# Controllability of Linear Fractional Systems with Delay in Control

Authors: Mina Ghasemi; Kameleh Nassiri
Journal: Journal of Function Spaces
 (2022)

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## Abstract

This paper discusses the controllability of continuous-time linear fractional systems with control delay. The Atangana-Baleanu fractional derivative with the Caputo approach is used. First, the solution expression for a linear fractional system is obtained. Then, the corresponding fractional delay controllability Gramian matrix is defined, and its non-singularity as necessary and sufficient conditions for the controllability is proved. Finally, another equivalent condition based on the matrix rank formed by the coefficients matrices of the original system is provided that is much easier to check.


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## Body

## 1. Introduction

Fractional calculus, with its long-memory property, is an excellent tool for modeling systems. In fact, some dynamic processes in many practical systems (biological, electrochemical, viscoelastic, etc.) are fractional [1, 2]. Fractional-order delay differential equations are applicable for establishing a very realistic model of some processes and systems with memory. The achievements of many researchers about the occurrence of delay in practical systems are presented in [3–5].Controllability, as one of the dynamic properties of fractional systems, plays a major role in modern control theory and engineering. Controllability of linear systems is established in [6]. Balachandran et al. [7] obtained sufficient conditions for the controllability of nonlinear fractional dynamical systems. The necessary and sufficient conditions for global relative controllability of linear fractional systems containing both lumped constant delay in state variables and distributed delays in admissible controls are presented by Klamka [8].The most well-known fractional derivatives are Riemann-Liouville and Caputo [9, 10]. The other fractional derivatives that look like the Riemann-Liouville and Caputo ones are presented in [11, 12]. The aforementioned operators with singular kernels have difficulties in the management of many physical phenomena. In 2015, Caputo and Fabrizio presented a new definition of fractional differential operator with exponential kernel [13]. Atangana and Baleanu suggested a generalized fractional derivative with a non-singular kernel containing the Mittag-Leffler function, in 2016 [14]. In [15], some controllability criteria of fractional systems involving the Atangana-Baleanu fractional derivative in Caputo sense are provided.In the current study, we investigate the controllability of linear fractional systems with control delay(1)DABCαxt=Axt+But+Cut−h,t≥0,x0=x0,ut=υt,−h≤t≤0,where DABCα is the Atangana-Baleanu fractional derivative using the Caputo approach of order 0<α<1; xt∈ℝn is a state vector; ut∈ℝm is a control vector; A∈ℝn×n, B,C∈ℝn×m are constant matrices; h>0 is the time control delay; and υt is the initial control function. The controllability of fractional systems with delay in control is investigated by Wei [16]. In [17], the constrained controllability of linear fractional control systems with multiple delays in control is discussed. The controllability of linear fractional systems with multiple variable delays and distributed delay in admissible control under Caputo derivative are analyzed by Klamka [18].The main novelties of this paper are that we present the necessary and sufficient conditions for the controllability of linear fractional systems with control delay under the Atangana-Baleanu fractional derivative using the Caputo approach. We show that these conditions are equivalent to the non-singularity of the controllability Gramian matrix and the full-rank property of a suitably defined matrix.The following are the major contributions of the paper:(i)
The solution expression of continuous-time linear fractional systems with control delay, involving the Atangana-Baleanu derivative using the Caputo approach, is obtained(ii)
The fractional delay controllability Gramian matrix is defined to deal with the controllability problem. We also show that its non-singularity is equivalent to the controllability of described system(iii)
The controlut, which steers the considered system from any admissible initial state and initial control to any state, is introduced(iv)
Another controllability criterion based on the rank of the matrixK is provided in Theorem 13The following notations will be used throughout this paper. LetA∈ℝn×n. The symbol A∗ is used for the transpose of matrix A. Assuming that I−1−αA is non-singular, we set A^=I−1−αA−1. The column space and the null space of matrix A are denoted as ImA and KerA, respectively.The structure of this study is as follows: Section2 is dedicated to a brief overview of fractional calculus. In Section 3, some controllability criteria of a linear fractional system with control delay are examined. Finally, a brief conclusion is provided in Section 4.


## 2. Preliminaries

Letn−1<α≤n,α∈ℝ+,n∈ℕ and g be a suitable function. In what follows, we recall some basic concepts of fractional calculus. For more details, see [9, 14, 19–23].Definition 1.
The Riemann-Liouville fractional integral and derivative of orderα of g are given by
(2)IαRLtαgt=1Γα∫att−θα−1gθdθ,DαRLtαgt=1Γn−αddtn∫atgθt−θα+1−ndθ.For0<α<1, the Riemann-Liouville fractional derivative of order α of g is
(3)DαRLtαgt=1Γ1−αddt∫atgθt−θαdθ.Definition 2.
The Caputo fractional derivative of orderα of g is defined as
(4)DαCtαgt=1Γn−α∫atgnθt−θα+1−ndθ.
When0<α<1, the Caputo fractional derivative of order α of g can be written as
(5)DαCtαgt=1Γ1−α∫atg′θt−θαdθ.Lemma 3.
The following equality holds true for Convolution operator in Riemann-Liouville sense:(6)D0RLtα∫0tϕt−θφθdθ=∫0tφt−θD0RLθαϕθdθ+φtlimθ⟶0+I0RLθ1−αϕθ.
Moreover, ifg0=0, then D0RLtαgt=D0Ctαgt.Lemma 4.
Let the functiongt has the Laplace transform, then the Laplace transform of the Caputo fractional derivative is
(7)LD0Ctαgts=sαLgts−∑k=0n−1sα−k−1gk0,n−1<α≤n.
For0<α≤1, the preceding equation becomes
(8)LD0Ctαgts=sαLgts−sα−1g0.Definition 5.
The well-known Mittag-Leffler function with two parameters is defined as(9)Eα,βz=∑j=0∞zjΓαj+β,α>0,β>0.Lemma 6.
LetRes>a1/α, then
(10)Ltβ−1Eα,β±atαs=sα−βsα∓a.Lemma 7.
Letβ>0 and ν>0, then
(11)1Γν∫0tt−θν−1Eα,βλθαθβ−1dθ=tβ+ν−1Eα,β+νλtα.
For the matrix Mittag-Leffler function, similar equations are provided.Definition 8.
The Atangana-Baleanu fractional derivative using the Caputo approach ofg∈H1a,b, b>a, and α∈0,1 is given by
(12)DaABCtαgt=Bα1−α∫atg′θEα−αt−θα1−αdθ,where the normalization function Bα is satisfied B0=B1=1. Throughout this paper, Bα=1.Lemma 9.
Let the functiongt has the Laplace transform, then the Laplace transform of the Atangana-Baleanu fractional derivative in Caputo sense is
(13)LD0ABCtαgts=Bα1−αsαLgts−sα−1g0sα+α/1−α.
For brevity’s sake,DCα and DABCα are used instead of D0Ctα and D0ABCtα, respectively.


## 3. Controllability Problem

In this section, we firstly present the solution expression of system (1). Then, defining the controllability Gramian matrix, the necessary and sufficient conditions for the controllability of this system are established.Theorem 10.
Let the initial conditions bex0=x0∈ℝn, ut∈ℝm, and there exist DCαuθ and DCαυθ. The solution of linear fractional system with control delay (1) is
(14)xt=Ψα,1αA^A,tx0+∫0t−hΨα,ααA^A,t−θA^B+Ψα,ααA^A,t−θ−hA^C×αuθ+1−αDCαuθdθ+∫t−htΨα,ααA^A,t−θA^Bαuθ+1−αDCαuθdθ+∫−h0Ψα,ααA^A,t−θ−hA^Cαυθ+1−αDCαυθdθ,where
(15)Ψα,βA,t=∑j=0∞Ajtαj+β−1Γαj+β=tβ−1∑j=0∞AjtαjΓαj+β=tβ−1Eα,βAtα.Proof.
Taking Laplace transform of system (1), we have
(16)sαLxts−sα−1x0sα1−α+α=ALxts+BLuts+CLut−hs,which can be rewritten as
(17)I−1−αAsαLxts−αALxts=sα−1x0+1−αBsαLuts+αBLuts+1−αCsαLut−hs+αCLut−hs.
Then, pre-multiplying both sides of (17) by the matrix A^ gives
(18)sαI−αA^ALxts=sα−1A^x0+1−αA^BsαLuts+αA^BLuts+1−αA^CsαLut−hs+αA^CLut−hs.
The above equation can be written as(19)Lxts=sαI−αA∧A−1sα−1A^x0+1−αsαI−αA∧A−1A^BsαLuts+αsαI−αA∧A−1A^BLuts+1−αsαI−αA∧A−1A^CsαLut−hs+αsαI−αA∧A−1A^CLut−hs.
By adding and subtracting1−αsαI−αA∧A−1A^Bsα−1u0 and 1−αsαI−αA∧A−1A^Csα−1u−h, we have
(20)Lxts=sαI−αA∧A−1sα−1A^x0+1−αsαI−αA∧A−1A^BsαLuts−sα−1u0+1−αsαI−αA∧A−1A^Bsα−1u0+αsαI−αA∧A−1A^BLuts+1−αsαI−αA∧A−1A^CsαLut−hs−sα−1u−h+1−αsαI−αA∧A−1A^Csα−1u−h+αsαI−αA∧A−1A^CLut−hs.
From Lemma4 and taking the inverse Laplace transform, we obtain
(21)xt=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^BLuts+1−αL−1sαI−αA∧A−1A^CLCDαut−h+1−αL−1sαI−αA∧A−1A^Csα−1u−h+αL−1sαI−αA∧A−1A^CLut−hs.
Finally, applying the Convolution theorem, Lemma6 and equation (21), we get
(22)xt=Eα,1αA^AtαA^x0+1−αBu0+1−αCu−h+∫0tt−θα−1Eα,ααA^At−θαA^Bαuθ+1−αDCαuθdθ+∫0tt−θα−1Eα,ααA^At−θαA^Cαuθ−h+1−αDCαuθ−hdθ=Eα,1αA^AtαA^x0+1−αBu0+1−αCu−h+∫0tt−θα−1Eα,ααA^At−θαA^Bαuθ+1−αDCαuθdθ+∫−ht−ht−θ−hα−1Eα,ααA^At−θ−hαA^Cαuθ+1−αDCαuθdθ.
SinceAx0+Bu0+Cu−h=0, we have A^x0+1−αBu0+1−αCu−h=x0, and the equation (22) result in (14).Definition 11.
The system (1) is controllable on 0,t1, if for every admissible initial state x0, initial control ut, and x1, there exists a control ut∈ℝm defined on 0,t1 such that the corresponding solution of (1) satisfies xt1=x1.
Corresponding to system (1), the controllability Gramian matrix is described as
(23)Wt1=∫0t1−hΨα,ααA^A,t1−θA^B+Ψα,ααA^A,t1−θ−hA^C×Ψα,ααA∧A,t1−θA∧B+Ψα,ααA∧A,t1−θ−hA∧C∗dθ+∫t1−ht1Ψα,ααA^A,t1−θA^BΨα,ααA∧A,t1−θA∧B∗dθ.
Now, we present the main results of this paper in two following theorems.Theorem 12.
The linear fractional system with control delay (1) is controllable on 0,t1, if and only if the controllability Gramian matrix Wt1 is non-singular.Proof.
First, to prove the sufficiency, suppose thatWt1 is non-singular.
Let(24)ut=11−α∫0tΨα,α−γ,t−θB∗A∧∗Ψα,ααA∗A∧∗,t1−θ+C∗A∧∗Ψα,ααA∗A∧∗,t1−θ−h×Wt1−1x1−Ψα,1αA^A,t1x0dθ,0≤t≤t1−h,B∗A∧∗1−α∫0tΨα,α−γ,t−θΨα,ααA∗A∧∗,t1−θ×Wt1−1x1−Ψα,1αA^A,t1x0dθ,t1−h<t≤t1,0,−h≤t≤0,where γ=α/1−α. It is demonstrated that ut has the Caputo derivative and satisfies in
(25)αut+1−αDCαut=Ψα,ααA∧A,t1−tA∧B+Ψα,ααA∧A,t1−t−hA∧C∗×Wt1−1x1−Ψα,1αA^A,t1x0,0≤t≤t1−h,B∗A∧∗Ψα,ααA∗A∧∗,t1−tWt1−1x1−Ψα,1αA^A,t1x0,t1−h<t≤t1,0,−h≤t≤0.
To prove this, first0≤t≤t1−h is considered. Since u0=0, then from Lemma 3, DRLαut=DCαut. Taking Caputo fractional derivative of ut, we have
(26)DCαut=DCα11−α∫0tB∗A∧∗Ψα,α−γ,t−θΨα,ααA∗A∧∗,t1−θydθ+DCα11−α∫0tC∗A∧∗Ψα,α−γ,t−θΨα,ααA∗A∧∗,t1−θ−hydθ=B∗A∧∗1−α∫0tΨα,ααA∗A∧∗,t1−t−θDRLαΨα,α−γ,θydθ+B∗A∧∗1−αΨα,ααA∗A∧∗,t1−tlimθ⟶0+IRL1−αΨα,α−γ,θy+C∗A∧∗1−α∫0tΨα,ααA∗A∧∗,t1−t−θ−hDRLαΨα,α−γ,θydθ+C∗A∧∗1−αΨα,ααA∗A∧∗,t1−t−hlimθ⟶0+IRL1−αΨα,α−γ,θy,where y=Wt1−1x1−Ψα,1αA^A,t1x0. Following Lemma 7, we get
(27)IRL1−αΨα,α−γ,θ=Eα,1−γθα,(28)DRLαΨα,α−γ,θ=ddθEα,1−γθα.
Equation (28) is easily equivalent to
(29)DRLαΨα,α−γ,θ=ddθEα,1−γθα=ddθ∑j=0∞−γθαjΓjα+1=−γθα−1∑j=0∞−γθαjΓjα+α=−γθα−1Eα,α−γθα.
The equations (26), (27), and (29) result in
(30)DCαut=−γB∗A∧∗1−α∫0tΨα,ααA∗A∧∗,t1−t−θΨα,α−γ,θydθ+B∗A∧∗1−αΨα,ααA∗A∧∗,t1−ty−γC∗A∧∗1−α∫0tΨα,ααA∗A∧∗,t1−t−θ−hΨα,α−γ,θydθ+C∗A∧∗1−αΨα,ααA∗A∧∗,t1−t−hy=−γ1−α∫0tΨα,α−γ,θB∗A∧∗Ψα,ααA∗A∧∗,t1−t−θ+C∗A∧∗Ψα,ααA∗A∧∗,t1−t−θ−hydθ+11−αB∗A∧∗Ψα,ααA∗A∧∗,t1−t+C∗A∧∗Ψα,ααA∗A∧∗,t1−t−hy.
Now, from (24) and (30), we get the result for 0≤t≤t1−h. Similarly, the desired results for t1−h<t≤t1 and −h≤t≤0 are achieved.
From (14) and (25), we obtain
(31)xt1=Ψα,1αA^A,t1x0+∫0t1−hΨα,ααA^A,t1−θA^B+Ψα,ααA^A,t1−θ−hA^C×αuθ+1−αDCαuθdθ+∫t1−ht1Ψα,ααA^A,t1−θA^Bαuθ+1−αDCαuθdθ,=∫0t1−hΨα,ααA^A,t1−θA^B+Ψα,ααA^A,t1−θ−hA^C×Ψα,ααA∧A,t1−θA∧B+Ψα,ααA∧A,t1−θ−hA∧C∗×Wt1−1x1−Ψα,1αA^A,t1x0dθ+∫t1−ht1Ψα,ααA^A,t1−θA^BΨα,ααA∧A,t1−θA∧B∗×Wt1−1x1−Ψα,1αA^A,t1x0dθ,=Ψα,1αA^A,t1x0+Wt1Wt1−1x1−Ψα,1αA^A,t1x0=x1,which demonstrates that system (1) is controllable.
Now, to prove the necessity, suppose that system (1) is controllable. If Wt1 is singular, then a vector y≠0 exists such that
(32)y∗Wt1y=0,that is
(33)0=y∗∫0t1−hΨα,ααA^A,t1−θA^B+Ψα,ααA^A,t1−θ−hA^C×Ψα,ααA∧A,t1−θA∧B+Ψα,ααA∧A,t1−θ−hA∧C∗dθy+y∗∫t1−ht1Ψα,ααA^A,t1−θA^BΨα,ααA∧A,t1−θA∧B∗dθy.
This equation is clearly equivalent to(34)0=∫0t1−hy∗Ψα,ααA∧A,t1−θA∧B+Ψα,ααA∧A,t1−θ−hA∧C2dθ+∫t1−ht1y∗Ψα,ααA∧A,t1−θA∧B2dθ.
It follows from the above equation(35)0=y∗Ψα,ααA^A,t1−θA^B+Ψα,ααA^A,t1−θ−hA^C,0≤θ≤t1−h,0=y∗Ψα,ααA^A,t1−θA^B,t1−h<θ≤t1.
Letx0=Ψα,1αA∧A,t1−1y. According to the assumption of controllability, a control ut exists on 0,t1 such that xt1=0. Consequently,
(36)0=Ψα,1αA^A,t1Ψα,1αA∧A,t1−1y+∫0t1−hΨα,ααA^A,t1−θA^B+Ψα,ααA^A,t1−θ−hA^C×αuθ+1−αDCαuθdθ+∫t1−ht1Ψα,ααA^A,t1−θA^Bαuθ+1−αDCαuθdθ.
Then,(37)0=y∗y+∫0t1−hy∗Ψα,ααA^A,t1−θA^B+Ψα,ααA^A,t1−θ−hA^C×αuθ+1−αDCαuθdθ+∫t1−ht1y∗Ψα,ααA^A,t1−θA^Bαuθ+1−αDCαuθdθ.
It follows from the equations (35) and (37) that y∗y=0. So, a contradiction is obtained. Therefore, Wt1 is non-singular.Theorem 13.
The linear fractional system with control delay (1) is controllable on 0,t1, if and only if the matrix
(38)K=BAB⋯An−1BCAC⋯An−1C,is full-rank.Proof.
Suppose(39)K^≔A^BA^AA^B⋯A∧An−1A^BA^CA^AA^C⋯A∧An−1A^C.
We prove thatImK^=ImWt1. To accomplish this, consider the set of reachable states of system (1) as
(40)Rt=η∈Rn:thereexistscontrolusuchthatxt=η,from initial conditions x0=0 and ut=υt=0,−h≤t≤0. We prove that for every t>0, Rt=ImK^=ImWt1. The proof is in three steps.
The first step is to demonstrate thatRt⊂ImK^. Let η∈Rt, then a control ut exists such that
(41)η=∫0t−hΨα,ααA^A,t−θA^B+Ψα,ααA^A,t−θ−hA^C×αuθ+1−αDCαuθdθ+∫t−htΨα,ααA^A,t−θA^Bαuθ+1−αDCαuθdθ,which is equal to
(42)η=∫0tΨα,ααA^A,t−θA^Bαuθ+1−αDCαuθdθ+∫0t−hΨα,ααA^A,t−θ−hA^Cαuθ+1−αDCαuθdθ=limL⟶∞∑j=0LA∧AjA^Bβj+A^Cγj,where
(43)βj=∫0tαjt−θj+1α−1Γαj+ααuθ+1−αDCαuθdθ,γj=∫0t−hαjt−θ−hj+1α−1Γαj+ααuθ+1−αDCαuθdθ,for j=0,1,⋯,L.
Equation (42) can be written as the following product:
(44)η=limL⟶∞A^BA^AA^B⋯A∧ALA^BA^CA^AA^C⋯A∧ALA^Cβ0β1⋮βLγ0γ1⋮γL.
By Cayley-Hamilton theorem,η∈ImK^. Therefore, Rt⊂ImK^.
The second step is to demonstrate thatImK^⊂ImWt1. We prove equivalently that ImWt1⊥⊂ImK∧⊥, where ImWt1⊥ and ImK∧⊥ are the orthogonal complements of ImWt1 and ImK^, respectively. We demonstrate that if η∈ImWt1⊥=KerWt1∗=KerWt1, then η∈ImK∧⊥. Let η∈KerWt1, then
(45)η∗Wt1η=0,that is
(46)0=∫0t1−hη∗Ψα,ααA∧A,t1−θA∧B+Ψα,ααA∧A,t1−θ−hA∧C2dθ+∫t1−ht1η∗Ψα,ααA∧A,t1−θA∧B2dθ.
The above equation is clearly equivalent to(47)0=η∗Ψα,ααA^A,t1−θA^B+Ψα,ααA^A,t1−θ−hA^C,0≤θ≤t1−h,0=η∗Ψα,ααA^A,t1−θA^B,t1−h<θ≤t1.
For the second equation of (47) with y=t1−θα, we have
(48)η∗∑j=0∞αjA∧AjyjΓαj+αA^B=0.
Taking derivativej times j=0,1,2,⋯,n−1, with respect to y and taking the limit y⟶0+, it follows
(49)η∗A∧AjA^B=0,j=0,1,⋯,n−1.
Then when0≤θ≤t1−h, from the Cayley-Hamilton theorem, we have
(50)η∗Ψα,ααA^A,t1−θA^B=∑j=0∞αjt1−θj+1α−1Γαj+αη∗A∧AjA^B=∑j=0n−1αjt1−θj+1α−1Γαj+αη∗A∧AjA^B=0.
The preceding equation and the first equation of (47) imply that
(51)0=η∗Ψα,ααA^A,t1−θ−hA^C=η∗∑j=0∞αjA∧Ajt1−θ−hj+1α−1Γαj+αA^C⇒η∗∑j=0∞αjA∧Ajt1−θ−hjαΓαj+αA^C=0.
Settingy=t1−θ−hα, we can write
(52)η∗∑j=0∞αjA∧AjyjΓαj+αA^C=0.
As mentioned procedure, differentiatingj times with respect to y and taking the limit y⟶0+ result in
(53)η∗A∧AjA^C=0,j=0,1,⋯,n−1.
From (49) and (53), we obtain
(54)η∗A^BA^AA^B⋯A∧An−1A^BA^CA^AA^C⋯A∧An−1A^C=η∗K^=0⇒K∧∗η=0.
It follows thatη∈KerK∧∗=ImK∧⊥. Therefore, ImK^⊂ImWt1.
The third step is to demonstrate thatImWt1⊂Rt. Let x1∈ImWt1, then a vector y exists such that x1=Wt1y.
Let(55)ut=11−α∫0tΨα,α−β,t−θB∗A∧∗Ψα,ααA∗A∧∗,t1−θ+C∗A∧∗Ψα,ααA∗A∧∗,t1−θ−hydθ,0≤t≤t1−h,B∗A∧∗1−α∫0tΨα,α−β,t−θΨα,ααA∗A∧∗,t1−θydθ,t1−h<t≤t1,0,−h≤t≤0.
It is possible to show thatDαut exists and
(56)αut+1−αCDαut=Ψα,ααA∧A,t1−tA∧B+Ψα,ααA∧A,t1−t−hA∧C∗y,0≤t≤t1−h,B∗A∧∗Ψα,ααA∗A∧∗,t1−ty,t1−h<t≤t1,0,−h≤t≤0.
Then, from Theorem10 with x0=0, we obtain
(57)xt1=∫0t1−hΨα,ααA^A,t1−θA^B+Ψα,ααA^A,t1−θ−hA^C×αuθ+1−αDCαuθdθ+∫t1−ht1Ψα,ααA^A,t1−θA^Bαuθ+1−αDCαuθdθ,=∫0t1−hΨα,ααA^A,t1−θA^B+Ψα,ααA^A,t1−θ−hA^C×Ψα,ααA∧A,t1−θA∧B+Ψα,ααA∧A,t1−θ−hA∧C∗ydθ+∫t1−ht1Ψα,ααA^A,t1−θA^BΨα,ααA∧A,t1−θA∧B∗ydθ,=Wt1y=x1.
It follows thatx1∈Rt. Therefore, ImWt1⊂Rt.
Taking into account the three preceding steps,ImK^=ImWt1 is concluded. Since, the matrix A^ is commutative with A, we can write
(58)K^=K100K1K,where
(59)K1=A^0⋯00A∧2⋯0⋮⋮⋱⋮00⋯A∧n.
So,ImK=ImK^=ImWt1. Therefore, the matrix K is full-rank if and only if the Gramian matrix Wt1 is non-singular. Now, from Theorem 12, the desired result is achieved.


## 4. Conclusion

In this paper, we investigated the controllability problem of linear fractional system with delay in control, involving the Atangana-Baleanu derivative in Caputo sense. The solution expression of such a system has been stated. Then, we introduced the fractional delay controllability Gramian matrix. According to the controllability Gramian matrix, the necessary and sufficient conditions for the controllability of a linear fractional system with control delay have been presented. Moreover, the desired control has been provided. We also established another controllability criterion based on the rank of the matrix presented in Theorem13.As future recommendations, the controllability of semilinear or generally nonlinear fractional systems with different types of delays not only in admissible controls but also in the state variables under the Atangana-Baleanu derivative can be considered. Another important issue to handle is the controllability of fractional systems with different orders of derivatives under the Atangana-Baleanu derivative.



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*Source: 5539770-2022-07-28.xml*