# Stability Analysis of Distributed Order Fractional Differential Equations

Authors: H. Saberi Najafi; A. Refahi Sheikhani; A. Ansari
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We analyze the stability of three classes of distributed order fractional differential equations (DOFDEs) with respect to the nonnegative density function. In this sense, we discover a robust stability condition for these systems based on characteristic function and new inertia concept of a matrix with respect to the density function. Moreover, we check the stability of a distributed order fractional WINDMI system to illustrate the validity of proposed procedure.


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## Body

## 1. Introduction

The fractional differential operator of distributed order(1.1)doDα=∫lub(α)dαdtαdα,u>l≥0,b(α)≥0
is a generalization of the single order soDα=dα/dtα which by considering a continuous or discrete distribution of fractional derivative is obtained.The idea of fractional derivative of distributed order is stated by Caputo [1] and later developed by Caputo himself [2, 3], Bagley and Torvik [4, 5]. Other researchers used this idea, and interesting reviews appeared to describe the related mathematical models of partial fractional differential equation of distributed order.For example, Diethelm and Ford [6] used a numerical technique along with its error analysis to solve the distributed order differential equation and analyze the physical phenomena and engineering problems, see [6] and references therein.Furthermore, some investigation on linear distributed order boundary value problems of form(1.2)∫0mb(α)Dαu(x,t)dα=B(D)u(x,t),D=ddx,t>0,x∈R,
with pseudodifferential operator B(D) and the Cauchy conditions(1.3)∂k∂tku(x,0+)=fk(x),k=0,1,…,m-1,
have been discussed [7–12].In particular cases, the characteristics of time-fractional diffusion equation of distributed order were studied for treatises in the sub-, normal, and superdiffusions.The fractional order applied to dynamical systems is of great importance in applied sciences and engineering [13–19]. The stability results of the fractional order differential equations (FODEs) systems have been a main goal in researches. For example, Matignon considers the stability of FODE system in control processing and Deng has studied the stability of FODE system with multiple time delays [20–23].Now, in this paper, we considerthe distributed order fractional differential equations systems (DOFDEs) with respect to the density function b(α)≥0 as follows:(1.4)doCDtαx(t)=Ax(t),x(0)=x0,0<α≤1,
where x(t)∈ℝn, A∈ℝn×n, and doCDtα=∫01b(α)soCDtαx(t)dα is the Caputo fractional derivative operator of distributed order with respect to the order-density function b(α).Since the solution of the above system is rather complicated similar to FODE systems, therefore, the study of stability for DOFDE is a main task.In this paper, we introduce three classes of DOFDE systems including(1)
distributed order fractional differential systems;(2)
distributed order fractional differential evolution systems with control vector;(3)
distributed order fractional differential evolution systems without control vector.For studying the stability of these classes of DOFDE systems, first, we introduce a characteristic function of a matrix with respect to the distribute functionB(s) where B(s)=∫01b(α)sαdα. Then, we establish a general theory based on new inertia concept for analyzing the stability of distributed order fractional differential equations. The concepts and theorems presented in this paper for DOFDE systems can be considered as generalizations of FODE and ODE systems [21, 24, 25].In Section2, we recall some basic definitions of the Caputo fractional derivative operator, the Mittag-Leffler function, and their elementary properties used in this paper. Section 3 contains the main definitions and theorems for checking the stability of DOFDE systems. Also, we study a distributed order fractional WINDMI system [26] generalized from fractional order to distributed order fractional. In Section 4, we introduce the distributed order fractional evolution systems(1.5)doCDtαx(t)=AdoCDtβx(t)+Bu(t),x(0)=x0,0<β<α≤1,
where u(t) is control vector, and generalize the results obtained in Section 3 for this case. Finally, the conclusions are given in the last section.


## 2. Elementary Definitions and Theorems

In this section, we consider the main definitions and properties of fractional derivative operators ofsingle and distribute order and the Mittag-Leffler function. Also, we recall two important theorems in inverse of the Laplace transform.


### 2.1. Fractional Derivative of Single and Distributed Order

The fractional derivative of single order off(t) in the Caputo sense is defined as [16, 27]
(2.1)soCDtαf(t)=1Γ(m-α)∫0tf(m)(τ)(t-τ)α-m+1dτ,
for m-1<α≤m,m∈ℕ,t>0. The Caputo's definition has the advantage of dealing properly with initial value problems in which the initial conditions are given in terms of the field variables and their integer order which is the case in most physical processes. Fortunately, the Laplace transform of the Caputo fractional derivative satisfies
(2.2)L{soCDtαf(t)}=sαL{f(t)}-∑k=0m-1f(k)(0+)sα-1-k,
where m-1<α≤m and s is the Laplace variable. Now, we generalize the above definition in the fractional derivative of distributed order in the Caputo sense with respect to order-density function b(α)≥0 as follows:
(2.3)doCDtαf(t)=∫m-1mb(α)doCDtαf(t)dα,
and the Laplace transform of the Caputo fractional derivative of distributed order satisfies
(2.4)L{doCDtαf(t)}=∫m-1mb(α)[sαF(s)-∑k=0m-1sα-1-kf(k)(0+)]dα=B(s)F(s)-∑k=0m-11sk+1B(s)f(k)(0+),
where
(2.5)B(s)=∫m-1mb(α)sαdα.


### 2.2. Mittag-Leffler Function

The one-parameter Mittag-Leffler functionEα(z) and the two-parameter Mittag-Leffler function Eα,β(z), which are relevant for their connection with fractional calculus, are defined as(2.6)Eα(z)=∑j=0∞zjΓ(αj+1),α>0,z∈C,(2.7)Eα,β(z)=∑j=0∞zjΓ(αj+β)α,β>0,z∈C.
One of the applicable relations in this paper is the Laplace transforms of the Mittag-leffler function given by(2.8)L(tβ-1Eα,β(λtα))=sα-β(sα-λ),R(s)>|λ|1/α.


### 2.3. Main Theorems about Inverse of the Laplace Transform

Theorem 2.1 (Schouten-Vanderpol Theorem [28]).
Suppose that the functionsF(s),ϕ(s) are analytic in the half plane ℜ(s)>s0, then, the Laplace transform inversion of F(ϕ(s)) can be obtained as
(2.9)L-1{F(ϕ(s))}=∫0+∞f(τ)L-1{e-ϕ(s)τ;t}dτ,
where f(t) is the Laplace transform inversion of the function F(s).Theorem 2.2 (Titchmarsh Theorem [29]).
LetF(s) be an analytic function which has a branch cut on the real negative semiaxis; furthermore, F(s) has the following properties:
(2.10)F(s)=O(1),|s|⟶∞,F(s)=O(1|s|),|s|⟶0,
for any sector |arg(s)|<π-η where 0<η<π. Then, the Laplace transform inversion f(t) can be written as the the Laplace transform of the imaginary part of the function F(re-iπ) as follows:
(2.11)f(t)=L-1{F(s);t}=1π∫0∞e-rtI(F(re-iπ))dr.Theorem 2.3 (Final Value Theorem [28]).
LetF(s) be the Laplace transform of the function f(t). If all poles of sF(s) are in the open left-half plane, then,
(2.12)lim⁡t→∞f(t)=lim⁡s→0sF(s).


## 2.1. Fractional Derivative of Single and Distributed Order

The fractional derivative of single order off(t) in the Caputo sense is defined as [16, 27]
(2.1)soCDtαf(t)=1Γ(m-α)∫0tf(m)(τ)(t-τ)α-m+1dτ,
for m-1<α≤m,m∈ℕ,t>0. The Caputo's definition has the advantage of dealing properly with initial value problems in which the initial conditions are given in terms of the field variables and their integer order which is the case in most physical processes. Fortunately, the Laplace transform of the Caputo fractional derivative satisfies
(2.2)L{soCDtαf(t)}=sαL{f(t)}-∑k=0m-1f(k)(0+)sα-1-k,
where m-1<α≤m and s is the Laplace variable. Now, we generalize the above definition in the fractional derivative of distributed order in the Caputo sense with respect to order-density function b(α)≥0 as follows:
(2.3)doCDtαf(t)=∫m-1mb(α)doCDtαf(t)dα,
and the Laplace transform of the Caputo fractional derivative of distributed order satisfies
(2.4)L{doCDtαf(t)}=∫m-1mb(α)[sαF(s)-∑k=0m-1sα-1-kf(k)(0+)]dα=B(s)F(s)-∑k=0m-11sk+1B(s)f(k)(0+),
where
(2.5)B(s)=∫m-1mb(α)sαdα.


## 2.2. Mittag-Leffler Function

The one-parameter Mittag-Leffler functionEα(z) and the two-parameter Mittag-Leffler function Eα,β(z), which are relevant for their connection with fractional calculus, are defined as(2.6)Eα(z)=∑j=0∞zjΓ(αj+1),α>0,z∈C,(2.7)Eα,β(z)=∑j=0∞zjΓ(αj+β)α,β>0,z∈C.
One of the applicable relations in this paper is the Laplace transforms of the Mittag-leffler function given by(2.8)L(tβ-1Eα,β(λtα))=sα-β(sα-λ),R(s)>|λ|1/α.


## 2.3. Main Theorems about Inverse of the Laplace Transform

Theorem 2.1 (Schouten-Vanderpol Theorem [28]).
Suppose that the functionsF(s),ϕ(s) are analytic in the half plane ℜ(s)>s0, then, the Laplace transform inversion of F(ϕ(s)) can be obtained as
(2.9)L-1{F(ϕ(s))}=∫0+∞f(τ)L-1{e-ϕ(s)τ;t}dτ,
where f(t) is the Laplace transform inversion of the function F(s).Theorem 2.2 (Titchmarsh Theorem [29]).
LetF(s) be an analytic function which has a branch cut on the real negative semiaxis; furthermore, F(s) has the following properties:
(2.10)F(s)=O(1),|s|⟶∞,F(s)=O(1|s|),|s|⟶0,
for any sector |arg(s)|<π-η where 0<η<π. Then, the Laplace transform inversion f(t) can be written as the the Laplace transform of the imaginary part of the function F(re-iπ) as follows:
(2.11)f(t)=L-1{F(s);t}=1π∫0∞e-rtI(F(re-iπ))dr.Theorem 2.3 (Final Value Theorem [28]).
LetF(s) be the Laplace transform of the function f(t). If all poles of sF(s) are in the open left-half plane, then,
(2.12)lim⁡t→∞f(t)=lim⁡s→0sF(s).


## 3. Stability Analysis of Distributed Order Fractional Systems

In this section, we generalize the main stability properties for the linear system of distributed order fractional differential equations in the following form:(3.1)doCDtαx(t)=Ax(t),x(0)=x0,0<α≤1,
where x∈ℝn, the matrix A∈ℝn×n, and doCDtα=∫01b(α)soCDtαx(t)dα is the Caputo fractional derivative operator of distributed order with respect to order-density function b(α)≥0. At first, we obtain the general solution of the system (3.1), and, next, we express the main theorem for checking the stability of this system.By implementation of the Laplace transform on the above system and using the initial condition and relation (2.4), we have(3.2)B(s)x(s)=Ax(s)+1/sB(s)x(0),B(s)=∫01b(α)sαdα,x(s)=B(s)s[B(s)I-A]x(0)=B(s)I-A+As[B(s)I-A]x(0)=1sx(0)+As[B(s)I-A]x(0).
Now, by applying the inverse of Laplace transform on the both sides of above relation, we have(3.3)x(t)=x(0)+L-1{As[B(s)I-A]x(0)}=x(0)+∫0tL-1{1[B(s)I-A]}Ax(0)dt,
which according to the Schouten-Vanderpol and Titchmarsh theorems we get(3.4)L-1{1B(s)I-A}=∫0∞eAτL-1{e-B(s)τ;t}dτ,(3.5)L-1{e-B(s)τ;t}=-1π∫0∞e-rtI(e-B(s)τ)dr=-1π∫0∞e-rt[e-ρcos⁡πγsin⁡(ρsin⁡πγ)]dr,
where B(s)=ρcosπγ+iρsinπγ, ρ=|B(s)|, γ=(1/π)arg[B(s)], and r=eiπ.Finally, by using (3.4) and (3.5), the general solution of the distributed order fractional systems (3.1) is written by(3.6)x(t)=x(0)+1π∫0t∫0∞∫0∞e-rt+Aτ-ρcos⁡πγsin⁡(ρsin⁡πγ)Ax(0)drdτdt.Theorem 3.1.
The distributed order fractional system of (3.1) is asymptotically stable if and only if all roots of det(B(s)I-A)=0 have negative real parts.Proof.
According to the relation (3.2), we have
(3.7)[B(s)I-A]sX(s)=B(s)x(0),
if all roots of the det(B(s)I-A)=0 lie in open left half complex plane (i.e., ℜ(s)<0), then, we consider (3.7) in ℜ(s)≥0. In this restricted area, the relation (3.7) has a unique solution sX(s)=(sX1(s),sX2(s),…,sXn(s)). Since lims→0B(s)=0, so we have
(3.8)lim⁡s→0,R(s)≥0sXi(s)=0,i=1,2,…,n,
which from the final value Theorem 2.3, we get
(3.9)lim⁡t→∞x(t)=lim⁡t→∞(x1(t),x2(t),…,xn(t))=lim⁡s→0,R(s)≥0(sX1(s),sX2(s),…,sXn(s))=0.
The above result shows that the system (3.1) is asymptotically stable.Definition 3.2.
The value ofdet(B(s)I-A) is the characteristic function of the matrix A with respect to the distributed function B(s), where B(s)=∫01b(α)sαdα is the distributed function with respect to the density function b(α)≥0.Definition 3.3.
The eigenvalues ofA with respect to the distributed function B(s) are the roots of the characteristic function of A.The inertia of a matrix is the triplet of the numbers of eigenvalues of A with positive, negative, and zero real parts. In this section, we generalize the inertia concept for analyzing the stability of linear distributed order fractional systems. According to the Theorem (3.1), the transient responses of the system (3.1) are governed by the region where the roots of det(B(s)I-A)=0 are located in the complex plane.Definition 3.4.
The inertia of a matrixA of order n respect to the order distributed function B(s) is the triplet
(3.10)InB(s)(A)=(πB(s)(A),νB(s)(A),δB(s)(A)),
where πB(s)(A), νB(s)(A), and δB(s)(A) are, respectively, the number of roots of det(B(s)I-A)=0 with positive, negative, and zero real parts where B(s)=∫01b(α)sαdα.Definition 3.5.
The matrixA is called a stable matrix with respect to the order distributed function B(s), if all of the eigenvalue of A with respect to the distributed function B(s) have negative real parts.Theorem 3.6.
The linear distributed order fractional system (3.1) is asymptotically stable if and only if any of the following equivalent conditions holds.  (1)
The matrixA is stable with respect to the distribute function B(s).(2)
πB(s)(A)=δB(s)(A)=0.(3)
All rootss of the characteristic function of A with respect to the distributed function B(s) satisfy |arg(s)|>π/2.Proof.
According to Theorem3.1 and the above definitions, proof can be easily obtained.Remark 3.7.
In special case, ifb(α)=δ(α-β), where 0<β≤1 and δ(x) is the Dirac delta function, then, we have the following linear system of fractional differential equations:
(3.11)dβdtβx(t)=Ax(t),x(0)=x0,
and B(s)=sβ. Also, the characteristic matrix and characteristic equation of (3.11) are reduced to sβI-A and det(sβI-A)=0, respectively. Let λ be sβ, then s=λ1/β, and, by using Theorem 3.6, we have |arg(λ1/β)|>π/2. Thus, all the roots λ of equation det(λI-A)=0 satisfy |arg(λ)|>βπ/2. This result is Theorem  2 of [22]. Here, we can very easily prove it by using Theorem 3.6 of the present paper. Particularly, if β=1, then, we have a linear system ẋ(t)=Ax(t). In this case, B(s)=s and the characteristic function of (3.1) are detsI-A. Also, the inertia of matrix A is a triplet (π(A),ν(A),δ(A)), where π(A), ν(A),  and δ(A) are, respectively, the number of eigenvalues of A with positive, negative, and zero real parts. This result is a special case of definition (3.4), which agrees with the typical definitions for typical differential equations.Example 3.8.
The solar-wind-driven magnetosphere-ionosphere (WINDMI) system is a complex driven-damped dynamical system which exhibits a variety of dynamical states that include low-level steady plasma convection, episodic releases of geotail stored plasma energy into the ionosphere known broadly as substorms, and states of continuous strong unloading [30, 31]. If we consider the integer-order WINDMI model as follows:
(3.12)dx1dt=x2,dx2dt=x3,dx3dt=-ax3-x2+b-ex1,
where x1, x2, and x3 are variables and a, b are positive constants, the corresponding distributed order fractional WINDMI system (3.12) can be written in the form:
(3.13)∫01b(α)dαdtαx1(t)dα=x2,∫01b(α)dαdtαx2(t)dα=x3,∫01b(α)dαdtαx3(t)dα=-ax3-x2+b-ex1,
where b(α)≥0 is the density function. As a generalization of nonlinear autonomous FODE into nonlinear autonomous DOFDE, the linearized form of the system (3.13) at the equilibrium point x̂=(lnb,0,0), that is, doCDtαx̂(t)=F(x̂)=0, can be written in the form
(3.14)doCDtαx(t)=Ax(t),
where (t)=(x1(t),x2(t),x3(t)),doCDtαx(t)=∫01b(α)soCDtαx(t)dα,and A=(∂F/∂x)|x=x̂,which is the Jacobian matrix at the equilibrium point [32], is given by
(3.15)A=(010001-b-1-a).
Now, for analyzing the stability of the nonlinear autonomous DFODE, we compute Inb(α)(A) in the case that the density function varies. The results are shown in Table 1 for some parameters a and b.Table 1
Stability analysis of distributed order fractional WINDMI system.
Parametersb(α)=δ(α-β)b(α)=δ(α-β1)+δ(α-β2)b(α)=2αβ=1β=.95β=.65β1=.5,β2=.85β1=.1,β2=.9a=0,b=0(0,0,3)(0,2,1)(0,2,1)(0,2,1)(0,2,1)(2,0,0)a=0,b=1(2,1,0)(1,0,0)(0,1,0)(0,1,0)(0,1,0)(1,0,0)a=1,b=1(0,1,2)(0,2,0)(0,2,0)(0,2,0)(0,2,0)(3,0,0)a=1,b=0(0,2,1)(0,2,1)(0,0,1)(0,0,1)(0,2,1)(2,0,0)a=1,b=0.001(0,3,0)(0,1,0)(0,0,0)(0,0,0)(0,1,0)(1,0,0)


## 4. Distributed Order Fractional Evolution Systems

In this section, as a generalization of the previous systems, we consider the systems of distributed order fractional differential evolution equations and state two theorems in stability of these systems.Theorem 4.1.
Consider linear system of distributed order fractional differential evolution equations,(4.1)doCDtαx(t)=AdoCDtβx(t),x(0)=x0,0<β<α≤1,
where A∈ℝp×p, doCDtαx(t)=∫01b1(α)soCDtαx(t)dα, and doCDtβx(t)=∫01b2(β)soCDtβx(t)dβ. Also, B1(s)=∫01b1(α)sαdα and B2(s)=∫01b2(β)sβdβ. The system (4.1) is stable if and only if all roots of characteristic function of matrix A with respect to the distributed function B1(s)/B2(s) have negative real parts.Proof.
Taking the Laplace transform on both sides of (4.1) gives
(4.2)B1(s)X(s)-B1(s)sx(0)=A[B2(s)X(s)-B2(s)sx(0)],[B1(s)I-AB2(s)]X(s)=1s[B1(s)-AB2(s)]x(0),[B1(s)B2(s)I-A](sX(s)-x(0))=0.
If all roots of characteristic function of matrix A with respect to the distributed function B1(s)/B2(s) have negative real parts,that is, ℜ(s)<0, then, we consider (4.2) in ℜ(s)≥0. In this restricted area by using final-value theorem of Laplace transform, we have
(4.3)lim⁡t→∞x(t)=lim⁡s→0,R(s)≥0sX(s)=x0.Theorem 4.2.
Consider the linear system of distributed order fractional differential evolution equations(4.4)doCDtαx(t)=AdoCDtβx(t)+Bu(t),x(0)=x0,0<β<α≤1,
with the same hypotheses described in Theorem 4.1 where B∈ℝn×n and u(t) is a control vector.The linear distributed order fractional system (4.4) is stabilizable if and only if there exists a linear feedback u(t)=YdoCDtβx(t), with Y∈ℝn×n, such that A+BY is stable with respect to the distributed function B1(s)/B2(s).Proof.
The proof can be easily expressed similar to Theorem4.1.Remark 4.3.
Ifb1(α)=δ(α-α1) and b2(β)=δ(β-β1) where 0<β1<α1≤1 then (4.4) is reduced to the following linear system of fractional differential equations:
(4.5)dα1dtα1x(t)=Adβ1dtβ1x(t)+Bu(t),x(0)=x0,0<β1<α1≤1.
By applying the Laplace transform on the above system and using the initial condition, we have
(4.6)sα1X(s)-sα1-1x(0)=A[sβ1X(s)-sβ1-1x(0)]+BU(s),
where X(s) is the Laplace transform of x(t), U(s) is the Laplace transform of u(t), and B(s)=∫01b(α)sαdα. Thus, we can write X(s) as,
(4.7)X(s)=BU(s)+sα1-1x(0)s1αI-As1β+sβ1-1Ax(0)s1αI-As1β=s-β1sα1-β1I-ABU(s)+sα1-β1-1sα1-β1I-Ax(0)+s-1sα1-β1I-AAx(0).
Applying the inverse Laplace transform to (4.7) and using property (2.8), we get
(4.8)x(t)=∫0t(t-x)α1-1Eα1-β1,α1(A(t-x)α1-β1)Bu(x)dx+Eα1-β1,1(Atα1-β1)x(0)+tα1-β1Eα1-β1,α1-β1+1(Atα1-β1)Ax(0).
Therefore, (4.5) is asymptotically stable if all eigenvalues of A with respect to the distributed function B1(s)/B2(s)=sα1-β1 have negative real parts which is a special case of Theorem 4.2.


## 5. Conclusions and Future Works

In this work, we introduced three classes of the distributed order fractional differential systems, the distributed order fractional differential evolution systems with control vector, and the distributed order fractional differential evolution systems without control vector. The analysis of the asymptotically stability for such systems based on Theorem3.1 and several interesting stability criteria are derived according to Theorem 3.6. Moreover, a numerical example was given to verify the effectiveness of the proposed schemes.In view of the above result, for future works, our attention may be focused on generalizing the numerical methods for computing the eigenvalues of a matrix with respect to the distributed function. The proposed algorithms in [33–35] for computing the eigenvalues of a matrix may be effective in this case.



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