# Stability of Fractional Differential Equations with New Generalized Hattaf Fractional Derivative

Authors: Khalid Hattaf
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This 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. The 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 and Mittag–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.


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## Body

## 1. Introduction

Fractional differential equations (FDEs) are recently developed in order to describe and model the dynamics of systems having memory or hereditary properties. These 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). The 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). The 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.The 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]. They 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. The 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.The 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].The 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 [16–18]. 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.The outline of this paper is organized as follows. After an introductory part, Section2 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.


## 2. 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.Definition 1 (see [12]).
Letα∈0,1, β,γ>0, and f∈H1a,b. The GHF derivative of order α in Caputo sense of the function ft with respect to the weight function wt is defined as follows:(1)DCa,t,wα,β,γft=Nα1−α1wt∫atEβ−μαt−τγddτwfτdτ,where w∈C1a,b, w,w′>0 on [a, b], Nα is a normalization function obeying N0=N1=1, μα=α/1−α, and Eβt=∑k=0+∞tk/Γβk+1 is the Mittag–Leffler function of parameter β.
The GHF derivative introduced in the above definition generalizes and extends many special cases available in the literature. For instance, whenwt=1 and β=γ=1, (1) reduced to the Caputo–Fabrizio fractional derivative [13] given by(2)DCa,t,1α,1,1ft=Nα1−α∫atexp−μαt−τf′τdτ.
Whenwt=1 and β=γ=α, (1) reduced to the Atangana–Baleanu fractional derivative [14] given by(3)DCa,t,1α,α,αft=Nα1−α∫atEα−μαt−ταf′τdτ.
Furthermore, the weighted Atangana–Baleanu fractional derivative [15], given by(4)DCa,t,wα,α,αft=Nα1−α1wt∫atEα−μαt−ταddτwfτdτ,is a special case of GHF derivative; it suffices to take β=γ=α.
Considering the importance of weighted fractional derivatives to write and solve many integral equations in an elegant way, the functionw has been introduced in equation (1). For instance, we consider the following integral equation:(5)λ∫0te−λt−τEβ−ρt−τγfτdτ+∫0te−λt−τEβ−ρt−τγf′τdτ=Λ−δft,where λ,ρ,Λ,δ>0. In terms of the GHF operator, this equation can be written as follows:(6)DCa,t,wα,β,γft=A−μft,where wt=eλt, Nα=1, α=ρ/1+ρ, A=Λ/1−α, and μ=δ/1−α. Similar to the example of HIV infection presented in [12], the solution of the above integral equation when γ=β is given by(7)ft=Aμ+Nαw0aαwtf0−AμEβ−αμaαtβ−ANαμaαwtEβ−αμaαtβ∗w′t,where aα=Nα+μ1−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.
Letx.∈IRn be a continuously differentiable function and P∈IRn×n be a symmetric positive definite matrix. Then, for any time t≥t0, we have(8)DCt0,t,1α,β,γxtTPxt≤2xtTPCDt0,t,1α,β,γxt.Proof.
Similar to [19, 20], we consider the following function:(9)gt=CDt0,t,1α,β,γxtTPxt−2xtTPCDt0,t,1α,β,γxt.
Then,(10)gt=Nα1−α∫t0tEβ−μαt−τγ2xτTPx˙τ−2xtTPx˙τdτ=Nα1−α∫t0tEβ−μαt−τγ2yτTPy˙τdτ=Nα1−α∫t0tEβ−μαt−τγyτTPyτ′dτ,where yτ=xτ−xt. Integrating by parts, we obtain(11)gt=Nα1−αEβ−μαt−τγyτTPyτ|τ=t0τ=t−αγNα1−α2∫t0tt−τγ−1Eβ,β+12−μαt−τγyτTPyτdτ.
Sincelimτ⟶tEβ−μαt−τγyτTPyτ=ytTPyt=0, we have(12)gt=−Nα1−αEβ−μαt−t0γyt0TPyt0−αγNα1−α2∫t0tt−τγ−1Eβ,β+12−μαt−τγyτTPyτdτ.
This follows thatgt≤0, for all t≥t0, and the proof is completed.Remark 1.
It is important to note that the above lemma extends the recent results presented in Lemma2 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, denoteDCa,t,wα,β,β by Da,wα,β. By [12], the generalized fractional integral associated to Da,wα,β is given by the following definition.Definition 2 (see [12]).
The generalized fractional integral operator associated toDa,wα,β is defined by(13)ℐa,wα,βft=1−αNαft+αNαRLℐa,wβft,where ℐRLa,wβ is the standard weighted Riemann–Liouville fractional integral of order β defined (14)ℐRLa,wβft=1Γβ1wt∫att−τβ−1wτfτdx.Remark 2.
The Atangana–Baleanu fractional integral operator is a particular case of (7), and it suffices to take wt=1 and β=α.
Now, we recall an important theorem that we will need in the following. This theorem extends the Newton–Leibniz formula introduced in [22, 23].Theorem 1 (see [20]).
Letα∈0,1, β>0, and f∈H1a,b. Then, we have the following properties:(15)ℐa,wα,βDa,wα,βft=ft−wafawt,(16)Da,wα,βℐa,wα,βft=ft−wafawt.On the contrary, we need the following results.Lemma 2.
Lety.∈IRn. The solution of the following Cauchy problem(17)D0,wα,βyt=λyt+ut,with initial boundary condition y0=y0, is given by(18)yt=Nαw0y0aαwtEβαλaαtβ+1−αaαut+NαλaαwtddtEβαλaαtβ∗wut,where aα=Nα−λ1−α≠0 and λ≠0.Proof.
By (11), we have(19)wtD0,wα,βyt=λwtyt+wtut.
By applying Laplace transform and using Theorem 2 in [12], we obtain(20)ℒwtyt=1−αsβ+μαNα−λ1−αsβ−λαℒwtut+Nαw0y0sβ−1Nα−λ1−αsβ−λα.
Then,(21)ℒwtyt=Nαw0y0aαsβ−1sβ−αλ/aα+1−αaαsβ−1sβ−αλ/aαsℒwtut+αaα1sβ−αλ/aαℒwtut=Nαw0y0aαℒEβαλaαtβ+1−αaαℒEβαλaαtβℒwu′t+w0u0+1λℒddtEβαλaαtβℒwtut.
The passage to the inverse Laplace gives(22)wtyt=Nαw0y0aαEβαλaαtβ+1−αaαwtut+NαλaαddtEβαλaαtβ∗wut.
From integration by parts, we have(23)ddtEβαλaαtβ∗wut=Eβαλaαtβwu0−wut+Eβαλaαtβ∗wu′t.
Hence,(24)yt=Nαw0y0aαwtEβαλaαtβ+1−αaαut+NαλaαwtddtEβαλaαtβ∗wut.
This completes the proof.Remark 3.
By using (12), the solution of (11) can be rewritten as follows:(25)yt=λy0+u0Nαw0λaαwtEβαλaαtβ−1λut+NαλaαwtEβαλaαtβ∗wu′t.Corollary 1.
Letλ>0 and ft be a function satisfying the following inequality:(26)D0,wα,βft≤−λft.Then,(27)ft≤f0Eβ−αλtβNα+λ1−α.Proof.
From (14), we deduce that there exists a nonnegative function ut such that(28)D0,wα,βft=−λft−ut.
By applying Lemma2, we obtain(29)ft=Nαw0f0Nα+λ1−αwtEβ−αλtβNα+λ1−α−1−αutNα+λ1−α+NαλNα+λ1−αwtddtEβ−αλtβNα+λ1−α∗wut=Nαw0f0Nα+λ1−αwtEβ−αλtβNα+λ1−α−1−αutNα+λ1−α−αβNαλNα+λ1−α2wttβ−1Eβ,β+12−αλtβNα+λ1−α∗wut.
Then,(30)ft≤Nαw0f0Nα+λ1−αwtEβ−αλtβNα+λ1−α.
SinceNα≤Nα+λ1−α and w0≤wt, we easily have (27).


## 3. 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:(31)D0,wα,βxt=ft,xt,where xt∈IRn is the state variable and f:0,+∞×Ω⟶IRn is a continuous locally Lipschitz function and Ω is a domain of IRn containing the origin x=0.System (31) is said to be autonomous if ft,x=fx. In this case, (31) becomes(32)D0,wα,βxt=fxt.First, we give some definitions that we will need in the following.Definition 3.
The trivial equilibrium pointx=0 of (31) is said to be stable if, for each ϵ>0, there exists a η>0 such that, for any initial condition xt0=x0 satisfying x0<η, the solution xt of (31) satisfies xt<ϵ, for all t≥t0. Furthermore, x=0 is said to be asymptotically stable if it is stable and limt⟶+∞xt=0.Definition 4.
A scalar functionVx is called a Lyapunov candidate function of autonomous system (32) if it is a positive definite in a neighborhood U of the origin, i.e., V0=0 and Vx>0, for all x∈U∖0. In addition, a scalar function Vt,x is a Lyapunov candidate function of nonautonomous system (31) if it is positive definite, i.e., Vt,0=0, for all t≥t0, and if there is a time-invariant function V0x which is positive definite such that Vt,x≥V0x, for all t≥t0.
Also, we introduce the definition of stability in the Mittag–Leffler sense.Definition 5.
The trivial solution of (31) is called Mittag–Leffler stable if(33)xt≤mxt0Eβ−λt−t0βν,where t0 is the initial time, λ≥0, ν>0, m0=0, mx≥0, and mx is locally Lipschitz on x∈IRn with the Lipschitz constant m0.Remark 4.
Mittag–Leffler stability generalizes the exponential stability and it implies asymptotic stability.Theorem 2.
Letx=0 be an equilibrium point for system (31). Let Vt,x:0,+∞×Ω⟶IR be a continuously differentiable function and locally Lipschitz with respect to x such that(34)kxp≤Vt,x,(35)D0,wα,βVt,x≤−qVt,x,where t≥0, x∈Ω, and k, p, and q are arbitrary positive constants. Then, x=0 is Mittag–Leffler stable. If (34) and (35) hold globally on IRn, then x=0 is globally Mittag–Leffler stable.Proof.
From (35) and according to Corollary 1, we deduce that(36)Vt,x≤V0,x0Eβ−qαtβNα+q1−α.
From (34), we obtain(37)kxtp≤V0,x0Eβ−qαtβNα+q1−α,which leads to(38)xt≤m1x0Eβ−λtβ1/p,where m1x=V0,x/k and λ=qα/Nα+q1−α. Clearly, m10=0 and m1x≥0. Since Vt,x is locally Lipschitz with respect to x, we deduce that m1x is locally Lipschitz on x. Therefore, the equilibrium x=0 is globally Mittag–Leffler stable.Remark 5.
Theorem2 generalizes the result of the exponential stability presented in Theorem 2 of [19]. Indeed, it suffices to take wt=1, β=γ=1, and k=1.Theorem 3.
Letx=0 be an equilibrium point for system (31). Let Vt,x:0,+∞×Ω⟶IR be a continuously differentiable function and locally Lipschitz with respect to x such that(39)k1xp≤Vt,x≤k2xpq,(40)D0,wα,βVt,x≤−k3xpq,where t≥0, x∈Ω, and k1, k2, k3, p, and q are arbitrary positive constants. Then, x=0 is Mittag–Leffler stable. If (39) and (40) hold globally on IRn, then x=0 is globally Mittag–Leffler stable.Proof.
According to (39) and (40), we have(41)D0,wα,βVt,x≤−k3k2Vt,x.
By applying Corollary1, we have(42)Vt,x≤V0,x0Eβ−κtβ,where κ=k3α/k2Nα+k31−α. By (39), we obtain(43)k1xtp≤V0,x0Eβ−κtβ,which implies that(44)xt≤m2x0Eβ−κtβ1/p,where m2x=V0,x/k1. This completes the proof.Remark 6.
Theorem3 extends the result of the Mittag–Leffler stability presented in Theorem 5.1 of [7] for the GHF derivative with nonsingular kernel. Moreover, the result given in Theorem 4 of [19] for Caputo–Fabrizio fractional derivative is recovered when wt=1, β=γ=1, p=2, and q=1.


## 4. 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:(45)D0,wα,βxt=Axt,where xt∈IRn is the state variable and A∈IRn×n.
To establish the stability of (45), we define the Lyapunov candidate function as follows:(46)Vt,x=xTPx,where P∈IRn×n is a symmetric positive definite matrix. Hence,(47)λminPx2≤Vt,x≤λmaxPx2,where λminP and λmaxP are the minimum and the maximum eigenvalues of the matrix P, respectively. Since P is positive definite, we have λminP>0 and λmaxP>0.
According to Lemma1, the GHF derivative of the Lyapunov function V along the trajectories of (45) satisfies(48)D0,1α,βVt,x≤2xTPD0,wα,βx=xTATP+PAx=−xTQx,where Q=−ATP+PA. It is obvious that Q is a symmetric matrix. Assume that Q is a positive definite matrix and let λminQ be the minimum of its positive eigenvalues. Then, we have(49)D0,1α,βVt,x≤−λminQx2.
By applying Theorem3, we deduce that the trivial solution of system (45) is globally Mittag–Leffler stable under the condition that Q is positive definite. This 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:(50)D0,1α,βSt=A−νSt−ℱSt,ItIt,D0,1α,βIt=ℱSt,ItIt−ν+d+rIt,D0,1α,βRt=rIt−νRt,where St, It, and Rt 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 ℱS,II. The natural death rate in all classes is denoted by ν, 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. Then, model (50) can be rewritten by the following system:(51)D0,1α,βSt=A−νSt−ℱSt,ItIt,D0,1α,βIt=ℱSt,ItIt−ν+d+rIt.
As in [24], we assume that the general incidence ℱ is continuously differentiable in the interior of IR+2 and satisfies the following conditions:(52)Hℱ0,S=0,∂ℱ∂SS,I>0,∂ℱ∂IS,I≤0,for allS,I≥0.
It is clear thatE0=S0,0 is the disease-free equilibrium of (51), where S0=A/ν. Then, the basic reproduction number of (51) is defined as follows:(53)ℛ0=ℱS0,0ν+d+r,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 ℛ0>1.
LetΩ=S,I∈IR+2:S≤S0. Assume that ℛ0<1 and consider the following Lyapunov function:(54)LS,I=ϖS0−S+I,where ϖ=1−ℛ0/2ℛ0. Obviously, L is a candidate Lyapunov function. Indeed, we have LE0=0 and LS,I>0, for all S,I∈Ω∖E0. Moreover, we have(55)D0,1α,βLS,I=−ϖD0,1α,βS+D0,1α,βI≤−ϖνS0−S−δ1−ℛ02I,where δ=ν+d+r. Hence,(56)D0,1α,βLS,I≤−ϱ1LS,I,where ϱ1=minν,δ1−ℛ0/2.
LetX=S,I∈IR2 with the norm X=S+I. Then,(57)ϱ2X−E0≤LS,I,where ϱ2=minϖ,1. By applying Theorem 2, we conclude that the disease-free equilibrium E0 of (51) is Mittag–Leffler stable in Ω when ℛ0<1.



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