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

Authors: MohammadHossein Derakhshan; Azim Aminataei
Journal: Abstract and Applied Analysis
 (2020)

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## 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.


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## Body

## 1. Introduction

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
(1)CD,cμ,ωγxt=fxt,t,where cμ is an absolutely integrable function in the interval μ∈0,1 and CDρ,cμ,ωγ is a distributed-order fractional differential/integral operator in the sense of a given fractional differential/integral operator of order cμ 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.(2)0+Itαft=1Γα∫0tfτt−τα−1dτ,t>0,0<α<1,0+Dtαft=1Γ1−αddt∫0tfτt−τ−αdτ,t>0,0<α<1,where Γz=∫0∞xz−1e−xdx,Rz>0 is the gamma function. Also, for the absolutely continuous function f, the Caputo fractional derivatives of order α is defined as follows:
(3)0+CDtαft=0+It1−αddtft=1Γ1−α∫0tt−τ−αf′τdτ.(4)Eρ,μ,ω,0+γft=∫0tt−τμ−1Eρ,μγωt−τρfτdτ,where Eρ,μγ is the generalized Mittag-Leffler function introduced by Prabhakar in 1971 [25]:
(5)Eρ,μγz=1Γγ∑n=0∞Γγ+nn!Γρn+μzn,Rρ,Rμ>0.(6)Dρ,μ,ω,0+γft=dmdtmEρ,m−μ,ω,0+−γft,t>0.Definition 1.
(see [23, 24]). Let 0<α<1 and f∈L10,b, 0<t<b≤∞. Then, the Riemann-Liouville fractional integral and derivative of order α are defined asDefinition 2.
(see [20]). For m−1<μ≤m and function f∈L10,b, 0<t<b≤∞, the Prabhakar fractional integral is defined as follows:Definition 3.
(see [20]). For the function f∈L10,b, the Prabhakar fractional derivative is defined asAlso, analogous formulas for the Caputo-Prabhakar fractional derivative are given by(7)CDρ,μ,ω,0+γft=Eρ,m−μ,ω,0+−γdmdtmft,t>0,where m−1<Rμ<m and ρ,μ,ω,γ∈ℂ.
(8)LDρ,μ,ω,0+γft;s=sμ1−ωs−ργFs−∑k=0m−1skDρ,μ−k,ω,0+γf0+,where Fs is the Laplace transform of ft(9)Fs=∫0∞e−stftdt.(10)Ltμ−1Eρ,μ−γωtρ;s=s−μ1−ωs−ργ,∣ωsρ∣<1,Rμ>0.(11)ddtntμ−1Eρ,μγωtρ=tμ−n−1Eρ,μ−nγωtρ.(12)LCDρ,μ,ω,0+γft;s=sμ1−ωs−ργFs−sμ−m1−ωs−ργ∑k=0m−1sm−k−1fk0+,(13)LCDρ,μ,ω,0+γft;s=Ltm−μEρ,μ−γωtρ;sLdmdtmft;s=sμ1−ωs−ργFs−1−ωs−ργ∑k=0m−1sμ−k−1fk0+,and the proof is completed.
(14)limt→∞ft=lims→0sFs.(15)CDρ,cμ,ω,0+γxt=∫m−1mcμCDρ,μ,ω,0+γxtdμ,m−1<μ<m.Lemma 4.
(see [20]). The Laplace transform of the Prabhakar fractional derivative for m−1<Rμ<m is given byLemma 5.
The Laplace transform of the generalized Mittag-Leffler functiontμ−1Eρ,μγωtρis given by [20]Lemma 6.
(see [23]). Let ρ,μ,γ,ω∈ℂRρ>0,Rμ>0. Then, for any n∈ℕ, the generalized Mittag-Leffler function derivative is defined asLemma 7.
The Laplace transform of (7) is given byProof.
Using the definition ofCDρ,μ,ω,0+γft and equation (10), we obtainLemma 8.
(see [26]). Let Fs=Lft;s. If all poles of sFs are in the open left-half complex plane, thenDefinition 9.
The distributed-order fractional integral operator in the Caputo-Prabhakar sense with respect to an order density functioncμ≥0 is defined byThe Laplace transform of the Caputo-Prabhakar distributed-order derivative is obtained as(16)LCDρ,cμ,ω,0+γxt;s=∫m−1mcμsμ1−ωs−ργXs−1−ωs−ργ∑k=0m−1sμ−k−1xk0+dμ=Cs1−ωs−ργXs−1−ωs−ργ∑k=0m−11sk+1Csxk0+,where Xs is the Laplace transform of xt and Cs=∫m−1mcμsμdμ.(17)ft,x−ft,y≤Lx−y,for all t,x,t,y∈D. The constant L is called a Lipschitz constant for ft,x with respect to x on D.Definition 10.
A real-valued continuous functionft,x is said to satisfy a Lipschitz condition with respect to x on D=0,∞ provided there is a constant L 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(18)CDρ,cμ,ω,0+γxt=fxt,t,t∈0,∞,0<μ<1,x0=x0,where CDρ,cμ,ω,0+γxt<M,fxt,t∈L10,∞ and f is a real-value continuous function. Also. in the above, cμ is an absolutely integrable function and it satisfies ∫01cμsμdμ≠0,Rs>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)):
(19)xt=x0+∫0tL−111−ωs−ργCs;t−ξfξdξ.(20)12CDρ,μ,ω,0+γx2t≤xtCDρ,μ,ω,0+γxt,μ∈0,1.(21)xtCDρ,μ,ω,0+γxt−12CDρ,μ,ω,0+γx2t≥0,using equation (7) in Definition 3 for (21), it can be written as follows:
(22)CDρ,μ,ω,0+γxt=∫0tt−τ−μEρ,1−μ−γωt−τρẋτdτ,and in the same way,
(23)12CDρ,μ,ω,0+γx2t=∫0tt−τ−μEρ,1−μ−γωt−τρxτẋτdτ.Lemma 11.
Letx∈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 thatRelation (21) can be written as
(24)∫0tt−τ−μEρ,1−μ−γωt−τρxt−xτẋτdτ≥0.Let us define the auxiliary variableyτ=xt−xτ; in this way, expression (24) can be written as
(25)∫0tt−τ−μEρ,1−μ−γωt−τρyτẏτdτ≤0,defining
(26)du=yτẏτdτ⇒u=12y2τ,v=t−τ−μEρ,1−μ−γωt−τρ⇒dv=−t−τ−μ−1Eρ,−μ−γωt−τρdτ,and taking the integration by parts of (25) turns it into
(27)−y2τEρ,1−μ−γωt−τρ2t−τμτ=t+y20Eρ,1−μ−γωtρ2tμ+12∫0ty2τEρ,2−μ−γωt−τρt−τμ+1dτ≥0.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 limτ→t−y2τEρ,1−μ−γωt−τρ/2t−τμ and its value is zero, then we have
(28)limτ→t−y2τEρ,1−μ−γωt−τρ2t−τμ=−12Γ1−μlimτ→txt−xτ2t−τμ,since it results in 0/0, by applying the L’Hôpital rule on (3-10), we obtain
(29)−12Γ1−μlimτ→t−2ẋτxt+2xτẋτ−μt−τμ−1=−12Γ1−μlimτ→t−2ẋτxt+2xτẋτt−τ1−μ−μ=0.Thus, relation (27) is reduced to
(30)y20Eρ,1−μ−γωtρ2tμ+12∫0ty2τEρ,2−μ−γωt−τρt−τμ+1dτ≥0,(31)y202Γ−γ∑n=0∞Γ−γ+nωnn!Γρn+1−μtρn−μ+12Γ−γ∑n=0∞Γ−γ+nωnn!Γρn+1−μ∫0ty2τt−τρn−μ−1≥0.Due tot≥τ,t≥0 and features of the gamma function, equation (31) is clearly true, and this concludes the proof.
(32)12CDρ,μ,ω,0+γxTtxt≤xTtCDρ,μ,ω,0+γxt,μ∈0,1.(33)12CDρ,cμ,ω,0+γxTtxt≤xTtCDρ,cμ,ω,0+γxt,μ∈0,1.(34)CDρ,cμ,ω,0+γxt=CDρ,cμ,ω,0+γyt+Mt.Remark 12.
Lemma11 is valid for xt∈ℝnLemma 13.
Letxt be defined as in Remark 12. Then, for any t≥t0, the following relationship holds:Proof.
Multiplying both sides of (20) by cμ≥0 and integrating with respect to μ in the interval (0,1), the desired result is obtained.Lemma 14.
Letμ∈0,1 and cμ is such that the operator ∫0tL−11/1−ωs−ργCs;t−ξ.dξ takes nonnegative functions into nonnegative functions. If CDρ,cμ,ω,0+γxt≥CDρ,cμ,ω,0+γyt and x0=y0, then xt≥yt.Proof.
Adding up a nonnegative functionMt to the right-hand side of the inequality CDρ,cμ,ω,0+γxt≥CDρ,cμ,ω,0+γyt, we haveUsing formula (16) and taking the Laplace transform of (34), we have
(35)1−ωs−ργCsxs−1−ωs−ργCssx0=1−ωs−ργCsys−1−ωs−ργCssy0+Ms.Thus,(36)xs=ys+Ms1−ωs−ργCs.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
(37)xt=yt+∫0tL−111−ωs−ργCs;t−ξMξdξ.The second term of the right-hand side of (37) is nonnegative, because L−11/1−ωs−ργCs;t−ξ,Mξ are nonnegative, then xt≥yt.According to Lemma14, the following corollary is obtained.
(38)α1∥x∥a≤Vxt,t≤α2∥x∥ab,(39)CDρ,cμ,ω,0+γVxt,t≤−α3∥x∥ab,where μ∈0,1, a,b,α1,α2,α3>0. The distributed-order fractional system of (18) is asymptotically stable in x=0 if the roots of 1−ωs−ργCs+α3/α2 are in the open left-half complex plane, and cμ is such that L−11/1−ωs−ργCs+α3/α2;t−ξ≥0,∀t≥0.
(40)CDρ,cμ,ω,0+γVxt,t≤−α3α2Vxt,t.Corollary 15.
Letμ∈0,1, 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 xTfxt,t≤0.Theorem 16.
Letx=0 be an equilibrium point for system (18). Let that there exists a Lyapunov function Vxt,t satisfyingProof.
Using equations (38) and (39), we can getAdding up a nonnegative functionMt to the left-hand side of the last inequality, we have
(41)CDρ,cμ,ω,0+γVxt,t+Mt=−α3α2Vxt,t.By applying the Laplace transform on both sides of (41), we have
(42)1−ωs−ργCsVxs,s−1−ωs−ργCssV0+Ms=−α3α2Vxs,s,and solving for Vs:
(43)Vs=Cs1−ωs−ργV0s1−ωs−ργCs+α3/α2−Ms1−ωs−ργCs+α3/α2.By applying the inverse of the Laplace transform on both sides of the above relation (43), we obtain
(44)Vt=L−1Cs1−ωs−ργV0s1−ωs−ργCs+α3/α2−L−1Ms1−ωs−ργCs+α3/α2.We can rewrite the second term of the right-hand side of (44) as
(45)L−1Ms1−ωs−ργCs+α3/α2=Mt∗gt=∫0tMξgt−ξdξ,where gt=L−11/1−ωs−ργCs+α3/α2;t. Considering that Cs is such that gt≥0,∀t≥0, and Mt,∀t≥0, then
(46)Vt≤L−1Cs1−ωs−ργV0s1−ωs−ργCs+α3/α2.Using Lemma8 and the hypothesis for the function, we get
(47)limt→∞Vt≤limt→∞L−1Cs1−ωs−ργV0s1−ωs−ργCs+α3/α2=lims→0sCs1−ωs−ργV0s1−ωs−ργCs+α3/α2=0.Using equations (38) and (3-28) and considering that Vt≥0,∀t≥0, we can get
(48)limt→∞α1∥xt∥a≤limt→∞Vxt,t=0,since α1,a>0, then we obtain limt→∞∥xt∥=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.(49)∥∫0tL−111−ωs−ργCs;t−ξfxξ,ξdξ∥≤≤∫0tL−111−ωs−ργCs;t−ξ∥fxξ,ξ∥dξ.(50)xt=∫0tL−111−ωs−ργCs;t−ξfxξ,ξdξ,norming both sides of (50)
(51)∥xt∥=∥∫0tL−111−ωs−ργCs;t−ξfxξ,ξdξ∥,and applying inequality properties, we get
(52)∥xt∥≤∫0t∥L−111−ωs−ργCs;t−ξ∥∥fxξ,ξ∥dξ,since L−11/1−ωs−ργCs;t−ξ≥0, then we have
(53)∥xt∥≤∫0tL−111−ωs−ργCs;t−ξ∥fxξ,ξ∥dξ,setting (51) in (53) and we get inequality (49).
(54)γ1∥xt∥a≤Vxt,t≤γ2∥xt∥,(55)dVxt,tdt≤−γ3∥xt∥,t≥0,where a,γ1,γ2,γ3>0,CDρ,−1−cμ,ω,0+γxtt=0=0. And the distribution function 1−cμ satisfies the conditions of Theorem 16, then system (18) is asymptotically stable.
(56)CDρ,1−cμ,ω,0+γxt=CDρ,−cμ,ω,0+γdVxt,tdt.Lemma 17.
IfL−11/1−ωs−ργCs;t−ξ≥0 thenProof.
We definext as follows:Theorem 18.
Assume thatfxt,t satisfies a Lipschitz condition with respect to x on D. If x0=0,L−11/1−ωs−ργCs;t−ξ≥0 and there exists a Lyapunov function Vxt,t that satisfiesProof.
By properties of the distributed derivativeSetting (56) in (55), we have
(57)CDρ,1−cμ,ω,0+γVxt,t≤−γ3CDρ,−cμ,ω,0+γ∥xt∥.LetL be a Lipschitz constant for ft,x with respect to x on D, and using the Lipschitz condition, then we have from (57)
(58)CDρ,1−cμ,ω,0+γVxt,t≤−γ3LCDρ,−cμ,ω,0+γ∥fxt,t∥,using Lemma 17 and considering that Dρ,−cμ,ω,0+γfxt,t=xt we have
(59)CDρ,1−cμ,ω,0+γVxt,t≤−γ3L∥CDρ,−cμ,ω,0+γfxt,t∥≤−γ3L∥xt∥.Consideringcμ=1−cμ,α3=γ3/L,b=a−1, 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.(60)CDρ,cμ,ω,0+γx1t=−5x1t+exptx2t,(61)CDρ,cμ,ω,0+γx2t=−5x2t−exptx1t.Example 19.
Consider the following system of a fractional distributed order, whenμ∈0,1To use Theorem16, first we show that L−11/1−ωs−ργCs+α3/α2≥0, for all t≥0 is hold, then letting α1=1/4,α2=1, and α3=4, we obtain
(62)1−ωs−ργCs+α3α2=4−s2/31−ωs−ργ,(63)14−s2/31−ωs−ργ=14∑n=0∞s2/31−ωs−ργ4n,∣s2/31−ωs−ργ∣<1.Using Lemma5 on equation (63), we obtain
(64)L−114−s2/31−ωs−ργ=14∑n=0∞L−1s2/31−ωs−ργ4n=14∑n=0∞14nt−2/3n−1Eρ,−2/3n−nγωtρ,t≥0.For eacht≥0, inequality 1/4∑n=0∞1/4nt−2/3n−1Eρ,−2/3n−nγωtρ≥0 is hold. Then, the first part is established. Also, all the roots of this function 4−s2/31−ωs−ργ=0 are located in the open left-half complex plane and this roots can be obtained by s=reiθ. Now, let us consider the following Lyapunov candidate function:
(65)α1∥xt∥2≤Vx1t,x2t=12x12t+12x22t≤∥xt∥2.Using Lemma13 for (65), we obtain
(66)CDρ,cμ,ω,0+γVx1t,x2t=12CDρ,cμ,ω,0+γx12t+12CDρ,cμ,ω,0+γx22t≤x1tCDρ,cμ,ω,0+γx1t+x2tCDρ,cμ,ω,0+γx2t.Substituting system (60) in (66), then we obtain
(67)CDρ,cμ,ω,0+γVx1t,x2t≤−5x1t2+x2t2≤−α3∥xt∥2.By Theorem16, 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.
(68)CDρ,cμ,ω,0+γx1t=−2x1tx22t+x1t,(69)CDρ,cμ,ω,0+γx2t=−2−x2tx12t+x2t.Example 20.
In this example, we consider the following nonlinear system of fractional distributed order whenμ∈0,1:Figure 1
The numerical results for system (60) at cμ=−δμ−2/3 for Example 19.Figure 2
Stability diagram of Example19 for x1t and x2t.With the same process as Example19, we consider the following Lyapunov candidate function:
(70)α1∥xt∥2≤Vx1t,x2t=12x12t+12x22t≤∥xt∥2.Using Lemma13 for (70), we obtain
(71)CDρ,cμ,ω,0+γVx1t,x2t=12CDρ,cμ,ω,0+γx12t+12CDρ,cμ,ω,0+γx22t≤x1tCDρ,cμ,ω,0+γx1t+x2tCDρ,cμ,ω,0+γx2t.Substituting system (68) in (71), we then have
(72)CDρ,cμ,ω,0+γVx1t,x2t≤−2x1tx1tx22t+x1t−2x2t−x2tx12t+x2t≤−α3∥xt∥2.By Theorem16, 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.Figure 3
The numerical results for system (60) at cμ=−δμ−2/3 for Example 20.Figure 4
Stability simulation of Example20 for x1t and x2t.


## 5. Conclusion

In this paper, we focus on the distributed-order linear and nonlinear time-varying systems containing Caputo-Prabhakar fractional derivative of ordercμ. 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.



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