Asymptotic Stability of Distributed-Order Nonlinear Time-Varying Systems with the Prabhakar Fractional Derivatives

In recent years, distributed-order fractional calculus has played a significant role in many areas of science, engineering, and mathematics [1–5]. 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


Introduction
In recent years, distributed-order fractional calculus has played a significant role in many areas of science, engineering, and mathematics [1][2][3][4][5]. 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 cðμÞ is an absolutely integrable function in the interval μ ∈ ½0, 1 and C D γ ρ,cðμÞ,ω is a distributed-order fractional differential/integral operator in the sense of a given fractional differential/integral operator of order cðμÞ 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.

Preliminaries
In this section, we recall some definitions and lemmas which are used in the next sections.
where FðsÞ is the Laplace transform of f ðtÞ and the proof is completed. The Laplace transform of the Caputo-Prabhakar distributed-order derivative is obtained as where XðsÞ is the Laplace transform of xðtÞ and CðsÞ = Ð m m−1 cðμÞs μ dμ: for all ðt, xÞ, ðt, yÞ ∈ D: The constant L is called a Lipschitz constant for f ðt, xÞ with respect to x on D: Definition 10. A real-valued continuous function f ðt, xÞ is said to satisfy a Lipschitz condition with respect to x on D = ½0,∞Þ provided there is a constant L such that

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 C D γ ρ,cðμÞ,ω,0 + xðtÞ < M, f ðxðtÞ, tÞ ∈ L 1 ½0,∞ and f is a real-value continuous function. Also. in the above, cðμÞ is an absolutely integrable function and it satisfies Ð 1 0 cðμÞs μ dμ ≠ 0, RðsÞ > 0. 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 x ∈ ℝ be a continuous and derivable function. Then, for any time instant t ≥ 0, we have: Proof. Proving that expression (20) is true, to prove that Relation (21) can be written as Let us define the auxiliary variable yðτÞ = xðtÞ − xðτÞ; in this way, expression (24) can be written as 3 Abstract and Applied Analysis and taking the integration by parts of (25) Let us check the first term of relation (27) which has an indetermination at τ = t, so let us analyze the corresponding limit. Now, we show that there exists lim τ→t − ½y 2 ðτÞ E −γ ρ,1−μ ðωðt − τÞ ρ Þ/2ðt − τÞ μ and its value is zero, then we have since it results in 0/0, by applying the L'Hôpital rule on (3-10), we obtain Thus, relation (27) is reduced to Due to t ≥ τ, t ≥ 0 and features of the gamma function, equation (31) is clearly true, and this concludes the proof.
Remark 12. Lemma 11 is valid for xðtÞ ∈ ℝ n Lemma 13. Let xðtÞ be defined as in Remark 12. Then, for any t ≥ t 0 , the following relationship holds: Proof. Multiplying both sides of (20) by cðμÞ ≥ 0 and integrating with respect to μ in the interval (0,1), the desired result is obtained.
Proof. Adding up a nonnegative function MðtÞ to the righthand side of the inequality C D γ ρ,cðμÞ,ω,0 + xðtÞ≥ C D γ ρ,cðμÞ,ω,0 + yðtÞ, 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 L −1 f1/ð1 − ωs −ρ Þ γ CðsÞ ; t − ξg, MðξÞ are nonnegative, then xðtÞ ≥ yðtÞ: According to Lemma 14, the following corollary is obtained.
By applying the Laplace transform on both sides of (41), we have and solving for VðsÞ: 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 gðtÞ = L −1 f1/ðð1 − ωs −ρ Þ γ CðsÞ + α 3 /α 2 Þ ; tg: Considering that CðsÞ is such that gðtÞ ≥ 0, ∀t ≥ 0, and MðtÞ, ∀t ≥ 0, then Using Lemma 8 and the hypothesis for the function, we get Using equations (38) and  and considering that V ðtÞ ≥ 0, ∀t ≥ 0, we can get since α 1 , a > 0, then we obtain lim t→∞ ∥xðtÞ∥ = 0: The proof is complete.
The following lemma allows us to determine asymptotic stability by analyzing the integer order derivative of an appropriate Lyapunov function.

Numerical Results
In this section, two numerical examples for the distributedorder 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 μ ∈ ð0, 1Þ To use Theorem 16, first we show that L −1 f1/ðð1 − ωs −ρ Þ γ CðsÞ + α 3 /α 2 Þg ≥ 0, for all t ≥ 0 is hold, then letting α 1 = 1/4,α 2 = 1, and α 3 = 4, we obtain Using Lemma 5 on equation (63), we obtain Then, the first part is established. Also, all the roots of this function 4 − s 2/3 ð1 − ωs −ρ Þ γ = 0 are located in the open left-half complex plane and this roots can be obtained by s = re iθ . 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 μ ∈ ð0, 1Þ:

Abstract and Applied Analysis
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.

Conclusion
In this paper, we focus on the distributed-order linear and nonlinear time-varying systems containing Caputo-Prabhakar fractional derivative of order cðμÞ. With the expansion the Lyapunov direct method to the distributedorder 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.

Conflicts of Interest
The authors declare that they have no conflicts of interest.