hadamard fractional differential system is presented and sufficient conditions for the (generalized) stability and the (generalized) Mittag-Leffler stability are obtained.
Section snippets
Preliminaries
First of all, we summarize some important definitions and related lemmas. Definition 2.1 The Hadamard fractional integral of order α for a function g is defined asprovided the integral exists. Definition 2.2 The Hadamard fractional derivative of fractional order α for a function is defined aswhere [α] denotes the integer part of the real number α and . Definition 2.3 The Caputo-type Hadamard fractional derivative of[4]
[4]
[4]
Stability of Caputo-type Hadamard fractional system
Consider the stability of the following Caputo-type Hadamard fractional differential systemwith the initial condition where be a domain containing the origin.
First, the general Caputo-type Hadamard fractional comparison principle will be presented. Here, we always assumes in the paper that there exists a unique continuously differentiable solution x(t) to Eq. (3.1) with the initial condition x0.
Comparison results will be used
Example
Example 4.1 For the Caputo-type Hadamard fractional order systemwhere γ ∈ (0, 1) and f(t, x) satisfies Lipschitz condition, and f(t, x) ≤ 0. Let the Lyapunov candidate be . ThenThe solution of the Caputo-type Hadamard fractional differential equationis given by . Thus, the zero solution of Eq. (4.2) is Mittag-Leffler stable. By Theorem 3.2, the zero
References (38)
- et al.
Compositions of hadamard-type fractional integration operators and the semigroup property
J. Math. Anal. Appl.
(2002) - et al.
Qualitative analysis for solutions of a certain more generalized two-dimensional fractional differential system with hadamard derivative
Appl. Math. Comput.
(2015) - et al.
A coupled system of hadamard type sequential fractional differential equations with coupled strip conditions
Chaos Solit. Fract.
(2016) - et al.
Explicit iteration to hadamard fractional integro-differential equations on infinite domain
Adv. Differ. Equ.
(2016) - et al.
Nonlocal hadamard fractional boundary value problem with hadamard integral and discrete boundary conditions on a half-line
J. Comput. Appl. Math.
(2018) - et al.
Successive iterations and positive extremal solutions for a hadamard type fractional integro-differential equations on infinite domain
Appl. Math. Comput.
(2017) - et al.
Finite-time stability analysis of fractional order time-delay systems: Gronwalls approach
Math. Comput. Model.
(2009) - et al.
Mittag-leffler stability of fractional order nonlinear dynamic systems
Automatica
(2009) - et al.
Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability
Comput. Math. Appl.
(2010) - et al.
Lmi stability conditions for fractional order systems
Comput. Math. Appl.
(2010)
Mittag-Leffler stability of nonlinear fractional neutral singular systems
Commun. Nonlinear. Sci. Numer. Simulat.
Generalized fractional calculus and applications
Pitman Research Notes in Mathematics No. 301 Longman
Fractional Differential Equations
Fractional Calculus in Bioengineering
Theory and applications of fractional differential equations
North-Holland Mathematics Studies, 204
Theory of Fractional Dynamic Systems
Fractional calculus models and numerical methods
Essai sur I’etude des fonctions donnees par leur developpment de taylor
J. Mat. Pure. Appl. Ser.
Cited by (50)
Asymptotic stability and fold bifurcation analysis in Caputo–Hadamard type fractional differential system
2024, Chinese Journal of PhysicsQualitative Behaviour of a Caputo Fractional Differential System
2023, Qualitative Theory of Dynamical SystemsExistence of solutions to nonlinear Katugampola fractional differential equations with mixed fractional boundary conditions
2023, Mathematical Methods in the Applied SciencesOn coupled nonlinear evolution system of fractional order with a proportional delay
2023, Mathematical Methods in the Applied SciencesMittag-Leffler stability analysis for time-fractional hyperbolic systems with space-dependent reactivity using backstepping-based boundary control
2023, International Journal of Modelling, Identification and Control