# A Note on the Stability Analysis of Fuzzy Nonlinear Fractional Differential Equations Involving the Caputo Fractional Derivative

Authors: Ali El Mfadel; Said Melliani; M’hamed Elomari
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## Abstract

In this paper, we present and establish a new result on the stability analysis of solutions for fuzzy nonlinear fractional differential equations by extending Lyapunov’s direct method from the fuzzy ordinary case to the fuzzy fractional case. As an application, several examples are presented to illustrate the proposed stability result.


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## Body

## 1. Introduction

The theory of fuzzy fractional differential equations was initiated by Agarwal et al. in [1] who proposed the concept of solutions for fractional differential equations with uncertainties, and they considered the Riemann–Liouville differentiability to solve the equations which is a combination of the Hukuhara difference and Riemann–Liouville derivative. Arshad and Lupulescu in [2, 3] developed the theory of the fractional calculus for interval-valued functions and proved some results on the existence and uniqueness of the solution for the fuzzy fractional differential equations. Later, Alikhani and Bahrami in [4] proved the existence and uniqueness results for nonlinear fuzzy fractional integral and integrodifferential equations by using the method of upper and lower solutions. The authors in [5–7] discussed the concepts about the generalized Hukuhara fractional Riemann–Liouville and Caputo differentiability of fuzzy-valued functions, and the equivalence between the fuzzy fractional differential equation and fuzzy fractional integral equation was discussed in [8]. On the contrary, numerical solutions of interval-valued fractional nonlinear differential equations were investigated in [9, 10].Recently, fractional calculus was introduced to the stability analysis of nonlinear systems. The stability notion is one of the most important issues for differential equations, although this problem is over many years. Stability of fractional differential systems has attracted increasing interest. For example, the earliest study on the stability of fractional differential equations started in [11]. For more details about the stability results and the methods available to analyze the stability of fractional differential equations, the reader may refer to the recent papers [12–14] and the references therein.In this paper, our aim is to study the stability result of the following fuzzy nonlinear fractional differential equation:(1)cD0+qut=ft,ut;t∈J=0,+∞,u0=u0,by extending Lyapunov’s direct method from the ordinary fuzzy case to the fractional fuzzy case such that f:J×E1⟶E1 is a fuzzy continuous function in t and locally Lipschitz in u and D0+qc is the Caputo fractional derivative of ut at order 0<q<1.Our paper is organized as follows: Section2 gives some basic definitions, lemmas, and theorems as preliminaries of the fuzzy set theory and fuzzy fractional calculus. In Section 3, we give some sufficient criteria to guarantee the stability of the trivial solution for fuzzy nonlinear fractional differential equation (1), and we present an example to illustrate the proposed stability result. Finally, in Section 4, this work is concluded by conclusion and future works.


## 2. Preliminaries

Here, we review some essential facts from fuzzy fractional calculus, basic definitions of a fuzzy number, and fuzzy concepts.Definition 1 (see [15]).
A fuzzy number is mappingu:ℝ⟶0,1 such that(1)
u is upper semicontinuous(2)
u is normal; that is, there exists x0∈ℝ such that ux0=1(3)
u is fuzzy convex; that is, uλx+1−λy≥minux,uy for all x,y∈ℝ and λ∈0,1(4)
x∈ℝ,ux>0¯ is compactTheα−cut of a fuzzy number u is defined as follows:(2)uα=x∈ℝ\ux≥α.Moreover, we can also present theα−cut of fuzzy number u by uα=ulα,urα.Notations:We denote byE1 the collection of all fuzzy numbers.We also denote by0E1 the fuzzy zero defined by(3)0E1x=1,ifx=0,0,elswhere.Example 1.
Letu be a fuzzy number defined by the following function:(4)μux=x−1,x∈1,2,−x+3,x∈2,3,0,elswhere.
Then, we haveu1=2.Definition 2 (see [8]).
Letu∈E1 and α∈0,1; we define the diameter of the α−level set of the fuzzy set u as follows:(5)duα=ur−ul.We denote byCJ,E1 the space of all fuzzy-valued functions which are continuous on J and ℙcℝ the space of all the compact subsets of ℝ.Definition 3 (see [8]).
The generalized Hukuhara difference of two fuzzy numbersu,v∈E1 is defined as follows:(6)u⊖gHv=w⟺iu=v+w,oriiv=u+−1w.Property 1 (see [8]).
Ifu∈E1 and v∈E1, then the following properties hold:(1)
Ifu⊖gHv exists, then it is unique(2)
u⊖gHu=0E1(3)
u+v⊖gHv=u(4)
u⊖gHv=0E1⟺u=vDefinition 4 (see [15]).
According to Zadeh’s extension principle, the addition onE1 is defined by(7)u⊕vz=supz=x+yminux,vy.
And scalar multiplication of a fuzzy number is given by(8)k⊙ux=uxk,k>0,0E1,k=0.Remark 1 (see [16]).
Letu,v∈E1 and α∈0,1; then, we have(9)u+vα=uα+vα,u−vα=u1α−v2α,u2α−v1α,kuα=kuα=λu1α,λu2α,ifλ≥0,λu2α,λu1α,ifλ<0,uvα=minu1αv1α,u1αv2α,u2αv1α,u2αv2α,maxu1αv1α,u1αv2α,u2αv1α,u2αv2α.Definition 5 (see [16]).
Letu,v∈E1 with α∈0,1; then, the Hausdorff distance between u and v is given by(10)Du,v=supα∈0,1duα,vα,where d is the Hausdorff metric defined in PcR.Proposition 1 (see [17]).
D is a metric on E1 and has the following properties:(1)
E1;D is a complete metric space(2)
Du+w,v+w=Du,v(3)
Dku,kv=kDu,v, ∀u,v∈E1 and k∈R(4)
Du+w,v+z≤Du,v+Dw,z, ∀u,v,w,z∈E1Definition 6 (see [18]).
Letf:J⟶E1 and t0∈J. We say that f is Hukuhara differentiable at t0 if there exists f′t0∈E1 such that(11)f′t0=limh⟶0+ft0+h⊖gHft0h=limh⟶0−ft0⊖gHft0−hh.Remark 2.
Letf:J⟶E1 be a fuzzy function such that fxα=f¯x;α,f¯x;α for each α∈0,1; then,(12)f′xα=f′¯x;α,f′¯x;α.Definition 7 (see [17]).
F:J⟶E1 is strongly measurable if ∀α∈0,1, the set-valued mapping Fα:J⟶PcR defined by Fαt=Ftα is Lebesgue measurable.
A functionF:J⟶E1 is called integrably bounded if there exists an integrable function h such that x<ht, ∀x∈F0t.Definition 8 (see [19]).
LetF:J⟶E1. The integral of F on J denoted by ∫JFtdt is given by(13)∫JFtdtα=∫JFαtdt=∫Jftdt|f:J⟶ℝis a measurable selection forFα,for all α∈0,1.Proposition 2 (see [17]).
LetF:J⟶E1 be a fuzzy function. If F is strongly measurable and integrably bounded, then it is integrable.


### 2.1. Fractional Integral and Fractional Derivative of the Fuzzy Function

Proposition 3 (see [17]).
Ifu∈E1, then the following properties hold:(1)
uβ⊂uαif0≤α≤β.(2)
Ifαn⊂0,1 is a nondecreasing sequence which converges to α, then(14)uα=∩n≥1uαn.Conversely, ifAα=u1α,u2α;α∈0,1 is a family of closed real intervals verifying (1) and (16), then Aα defined a fuzzy number u∈E1 such that uα=Aα.Let0<q<1; the fractional integral of order q of a real function g:J⟶R is given by(15)I0+qgt=1Γq∫0tt−sq−1gsds.Letft∈LJ,E1 such that ft=f1αt,f2αt. Suppose that f1α,f2α∈LJ,ℝ for all α∈0,1, and let(16)Aα=1Γq∫0tt−sq−1f1αsds,1Γq∫0tt−sq−1f2αsds,where Γ⋅ is the Euler gamma function.We have the following lemma.Lemma 1 (see [1]).
The familyAα;α∈0,1 given by (16) defined a fuzzy number u∈E1 such that uα=Aα.Definition 9 (see [6]).
Letft∈LJ,E1.
The fuzzy fractional integral of orderq∈0,1 of f denoted by(17)I0+qft=1Γq∫0tt−sq−1fsdsis defined by(18)I0+qftα=I0+qflt;α,I0+qfrt;α.Proposition 4 (see [6]).
Letf,g∈LJ,E1,q∈0,1, and b∈E1; then, we have(1)
I0+qbft=bI0+qft(2)
I0+qf+gt=I0+qft+I0+qgt(3)
I0+q1I0+q2ft=I0+q1+q2ft, where q1,q2∈0,12Example 2.
Letut=u∈E1 such that uα=uα1,uα2; then,(19)I0+qutα=1Γq∫0tt−sq−1uα1sds,1Γq∫0tt−sq−1uα2sds,I0+qutα=tqΓq+1uα1,uα2,I0+qutα=tqΓq+1uα.Definition 10 (see [6]).
Letf∈CJ,E1∩LJ,E1. The function f is called fuzzy Caputo fractional differentiable of order 0<q<1 at t if there exists an element D0+qcft∈E1 such that(20)D0+qcft=1Γq∫0tt−sq−1f′sds.Remark 3 (see [6]).
Sinceftα=flt;α,frt;α for each α∈0,1,(21)D0+qcftα=D0+qcflt;α,Dqcfrt;α,where(22)D0+qcflt;α≔1Γq∫0tt−sq−1fl′s,αds,D0+qcfrt;α≔1Γq∫0tt−sq−1fr′s,αds.Example 3.
Letut=u∈E1. If uα=uα1,uα2, then(23)D0+qcutα=1Γq∫0tt−sq−1uα1′ds,1Γq∫0tt−sq−1uα2′ds,D0+qcutα=0,D0+qcut=0E1.


### 2.2. Laplace Transform of the Caputo Fractional Derivative

For establishing the Laplace transform of the Caputo fractional derivative [20], we write the Caputo derivative under the form(24)D0+βcft≔In−βfnt,where β>0 such that n<β<n+1.By using the formula of the Laplace transform of the Riemann–Liouville fractional integral, we have(25)LDαcft,s≔LIn−βfnt,s=sβ−nGs,where Gs is given by(26)Gs=snFs−∑k=0n−1sn−k−1fk0=snFs−∑k=0n−1skfn−k−10.Finally, the Laplace transform of the Caputo fractional derivative is given by(27)LDβcft,s=sβFs−∑k=0n−1sβ−k−1fk0.Remark 4 (see [20]).
If0<β<1, then we have(28)LDβcft,s=sβFs−f0.


### 2.3. Laplace Transform of the Mittag-Leffler Function

The Mittag-Leffler function is an important function that finds widespread use in the world of fractional calculus. Just as the exponential naturally arises out of the solution to integer-order differential equations, the Mittag-Leffler function plays an analogous role in the solution of noninteger-order differential equations.Definition 11 (see [20]).
We recall that the Mittag-Leffler function is given by(29)Eαz=∑k=0+∞zkΓαk+1,α∈ℂ,Rα>0,z∈ℂ.
And the general form is given by(30)Eα,βz=∑k=0+∞zkΓαk+β,β,α∈ℂ,Rβ>0,Rα>0,z∈ℂ.
Then, we have(31)LEαλtα;s=sα−1sα−λ,s>λ1/α.
Indeed, fors>λ, using the series expansion of the exponential function, we have(32)1s−λ=∈0+∞e−steλtdt=∑k=0∞λkk!∈0+∞e−sttkdt=∑k=0∞λkk!k!sk+1=∑k=0∞λksk+1.
Then, similarly, for the Mittag-Leffler function, we obtain(33)LEαλtα;s=∑k=0∞λkΓαk+1∈0+∞e−sttαkdt,LEαλtα;s=∑k=0∞λkΓαk+1Γαk+1sαk+1,LEαλtα;s=∑k=0∞λksαk+1=sα−1∑k=0∞λksαk+1,LEαλtα;s==sα−1sα−λ,s>λ1/α.
The Laplace transform of the Mittag-Leffler function in two parametersα,β∈ℂ is given by(34)Ltβ−1Eα,β−λtα;s=sα−βsα+λ,Rs>λ1/α.


## 2.1. Fractional Integral and Fractional Derivative of the Fuzzy Function

Proposition 3 (see [17]).
Ifu∈E1, then the following properties hold:(1)
uβ⊂uαif0≤α≤β.(2)
Ifαn⊂0,1 is a nondecreasing sequence which converges to α, then(14)uα=∩n≥1uαn.Conversely, ifAα=u1α,u2α;α∈0,1 is a family of closed real intervals verifying (1) and (16), then Aα defined a fuzzy number u∈E1 such that uα=Aα.Let0<q<1; the fractional integral of order q of a real function g:J⟶R is given by(15)I0+qgt=1Γq∫0tt−sq−1gsds.Letft∈LJ,E1 such that ft=f1αt,f2αt. Suppose that f1α,f2α∈LJ,ℝ for all α∈0,1, and let(16)Aα=1Γq∫0tt−sq−1f1αsds,1Γq∫0tt−sq−1f2αsds,where Γ⋅ is the Euler gamma function.We have the following lemma.Lemma 1 (see [1]).
The familyAα;α∈0,1 given by (16) defined a fuzzy number u∈E1 such that uα=Aα.Definition 9 (see [6]).
Letft∈LJ,E1.
The fuzzy fractional integral of orderq∈0,1 of f denoted by(17)I0+qft=1Γq∫0tt−sq−1fsdsis defined by(18)I0+qftα=I0+qflt;α,I0+qfrt;α.Proposition 4 (see [6]).
Letf,g∈LJ,E1,q∈0,1, and b∈E1; then, we have(1)
I0+qbft=bI0+qft(2)
I0+qf+gt=I0+qft+I0+qgt(3)
I0+q1I0+q2ft=I0+q1+q2ft, where q1,q2∈0,12Example 2.
Letut=u∈E1 such that uα=uα1,uα2; then,(19)I0+qutα=1Γq∫0tt−sq−1uα1sds,1Γq∫0tt−sq−1uα2sds,I0+qutα=tqΓq+1uα1,uα2,I0+qutα=tqΓq+1uα.Definition 10 (see [6]).
Letf∈CJ,E1∩LJ,E1. The function f is called fuzzy Caputo fractional differentiable of order 0<q<1 at t if there exists an element D0+qcft∈E1 such that(20)D0+qcft=1Γq∫0tt−sq−1f′sds.Remark 3 (see [6]).
Sinceftα=flt;α,frt;α for each α∈0,1,(21)D0+qcftα=D0+qcflt;α,Dqcfrt;α,where(22)D0+qcflt;α≔1Γq∫0tt−sq−1fl′s,αds,D0+qcfrt;α≔1Γq∫0tt−sq−1fr′s,αds.Example 3.
Letut=u∈E1. If uα=uα1,uα2, then(23)D0+qcutα=1Γq∫0tt−sq−1uα1′ds,1Γq∫0tt−sq−1uα2′ds,D0+qcutα=0,D0+qcut=0E1.


## 2.2. Laplace Transform of the Caputo Fractional Derivative

For establishing the Laplace transform of the Caputo fractional derivative [20], we write the Caputo derivative under the form(24)D0+βcft≔In−βfnt,where β>0 such that n<β<n+1.By using the formula of the Laplace transform of the Riemann–Liouville fractional integral, we have(25)LDαcft,s≔LIn−βfnt,s=sβ−nGs,where Gs is given by(26)Gs=snFs−∑k=0n−1sn−k−1fk0=snFs−∑k=0n−1skfn−k−10.Finally, the Laplace transform of the Caputo fractional derivative is given by(27)LDβcft,s=sβFs−∑k=0n−1sβ−k−1fk0.Remark 4 (see [20]).
If0<β<1, then we have(28)LDβcft,s=sβFs−f0.


## 2.3. Laplace Transform of the Mittag-Leffler Function

The Mittag-Leffler function is an important function that finds widespread use in the world of fractional calculus. Just as the exponential naturally arises out of the solution to integer-order differential equations, the Mittag-Leffler function plays an analogous role in the solution of noninteger-order differential equations.Definition 11 (see [20]).
We recall that the Mittag-Leffler function is given by(29)Eαz=∑k=0+∞zkΓαk+1,α∈ℂ,Rα>0,z∈ℂ.
And the general form is given by(30)Eα,βz=∑k=0+∞zkΓαk+β,β,α∈ℂ,Rβ>0,Rα>0,z∈ℂ.
Then, we have(31)LEαλtα;s=sα−1sα−λ,s>λ1/α.
Indeed, fors>λ, using the series expansion of the exponential function, we have(32)1s−λ=∈0+∞e−steλtdt=∑k=0∞λkk!∈0+∞e−sttkdt=∑k=0∞λkk!k!sk+1=∑k=0∞λksk+1.
Then, similarly, for the Mittag-Leffler function, we obtain(33)LEαλtα;s=∑k=0∞λkΓαk+1∈0+∞e−sttαkdt,LEαλtα;s=∑k=0∞λkΓαk+1Γαk+1sαk+1,LEαλtα;s=∑k=0∞λksαk+1=sα−1∑k=0∞λksαk+1,LEαλtα;s==sα−1sα−λ,s>λ1/α.
The Laplace transform of the Mittag-Leffler function in two parametersα,β∈ℂ is given by(34)Ltβ−1Eα,β−λtα;s=sα−βsα+λ,Rs>λ1/α.


## 3. Main Result

The existence and uniqueness results of the solution for problem (1) are discussed in [21] by using the continuity and Lipschitz conditions of the function f.In this paper, we suppose thatft,0E1=0E1; this fact means that the fuzzy zero function is a solution of problem (1). We extend the Lyapunov direct method of stability to introduce Mittag-Leffler stability analysis of the fuzzy trivial solution u=0E1 for problem (1).Definition 12.
A fuzzy numberue is an equilibrium point of system (1) if and only if ft,ue=0E1.Remark 5.
For convenience, we state all definitions and theorems for the case when the equilibrium point is the originue=0E1.
There is no loss of generality in doing so because any equilibrium point can be shifted to the origin via a change of variables.
Suppose that the fuzzy equilibrium point for (1) is ue≠0E1, and consider the change of variable U=u⊖gHue. The fractional derivative of Ut is given by(35)D0+qcUt=D0+qcut⊖gHue=ft,ut+ue=Ft,ut,where Ft,0E1=0E1 and the system has equilibrium at the origin.Definition 13.
The trivial solution0E1 of problem (1) is said to be(1)
Stable iff for anyϵ>0, there exists δ>0 such that(36)Du0,0E1<δ⇒Dut,0E1<ϵ,∀t≥0.(2)
Asymptotically stable ifflimt⟶+∞Dut,0E1=0.Definition 14.
The solution of (1) is said to be Mittag-Leffler stable if(37)Dut,0E1≤mu0Eq−λtqb,where 0<q<1, λ≥0, b>0, m0=0, and mu≥0 and mu is locally Lipschitz on u with Lipschitz constant m0.Remark 6.
The Mittag-Leffler stability implies the stability of Lyapunov.Theorem 1.
LetVt,ut:0,+∞×E1⟶ℝ be a continuously differentiable function and locally Lipschitz with respect to u such that(38)Dac1ut,0E1≤Vt,ut≤Dabc2ut,0E1,(39)D0+βcVt,ut≤Dab−c3ut,0E1,where t≥0,β∈0,1,andc1,c2,c3,a, and b are positive constants; then, the trivial solution of (1) is Mittag-Leffler stable.Proof.
From equations (38) and (39), we obtain(40)D0+βcVt,ut≤−c3c2Vt,ut.
There exists a positive functionkt satisfying(41)D0+βcVt,ut+kt=−c3c2Vt,ut.
By using the Laplace transform (27), we get(42)SβLVt,ut;s−Sβ−1V0,u0+Lkt;s=−c3c2LVt,ut;s.
It follows that(43)LVt,ut;s=Sβ−1V0,u0−Lkt;sSβ+c3/c2.
By using the inverse of the Laplace transform, we obtain(44)Vt,ut=Eβ−c3c2tβV0,u0−kt∗t1−βEβ,β−c3c2tβ.
SinceEβ,β−c3/c2tβ is a nonnegative function (see [20]), it follows that(45)Vt,ut≤Eβ−c3c2tβV0,u0,Dut,0E1≤V0c1Eβ−c3c2tβ1/a,where V0=V0,u0.
Letm=V0/c1; then, we have(46)Dut,0E1≤mEβ−c3c2tβ1/a,where m=0 if and only if u0=0E1.
SinceVt,u is locally Lipschitz with respect to x, it follows that m=V0,u0/c1 is Lipschitz with respect to u0 and m0=0 which imply the Mittag-Leffler stability of system (1).


### 3.1. Illustrative Example

The following example is used to demonstrate the applicability of our stability result. We consider the following fuzzy nonlinear system:(47)cD0+qut=ft,ut,u0=u0,where 0<q<1, ue=0E1 is the equilibrium point of problem (47), and f:J×E1⟶E1 is a fuzzy Lipschitz function with Lipschitz constant k>0.We suppose that there exists a Lyapunov functionVt,ut satisfying the following conditions:(48)Dc1ut,0E1≤Vt,ut≤Dc2ut,0E1,(49)V′t,u≤D−c3u,0E1,where c1, c2, and c3 are positive constants and V′t,u=dVt,ut/dt.By using inequalities (48) and (49), we can write(50)D0+1−qcVt,ut=I0+qV′t,u≤I0+q−c3Dut,0E1.Since the fuzzy functionf:J×E1⟶E1 is Lipschitz with a real constant k>0, it follows that(51)Dft,u,0E1≤k,Dut,0E1.Thus,(52)D0+1−qcVt,ut≤−c3kI0+qDft,u,0E1,D0+1−qcVt,ut≤−c3kDI0+qft,u,0E1,D0+1−qcVt,ut≤−c3kDut,0E1.Finally, by applyingc1,c2, and −c3/k in Theorem 1, we get(53)Dut,0E1≤V0c1E1−q−c3c2kt1−q,where V0=V0,u0.Finally, all the conditions of Theorem1 are satisfied; thus, it follows that problem (47) is Mittag-Leffler stable.


## 3.1. Illustrative Example

The following example is used to demonstrate the applicability of our stability result. We consider the following fuzzy nonlinear system:(47)cD0+qut=ft,ut,u0=u0,where 0<q<1, ue=0E1 is the equilibrium point of problem (47), and f:J×E1⟶E1 is a fuzzy Lipschitz function with Lipschitz constant k>0.We suppose that there exists a Lyapunov functionVt,ut satisfying the following conditions:(48)Dc1ut,0E1≤Vt,ut≤Dc2ut,0E1,(49)V′t,u≤D−c3u,0E1,where c1, c2, and c3 are positive constants and V′t,u=dVt,ut/dt.By using inequalities (48) and (49), we can write(50)D0+1−qcVt,ut=I0+qV′t,u≤I0+q−c3Dut,0E1.Since the fuzzy functionf:J×E1⟶E1 is Lipschitz with a real constant k>0, it follows that(51)Dft,u,0E1≤k,Dut,0E1.Thus,(52)D0+1−qcVt,ut≤−c3kI0+qDft,u,0E1,D0+1−qcVt,ut≤−c3kDI0+qft,u,0E1,D0+1−qcVt,ut≤−c3kDut,0E1.Finally, by applyingc1,c2, and −c3/k in Theorem 1, we get(53)Dut,0E1≤V0c1E1−q−c3c2kt1−q,where V0=V0,u0.Finally, all the conditions of Theorem1 are satisfied; thus, it follows that problem (47) is Mittag-Leffler stable.


## 4. Conclusion and Future Works

In this paper, we studied the stability result of fuzzy fractional differential equations by introducing Mittag-Leffler stability notion. Indeed, we discussed some sufficient criteria to demonstrate the stability of the trivial solution of the proposed system.Our future work is to establish the Mittag-Leffler stability of intuitionistic fuzzy multivariable nonlinear fractional systems.



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