Stability analysis of nonlinear Hadamard fractional differential system

doi:10.1016/j.jfranklin.2018.12.033

Journal of the Franklin Institute

Volume 356, Issue 12, August 2019, Pages 6538-6546

https://doi.org/10.1016/j.jfranklin.2018.12.033 Get rights and content

Abstract

The stability of the zero solution of a class of nonlinear Hadamard type fractional differential system is investigated by utilizing a new fractional comparison principle. The novelty of this paper is based on some new fractional differential inequalities along the given nonlinear Hadamard fractional differential system. A comparison principle employing the new fractional differential inequality for scalar Hadamard fractional differential system is presented. Based on the new comparison principle, some sufficient conditions for the (generalized) stability and the (generalized) Mittag-Leffler stability are given.

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

[4]

The Hadamard fractional integral of order α for a function g is defined as $^{H} I^{α} g (t) = \frac{1}{Γ (α)} \int_{1}^{t} {(\log \frac{t}{s})}^{α - 1} \frac{g (s)}{s} d s, α > 0,$ provided the integral exists.

Definition 2.2

[4]

The Hadamard fractional derivative of fractional order α for a function $g : [1, \infty) \to R$ is defined as $^{H} D^{α} g (t) = \frac{1}{Γ (n - α)} {(t \frac{d}{d t})}^{n} \int_{1}^{t} {(\log \frac{t}{s})}^{n - α - 1} \frac{g (s)}{s} d s, n - 1 < α < n, n = [α] + 1,$ where [α] denotes the integer part of the real number α and $\log (\cdot) = \log_{e} (\cdot)$ .

Definition 2.3

[4]

The Caputo-type Hadamard fractional derivative of

Stability of Caputo-type Hadamard fractional system

Consider the stability of the following Caputo-type Hadamard fractional differential system $_{C}^{H} D_{t_{0}}^{γ} x (t) = f (t, x),$ with the initial condition $x (t_{0}) = x_{0},$ where $t_{0} \geq 1, 0 < γ < 1, f \in C ([1, + \infty) \times D, R^{n}), f (t, 0) \equiv 0, D \in R^{n}$ 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 system $_{C}^{H} D_{t_{0}}^{γ} | x (t) | = - | x (t) | + f (t, x),$ where γ ∈ (0, 1) and f(t, x) satisfies Lipschitz condition, $f (t, 0) = 0$ and f(t, x) ≤ 0. Let the Lyapunov candidate be $V (t, x) = | x |$ . Then $_{C}^{H} D_{t_{0}}^{γ} V (t, x (t)) = - V (t, x (t)) + f (t, x) \leq - V (t, x (t)) .$ The solution of the Caputo-type Hadamard fractional differential equation $_{C}^{H} D_{t_{0}}^{γ} u (t) = - u, u (t_{0}) = V (t_{0}, x (t_{0})) = | x (t_{0}) |$ is given by $u (t) = u (t_{0}) E_{γ} (- {(\log \frac{t}{t_{0}})}^{γ})$ . Thus, the zero solution $u = 0$ of Eq. (4.2) is Mittag-Leffler stable. By Theorem 3.2, the zero

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^☆: Supported by National Natural Science Foundation of China (No.11501342) and the Natural Science Foundation for Young Scientists of Shanxi Province, China(No.201701D221007). All authors equally contributed this manuscript and approved the final version.

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Stability analysis of nonlinear Hadamard fractional differential system☆