Journal of Mathematics

Volume 2018, Article ID 1723481, 10 pages

https://doi.org/10.1155/2018/1723481

## On Stability of Nonautonomous Perturbed Semilinear Fractional Differential Systems of Order

Mathematics Department, Al-Azhar University-Gaza, State of Palestine

Correspondence should be addressed to Mohammed M. Matar; moc.liamtoh@rattam_demmahom

Received 13 May 2018; Accepted 25 July 2018; Published 5 August 2018

Academic Editor: Morteza Khodabin

Copyright © 2018 Mohammed M. Matar and Esmail S. Abu Skhail. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We study the Mittag-Leffler and class-K function stability of fractional differential equations with order . We also investigate the comparison between two systems with Caputo and Riemann-Liouville derivatives. Two examples related to fractional-order Hopfield neural networks with constant external inputs and a marine protected area model are introduced to illustrate the applicability of stability results.

#### 1. Introduction

Recently, the topic of fractional differential equations has attracted many researchers which plays a main role in various applications. Many areas such as physics, biophysics, aerodynamics, control theory, viscoelasticity, capacitor theory, electrical circuit, description of memory, and hereditary properties (see [1–4]) used the fractional models instead of classical models. Recently, stability of fractional differential systems has attracted many authors for more investigation in this topic [5]. In 1996, Matignon [6] firstly studied the stability of linear fractional differential systems with the Caputo derivative. Since then, many researchers have done further studies on the stability of linear fractional differential systems (see [7, 8]). In nonlinear systems, Lyapunov’s direct method provides a way to analyze the stability as Mittag-Leffler stability of a system without explicitly solving the differential equations (see [9–11]). The method generalizes the idea which shows that the system is stable if there are some Lyapunov function candidates for the system. Lyapunov’s direct method is a sufficient condition to show the stability of the systems, which means that the systems may still be stable even if one cannot find Lyapunov’s function candidate to conclude the system stability property. For the nonlinear fractional differential systems, the stability analysis is much more difficult and only a few are available.

Some authors (see [12–14]) studied the following nonlinear fractional differential system:with initial values , where . Then, some literatures on the stability of linear fractional differential systems with have appeared (see [15, 16]). However, not all the fractional differential systems have fractional orders in . There exist fractional models which have fractional orders lying in , for example, super-diffusion. In [17], the authors study the stability of* n*-dimensional linear fractional differential systemsof order and initial conditionsThe corresponding perturbed system iswhich is investigated by authors in [17].

For fractional-order Hopfield neural networks in applications to parallel computation and signal processing, it is required that there be a well-defined computable solution for all possible initial states. From a mathematical viewpoint, the stability analysis of a unique equilibrium point is very necessary and valuable. Recently, several important and interesting results for the stability analysis of fractional-order Hopfield neural networks have been gained [4].

The marine protected area model [18] can describe the ecological linkage between the reserve and fishing ground by the autonomous linear systemHere, is the natural mortality (death) rate, is the dispersal rate, and is the harvesting rate. The corresponding fractional behavior of such system may exist in theoretical view as the fractional order .

Motivated by these articles, we investigate the stability of the nonautonomous perturbed fractional semilinear system of the formwhere , , , , and is a given continuous function such that . The fractional derivative is either Caputo or Riemann-Liouville.

#### 2. Preliminaries and Notations

We introduce in this section several definitions and results that will be used later in the sequel.

*Definition 1 ([19]). *The Riemann-Liouville fractional integral of a function is defined by

*Definition 2 ([19]). *The Riemann-Liouville fractional derivative of order , is defined bywhere is the ordinary differential operator.

Theorem 3 ([19]). *If and , then the equalityholds everywhere on , where .*

*Definition 4 ([20]). *The Caputo derivative (left-sided) of a function , , is defined as

Lemma 5 ([21]). *Let ; then*

*Definition 6. *The Laplace transform of a real function is defined as The Laplace transform of the Caputo fractional derivative is given byThe Laplace transform of the Riemann-Liouville fractional derivative is given by [20]The Laplace transform of is given byThe convolution of the functions and is defined asThe inverse Laplace transform of the product of the functions and is given by

Lemma 7 ([20]). *The Laplace transform of the Mittag-Leffler function is given byWe conclude thatThe Mittag-Leffler function [21] is very important in solving fractional differential equations and in many applications. The exponential function as special case of Mittag-Leffler which is frequently used in the solution of integer order differential equations. The Mittag-Leffler function of one parameter is defined asThe Mittag-Leffler function of two parameters is defined bywhich both satisfy the fact that , and *

Lemma 8 ([22]). *The system is asymptotically stable if where , and are the eigenvalues of matrix . The stable and unstable regions for and are shown in Figures 1 and 2, respectively.*