#### Abstract

In this article, we survey the Lyapunov direct method for distributed-order nonlinear time-varying systems with the Prabhakar fractional derivatives. We provide various ways to determine the stability or asymptotic stability for these types of fractional differential systems. Some examples are applied to determine the stability of certain distributed-order systems.

#### 1. Introduction

In recent years, distributed-order fractional calculus has played a significant role in many areas of science, engineering, and mathematics [15]. For the first time in 1969, the distributed-order fractional calculus with the Caputo fractional derivatives was surveyed by Caputo [6]. Later, other research on the distributed-order fractional derivatives was presented. For example, Fernández-Anaya et al. [7] studied asymptotic stability of distributed-order nonlinear dynamical systems with the Caputo fractional derivative. Moreover, Duong et al. [3] studied the deterministic analysis of distributed-order systems using operational matrix. A new method for obtaining the numerical solution of distributed-order time-fractional-subdiffusion equations (DO-TFSDE) of the fourth order is studied in [8], and solving a two-dimensional distributed-order time-fractional fourth-order partial differential equation by using of the space-time Petrov-Galerkin spectral method is studied in [9]. The stability of distributed-order fractional differential systems with respect to the nonnegative density function has also been studied [10, 11]. We define fractional distributed-order nonautonomous systems of the form where is an absolutely integrable function in the interval and is a distributed-order fractional differential/integral operator in the sense of a given fractional differential/integral operator of order which discusses about the stability or asymptotic stability for these systems. Our interest in choosing this type of derivative is related to the three-parameter Mittag-Leffler function. One useful application of the three-parameter Mittag-Leffler function in mathematics has been related to their importance in fractional calculus as a model of complex susceptibility in the response of disordered materials and heterogeneous systems [12], in the response in anomalous dielectrics of Havriliak-Negami type [13], in fractional viscoelasticity [14], in the discussion of stochastic processes [15], in probability theory [16], in the description of dynamical models of spherical stellar systems [17], in the polarization processes in Havriliak-Negami models [13, 18], and in fractional or integral differential equations [19]. In this paper, we intend to survey the stability or asymptotic stability analysis of a distributed-order fractional differential/integral operator containing the Prabhakar fractional derivatives. This type of fractional derivative was introduced by Garra et al. [20] in that it is considered in terms of the generalized Mittag-Leffler function and can be considered as a generalization of the most popular definitions of fractional derivatives. In the field of stability and asymptotic stability, several papers have been published as follows: in [21], the Hyers-Ulam stability of the linear and nonlinear differential equations of fractional order with Prabhakar derivative by using the Laplace transform method is studied and the authors show that the fractional equation introduced is Hyers-Ulam stable, and in [22], the authors obtained the stability regions of differential systems of fractional order with the Prabhakar fractional derivatives. For this purpose, in Section 2, we recall some definitions and lemmas in generalized fractional calculus. In Section 3, we introduce the distributed-order nonlinear time-varying systems containing the Prabhakar fractional derivative and discuss about the stability analysis of these types of fractional differential systems. In Section 4, we plot two examples in order to show the performance and accuracy of the proposed method.

#### 2. Preliminaries

In this section, we recall some definitions and lemmas which are used in the next sections. where is the gamma function. Also, for the absolutely continuous function , the Caputo fractional derivatives of order is defined as follows: where is the generalized Mittag-Leffler function introduced by Prabhakar in 1971 [25]:

Definition 1. (see [23, 24]). Let and , . Then, the Riemann-Liouville fractional integral and derivative of order are defined as

Definition 2. (see [20]). For and function , , the Prabhakar fractional integral is defined as follows:

Definition 3. (see [20]). For the function , the Prabhakar fractional derivative is defined as

Also, analogous formulas for the Caputo-Prabhakar fractional derivative are given by where and . where is the Laplace transform of and the proof is completed.

Lemma 4. (see [20]). The Laplace transform of the Prabhakar fractional derivative for is given by

Lemma 5. The Laplace transform of the generalized Mittag-Leffler function is given by [20]

Lemma 6. (see [23]). Let . Then, for any , the generalized Mittag-Leffler function derivative is defined as

Lemma 7. The Laplace transform of (7) is given by

Proof. Using the definition of and equation (10), we obtain

Lemma 8. (see [26]). Let If all poles of are in the open left-half complex plane, then

Definition 9. The distributed-order fractional integral operator in the Caputo-Prabhakar sense with respect to an order density function is defined by

The Laplace transform of the Caputo-Prabhakar distributed-order derivative is obtained as where is the Laplace transform of and for all The constant is called a Lipschitz constant for with respect to on

Definition 10. A real-valued continuous function is said to satisfy a Lipschitz condition with respect to on provided there is a constant such that

#### 3. The Distributed-Order Fractional Integral Operator

In this section, we state the stability and asymptotic stability of the distributed-order nonlinear time-varying systems as where and is a real-value continuous function. Also. in the above, is an absolutely integrable function and it satisfies . Assuming the above conditions are satisfied for the system (18), in this case, to prove the existence and uniqueness of system (18), we can perform a process similar to [4], and assuming that the system solution will be as follows, these solutions are obtained by taking the Laplace transform from both sides of system ((18)): using equation (7) in Definition 3 for (21), it can be written as follows: and in the same way,

Lemma 11. Let be a continuous and derivable function. Then, for any time instant , we have:

Proof. Proving that expression (20) is true, to prove that

Relation (21) can be written as

Let us define the auxiliary variable ; in this way, expression (24) can be written as defining and taking the integration by parts of (25) turns it into

Let us check the first term of relation (27) which has an indetermination at , so let us analyze the corresponding limit. Now, we show that there exists and its value is zero, then we have since it results in , by applying the L’Hôpital rule on (3-10), we obtain

Thus, relation (27) is reduced to

Due to and features of the gamma function, equation (31) is clearly true, and this concludes the proof.

Remark 12. Lemma 11 is valid for

Lemma 13. Let be defined as in Remark 12. Then, for any , the following relationship holds:

Proof. Multiplying both sides of (20) by and integrating with respect to in the interval (0,1), the desired result is obtained.

Lemma 14. Let and is such that the operator takes nonnegative functions into nonnegative functions. If and then .

Proof. Adding up a nonnegative function to the right-hand side of the inequality we have

Using formula (16) and taking the Laplace transform of (34), we have

Thus,

At this point, by applying the inverse of the Laplace transform on both sides of the above relation (36) and using the convolution theorem, we then obtain

The second term of the right-hand side of (37) is nonnegative, because are nonnegative, then

According to Lemma 14, the following corollary is obtained. where , The distributed-order fractional system of (18) is asymptotically stable in if the roots of are in the open left-half complex plane, and is such that .

Corollary 15. Let , and the features of Lemma 14 and Lemma 13 are established. Then, the origin of system (18) is stable in that the origin is the equilibrium point, if .

Theorem 16. Let be an equilibrium point for system (18). Let that there exists a Lyapunov function satisfying

Proof. Using equations (38) and (39), we can get

Adding up a nonnegative function to the left-hand side of the last inequality, we have

By applying the Laplace transform on both sides of (41), we have and solving for :

By applying the inverse of the Laplace transform on both sides of the above relation (43), we obtain

We can rewrite the second term of the right-hand side of (44) as where Considering that is such that , and then

Using Lemma 8 and the hypothesis for the function, we get

Using equations (38) and (3-28) and considering that , we can get since , then we obtain The proof is complete.

The following lemma allows us to determine asymptotic stability by analyzing the integer order derivative of an appropriate Lyapunov function. norming both sides of (50) and applying inequality properties, we get since , then we have setting (51) in (53) and we get inequality (49). where And the distribution function satisfies the conditions of Theorem 16, then system (18) is asymptotically stable.

Lemma 17. If then

Proof. We define as follows:

Theorem 18. Assume that satisfies a Lipschitz condition with respect to on . If and there exists a Lyapunov function that satisfies

Proof. By properties of the distributed derivative

Setting (56) in (55), we have

Let be a Lipschitz constant for with respect to on , and using the Lipschitz condition, then we have from (57) using Lemma 17 and considering that we have

Considering it follows from Theorem 16 that the system is asymptotically stable.

#### 4. Numerical Results

In this section, two numerical examples for the distributed-order linear and nonlinear systems are presented to verify the efficiency of the proposed method.

Example 19. Consider the following system of a fractional distributed order, when

To use Theorem 16, first we show that for all is hold, then letting and , we obtain

Using Lemma 5 on equation (63), we obtain

For each , inequality is hold. Then, the first part is established. Also, all the roots of this function are located in the open left-half complex plane and this roots can be obtained by . Now, let us consider the following Lyapunov candidate function:

Using Lemma 13 for (65), we obtain

Substituting system (60) in (66), then we obtain

By Theorem 16, we can conclude that the origin of (60) is asymptotically stable. Figures 1 and 2 demonstrate the behavior of system (60) for a short time scale.

Example 20. In this example, we consider the following nonlinear system of fractional distributed order when :

With the same process as Example 19, we consider the following Lyapunov candidate function:

Using Lemma 13 for (70), we obtain

Substituting system (68) in (71), we then have

By Theorem 16, we can conclude that the origin of (68) is asymptotically stable. Figures 3 and 4 demonstrate the behavior of system (68) for a short time scale.

#### 5. Conclusion

In this paper, we focus on the distributed-order linear and nonlinear time-varying systems containing Caputo-Prabhakar fractional derivative of order . With the expansion the Lyapunov direct method to the distributed-order case, we state that stability and asymptotic stability results in this kind of systems. Also, in this paper, Lemma 11 is a generalization of Lemma 1 in [27], Theorem 16 is a generalization of Theorem 3 in [7], Lemma 17 is a generalization of Lemma 4 in [7], and Theorem 18 is a generalization of Theorem 4 in [7]. In order to demonstration the validity and applicability of the obtained results in this paper, two examples are shown.