Stability Analysis of Distributed Order Fractional Differential Equations

and Applied Analysis 3 fractional order to distributed order fractional. In Section 4, we introduce the distributed order fractional evolution systems C doD α t x t A C doD β t x t Bu t , x 0 x0, 0 < β < α ≤ 1, 1.5 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 of single 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 of f t in the Caputo sense is defined as 16, 27 C soD α t f t 1 Γ m − α ∫ t 0 f m τ t − τ α−m 1 dτ, 2.1 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 L { C soD α t f t } sαL{f t } − m−1 ∑ k 0 f k 0 sα−1−k, 2.2 wherem−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: C doD α t f t ∫m m−1 b α CdoD α t f t dα, 2.3 and the Laplace transform of the Caputo fractional derivative of distributed order satisfies L { C doD α t f t } ∫m m−1 b α [ sF s − m−1 ∑ k 0 sα−1−kf k 0 ] dα


Introduction
The fractional differential operator of distributed order is a generalization of the single order so D α d α /dt α 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.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][14][15][16][17][18][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 consider the distributed order fractional differential equations systems (DOFDEs) with respect to the density function b α ≥ 0 as follows: where x t ∈ R n , A ∈ R n×n , and C do D α t 1 0 b α C so D α t x 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 function B s where B s 1 0 b α s α dα.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 Section 2, 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 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.

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

Fractional Derivative of Single and Distributed Order
The fractional derivative of single order of f t in the Caputo sense is defined as 16, 27 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 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: and the Laplace transform of the Caputo fractional derivative of distributed order satisfies 2.5

Mittag-Leffler Function
The one-parameter Mittag-Leffler function E α z and the two-parameter Mittag-Leffler function E α,β z , which are relevant for their connection with fractional calculus, are defined as One of the applicable relations in this paper is the Laplace transforms of the Mittag-leffler function given by

Main Theorems about Inverse of the Laplace Transform
Theorem 2.1 Schouten-Vanderpol Theorem 28 .Suppose that the functions F s , φ s are analytic in the half plane R s > s 0 , then, the Laplace transform inversion of F φ s can be obtained as where f t is the Laplace transform inversion of the function F s .
Theorem 2.2 Titchmarsh Theorem 29 .Let F s be an analytic function which has a branch cut on the real negative semiaxis; furthermore, F s has the following properties: 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:

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: where x ∈ R n , the matrix A ∈ R n×n , and C do D α t 1 0 b α C so D α t x 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
Now, by applying the inverse of Laplace transform on the both sides of above relation, we have which according to the Schouten-Vanderpol and Titchmarsh theorems we get Finally, by using 3.4 and 3.5 , the general solution of the distributed order fractional systems 3.1 is written by The above result shows that the system 3.1 is asymptotically stable.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 3.5.The matrix A 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 matrix A is stable with respect to the distribute function B s .
3 All roots s of the characteristic function of A with respect to the distributed function B s satisfy Proof.According to Theorem 3.1 and the above definitions, proof can be easily obtained.
Remark 3.7.In special case, if b 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: and B s s β .Also, the characteristic matrix and characteristic equation of 3.11 are reduced to s β I − A and det s β I − A 0, respectively.Let λ be s β , then s λ 1/β , and, by using Theorem 3.6, we have | arg λ 1/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 det sI − 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
where x 1 , x 2 , and x 3 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
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 x ln b, 0, 0 , that is, C do D α t x t F x 0, can be written in the form , which is the Jacobian matrix at the equilibrium point 32 , is given by

3.15
Now, for analyzing the stability of the nonlinear autonomous DFODE, we compute In b α A in the case that the density function varies.The results are shown in Table 1 for some parameters a and b.

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.

4.7
Applying the inverse Laplace transform to 4.7 and using property 2.8 , we get

4.8
Therefore, 4.5 is asymptotically stable if all eigenvalues of A with respect to the distributed function B 1 s /B 2 s s α 1 −β 1 have negative real parts which is a special case of Theorem 4.2.

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 Theorem 3.1 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.

Furthermore, some investigation
on linear distributed order boundary value problems of formm 0 b α D α u x, t dα B D u x, t , D d dx , t > 0, x ∈ R, 1.2with pseudodifferential operator B D and the Cauchy conditions∂ k ∂t k u x, 0 f k x , k 0, 1, . . ., m − 1, 1.3have been discussed 7-12 .

Definition 3 . 2 . 1 0
The value of det B s I − A is the characteristic function of the matrix A with respect to the distributed function B s , where B s b α s α dα is the distributed function with respect to the density function b α ≥ 0. Definition 3.3.The eigenvalues of A with respect to the distributed function B s are the roots of the characteristic function of A.

Definition 3 . 4 . 1 0
The inertia of a matrix A of order n respect to the order distributed function B s is the tripletIn B s A π B s A , ν B s A , δ B s A , 3.10where π B s A , ν B s A , and δ B 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 b α s α dα.

Remark 4 . 3 .
If b 1 α δ α − α 1 and b 2 β δ β − β 1 where 0 < β 1 < α 1 ≤ 1 then 4.4 is reduced to the following linear system of fractional differential equations: Final Value Theorem 28 .Let F s be the Laplace transform of the function f t .If all poles of sF s are in the open left-half plane, then, I F re −iπ dr.2.11 Theorem 2.3 Aτ−ρ cos πγ sin ρ sin πγ Ax 0 dr dτ dt. 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 sX 1 s , sX 2 s , . . ., sX n s .Since lim s → 0 B s t → ∞ x 1 t , x 2 t , ..., x n t lim s → 0,R s ≥0sX 1 s , sX 2 s , . .., sX n s 0.3.9 The system 4.1 is stable if and only if all roots of characteristic function of matrix A with respect to the distributed function B 1 s /B 2 s have negative real parts.

Table 1 :
Stability analysis of distributed order fractional WINDMI system.
, with Y ∈ R n×n , such that A BY is stable with respect to the distributed functionB 1 s /B 2 s .Proof.The proof can be easily expressed similar to Theorem 4.1.
.2If all roots of characteristic function of matrix A with respect to the distributed function B 1 s /B 2 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 lim