# Finite-Time Stability of Atangana–Baleanu Fractional-Order Linear Systems

Authors: Jiale Sheng; Wei Jiang; Denghao Pang
Journal: Complexity
 (2020)

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## Abstract

This paper investigates a fractional-order linear system in the frame of Atangana–Baleanu fractional derivative. First, we prove that some properties for the Caputo fractional derivative also hold in the sense of AB fractional derivative. Subsequently, several sufficient criteria to guarantee the finite-time stability and the finite-time boundedness for the system are derived. Finally, an example is presented to illustrate the validity of our main results.


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## Body

## 1. Introduction

The topic of fractional calculus is more than three hundred years old. The number of applications of fractional calculus rapidly grows in recent decades. Fractional differential equations (FDEs) [1–7] have been successfully confirmed to be useful tools in various fields such as electrical circuits, diffusion, economy, and control problem.Recently, many authors try to find new fractional operators with different kernels in order to better describe these phenomena. In 2015, Caputo and Fabrizi [8] present a new definition of fractional derivative with exponential kernel which is called CF fractional derivatives. In 2016, based on CF fractional derivatives, Atangana and Baleanu [9] introduced another new definition of fractional derivatives called AB fractional derivatives with a nonlocal and nonsingular kernel which are built by the generalized Mittag–Leffler function. In addition, the study in [9] derived the fractional integral associate to AB derivatives by taking the inverse Laplace transform and using the convolution theorem. The detail definitions of ABR fractional derivative, ABC fractional derivative, and AB fractional integral will be introduced in Section 2.On the other hand, many basic properties of AB fractional differintegral have already been studied in recent years [10–12], such as integration by parts, mean value theorem and Taylor’s theorem, semigroup property, and product rule and chain rule. And some problems about existence of solution, stability, controllability, and optimal control are also studied by several researchers [13–18]. Having the advantage of nonlocal and nonsingular kernel, AB fractional derivatives have been widely applied in many fields such as diffusion equation [19], electromagnetic waves in dielectric media [20], chaos [21], and circuit model [22].Finite-time stability (FTS) and finite-time boundedness (FTB) analysis is an important problem in control theory [23–29]. Since it always considers the behavior of systems in finite time, it is very useful in many practical application systems. Moreover, it should be noted that FTS and Lyapunov asymptotic stability are independent concepts. In [23], the authors have introduced the definition of FTS and FTB and given the sufficient condition for the FTB of linear time-invariant (LTI) systems with integer order. The FTS and FTB for the fractional-order systems with α∈0,1 have been given in [24]. Motivated by the above works, this paper studies the FTS and FTB for the class of fractional-order LTI systems in the sense of this new type of AB fractional derivatives. First, we establish one new property of this new type of AB fractional derivatives. Second, we introduce the concept of FTS and FTB for the fractional-order LTI system in the sense of AB fractional derivative corresponding with integer-order system and provide the sufficient conditions which guarantee FTS and FTB for a class of fractional-order LTI systems in the sense of AB fractional derivative. Third, the problem of designing feedback controllers to keep FTS and FTB for fractional closed-loop systems in the sense of AB is studied.Throughout this paper,ℜn denotes the n-dimensional Euclidean space, ℜm×n is m×n real matrix, AT is the transpose of matrix A, A−1 is the inverse of matrix A, A>0<0 indicates that the matrix A is positive (negative) definite, and λmaxA and λminA are the maximum and minimum eigenvalues of matrix A.


## 2. Preliminaries

In this section, we present the definition of AB fractional derivatives and integral.In addition, some properties of AB fractional calculus and Mittag–Leffler function are introduced.Definition 1.
(see [9]). Let 0<α<1, f∈L10,T, and T>0. Then, the AB fractional integral operator ItαAB is defined by(1)ItαABft=1−αBαft+αBαItαft,where Itαft=1/Γα∫0tt−sα−1fsds is the classical Riemann–Liouville fractional integral, Bα denotes a real-valued normalization function satisfying Bα>0, and B0=B1=1.Definition 2.
(see [9]). Let 0<α<1, f∈L10,T, and T>0. Then, the ABR fractional derivative of order α is defined by(2)DtαABRft=Bα1−αddt∫0tEα−α1−αt−sαfsds,where Eα is one-parameter Mittag–Leffler function denoted by Eαt=∑n=0∞tn/Γαn+1.Definition 3.
(see [9]). Let 0<α<1, f∈L10,T, and T>0. Then, the ABC fractional derivative of order α is defined by(3)ItαABCft=Bα1−α∫0tEα−α1−αt−sαf′sds.
In [9, 12], the authors have introduced some basic properties of AB fractional differintegral which will be used in this paper. For 0<α<1, the relation between AB integral operators and AB differential operators is given as(4)ItαABDtαABRft=ft,ItαABDtαABRft=ft−f0.
The Laplace transform of AB fractional derivative is defined as follows:(5)LDtαABCft;s=Bα1−α⋅sαFs−sα−1f0sα+α/1−α,where Fs=Lft;s=∫0∞e−stftds.
In addition, the Mittag–Leffler function which plays an important role in AB fractional derivative will appear frequently in this paper. It is indispensable to introduce some more properties of the Mittag–Leffler function. The generalized Mittag–Leffler functions (two parameters) are denoted byEα,βt=∑n=0∞tn/Γαn+β, α,β>0, and t∈ℜ. For β=1, Eα,1t=Eαt. The Laplace transform of the Mittag–Leffler function is given as(6)Ltβ−1Eα,β−λtα;s=sα−βsα+λ.
In [30], Wang et al. provide some properties which are useful to prove Lemma 2. It is easy to note that Eα′t=1/αEα,αt. So, we have(7)λt−sα−1Eα,α−λt−sα=ddtEα−λt−sα.And for 0<α<1, the functions Eα⋅ and Eα,α⋅ are nonnegative, and for t=0, we have Eα0=1.


## 3. Lemmas

This section will introduce two important lemmas. In [29], Ye et al. have obtained the generalized Gronwall inequality in the sense of Caputo derivative which has wide applications in fractional differential equations. On this basis, Jarad et al. [15] proposed the Gronwall inequality in the frame of AB fractional derivative.Lemma 1.
Suppose thatα>0, gt1−1−α/Bαht−1 is a nonnegative, nondecreasing, and locally integrable function on a,b, αht/Bα1−1−α/Bαht−1 is nonnegative and bounded on a,b, and xt is nonnegative and locally integrable on a,b with(8)xt≤gt+htItαABxt.Then,(9)xt≤gtBαBα−1−αhtEααhtt−aαBα−1−αht.In [31], Norelys et al. present a new property for the Caputo fractional derivative that is 1/2Dtαt0Cx2t≤xtDtαt0Cxt, t≥t0 holds for a continuous and derivable function xt∈ℜ, and 0<α<1. According to this, we can prove that this property also holds in the sense of AB fractional derivative.Lemma 2.
Let0<α<1 and xt∈CJ;ℜ be a continuous and differentiable function. Then, for any t∈J, the following inequality holds:(10)12DtαABCx2t≤xtDtαABCxt.Proof.
By the definition of AB fractional derivative, we have(11)xtDtαABCxt=Bα1−α∫0tEα−αt−sα1−αxtx′sds,12DtαABCx2t=Bα1−α∫0tEα−αt−sα1−αxsx′sds.
Equation (10) is equivalent to(12)Bα1−α∫0tEα−αt−sα1−αxt−xsx′sds≥0.
Defineys=xt−xs. Then, equation (12) can be reformulated as(13)−Bα1−α∫0tEα−αt−sα1−αysy′sds≥0.
Letu=1/2y2 and v=Bα/1−αEα−αt−sα/1−α. By the integration by parts, equation (13) is equivalent to(14)−Bαy2s21−αEα−αt−sα1−αs=t+Bαy2021−αEα−αtα1−α+Bα21−α∫0ty2sαt−sα−1Eα,α−αt−sα1−αds≥0.
Indeed, equation (14) holds obviously since Eα0=1 and yt=0 deduce that the first term equals zero and the nonnegativity of y2s, y20, Eα⋅, and Eα,α⋅ leads to the nonnegativity of the latter two terms.
From Lemma2, we can easily obtain the following two results.Corollary 1.
Let0<α<1 and xt∈CJ;ℜn be a continuous and differentiable function on J. Then, for any t∈J, the following inequality holds:(15)12DtαABCxTtxt≤xTtDtαABCxt.Proof.
Assume thatxt=x1t,x2t,…,xntT and xit∈CJ,ℜ, i=1,2,…,n. By Lemma 2, we have(16)12DtαABCxTtxt=12∑i=0nDtαABCxitxit≤∑i=0nxitDtαABCxit=xTtDtαABCxt.Corollary 2.
LetU∈ℜn×n be a symmetric positive definite matrix, 0<α<1, and xt∈CJ;ℜn be a continuous and differentiable function on J. Then, for any t∈J, the following inequality holds:(17)12DtαABCxTtUxt≤xTtUDtαABCxt.Proof.
From matrix theory, we know that there exists a nonsingular matrixV such that U=VTV. Define yt=Vxt; then,(18)12DtαABCxTtUxt=12DtαABCyTtyt≤yTtDtαABCyt=xTtUDtαABCxt.


## 4. Main Results

In this section, the concept of FTS and FTB for fractional-order LTI systems in the sense of AB fractional derivative corresponding with integer-order systems is introduced. Several sufficient conditions of FTS and FTB for such systems are given.Consider the following fractional order LTI system in the sense of AB fractional derivative:(19)DtαABCxt=Axt,t∈J≔0,T,x0=x0,where 0<α<1, x⋅∈ℜn, and A∈ℜn×n is a constant matrix.Definition 4.
System (19) is said to be of finite-time stability with respect to r1,r2,R,T, with positive scalars r1,r2,T, r2>r1, and a matrix R>0, if(20)xT0Rx0≤r1⟹xTtRxt≤r2,t∈0,T.Theorem 1.
Assume that there exists a scalarγ>0 and a matrix V∈ℜn×n, V>0, satisfying(21)UA+ATU<γU,(22)λmaxVBαλminVBα−γ1−αEααγTαBα−γ1−α<r2r1,where U=R1/2VR1/2. Then, system (19) is FTS with respect to r1,r2,R,T.Proof.
Define the functionVxt=xTtUxt. From Corollary 2 and condition equation (21), we have(23)DtαABCVxt≤xTtUA+ATUxt≤γVxt.
Then, there exists a nonnegative functionMt satisfying(24)DtαABCVxt+11−α∫0tt−τα−1Eα,α−αt−τα1−αMτdτ=γVxt.
Taking the Laplace transform on both sides of equality (24), we obtain(25)BαsαVxs−sα−1Vx01−αsα+α+Ms1−αsα+α=γVxt.
It can be reformulated that(26)Vxs=BαBα−γ1−α⋅Vx0sα−1sα−αγ/Bα−γ1−α−1Bα−γ1−α⋅Mssα−αγ/Bα−γ1−α.
Applying the inverse Laplace transform, one has(27)Vxt=BαVx0Bα−γ1−αEααγtαBα−γ1−α−1Bα−γ1−α∫0tt−τα−1Eα,ααγt−ταBα−γ1−αMτdτ.
Due to the nonnegativity of the second term in equation (27), it is easy to obtain(28)Vxt≤BαVx0Bα−γ1−αEααγTαBα−γ1−α.
TakingVxt=xTtUxt and U=R1/2VR1/2 into equation (28) yields(29)xTtR1/2VR1/2xt≤BαxT0R1/2VR1/2x0Bα−γ1−αEααγTαBα−γ1−α.
This implies that(30)xTtRxt≤λmaxVλminV⋅BαxT0Rx0Bα−γ1−αEααγTαBα−γ1−α.
According to condition equation (22), we can conclude that(31)xT0Rx0≤r1⟹xTtRxt≤r2.
For system (19), we also have the following results similar to Theorem 1.Corollary 3.
Assume that there exists a scalarγ>0 and a matrix V∈ℜn×n, V>0, satisfying(32)UAT+AU<γU,λmaxVBαλminVBα−γ1−αEααγTαBα−γ1−α<r2r1,where U=R1/2VR1/2. Then, system (19) is FTS with respect to r1,r2,R,T.Based on system (19), we will consider the finite-time stabilization of fractional-order systems in the AB fractional derivative sense with control function:(33)DtαABCxt=Axt+But,t∈J,x0=x0,where ut∈ℜm is a control function and B∈ℜn×m is a constant matrix. We assume that all the state variables are available for state feedback. The problem is to design a state feedback controller ut=Kxt, where K∈ℜm×n is the control gain matrix to be designed, such that the closed-loop system is(34)DtαABCxt=Acxt,t∈J,x0=x0.where Ac=A+BK, being FTS with respect to r1,r2,R,T.Theorem 2.
Assume that there exists a scalarγ>0, a matrix V∈ℜn×n, V>0, and a matrix L∈ℜm×n satisfying(35)AU+UAT+BL+LTBT<γU,(36)λmaxVBαλminVBα−γ1−αEααγTαBα−γ1−α<r2r1,where U=R1/2VR1/2. Then, under the feedback controller ut=Kxt=LU−1xt, system (33) is FTS with respect to r1,r2,R,T.Proof.
Apply the state feedback controllerut=LU−1xt to system (33) such that the corresponding closed-loop system is(37)DtαABCxt=A+BLU−1xt.
By Corollary3, it is clear that system (34) is FTS under the condition equation (35) and equation (36).
Next, we will study the problem of FTB for fractional-order LTI systems in the frame of AB fractional derivative. Consider the following systems:(38)DtαABCxt=Axt+Dwt,t∈J,x0=x0,where wt∈ℜl is the disturbance input and satisfies wTtwt≤d, d≥0. D∈ℜn×l is a constant matrix.Definition 5.
System (38) is said to be of finite-time boundedness with respect to r1,r2,R,T,d, with positive scalars r1,r2,T, r2>r1, and a matrix R>0, if(39)xT0Rx0≤r1⟹xTtRxt≤r2,t∈0,T,∀w:wTtwt≤d.Theorem 3.
Assume that there exists a scalarγ>0 and two matrices U1∈ℜn×n, U1>0, and V∈ℜl×l, V>0, satisfying(40)AU+UAT−γUDVVDT−γV<0,(41)r1λminU1Kα+γdMαλminVKαEααγTαKα<r2λmaxU1,(42)Kα=Bα−γ1−α>0.where U=R−1/2U1R−1/2 and Mα=1−α/Bα+Tα/BαΓα. Then, system (38) is FTB with respect to r1,r2,R,T,d.Proof.
LetVxt=xTtU−1xt, and taking AB fractional derivative of Vxt, by Corollary 2, we have(43)DtαABCVxt≤xTtU−1Axt+Dwt+Axt+DwtTU−1xt=xTtwTtU−1A+ATU−1U−1DDTU−10xtwt.
By premultiplying and postmultiplying equation (40) by the symmetric positive definite matrix U−100V−1, we have(44)U−1A+AU−1−γU−1U−1DDTU−1−γV−1<0.
Then, combining equations (43) and (44) leads to(45)DtαABCVxt≤xTtwTtγU−100γV−1xtwt=γVxt+γwTtV−1wt≤γVxt+γdλminV.
Integrating with orderα from 0 to t in the frame of AB fractional integral on both sides of equality (45), then(46)Vxt≤Vx0+IABtαγdλminV+γIABtαVxt=Vx0+1−αBα⋅γdλminV+αγdBαΓαλminV∫0tt−sα−1ds+γIABtαVxt≤Vx0+γdMαλminV+γIABtαVxt,where Mα=1−α/Bα+Tα/BαΓα. By equation (42) and applying Lemma 1, we have(47)Vxt≤Vx0Bα−γ1−α+γdMαλminVBα−γ1−αEααγtαBα−γ1−α1−α=Vx0Kα+γdMαλminVKαEααγTαKα.
It follows fromU=R−1/2U1R−1/2 that(48)Vxt=xTtU−1xt=xTtR−1/2U1R−1/2xt≥xTtRxtλmaxU1,(49)Vx0=xT0U−1x0=xT0R−1/2U1R−1/2x0≤xTtRxtλminU1.
Combining equations (47), (48), and (49), it follows that(50)xTtRxtλmaxU1≤r1λminU1Kα+γdMαλminVKαEααγTαKα.
By the condition equations (41) and (50), it implies that(51)xTtRxt≤r2,t∈0,T.
Now, we consider the problem of designing state feedback controllersut=Kxt to stabilize the FTB problem for fractional-order LTI systems in the frame of AB fractional derivative:(52)DtαABCxt=Axt+But+Dwt,t∈J,x0=x0.Theorem 4.
Assume that there exist a scalarγ>0, two matrices U1∈ℜn×n, U1>0, and V∈ℜl×l, V>0, and a matrix L∈ℜm×n satisfying equations (41) and (42) and(53)AU+UAT+BL+LTBT−γUDVVDT−γV<0,where U=R−1/2U1R−1/2. Then, under the feedback controller ut=Kxt=LU−1xt, system (53) is FTS with respect to r1,r2,R,T,d.Proof.
Apply the state feedback controllerut=LU−1xt to system (53) such that the corresponding closed-loop system is(54)DtαABCxt=A+BLU−1xt+Dwt.
By Theorem3, it is clear that system (52) is FTB under the condition equations (41), (42), and (53).


## 5. Example

An example is presented to illustrate the main results.Example 1.
Consider the following fractional-order LTI system in the sense of AB fractional derivative with a disturbance defined by(55)Dt1/2ABCxt=11−11xt+1001wt,where the initial value x0=10T and the disturbance wt=sint−1costT. The parameters are given as c1=1, c2=20, d=2, T=10, R=I, and Bα=1. Applying the condition equations (40), (41), and (42) of Theorem 3 and setting γ=0.05, we can obtain the feasible matrices U1 and V as follows:(56)U1=0.97000.97,V=0.55000.55.
Therefore, system (55) is FTB with respect to 1,20,I,10,2. The state trajectory over 0−10s with the initial state x0=10T is shown in Figure 1. It is easy to see that system (55) is FTB with respect to 1,20,I,10,2 from Figure 2.Figure 1
The statext of system (55) versus time.Figure 2
The trajectory ofxTtRxt of system (55) versus time.


## 6. Conclusion

The contribution of this paper is to present one new property of AB fractional derivatives and to provide the sufficient conditions which guarantee FTS and FTB for a class of fractional-order LTI systems in the sense of AB fractional derivative, as well as to study the problem of designing feedback controllers. Because of the interesting nonsingular kernel of this new fractional derivative, there is much work on this type of fractional calculus that is worth thinking about and discussing in the future.



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