Stability of Fractional Differential Equations with New Generalized Hattaf Fractional Derivative

)is paper aims to study the stability of fractional differential equations involving the new generalized Hattaf fractional derivative which includes the most types of fractional derivatives with nonsingular kernels. )e stability analysis is obtained by means of the Lyapunov direct method. First, some fundamental results and lemmas are established in order to achieve the goal of this study. Furthermore, the results related to exponential andMittag–Leffler stability existing in recent studies are extended and generalized. Finally, illustrative examples are presented to show the applicability of our main results in some areas of science and engineering.


Introduction
Fractional differential equations (FDEs) are recently developed in order to describe and model the dynamics of systems having memory or hereditary properties. ese types of equations have been used and applied in various areas of science and engineering such as epidemiology [1], cancerology [2], viral immunology [3,4], and viscoelastic fluid flows [5], as well as adaptive control engineering [6].
It is well known that there are two main methods to analyze the stability of ordinary differential equations (ODEs). e first one is called the Lyapunov indirect method that aims to study the local stability by means of the linearization of a system around its steady state (equilibrium point).
e second method called the Lyapunov direct method consists to find or construct an appropriate auxiliary function, named a Lyapunov candidate function. Furthermore, the Lyapunov direct method provides a substantial tool for stability analysis of nonlinear systems. It can be used to determine the global dynamical behaviors of these systems without the need to solve explicitly the solutions of ODEs. e stability of FDEs has attracted the attention of several researchers. In 2010, Li et al. [7] studied the stability of nonlinear systems of FDEs involving the Caputo fractional derivative with singular kernel [8]. ey extended the Lyapunov direct method to the case of FDEs. In the same year, Sadati et al. [9] extended the Mittag-Leffler stability theorem for fractional nonlinear systems of FDEs with delay. e stability of a class of nonlinear systems of FDEs involving the Hadamard fractional derivative [10] was investigated in [11] by using a fractional comparison principle. e theory of the stability of FDEs involving fractional derivatives with nonsingular kernels is new, and it requires an important development in order to study the dynamical behaviors of several systems available in the literature and using such derivatives. For these reasons, the main purpose of this paper is to extend the Lyapunov direct method for systems of FDEs involving the new generalized Hattaf fractional (GHF) derivative [12], which covers the most famous fractional derivatives with nonsingular kernels existing in the literature such as the Caputo-Fabrizio fractional derivative [13], the Atangana-Baleanu fractional derivative [14], and the weighted Atangana-Baleanu fractional derivative [15]. e main advantage of using the GHF derivative is that it is a nonlocal operator and it has a nonsingular kernel formulated by the Mittag-Leffler function with a parameter different to the order of the fractional derivative. Furthermore, this operator is a weighted fractional derivative which can be used to solve various types of integral equations with elegant ways as in [16][17][18]. On the other hand, the novelties of this article are the study of the stability of FDEs with the new GHF operator by means of the Lyapunov direct method and the extension and generalization of the results related to exponential and Mittag-Leffler stability presented in [7,19], as well as the establishment of some interesting properties and inequalities of GHF derivative in order to easily prove the Lyapunov stability theorems and construct Lyapunov candidate functions of quadratic-type, which are frequently used for demonstrating the global stability of many fractional order systems. e outline of this paper is organized as follows. After an introductory part, Section 2 introduces the basic definitions and provides some lemmas and fundamental properties of the GHF derivative with nonsingular kernel in Caputo sense necessary to achieve the objective of this study. Section 3 is devoted to stability analysis. Finally, Section 4 presents some applications of our main results in the field of epidemiology as well as in the fractional linear systems theory.

Fundamental Results
In this section of the paper, we present the definitions and provide some fundamental results related to the GHF derivative with nonsingular kernel.
e GHF derivative introduced in the above definition generalizes and extends many special cases available in the literature. For instance, when w(t) � 1 and β � c � 1, (1) reduced to the Caputo-Fabrizio fractional derivative [13] given by When w(t) � 1 and β � c � α, (1) reduced to the Atangana-Baleanu fractional derivative [14] given by Furthermore, the weighted Atangana-Baleanu fractional derivative [15], given by is a special case of GHF derivative; it suffices to take β � c � α.
Considering the importance of weighted fractional derivatives to write and solve many integral equations in an elegant way, the function w has been introduced in equation (1). For instance, we consider the following integral equation: where λ, ρ, Λ, δ > 0. In terms of the GHF operator, this equation can be written as follows: where Similar to the example of HIV infection presented in [12], the solution of the above integral equation when c � β is given by where a α � N(α) + μ(1 − α).
In various areas of science and engineering, the method of constructing Lyapunov functions is often based on quadratic-type functions. So, we provide the following lemma that estimates the GHF derivative of these types of Lyapunov candidate functions. Lemma 1. Let x(.) ∈ IR n be a continuously differentiable function and P ∈ IR n×n be a symmetric positive definite matrix. en, for any time t ≥ t 0 , we have Proof. Similar to [19,20], we consider the following function: en, where Integrating by parts, we obtain Since is follows that g(t) ≤ 0, for all t ≥ t 0 , and the proof is completed.
□ Remark 1. It is important to note that the above lemma extends the recent results presented in Lemma 2 of [19] and Corollary 1 of [20]. Moreover, the results presented in Lemma 3.1 of [21] to estimate the Atangana-Baleanu Caputo derivative of quadratic Lyapunov functions is extended to the case of GHF derivative.
For simplicity, denote C D α,β,β a,t,w by D α,β a,w . By [12], the generalized fractional integral associated to D α,β a,w is given by the following definition.
Definition 2 (see [12]). e generalized fractional integral operator associated to D α,β a,w is defined by Remark 2.
e Atangana-Baleanu fractional integral operator is a particular case of (7), and it suffices to take w(t) � 1 and β � α. Now, we recall an important theorem that we will need in the following. is theorem extends the Newton-Leibniz formula introduced in [22,23].
Proof. By (11), we have By applying Laplace transform and using eorem 2 in [12], we obtain en,

Mathematical Problems in Engineering
e passage to the inverse Laplace gives From integration by parts, we have Hence, is completes the proof. □ Remark 3. By using (12), the solution of (11) can be rewritten as follows: (25)

Stability Analysis
In this section, we focus on the stability analysis of the fractional differential equations with the GHF derivative.
Consider the following fractional differential nonautonomous equation: where x(t) ∈ IR n is the state variable and f: [0, +∞) × Ω ⟶ IR n is a continuous locally Lipschitz function and Ω is a domain of IR n containing the origin (32) First, we give some definitions that we will need in the following.

Definition 3.
e trivial equilibrium point x � 0 of (31) is said to be stable if, for each ϵ > 0, there exists a η > 0 such that, for any initial condition x(t 0 ) � x 0 satisfying ‖x 0 ‖ < η, the solution x(t) of (31) satisfies ‖x(t)‖ < ϵ, for all t ≥ t 0 . Furthermore, x � 0 is said to be asymptotically stable if it is stable and lim t⟶+∞ x(t) � 0.
Also, we introduce the definition of stability in the Mittag-Leffler sense.

Remark 4.
Mittag-Leffler stability generalizes the exponential stability and it implies asymptotic stability.

Theorem 2. Let x � 0 be an equilibrium point for system (31). Let V(t, x): [0, +∞) × Ω ⟶ IR be a continuously differentiable function and locally Lipschitz with respect to x such that
where t ≥ 0, x ∈ Ω, and k, p, and q are arbitrary positive constants. en, x � 0 is Mittag-Leffler stable. If (34) and (35) hold globally on IR n , then x � 0 is globally Mittag-Leffler stable.
Proof. From (35) and according to Corollary 1, we deduce that From (34), we obtain which leads to where Clearly, m 1 (0) � 0 and m 1 (x) ≥ 0. Since V(t, x) is locally Lipschitz with respect to x, we deduce that m 1 (x) is locally Lipschitz on x. erefore, the equilibrium x � 0 is globally Mittag-Leffler stable.

Applications
In this section, we apply the main results obtained in this paper to investigate the stability of the following examples of fractional systems. Example 1. Consider the following fractional linear system: where x(t) ∈ IR n is the state variable and A ∈ IR n×n . To establish the stability of (45), we define the Lyapunov candidate function as follows: where P ∈ IR n×n is a symmetric positive definite matrix. Hence, where λ min (P) and λ max (P) are the minimum and the maximum eigenvalues of the matrix P, respectively. Since P is positive definite, we have λ min (P) > 0 and λ max (P) > 0. According to Lemma 1, the GHF derivative of the Lyapunov function V along the trajectories of (45) satisfies where Q � − (A T P + PA). It is obvious that Q is a symmetric matrix. Assume that Q is a positive definite matrix and let λ min (Q) be the minimum of its positive eigenvalues. en, we have By applying eorem 3, we deduce that the trivial solution of system (45) is globally Mittag-Leffler stable under the condition that Q is positive definite. is condition is satisfied when A is Hurwitz, i.e., all the eigenvalues of A have negative real parts.
Example 2. Consider the following fractional epidemic model: where S(t), I(t), and R(t) are the susceptible, infected, and recovered individuals at time t, respectively. Here, the susceptible individuals are recruited at a constant rate A and become infected by effective contact with infected individuals at rate F(S, I)I. e natural death rate in all classes is denoted by ], while d is the death rate due to the disease. Furthermore, r is the recovery rate of the infected individuals.
Obviously, the first two equations of (50) do not depend on the variable R. en, model (50) can be rewritten by the following system: As in [24], we assume that the general incidence F is continuously differentiable in the interior of IR 2 + and satisfies the following conditions: It is clear that E 0 � (S 0 , 0) is the disease-free equilibrium of (51), where S 0 � A/]. en, the basic reproduction number of (51) is defined as follows: which epidemiologically represents the number of secondary infections produced by a single infected individual throughout the period of infection when all individuals are uninfected. Based on the same technique in [24], we can easily prove that model (51) has another equilibrium when R 0 > 1.

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares that he has no conflicts of interest.