Stability Analysis of Distributed Order Fractional Differential Equations

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.

1. Introduction

The fractional differential operator of distributed order_do𝐷^𝛼=^𝑢𝑙𝑏(𝛼)𝑑^𝛼𝑑𝑡^𝛼𝑑𝛼,𝑢>𝑙≥0,𝑏(𝛼)≥0(1.1) is a generalization of the single order _so𝐷^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.

Furthermore, some investigation on linear distributed order boundary value problems of form^𝑚0𝑏(𝛼)𝐷^𝛼𝑢(𝑥,𝑡)𝑑𝛼=𝐵(𝐷)𝑢(𝑥,𝑡),𝐷=𝑑𝑑𝑥,𝑡>0,𝑥∈𝑅,(1.2) with pseudodifferential operator 𝐵(𝐷) and the Cauchy conditions𝜕^𝑘𝜕𝑡^𝑘𝑢𝑥,0⁺=𝑓_𝑘(𝑥),𝑘=0,1,…,𝑚−1,(1.3) 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 consider the distributed order fractional differential equations systems (DOFDEs) with respect to the density function 𝑏(𝛼)≥0 as follows:^𝐶do𝐷^𝛼𝑡𝑥(𝑡)=𝐴𝑥(𝑡),𝑥(0)=𝑥₀,0<𝛼≤1,(1.4) where 𝑥(𝑡)∈ℝ^𝑛, 𝐴∈ℝ^𝑛×𝑛, and ^𝐶do𝐷^𝛼𝑡=∫¹⁰𝑏(𝛼)^𝐶so𝐷^𝛼𝑡𝑥(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)=∫¹⁰𝑏(𝛼)𝑠^𝛼𝑑𝛼. 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 ^𝐶do𝐷^𝛼𝑡𝑥(𝑡)=𝐴^𝐶do𝐷^𝛽𝑡𝑥(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] ^𝐶so𝐷^𝛼𝑡𝑓(𝑡)=1Γ(𝑚−𝛼)^𝑡0𝑓^(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^𝐶so𝐷^𝛼𝑡𝑓(𝑡)=𝑠^𝛼ℒ{𝑓(𝑡)}−^𝑚−1_𝑘=0𝑓^(𝑘)(0+)𝑠^{𝛼−1−𝑘},(2.2) 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: ^𝐶do𝐷^𝛼𝑡𝑓(𝑡)=^𝑚𝑚−1𝑏(𝛼)^𝐶do𝐷^𝛼𝑡𝑓(𝑡)𝑑𝛼,(2.3) and the Laplace transform of the Caputo fractional derivative of distributed order satisfies ℒ^𝐶do𝐷^𝛼𝑡𝑓(𝑡)=^𝑚𝑚−1𝑏(𝛼)𝑠^𝛼𝐹(𝑠)−^𝑚−1_𝑘=0=𝐵(𝑠)𝐹(𝑠)−_𝑚−1𝑠^{𝛼−1−𝑘}𝑓^(𝑘)0⁺𝑑𝛼_𝑘=01𝑠^𝑘+1𝐵(𝑠)𝑓^(𝑘)0⁺,(2.4) where 𝐵(𝑠)=^𝑚𝑚−1𝑏(𝛼)𝑠^𝛼𝑑𝛼.(2.5)

2.2. 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 as𝐸_𝛼(𝑧)=^∞_𝑗=0𝑧^𝑗Γ(𝛼𝑗+1),𝛼>0,𝑧∈ℂ,(2.6)𝐸_𝛼,𝛽(𝑧)=^∞_𝑗=0𝑧^𝑗Γ(𝛼𝑗+𝛽)𝛼,𝛽>0,𝑧∈ℂ.(2.7) One of the applicable relations in this paper is the Laplace transforms of the Mittag-leffler function given byℒ𝑡^𝛽−1𝐸_𝛼,𝛽(𝜆𝑡^𝛼)=𝑠^𝛼−𝛽(𝑠^𝛼−𝜆),ℜ(𝑠)>||𝜆||^1/𝛼.(2.8)

2.3. 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)>s₀, then, the Laplace transform inversion of F(𝜙(s)) can be obtained as ℒ⁻¹{𝐹(𝜙(𝑠))}=^0+∞𝑓(𝜏)ℒ⁻¹𝑒^{−𝜙(𝑠)𝜏};𝑡𝑑𝜏,(2.9) 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: 𝐹𝐹(𝑠()𝑠=)=𝑂𝑂(11),|𝑠|⟶∞,|𝑠|,|𝑠|⟶0,(2.10) 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: 𝑓(𝑡)=ℒ⁻¹{𝐹(𝑠);𝑡}=1𝜋^∞0𝑒^−𝑟𝑡ℑ𝐹𝑟𝑒^−𝑖𝜋𝑑𝑟.(2.11)

Theorem 2.3 (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, lim_𝑡→∞𝑓(𝑡)=lim_𝑠→0𝑠𝐹(𝑠).(2.12)

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:^𝐶do𝐷^𝛼𝑡𝑥(𝑡)=𝐴𝑥(𝑡),𝑥(0)=𝑥₀,0<𝛼≤1,(3.1) where 𝑥∈ℝ^𝑛, the matrix 𝐴∈ℝ^𝑛×𝑛, and ^𝐶do𝐷^𝛼𝑡=∫¹⁰𝑏(𝛼)^𝐶so𝐷^𝛼𝑡𝑥(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 have𝐵(𝑠)𝑥𝑥((𝑠𝑠))==𝐴𝑥(𝑠𝐵)+(𝑠)1/𝑠𝐵(𝑠)𝑥(0),𝐵(𝑠)=¹⁰𝑏(𝛼)𝑠^𝛼𝑑𝛼,=1𝑠[𝐵(𝑠)𝐼−𝐴]𝑥(0)=𝐵𝑠[(𝐵𝑠)(𝐼𝑠−)𝐼𝐴−+𝐴𝐴]𝑥(0)𝑠𝑥(0)+𝑠[𝐵(𝑠𝐴)𝐼−𝐴]𝑥(0).(3.2) Now, by applying the inverse of Laplace transform on the both sides of above relation, we have𝑥(𝑡)=𝑥(0)+ℒ⁻¹=𝑥(0)+_𝑡0𝑠[𝐵(𝑠𝐴)𝐼−𝐴]𝑥(0)ℒ⁻¹[𝐵(𝑠)1𝐼−𝐴]𝐴𝑥(0)𝑑𝑡,(3.3) which according to the Schouten-Vanderpol and Titchmarsh theorems we getℒ⁻¹𝐵(𝑠)1𝐼−𝐴=^∞0𝑒^𝐴𝜏ℒ⁻¹𝑒^{−𝐵(𝑠)𝜏};𝑡𝑑𝜏,(3.4)ℒ⁻¹𝑒^{−𝐵(𝑠)𝜏};𝑡=−1𝜋^∞0=−1𝑒^−𝑟𝑡ℑ𝑒^{−𝐵(𝑠)𝜏}𝑑𝑟𝜋^∞0𝑒^−𝑟𝑡𝑒^{−𝜌cos𝜋𝛾}sin(𝜌sin𝜋𝛾)𝑑𝑟,(3.5) where 𝐵(𝑠)=𝜌cos𝜋𝛾+𝑖𝜌sin𝜋𝛾, 𝜌=|𝐵(𝑠)|, 𝛾=(1/𝜋)arg[𝐵(𝑠)], and 𝑟=𝑒^𝑖𝜋.

Finally, by using (3.4) and (3.5), the general solution of the distributed order fractional systems (3.1) is written by𝑥(𝑡)=𝑥(0)+1𝜋^𝑡0^∞0^∞0𝑒^{−𝑟𝑡+𝐴𝜏−𝜌cos𝜋𝛾}sin(𝜌sin𝜋𝛾)𝐴𝑥(0)𝑑𝑟𝑑𝜏𝑑𝑡.(3.6)

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 [𝐵(𝑠)𝐼−𝐴]𝑠𝑋(𝑠)=𝐵(𝑠)𝑥(0),(3.7) 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),…,𝑠𝑋_𝑛(𝑠)). Since lim_𝑠→0𝐵(𝑠)=0, so we have lim_{𝑠→0,ℜ(𝑠)≥0}𝑠𝑋_𝑖(𝑠)=0,𝑖=1,2,…,𝑛,(3.8) which from the final value Theorem 2.3, we get lim_𝑡→∞𝑥(𝑡)=lim_𝑡→∞𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)=lim_{𝑠→0,ℜ(𝑠)≥0}𝑠𝑋₁(𝑠),𝑠𝑋₂(𝑠),…,𝑠𝑋_𝑛(𝑠)=0.(3.9) The above result shows that the system (3.1) is asymptotically stable.

Definition 3.2. 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)=∫¹⁰𝑏(𝛼)𝑠^𝛼𝑑𝛼 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.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 matrix 𝐴 of order 𝑛 respect to the order distributed function 𝐵(𝑠) is the triplet 𝐼𝑛_𝐵(𝑠)(𝐴)=𝜋_𝐵(𝑠)(𝐴),𝜈_𝐵(𝑠)(𝐴),𝛿_𝐵(𝑠)(𝐴),(3.10) where 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)=∫¹⁰𝑏(𝛼)𝑠^𝛼𝑑𝛼.

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). (2)𝜋_𝐵(𝑠)(𝐴)=𝛿_𝐵(𝑠)(𝐴)=0. (3)All roots 𝑠 of the characteristic function of 𝐴 with respect to the distributed function 𝐵(𝑠) satisfy |arg(𝑠)|>𝜋/2.

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: 𝑑^𝛽𝑑𝑡^𝛽𝑥(𝑡)=𝐴𝑥(𝑡),𝑥(0)=𝑥₀,(3.11) and 𝐵(𝑠)=𝑠^𝛽. Also, the characteristic matrix and characteristic equation of (3.11) are reduced to 𝑠^𝛽𝐼−𝐴 and det(𝑠^𝛽𝐼−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 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: 𝑑𝑥₁𝑑𝑡=𝑥²𝑑𝑥₂,𝑑𝑡=𝑥³𝑑𝑥₃,𝑑𝑡=−𝑎𝑥³−𝑥₂+𝑏−𝑒^𝑥1,(3.12) 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: ¹⁰𝑏(𝛼)𝑑^𝛼𝑑𝑡^𝛼₁₀𝑥₁(𝑡)𝑑𝛼=𝑥₂,𝑏(𝛼)𝑑^𝛼𝑑𝑡^𝛼₁₀𝑥₂(𝑡)𝑑𝛼=𝑥₃,𝑏(𝛼)𝑑^𝛼𝑑𝑡^𝛼𝑥₃(𝑡)𝑑𝛼=−𝑎𝑥₃−𝑥₂+𝑏−𝑒^𝑥1,(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 𝑥̂=(ln𝑏,0,0), that is, ^𝐶do𝐷^𝛼𝑡𝑥̂(𝑡)=𝐹(𝑥̂)=0, can be written in the form ^𝐶do𝐷^𝛼𝑡𝑥(𝑡)=𝐴𝑥(𝑡),(3.14) where (𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡)), ^𝐶do𝐷^𝛼𝑡𝑥(𝑡)=∫¹⁰𝑏(𝛼)^𝐶so𝐷^𝛼𝑡𝑥(𝑡)𝑑𝛼, and 𝐴=(𝜕𝐹/𝜕𝑥)|_𝑥=𝑥̂, which is the Jacobian matrix at the equilibrium point [32], is given by 𝐴=−00𝑏−101−01𝑎.(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.

Table 1 

Stability analysis of distributed order fractional WINDMI system.

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, ^𝐶do𝐷^𝛼𝑡𝑥(𝑡)=𝐴^𝐶do𝐷^𝛽𝑡𝑥(𝑡),𝑥(0)=𝑥₀,0<𝛽<𝛼≤1,(4.1) where 𝐴∈ℝ^𝑝×𝑝, ^𝐶do𝐷^𝛼𝑡𝑥(𝑡)=∫¹⁰𝑏₁(𝛼)^𝐶so𝐷^𝛼𝑡𝑥(𝑡)𝑑𝛼, and ^𝐶do𝐷^𝛽𝑡𝑥(𝑡)=∫¹⁰𝑏₂(𝛽)^𝐶so𝐷^𝛽𝑡𝑥(𝑡)𝑑𝛽. Also, 𝐵₁(𝑠)=∫¹⁰𝑏₁(𝛼)𝑠^𝛼𝑑𝛼 and 𝐵₂(𝑠)=∫¹⁰𝑏₂(𝛽)𝑠^𝛽𝑑𝛽. 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 𝐵₁(𝑠)𝑋(𝑠)−𝐵¹(𝑠)𝑠𝑥(0)=𝐴𝐵²(𝑠)𝑋(𝑠)−𝐵²(𝑠)𝐵𝑠𝑥(0)₁,(𝑠)𝐼−𝐴𝐵₂(𝑠)𝑋(𝑠)=1𝑠𝐵¹𝐵(𝑠)−𝐴𝐵₂(𝑠)𝑥(0),₁(𝑠)𝐵₂(𝑠)𝐼−𝐴(𝑠𝑋(𝑠)−𝑥(0))=0.(4.2) 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 lim_𝑡→∞𝑥(𝑡)=lim_{𝑠→0,ℜ(𝑠)≥0}𝑠𝑋(𝑠)=𝑥₀.(4.3)

Theorem 4.2. Consider the linear system of distributed order fractional differential evolution equations ^𝐶do𝐷^𝛼𝑡𝑥(𝑡)=𝐴^𝐶do𝐷^𝛽𝑡𝑥(𝑡)+𝐵𝑢(𝑡),𝑥(0)=𝑥₀,0<𝛽<𝛼≤1,(4.4) with the same hypotheses described in Theorem 4.1 where 𝐵∈ℝ^𝑛×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)=𝑌^𝐶𝑑𝑜𝐷^𝛽𝑡𝑥(𝑡), with 𝑌∈ℝ^𝑛×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 Theorem 4.1.

Remark 4.3. If 𝑏₁(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: 𝑑^𝛼1𝑑𝑡^𝛼1𝑥(𝑡)=𝐴𝑑^𝛽1𝑑𝑡^𝛽1𝑥(𝑡)+𝐵𝑢(𝑡),𝑥(0)=𝑥₀,0<𝛽₁<𝛼₁≤1.(4.5) By applying the Laplace transform on the above system and using the initial condition, we have 𝑠^𝛼1𝑋(𝑠)−𝑠^𝛼1−1𝑥(0)=𝐴𝑠^𝛽1𝑋(𝑠)−𝑠^𝛽1−1𝑥(0)+𝐵𝑈(𝑠),(4.6) where 𝑋(𝑠) is the Laplace transform of 𝑥(𝑡), 𝑈(𝑠) is the Laplace transform of 𝑢(𝑡), and 𝐵(𝑠)=∫¹⁰𝑏(𝛼)𝑠^𝛼𝑑𝛼. Thus, we can write 𝑋(𝑠) as, 𝑋(𝑠)=𝐵𝑈(𝑠)+𝑠^𝛼1−1𝑥(0)𝑠^𝛼1𝐼−𝐴𝑠^𝛽1+𝑠^𝛽1−1𝐴𝑥(0)𝑠^𝛼1𝐼−𝐴𝑠^𝛽1=𝑠^−𝛽1𝑠^{𝛼1−𝛽1}𝐼−𝐴𝐵𝑈(𝑠)+𝑠𝑠^{𝛼1−𝛽1}𝐼−^{𝛼1−𝛽1−1}𝐴𝑥(0)+𝑠𝑠^{𝛼1−𝛽1}𝐼⁻¹−𝐴𝐴𝑥(0).(4.7) Applying the inverse Laplace transform to (4.7) and using property (2.8), we get 𝑥(𝑡)=+^𝑡0𝐸(𝑡−𝑥)^𝛼1−1_{𝛼1−𝛽1,1}𝐸_{𝛼1−𝛽1,𝛼1}𝐴(𝑡−𝑥)^{𝛼1−𝛽1}𝐵𝑢(𝑥)𝑑𝑥𝐴𝑡^{𝛼1−𝛽1}𝑥(0)+𝑡^{𝛼1−𝛽1}𝐸_{𝛼1−𝛽1,𝛼1−𝛽1+1}𝐴𝑡^{𝛼1−𝛽1}𝐴𝑥(0).(4.8) Therefore, (4.5) is asymptotically stable if all eigenvalues of 𝐴 with respect to the distributed function 𝐵₁(𝑠)/𝐵₂(𝑠)=𝑠^{𝛼1−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 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.