**1. Introduction**

The Fractional Order System (FOS) is a nonlinear system presented with a non-integer derivative. It is well established that mathematical models can be used to describe physical systems. These mathematical models are used to operate such systems in a variety of ways, including controlling, observing, and detecting. The faults and errors of modelization may affect the system quality and performance. Therefore, the use of Fractional derivatives can approach such a mathematical model to physical reality. This fact is proved in many real physical systems, see for example [1]. Recently, the fractional calculus has attracted the attention of many researchers and numerous works have been published in this context [2–11]. In fact, by using quantum calculus, the work in [6] deals with the extension of a hybrid fractional differential operator. Utilizing the local fractional Laplace variational iteration methods and the local fractional reduced differential transform, authors in [7] have obtained an approximation of the solutions for coupled Korteweg De Vries Equations. The application of these FOSs is numerous in different domain applications, whether in electricity [10], thermal [5], chemistry [11], signal processing [12], biology [13,14] or control theory, such as fault estimation [15], stabilization [16], observer design [16,17], optimal control [18], and asymptotic stability [19,20].

The study of FTS for the Fractional Order Time Delay Systems (FOTDSs) has been largely studied in the literature in the case of continuous and discrete time [21–30]. In [30], H. Ye et al., have shown a Generalized Gronwall Inequality (GGI). After that, authors in [25] have used the GGI to study the FTS for FOTDSs. The stability of neutral fractional order time delay systems with Lipschitz nonlinearities in finite time has been investigated by F. Du et al. in [23]. The finite-time stability of a class of fractional delayed neural networks with commensurate order between 0 and 1 was studied by the authors in [28]. Additionally, the authors in [26] have provided an analytical method based on the Laplace transform and the 'inf-sup' approach for evaluating the finite-time stability of singular fractionalorder switching systems with delay. The authors have proposed a constructive geometric design for switching laws based on the partitioning of the stability state regions in convex cones. The suggested technique allows for the development of novel delay-dependent

**Citation:** Ben Makhlouf, A.; Baleanu, D. Finite Time Stability of Fractional Order Systems of Neutral Type. *Fractal Fract.* **2022**, *6*, 289. https://doi.org/10.3390/ fractalfract6060289

Academic Editors: Ricardo Almeida and Yongguang Yu

Received: 23 March 2022 Accepted: 24 May 2022 Published: 26 May 2022

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adequate conditions for the system's regularity, impulse-free, and finite-time stability in terms of tractable matrix inequalities and Mittag–Leffler functions. A case study is offered to demonstrate the proposed method's efficacy. Using the Lyapunov method, Thanh et al. in [27] have investigated a novel FTS analysis of FOTDSs. By using Banach fixed point method, author in [21] has studied the FTS for FOTDSs. In the discrete case, one has the following references [22,24,29]. Indeed, authors in [24] have proposed a sufficient condition for ensuring the FTS for Nabla uncertain FOS. Furthermore, authors in [22] have established a new Gronwall Inequality and they have used it to study the FTS of a class of nonlinear fractional delay difference systems. Furthermore, in [29], the FTS of Caputo delta fractional difference equations is investigated. On a finite time domain, a generalized Gronwall inequality is given. For fractional differential equations, a finite-time stability condition is suggested. The concept is then generalized to discrete fractional cases. There are finite-time stable conditions for a linear fractional difference equation with constant delays. To support the theoretical result, one example is numerically shown.

Motivated by the above study, this article treats the FTS for FOS of neutral type by using a version of the Banach fixed point theorem and some properties of the Mittag–Leffler Function (MLF). The contribution of this work is summarized as follows:


The rest of the paper is organized as follows. The second section deals with some preliminaries. Some basic results related to fractional calculus, fixed point theory, as well as finite time stability are shown. In regards to the third section, the stability analysis of the suggested system (2), in the case of (*λ*<sup>1</sup> < *λ*2) and (*λ*<sup>1</sup> = *λ*2), is investigated and described. Note that the fixed point approach is used to demonstrate the main results. The fourth section is concentrated to show the validity of the proposed results. Two examples are suggested to demonstrate the efficiency of the main results. Finally, to end the work, a conclusion is presented in the fifth section showing the principle fundamentals of the work.
