# On Controllability of Fractional Continuous-Time Systems

Authors: Mina Ghasemi; Kameleh Nassiri
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The aim of this paper is to study the controllability of fractional systems involving the Atangana–Baleanu fractional derivative using the Caputo approach. In the first step, the solution of a linear fractional system is obtained. Then, based on the obtained solution, some necessary and sufficient conditions for the controllability of such a system will be presented. Afterwards, the controllability of a nonlinear fractional system will be analyzed, based on these results. Our tool for the presentation of the sufficient conditions of controllability in this part is Schauder fixed point theorem. In the last step, the analytical results are illustrated by numerical examples.


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## Body

## 1. Introduction

Fractional calculus, with its long history, has been used in a wide range of applications. The fractional-order systems, for instance, can describe some dynamic procedures in many physical systems such as heat conduction [1] and viscoelastic materials [2] and in biology [3], bioengineering [4], energy systems [5], and economics [6]. The long-memory property of fractional calculus makes it a potent instrument for describing and designing a specific set of nonlocal dynamic trends, associated with complex systems [7]. Formerly, there have been widespread discussions in the field of dynamic properties of fractional systems. The problem of stability, for example, has been studied in [8–11].Another important issue of fractional systems is the concept of controllability. The controllability of linear systems is established by Klamka [12]. The sufficient conditions of controllability of semilinear systems with multiple variable delays in control are formulated in [13]. Buedo-Fernández and Nieto [14] presented the necessary and sufficient conditions for the controllability of a linear fractional system, with constant coefficients using the Caputo fractional derivative, based on the Kalman matrix. In [15], the controllability of linear fractional dynamical systems with different order is investigated. The constrained controllability of continuous-time fractional-order control systems with multiple delays in control is considered in [16]. A computational procedure for the controllability of linear and nonlinear fractional dynamical systems of order 1<α≤2 is provided by Balachandran and Govindaraj [17]. Balachandran et al. [18] established a set of sufficient conditions for the controllability of nonlinear fractional dynamical systems. In [19], the controllability of nonlinear fractional delay dynamical systems with prescribed controls is considered. A variational approach to study the finite approximate controllability for Sobolev-type fractional semilinear evolution equations with nonlocal conditions in Hilbert spaces is extended by Mahmudov [20].So far, various types of fractional derivatives such as Riemann–Liouville and Caputo have been proposed [7, 21, 22]. Derivatives such as those mentioned above can not properly model some nonlocal phenomena due to having a singular kernel. To overcome the singularity problems in fractional derivative, Caputo and Fabrizio introduced a new fractional differential operator using the exponential function as a kernel [23]. In [24], a new formulation of time fractional optimal control problems governed by Caputo–Fabrizio fractional derivative is proposed. Also, Atangana and Baleanu proposed a new derivative based on the Mittag-Leffler function [25]. An open discussion is ongoing about the mathematical construction of the Atangana–Baleanu fractional operator in Caputo sense (ABC). Diethelm et al. [26] showed that it is possible to solve differential equations with nonsingular kernel derivatives only when a very restrictive and unnatural assumption is made on the initial condition. Giusti [27] indicated that this operator can be expressed as an infinite series involving Riemann–Liouville integrals. The authors in [28] showed that the ABC definition cannot be useful in modeling problems such as the fractional diffusion equation because the solutions obtained for these equations do not satisfy the initial condition. Ortigueira et al. [29] showed that the models involving the generalized fractional derivative with regular kernels poorly reflect the real-world data. In responses to these criticisms, Atangana and Gómez-Aguilar [30] emphasized the need to account for a fractional calculus approach without an imposed index law and with nonsingular kernels. Furthermore, Sabatier [31] showed that the papers [26, 27] are not correct and produce the wrong conclusion on the restriction imposed by nonsingular kernels. In a comment written by Baleanu [32], it has been shown that the opinions of Ortigueira et al. [29] are not consistent. Also, Atangana and Goufo [33] presented some interesting results to clarify the mistake and lack of understanding for those writing against derivatives with nonsingular kernels. Hristov [34] investigated the underlying physical meaning of the nonsingular kernel and also presented a collection of recent applications of fractional differentiation operators with nonsingular kernels.Motivated by the references mentioned above, we study the controllability of a nonlinear fractional system, represented as follows:(1)DαABCxt=Axt+But+ft,xt,ut,t∈0,Tx0=x0,where DαABC is the Atangana–Baleanu fractional derivative in Caputo sense of order 0<α<1; xt∈ℝn and ut∈ℝm are state and input vectors, respectively; and A∈ℝn×n and B∈ℝn×m are constant matrices. Also, f:0,T×ℝn×ℝm⟶ℝn and its Caputo derivative are continuous.The following notations are used throughout this paper. ForA∈ℝn×n, A∗ denotes the matrix transpose of A. We assume that I−1−αA is nonsingular and A^=I−1−αA−1. We use the notations ImA and KerA, respectively, for the column space and the null space of the matrix A. The symbols u0 and f0 are used for ut and ft,xt,ut, respectively, at t=0.This paper is organized as follows. Section2 presents some preliminary concepts of fractional calculus. In Section 3, we obtain the solution of a linear fractional system and analyze the controllability of such a system. In Section 4, we establish the sufficient conditions for the controllability of a nonlinear fractional system. The obtained results are numerically confirmed in Section 5. Eventually, the conclusion is stated in Section 6.


## 2. Preliminaries

We start with a brief overview of some mathematical preliminaries.Definition 1.
Letα∈ℝ+, n−1<α≤n, n∈ℕ, and f be a suitable function. The Caputo fractional derivative of order α is defined as [21, 35](2)DtαaCft=1Γn−α∫atfnτt−τα+1−ndτ.
Throughout this paper0<α<1; then,(3)DtαaCft=1Γ1−α∫atf′τt−ταdτ.Lemma 2.
Assuming that the Laplace transformFs of the function f exists, then the Laplace transform formula for the Caputo fractional derivative is [21](4)LDtα0Cft=sαFs−∑k=0n−1sα−k−1fk0,n−1<α≤n.
For0<α≤1, we have(5)LDtα0Cft=sαFs−sα−1f0.Definition 2.
The two-parameter Mittag-Leffler function, as an important function in fractional calculus, is defined as [36](6)Eα,βz=∑k=0∞zkΓαk+β,α>0,β>0.
In particular, ifβ=1, then(7)Eα,1z=Eαz=∑k=0∞zkΓαk+1.Lemma 2.
The Laplace transform of the functiontβ−1Eα,β±atα is [21](8)Ltβ−1Eα,β±atα=sα−βsα∓a,Res>a1/α.
In particular, forβ=1, equation (3) becomes(9)LEα,1±atα=sα−1sα∓a.
For Mittag-Leffler function in matrix form, similar equations are established.Definition 3.
Letf∈H1a,b, b>a, and α∈0,1. The Atangana–Baleanu fractional derivative in Caputo sense is given as [25](10)DtαaABCft=Bα1−α∫atf′xEα−αt−xα1−αdx,where Bα is a normalization function that can be any function satisfying B0=B1=1. Throughout this paper, we suppose Bα=1.Lemma 3.
Assuming that the Laplace transformFs of the function f exists, then the Laplace transform of the Atangana–Baleanu fractional derivative in Caputo sense is as follows [25]:(11)LDtα0ABCft=Bα1−αsαLft−sα−1f0sα+α/1−α.
Throughout this paper, for brevity, we useDαC and DαABC instead of Dtα0C and Dtα0ABC, respectively.


## 3. Linear System

The continuous-time linear fractional system with the Atangana–Baleanu fractional derivative in Caputo sense is presented as follows:(12)DαABCxt=Axt+But,t∈0,T,x0=x0,where xt∈ℝn and ut∈ℝm are state and input vectors, respectively, and A∈ℝn×n and B∈ℝn×m are constant matrices.In the following theorem, we present the solution of system (12).Theorem 1.
The solution of system (12), starting from x0=x0∈ℝn, ut∈ℝmt≥0, and assuming the existence of DαCuτ, is(13)xt=Eα,1αA^Atαx0+∫0tt−τα−1Eα,ααA^At−ταA^Bαuτ+1−αCDαuτdτ.Proof.
Taking Laplace transform of system (12), we have(14)sαXs−sα−1x0sα1−α+α=AXs+BUs,which may be rewritten as(15)I−1−αAsαXs−αAXs=sα−1x0+1−αBsαUs+αBUs.
Multiplying equation (15) by A^, we obtain(16)sαI−αA^AXs=sα−1A^x0+1−αA^BsαUs+αA^BUs.
Therefore, we can write(17)Xs=sαI−αA^A−1sα−1A^x0+1−αsαI−αA^A−1A^BsαUs+αsαI−αA^A−1A^BUs.
Then, adding and subtracting1−αsαI−αA^A−1A^Bsα−1u0 yields(18)Xs=sαI−αA^A−1sα−1A^x0+1−αsαI−αA^A−1A^BsαUs−sα−1u0+1−αsαI−αA^A−1A^Bsα−1u0+αsαI−αA^A−1A^BUs.
According to (5) and taking the inverse Laplace transform, we have(19)L−1Xs=L−1sαI−αA^A−1sα−1A^x0+1−αL−1sαI−αA^A−1A^BLCDαut+1−αL−1sαI−αA^A−1A^Bsα−1u0+αL−1sαI−αA^A−1A^BUs.
Finally, by applying the convolution theorem and equations (8), (9), and (19), we obtain(20)xt=Eα,1αA^AtαA^x0+1−αBu0+∫0tt−τα−1Eα,ααA^At−ταA^Bαuτ+1−αCDαuτdτ.
SinceAx0+Bu0=0, we have A^x0+1−αBu0=x0, and so the proof is complete.
First, we review the definition of controllability of the fractional dynamical system, in agreement with [37].Definition 4.
System (12) is controllable on 0,T, if for every x0,x1∈ℝn, there exists a control ut∈ℝm defined on 0,T such that the corresponding solution of (12) satisfies xT=x1.
Corresponding to system (12), we define the controllability Gramian matrix as follows:(21)WT=∫0TT−τα−1Eα,ααA^AT−ταA^BEα,ααA^AT−ταA^B∗dτ.
Then, we will present the controllability criteria of system (12).Theorem 2.
The continuous-time linear fractional system (12) is controllable on 0,T, if and only if the controllability Gramian matrix WT is nonsingular.Proof.
Sufficiency: suppose thatWT is nonsingular. For every u0 that satisfies Ax0+Bu0=0, we choose the control(22)ut=Eα,1−γtαu0+B∗A^∗1−α∫0tt−τα−1Eα,α−γt−ταEα,ααA∗A^∗T−τα×WT−1x1−Eα,1αA^ATαx0dτ,with γ=α/1−α. It can be shown that Dαut exists and ut satisfies in(23)αut+1−αCDαut=B∗A^∗Eα,ααA∗A^∗T−tαWT−1x1−Eα,1αA^ATαx0.
Combining (13) at t=T and (23), we obtain(24)xT=Eα,1αA^ATαx0+∫0TT−τα−1Eα,ααA^AT−ταA^BB∗A^∗Eα,ααA∗A^∗T−τα×WT−1x1−Eα,1αA^ATαx0dτ=Eα,1αA^ATαx0+WTWT−1x1−Eα,1αA^ATαx0=x1,which shows linear system (12) is controllable.
Necessity: assume that linear system (12) is controllable. If WT is singular, then there exists a vector z≠0such that z∗WTz=0, which is equivalent to(25)z∗∫0TT−τα−1Eα,ααA^AT−ταA^BEα,ααA^AT−ταA^B∗dτz=0.
The preceding equation implies that(26)z∗Eα,ααA^AT−ταA^B=0,on0,T.
Letx0=Eα,1αA^ATα−1z. According to controllability assumption, there exists a control ut on 0,T such that xT=0, which means(27)0=Eα,1αA^ATαEα,1αA^ATα−1z+∫0TT−τα−1Eα,ααA^AT−ταA^Bαuτ+1−αDαCuτdτ.
Multiplying equation (27) by z∗, we can write(28)0=z∗z+∫0TT−τα−1z∗Eα,ααA^AT−ταA^Bαuτ+1−αDαCuτdτ.
Equations (26) and (28) yield z∗z=0, which is a contradiction to z≠0. Thus, WT is nonsingular.
Now, we present another criterion for the controllability of system (12).Theorem 3.
The continuous-time linear fractional system (12) is controllable on 0,T, if and only if the Kalman matrix K=BAB⋯An−1B is full rank.Proof.
LetKABC:=A^BA^AA^B⋯A^An−1A^B. At first, we prove that ImKABC=ImWT. For this purpose, we consider Rt=ζ∈ℝn:there existsusuch thatxt=ζ as the set of reachable states based on system (12), with the zero initial condition. We show that for every t>0, Rt=ImKABC=ImWT. We will accomplish the proof in three steps.
In the first step, we prove thatRt⊂ImKABC. Let ζ∈Rt be every reachable state. Hence, there exists a control ut such that(29)ζ=∫0tt−τα−1Eα,ααA^At−ταA^Bαuτ+1−αDαCuτdτ,which is equal to(30)ζ=∫0tt−τα−1∑k=0∞αkA^Akt−τkαΓαk+αA^Bαuτ+1−αDαCuτdτ=limN⟶∞∑k=0NA^AkA^B∫0tt−τα−1αkt−τkαΓαk+ααuτ+1−αDαCuτdτ=limN⟶∞∑k=0NA^AkA^Bβk,where(31)βk=∫0tt−τα−1αkt−τkαΓαk+ααuτ+1−αDαCuτdτ,fork=0,1,…,N.
Writing (30) as a product, we have(32)ζ=limN⟶∞A^BA^AA^B⋯A^ANA^Bβ0β1⋮βN.
By Cayley–Hamilton theorem, any of these matricesA^B,A^AA^B,…,A^ANA^B is a linear combination of A^B,A^AA^B,…,A^An−1A^B. Hence, ζ∈ImKABC. Therefore, Rt⊂ImKABC.
In the second step, we prove thatImKABC⊂ImWT. We will show equivalently that ImWT⊥⊂ImKABC⊥, where ImWT⊥ and ImKABC⊥ are the orthogonal complements of ImWT and ImKABC, respectively. We need to show that if ζ∈ImWT⊥=KerWT∗=KerWT, then ζ∈ImKABC⊥. Let ζ∈KerWT. Thus, we have ζ∗WTζ=0, which is equivalent to(33)0=∫0TT−τα−1ζ∗Eα,ααA^AT−ταA^Bζ∗Eα,ααA^AT−ταA^B∗dτ=∫0TT−τα−1ζ∗Eα,ααA^AT−ταA^B2dτ.
Forτ∈0,T, T−τα∈0,Tα. Setting z=T−τα, we have(34)ζ∗Eα,ααA^AzA^B=ζ∗∑k=0∞αkA^AkzkΓαk+αA^B=0.
Differentiatingk times k=0,1,2,…, with respect to z and taking the limit when z⟶0+ implies that ζ∗A^AkA^B=0, for k=0,1,…,n−1. It follows that(35)ζ∗A^BA^AA^B⋯A^An−1A^B=ζ∗KABC=0⟹KABC∗ζ=0.
This gives us thatζ∈KerKABC∗=ImKABC⊥. Therefore, ImKABC⊂ImWT.
In the third step, we prove thatImWT⊂Rt. Let x1∈ImWT. Then, there exists y such that x1=WTy. For every u0 that satisfies Ax0+Bu0=0, we define the control ut as(36)ut=Eα,1−γtαu0+B∗A^∗1−α∫0tt−τα−1Eα,α−γt−ταEα,ααA∗A^∗T−ταydτ,with γ=α/1−α. It can be shown that Dαut exists and ut satisfies in(37)αut+1−αDαCut=B∗A^∗Eα,ααA∗A^∗T−tαy.
Then, the solution of system (12) with x0=0 at t=T is(38)xT=∫0TT−τα−1Eα,ααA^AT−ταA^Bαuτ+1−αCDαuτdτ=∫0TT−τα−1Eα,ααA^AT−ταA^BEα,ααA^AT−ταA^B∗ydτ=WT=x1.
Therefore,x1∈Rt. This proves that ImWT⊂Rt.
Considering all three previous steps, we obtainImKABC=ImWT. Since A^A=AA^, we have(39)KABC=A^0⋯00A^2⋯0⋮⋮⋱⋮00⋯A^nK.
So,ImK=ImKABC=ImWT. Therefore, the Kalman matrix K is full rank if and only if the Gramian matrix WT is nonsingular. Using Theorem 2, the proof is completed.
Using Theorem2 and Theorem 4, we obtain the necessary and sufficient conditions for the controllability of classical linear system [38]. It should be noted that the presented criterion in Theorem 3 is similar to [14, Theorem 2] and does not depend on α.


## 4. Nonlinear System

Consider the continuous-time nonlinear fractional system (1). Let Q be the Banach space of continuous ℝn×ℝm valued function, defined on 0,T with the following norm:(40)x,usup=xsup+usup,where xsup=supxt:t∈0,T and usup=suput:t∈0,T.Assume that the Caputo derivatives ofu and f exist. Then, for every z,v∈Q, a similar process of Theorem 1 can be used to show that the solution of the linear fractional system(41)DαABCxt=Axt+But+ft,zt,vt,x0=x0,is(42)xt=Eα,1αA^Atαx0+∫0tt−τα−1Eα,ααA^At−ταA^Bαuτ+1−αCDαuτdτ+∫0tt−τα−1Eα,ααA^At−ταA^×αfτ,zτ,vτ+1−αCDαfτ,zτ,vτdτ.In order to analyze the controllability of nonlinear system (1), we propose the following theorem that presents sufficient conditions for the controllability of such a system.Theorem 4.
Let the continuous functionf with its continuous Caputo derivative satisfy the condition(43)limx,usup⟶∞αft,x,u+1−αCDαft,x,ux,usup=0,uniformly for t∈0,T. If linear system (12) is controllable, then nonlinear system (1) is controllable on 0,T.Proof.
Consider the operatorP:Q⟶Q with Pz,v=x,u. For every u0 that satisfies Ax0+Bu0+f0=0, we define(44)ut=Eα,1−γtαu0+B∗A^∗1−α∫0tt−τα−1Eα,α−γt−ταEα,ααA∗A^∗T−ταdτ×WT−1x1−Eα,1αA^ATαx0−∫0TT−τα−1Eα,ααA^AT−ταA^×αfτ,zτ,vτ+1−αCDαfτ,zτ,vτdτ,with γ=α/1−α, and(45)xt=Eα,1αA^Atαx0+∫0tt−τα−1Eα,ααA^At−ταA^Bαuτ+1−αCDαuτdτ+∫0tt−τα−1Eα,ααA^At−ταA^×αfτ,zτ,vτ+1−αCDαfτ,zτ,vτdτ.
We are going to provide the conditions for using the Schauder fixed point theorem to show thatP has a fixed point. According to the continuity of the Mittag-Leffler function and from (44), there exist positive numbers c1 and d1, such that for t∈0,T, we have(46)ut≤c1+d1supαf+1−αCDαf,where(47)supαf+1−αDαCf=supαft,zt,vt+1−αCDαft,zt,vt:t∈0,T.
It should be noted thatDαut exists and ut satisfies in(48)αut+1−αDαCut=B∗A^∗Eα,ααA∗A^∗T−tαWT−1×x1−Eα,1αA^ATαx0−∫0TT−τα−1Eα,ααA^AT−ταA^×αfτ,zτ,vτ+1−αCDαfτ,zτ,vτdτ.
Similarly, from (45) and (48), there exist positive numbers c2 and d2 such that for t∈0,T,(49)xt≤c2+d2supαf+1−αCDαf.
Next, definingc=2maxc1,c2 and d=2maxd1,d2, we can write (46) and (49) as(50)ut≤12c+dsupαf+1−αCDαf,(51)xt≤12c+dsupαf+1−αDαCf.
From [39], since continuous function αft,x,u+1−αCDαft,x,u satisfies assumption (43), for every pair of positive constants c and d, there exists a positive constant r such that, if x,u≤r, then(52)c+dαft,x,u+1−αDαCft,x,u≤r,for allt∈0,T.
Next, we introduce the setQr=z,v∈Q:zsup≤r/2,vsup≤r/2. We show that P maps Qr into itself. For every z,v∈Qr, we have zsup≤r/2 and vsup≤r/2. Therefore, z,usup=zsup+vsup≤r/2+r/2=r, for all t∈0,T. It follows that c+dsupαf+1−αDαCf≤r. Then, from (50) and (51), we get ut≤r/2 and xt≤r/2. Hence, usup≤r/2 and xsup≤r/2. Since αf+1−αCDαf is continuous, the operator P is continuous. It should be noted that Qr is closed, bounded, and convex, so by the Schauder fixed point theorem [40, Theorem 4.14], P has a fixed point z,v∈Qr, such that Pz,v=z,v≡x,u. Therefore, we rewrite (17) and (18) as(53)ut=Eα,1−γtαu0+B∗A^∗1−α∫0tt−τα−1Eα,α−γt−ταEα,ααA∗A^∗T−ταdτ×WT−1x1−Eα,1αA^ATαx0−∫0TT−τα−1Eα,ααA^AT−ταA^×αfτ,xτ,uτ+1−αCDαfτ,xτ,uτdτ,and(54)xt=Eα,1αA^Atαx0+∫0tt−τα−1Eα,ααA^At−ταA^Bαuτ+1−αDαCuτdτ+∫0tt−τα−1Eα,ααA^At−τα×A^αfτ,xτ,uτ+1−αDαCfτ,xτ,uτdτ,which shows that xt is the solution of nonlinear system (1). Now, setting t=T in (54), we get xT=x1, so nonlinear system (1) is controllable.Proposition 1.
The values of positive numbersc1, d1, c2, and d2 used in Theorem 4 are calculated by(55)c1=a1+Tαα−1a2a31−αB∗A^∗WT−1x1+a4,d1=Tαα−1a32a21−αB∗WT−1A^2,c2=a4+Tαα−1a32A^2B2WT−1x1+a4,d2=Tαα−12a33A^3B2WT−1+Tαα−1a3A^,where(56)a1=supt∈0,TEα,1−γtαu0,a2=supt∈0,TEα,α−γT−tα,a3=supt∈0,TEα,ααA^AT−tα,a4=supt∈0,TEα,1αA^Atαx0.Proof.
According to (53) and (56), we can write(57)ut≤a1+11−αB∗A^∗tαα−1a2a3WT−1×x1+a4+Tαα−1a3A^supαf+1−αDαCf≤a1+Tαα−1a2a31−αB∗A^∗WT−1x1+a4+Tαα−1a32a21−αB∗A^∗WT−1A^supαf+1−αDαCf.
Defining(58)c1=a1+Tαα−1a2a31−αB∗A^∗WT−1x1+a4,d1=Tαα−1a32a21−αB∗WT−1A^2,yields(59)ut≤c1+d1supαf+1−αDαCf.
Similarly, it follows from (54) that(60)xt≤a4+Tαα−1a3A^Bαu+1−αDαCusup+Tαα−1a3A^supαf+1−αDαCf.
Using (48) yields(61)αut+1−αDαCut≤B∗A^∗a3WT−1×x1+a4+Tαα−1a3A^supαf+1−αDαCf.
Combining (60) and (61) results in(62)xt≤a4+Tαα−1a32A^BB∗A^∗WT−1x1+a4+Tαα−12a33A^2BB∗A^∗WT−1+Tαα−1a3A^supαf+1−αDαCf.
Then, defining(63)c2=a4+Tαα−1a32A^2B2WT−1x1+a4,d2=Tαα−12a33A^3B2WT−1+Tαα−1a3A^,we derive(64)xt≤c2+d2supαf+1−αDαCf.
It should be noted that due to the existence ofx′t or u′t or both of them in the calculation of the Caputo derivative of ft,xt,ut, we present the following proposition, which is an appropriate tool for examining condition (43).Proposition 2.
Assume that the notationsa2, a3, and a4 are described in Proposition 1. Then, the derivatives of xt and ut exist and are bounded.Proof.
According to (53), we can write(65)u′t=Eα,1′−γtαu0+B∗A^∗1−α∫0tα−1t−τα−2Eα,α−γt−τα+t−τα−1Eα,α′−γt−τα×Eα,ααA∗A^∗T−ταdτ×WT−1x1−Eα,1αA^ATαx0−∫0TT−τα−1Eα,ααA^AT−τα×A^αfτ,xτ,uτ+1−αDαCfτ,xτ,uτdτ.
Setting(66)a5=supt∈0,TEα,1′−γtαu0,a6=supt∈0,TEα,α′−γT−tα,we have(67)u′t≤a5+11−αB∗A^∗tα−1a2+tαα−1a6a3WT−1×x1+a4+Tαα−1a3A^supαf+1−αCDαf≤a5+Tα−1a2+Tαα−1a6a31−αB∗A^∗WT−1x1+a4+T2α−1α−1a2+T2αα−2a6a321−αB∗A^∗WT−1A^supαf+1−αCDαf.
Defining(68)c3=a5+Tα−1a2+Tαα−1a6a31−αB∗A^∗WT−1x1+a4,d3=T2α−1α−1a2+T2αα−2a6a321−αB∗WT−1A^2,yields(69)ut≤c3+d3supαf+1−αDαCf.
Similarly, it follows from (54) that(70)x′t=Eα,1′αA^Atαx0+∫0tα−1t−τα−2Eα,ααA^At−τα+t−τα−1Eα,α′αA^At−τα×A^Bαuτ+1−αDαCuτdτ+∫0tα−1t−τα−2Eα,ααA^At−τα+t−τα−1Eα,α′αA^At−τα×A^αfτ,xτ,uτ+1−αDαCfτ,xτ,uτdτ.
Setting(71)a7=supt∈0,TEα,α′αA^AT−tα,a8=supt∈0,TEα,1′αA^Atαx0,we obtain(72)x′t≤a7+Tα−1a3+Tαα−1a8A^Bαu+1−αDαCusup+Tα−1a3+Tαα−1a8A^supαf+1−αDαCf.
Then, using (61) yields(73)x′t≤a7+Tα−1a3+Tαα−1a8a3A^2B2WT−1x1+a4+T2α−1α−1a3+T2αα−2a8a32A^3B2WT−1+Tα−1a3+Tαα−1a8A^×supαf+1−αDαCf.
Defining(74)c4=a7+Tα−1a3+Tαα−1a8a3A^2B2WT−1x1+a4,d4=T2α−1α−1a3+T2αα−2a8a32A^3B2WT−1+Tα−1a3+Tαα−1a8A^,we derive(75)x′t≤c4+d4supαf+1−αDαCf.


## 5. Numerical Examples

In this section, we utilize some examples to illustrate the validity of our results.Example 1.
We consider the following linear system:(76)DαABCxt=Axt+But,x0=x0,where α=1/3, t∈0,2, and(77)A=2−1−12,B=23−13.
From (21), the controllability Gramian matrix is(78)W2=∫022−τα−1Eα,ααA^A2−ταA^BEα,ααA^A2−ταA^B∗dτ,where(79)Eα,ααA^A2−τα=∑k=0∞2−τ2kαΓ2k+1α∑k=0∞2−τ2k+1αΓ2k+2α∑k=0∞2−τ2k+1αΓ2k+2α∑k=0∞2−τ2kαΓ2k+1α.
Then, definingc1=∑k=0∞2−τ2kα/Γ2k+1α and c2=∑k=0∞2−τ2k+1α/Γ2k+2α yields(80)Eα,ααA^A2−τα=c1c2c2c1.
Therefore, (78) becomes(81)W2=∫022−τα−1c22c1c2c1c2c12dτ,which is nonsingular.
Also, the Kalman matrixK=1/325−1−4 is full rank. Hence, from Theorem 2 or Theorem 3, system (76) with control ut defined in (22) is controllable. We consider x1t and x2t as the solutions of system (76) for initial conditions x10 and x20, respectively. Here(82)x1t=x11tx21t,x2t=x12tx22t,x10=10,x20=−20.
The controlut steers system (76) from x10 and x20 to x12 and x22, respectively, where(83)x12=51,x22=42.
Now, consider the following system:(84)DαABCxt=Axt+But+0t,x0=x0,where A, B, α, and t are as above. According to the above description, this system is also controllable. Using Matlab, the trajectories of solutions are depicted in Figures 1 and 2.Figure 1
The trajectories of linear system (76).Figure 2
The trajectories of linear system (84).Example 2.
We consider the nonlinear system as follows:(85)DαABCxt=Axt+But+ft,xt,x0=x0,where ft,xt=01/1+x12t and A, B, α, and t are the same as in the Example 1. The Caputo derivative of f is(86)DαCft,xt=1Γ2/30∫0t−2x1τx1′τ/1+x12τ2t−τ1/3dτ.
From Proposition2, αf+1−αDαCf satisfies condition (43). Also, according to Example 1, the linear system is controllable. Therefore, nonlinear system (85) is controllable.


## 6. Conclusion

In this paper, we obtained the solution of the linear fractional system involving the Atangana–Baleanu derivative with the Caputo approach. According to the definition of controllability, we presented some necessary and sufficient conditions for the controllability of such a system based on the controllability Gramian matrix and the Kalman matrix. Using the Schauder fixed point theorem and the obtained results for the linear system, we also established sufficient conditions for the controllability of the nonlinear system. The reliability of the analytical process used in our work has been checked by numerical examples.



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