# Controllability of Differential Systems with the General Conformable Derivative

Authors: Ferrag Azouz; Djalal Boucenna; Abdellatif Ben Makhlouf; Lassaad Mchiri; Abbes Benchaabane
Journal: Complexity
 (2021)

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## Abstract

In this paper, the controllability of differential systems with the general conformable derivative is studied. By elaborating the rank criterion and the conformable Gram criterion, sufficient and necessary conditions to investigate that a linear general conformable system is null completely controllable are given. We obtain a full generalization to the general conformable fractional-order system case. In addition, Krasnoselskii’s fixed point theorem to obtain a complete controllability result for a semilinear general conformable system is applied.


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## Body

## 1. Introduction

The presentation of a nonlinear system with the noninteger derivative is named fractional-order system (FOS). There are many applications of FOSs on real-world fields, whether in signal processing, chemistry, electricity, thermal, or control theory, for example, observer design [1–3], finite-time stability [4], fault estimation [5], and asymptotic stability [6–8]. In fact, with regard to observer design, authors in [1] presented a fractional-order observer design for fractional-order nonlinear systems. Dealing with finite-time stability, Naifar et al. in [4] described the finite-time stability concept of linear fractional-order time-delay systems. In the context of fault estimation, Ibrahima N’Doye and Taous-Meriem Laleg-Kirati in [5] detailed the fractional-order adaptive fault estimation for a class of nonlinear fractional-order systems. Finally, with regard to asymptotic stability, in [6], authors presented the asymptotic stability of distributed-order nonlinear time-varying systems with the Prabhakar fractional derivatives.In recent years, in [9], the conformable fractional derivative has been defined. Recently, Abdeljawad [10] worked to improve such a derivative. Much research and descriptions are currently being carried out on it, and we give the following related works [11, 12]. Indeed, authors in [11] worked on the analytical solutions of the system of conformable time-fractional Robertson equations with 1D diffusion. Furthermore, authors in [12] detailed the stability analysis of conformable fractional-order nonlinear systems depending on a parameter. After that, a generalization of the classical conformable fractional derivative is investigated [13, 14]. In fact, the authors of [14] introduced a new class of fractional derivative called the general conformable fractional derivative. Furthermore, in [13], the authors proved some uniqueness results of the solution of the diffusion equation.Furthermore, there are many recent papers about the applications of the FOS in the real world such as the electrical engineering (see [3, 15, 16]).On the contrary, controllability theory is very important in control systems. Many related works have investigated the control systems’ complete controllability [17, 18]. In fact, Jiang in [17] presented the controllability of fractional control systems with control delay. Furthermore, authors in [18] worked on the controllability of a fractional linear time-invariant neutral dynamical system. Especially, numerous works from the above citations [15, 19–21] have described the complete controllability of conformable systems. In particular, Wang et al. [19] investigated the controllability of conformable semilinear systems. In addition, authors in [20] described the linear conformable differential system and its controllability.In this paper, the controllability of conformable linear and semilinear systems is introduced and described. As an example of comparison, the work presented in [19] used the classical conformable derivative to prove their main results, whereas in our case, one has made a full generalization to the general conformable fractional-order system case.The main contributions of this work are highlighted as follows:(i)
To the best of our knowledge, among all the existing works in the literature, there is no research development which treats the controllability of general conformable linear and semilinear systems(ii)
The validity of such a development for the general conformable fractional derivative will open several perspectives and solutions for the applications in control theory


## 2. Preliminaries

We start this section by recalling some definitions, some lemmas, and theorems [6, 13, 14, 22].Definition 1.
Assume a functionϕ which is defined on a,b; then, the general conformable derivative (GCD) starting from real a of ϕ is defined by(1)Taθ,ψaϕt=limε⟶0ϕt+εψat,θ−ϕtε,for all t>a, where θ∈0,1 and ψat,θ is a continuous nonnegative function that depends on t and satisfies(2)ψat,1=1,ψa.,θ1≠ψa.,θ2,whereθ1≠θ2andθ1,θ2∈0,1.
IfTaθ,ψaϕt exists, for every t∈a,c and for some c>a, and limt⟶a+Taθ,ψaϕt exists, then(3)Taθ,ψaϕa≔limt⟶a+Taθ,ψaϕt.Remark 1.
The GCD generalizes the classical derivativeθ=1 and the conformable derivative ψat,θ=t−a1−θ (see [6, 22] for a=0).Remark 2.
To further study the properties of the GCD, we suppose thatψat,θ>0 for all t>a and 1/ψa.,θ is integrable.Definition 2.
Let0<θ<1. The conformable fractional integral (CFI) of a function ϕ is defined by(4)Iaθ,ψaϕt=∫atϕxψax,θdx.Lemma 1.
Assume a continuous functionϕ defined on a,b. Then, for every t≥a, we have(5)Taθ,ψaIaθ,ψaϕt=ϕt.Lemma 2.
Assume thatϕ is an absolutely continuous function on a,b. Then, for all t≥a, we have(6)Iaθ,ψaTaθ,ψaϕt=ϕt−ϕa.Lemma 3.
Letρ1, ρ2, ρ3∈ℝ and the functions ϕ1, ϕ2:a,b⟶ℝ such that Taθ,ψaϕ1t and Taθ,ψaϕ2t exist on a,b. Then,(i)
Taθ,ψaρ1ϕ1+ρ2ϕ2t=ρ1Taθ,ψaϕ1t+ρ2Taθ,ψaϕ2t(ii)
Taθ,ψaρ3=0(iii)
Taθ,ψaϕ1ϕ2t=ϕ1tTaθ,ψaϕ2t+ϕ2tTaθ,ψaϕ1t(iv)
Taθ,ψaϕ1/ϕ2t=ϕ2tTaθ,ψaϕ1t−ϕ1tTaθ,ψaϕ2t/ϕ22t, for each t∈a,b such that ϕ2t≠0Remark 3.
Assume thatϑ∈ℝ∗. If gt:=Eθψaϑ,t,a=eϑ∫at1/ψax,θdx, then Taθ,ψagt=ϑgt and Iaθ,ψagt=1/ϑgs−1.


## 3. Linear Systems

Consider the following linear system:(7)Tα,ψt0xt=Axt+But,t∈t0,b,xt0=x0.The solution of problem (7) is given by(8)xt=Eαψt0A,t,t0xt0+∫t0t1ψt0s,αEαψt0A,t,t0Eαψt0−A,s,t0Busds,where(9)Eαψt0A,t,t0=expA∫t0t1ψt0s,αds.Definition 3.
System (7) is said to be controllable on t0,b if for every x0,x1∈ℝn, there is a control ut such that the solution xt of such a system satisfies xt0=x0 and xb=x1.We introduce a notation of a controllability Gram matrixW as follows:(10)W=∫t0b1ψt0s,αEαψt0A,b,t0Eαψt0−A,s,t0BEαψt0A,b,t0Eαψt0−A,s,t0B∗ds.Here,∗ denotes the matrix transpose.Theorem 1.
Control system (7) is controllable on t0,b if and only if the controllability Gram matrix W defined by (10) is positive definite, for some b>0.Proof.
Suppose thatW is positive definite; then, its inverse is well defined. Define the control function as follows:(11)ut=B∗Eαψt0−A∗,t,t0Eαψt0A∗,b,t0W−1x1−Eαψt0A,b,t0x0.
Substitutingt=b in (8) and using (11), we get(12)xb=Eαψt0A,b,t0x0+∫t0b1ψt0s,αEαψt0A,b,t0Eαψt0−A,s,t0×BB∗Eαψt0−A∗,s,t0Eαψt0A∗,b,t0W−1x1−Eαψt0A,b,t0x0ds=Eαψt0A,b,t0x0+WW−1x1−Eαψt0A,b,t0x0=x1.
Thus, system (7) is controllable.
Conversely, suppose that the controllability Gram matrixW is not positive definite. So, there exists a nonzero y such that y∗Wy=0; then,(13)y∗Wy=∫t0by∗1ψt0s,αEαψt0A,b,t0Eαψt0−A,s,t0BEαψt0A,b,t0Eαψt0−A,s,t0B∗yds=0,and it implies that(14)y∗Eαψt0A,b,t0Eαψt0−A,t,t0B=0,ont0,b.
Letx0=Eαψt0A,b,t0−1y. It follows from the above result that there is ut such that it steers x0 to the origin on t0,b. Then,(15)xb=Eαψt0A,b,t0x0+∫t0b1ψt0s,αEαψt0A,b,t0Eαψt0−A,s,t0Busds=y+∫t0b1ψt0s,αEαψt0A,b,t0Eαψt0−A,s,t0Busds.
Therefore,(16)y∗y+∫t0b1ψt0s,αy∗Eαψt0A,b,t0Eαψt0−A,s,t0Busds=0.
Using (14), we get y∗y = 0; this is a contradiction to y≠0. Thus, the matrix W is positive definite.Now, we consider a notation of a rank criterion as follows:(17)Γc=BAB…An−1B.Theorem 2.
Control system (7) is controllable on t0,b if and only if rank Γc=n.Proof.
Suppose that system (7) is not controllable. According to the above theorem, there exists a nonzero y such that(18)y∗Eαψt0A,b,t0Eαψt0−A,t,t0B=0,ont0,b.
Takingt=b in (18), we get y∗B=0. By the derivative of (18) and from ψt0t,α≠0, we get(19)y∗AEαψt0A,b,t0Eαψt0−A,t,t0B=0,ont0,b.
Takingt=b in (19), we obtain y∗AB=0. By repeating this operation n−1 times, we get(20)y∗B=0,y∗AB=0,…,y∗An−1B=0.
Thus,(21)y∗BAB…An−1B=0.
According toy≠0, we obtain rank Γc<n.
Conversely, assume that rankΓc<n; then, there is a nonzero y such that(22)y∗Gc=y∗BAB…An−1B=0.
Therefore,(23)y∗AiB=0,fori=0,1,…,n−1.
By the Cayley–Hamilton theorem, we get(24)y∗AiB=0,fori=0,1,2,….
It implies that, for allt∈t0,b,(25)y∗Eαψt0A,b,t0Ai∫t0t1/ψt0s,αdsii!B=0,fori=0,1,2,….
Therefore,(26)y∗Eαψt0A,b,t0Eαψt0−A,t,t0B=0,ont0,b.
From (26) and the definition of the matrix W, we obtain(27)y∗Wy=∫t0by∗1ψt0s,αEαψt0A,b,t0Eαψt0−A,t,t0BEαψt0A,b,t0Eαψt0−A,t,t0B∗yds=0,and hence, the matrix W is not positive definite. It follows from the above theorem that the system is not controllable.


## 4. Semilinear Systems

Consider the following system:(28)Tα,ψt0xt=Axt+ft,xt+But,t∈t0,b,xt0=x0.We introduce the following assumptions:(A1)
The function1/ψt0 is square integrable, and the operator W:L2J,ℝr⟶ℝn defined by(29)Wu=∫t0t1ψt0s,αEαψt0A,t,t0Eαψt0−A,s,t0Busdshas an inverse operatorW−1 such that W−1 takes values in L2J,ℝr\kerW.(A2)
f:J×ℝn⟶ℝn is a continuous function, and there exists Lf.∈CJ,ℝ+ such that(30)ft,x−ft,y≤Lftx−y,for allx,y∈ℝnandt∈J.Forxt∈CJ,ℝn, we define the control function uxt by(31)uxt=W−1x1−Eαψt0A,b,t0x0−∫t0b1ψt0s,αEαψt0A,b,t0Eαψt0−A,s,t0fs,xsdst,t∈J.Firstly, by using (31), we show that the operator P:CJ,ℝn⟶CJ,ℝn defined by(32)Pxt=Eαψt0A,t,t0x0+∫t0t1ψt0s,αEαψt0A,t,t0Eαψt0−A,s,t0fs,xsds+∫t0t1ψt0s,αEαψt0A,t,t0Eαψt0−A,s,t0Buxsdshas a fixed point x, which is a solution of system (28). To prove the relative controllability result for system (28), we use Krasnoselskii’s fixed point theorem. Let us consider that ℬr=x∈CJ,ℝn:xC≤r, and note that ℬr is a closed, bounded, and convex subset of CJ,ℝn.In the following, we consider(33)R=W−1ℒℝn,L2J,ℝr\kerW,NA=A,ℛf=supt∈Jft,0,H1=Eαψt0NA,b,t0x0+ℛfNAEαψt0NA,b,t0Eαψt0NA,b,t0−1,H2=1NAEαψt0NA,b,t0Eαψt0NA,b,t0−1LfCJ,ℝ+,H3=REαψt0NA,b,t0∫t0b1ψt02s,αEαψt02NA,s,t0ds1/2.Next, we consider the operatorP=P1+P1 such that(34)P1xt=Eαψt0A,t,t0x0+∫t0t1ψt0s,αEαψt0A,t,t0Eαψt0−A,s,t0Buxsds,t∈J,P2xt=∫t0t1ψt0s,αEαψt0A,t,t0Eαψt0−A,s,t0fs,xsds,t∈J.Theorem 3.
Suppose that (A1) and (A2) are satisfied. Then, system (28) is completely controllable if the following condition holds:(35)H21+H3B<1.Proof.
The proof will be given in several steps.Step 1.
We show thatP1x+P2y∈ℬr for any x,y∈ℬr.
According to (A1), (A2), and Remark 3 and using the fact eAt≤eAt, we obtain for any x∈ℬr and t∈J,(36)uxL2J,ℝr\kerW≤W−1ℒℝn,L2J,ℝr\kerW×x1−Eαψt0A,b,t0x0−∫t0b1ψt0s,αEαψt0A,b,t0Eαψt0−A,s,t0fs,xsds≤Rx1+REαψt0NA,b,t0x0+REαψt0NA,b,t0∫t0b1ψt0s,αEαψt0NA,s,t0fs,xs−fs,0+fs,0ds≤Rx1+REαψt0NA,b,t0x0+REαψt0NA,b,t0∫t0b1ψt0s,αEαψt0NA,s,t0Lfsxsds+REαψt0NA,b,t0∫t0b1ψt0s,αEαψt0NA,s,t0fs,0ds≤Rx1+REαψt0NA,b,t0x0+RNAEαψt0NA,b,t0Eαψt0NA,b,t0−1LfCJ,ℝ+xCJ,ℝnRℛfNAEαψt0NA,b,t0Eαψt0NA,b,t0−1≤Rx1+RH1+RH2r.
According to (A1), (A2), and Remark 3 and using the fact eAt≤eAt, we obtain for any x,y∈ℬr and t∈J,(37)P1xt+P2yt≤Eαψt0NA,t,t0x0+Eαψt0NA,t,t0∫t0t1ψt0s,αEαψt0NA,s,t0fs,ysds+Eαψt0NA,t,t0∫t0t1ψt0s,αEαψt0NA,s,t0B≤Eαψt0NA,t,t0x0+Eαψt0NA,t,t0∫t0t1ψt0s,αEαψt0NA,s,t0Lfsysdsuxsds+Eαψt0NA,t,t0∫t0t1ψt0s,αEαψt0NA,s,t0fs,0ds+Eαψt0NA,t,t0∫t0t1ψt02s,αEαψt02NA,s,t0ds1/2BuxL2J,ℝr≤Eαψt0NA,t,t0x0+1NAEαψt0NA,t,t0Eαψt0NA,t,t0−1LfCJ,ℝ+yCJ,ℝn+ℛfNAEαψt0NA,t,t0Eαψt0NA,t,t0−1+Eαψt0NA,t,t0∫t0t1ψt02s,αEαψt02NA,s,t0ds1/2BRx1+RH1+RH2r≤H11+H3B+H3Bx1+H21+H3Br≤r,for(38)r=H11+H3B+H3Bx11−H21+H3B.
Then,P1x+P2y∈ℬr for any x,y∈ℬr.Step 2.
We show thatP1 is a contraction mapping.
According to (A1), (A2), and Remark 3 and using the fact eAt≤eAt, we get for any x,y∈ℬr and t∈J,(39)ux−uyL2J,ℝr\kerW≤W−1ℒℝn,L2J,ℝr\kerW×∫t0b1ψt0s,αEαψt0A,b,t0Eαψt0−A,s,t0fs,xs−fs,ysds≤REαψt0NA,b,t0∫t0b1ψt0s,αEαψt0NA,s,t0Lfsxs−,ysds≤RNAEαψt0NA,b,t0Eαψt0NA,b,t0−1LfCJ,ℝ+x−yCJ,ℝn≤RH2x−yCJ,ℝn.
Therefore,(40)P1xt−P1yt≤Eαψt0NA,t,t0∫t0t1ψt0s,αEαψt0NA,s,t0Buxs−uysds≤Eαψt0NA,t,t0∫t0t1ψt02s,αEαψt02NA,s,t0ds1/2Bux−uyL2J,ℝr≤H2H3Bx−yCJ,ℝn.
Then,(41)P1x−P1y≤Kx−y,forK=H2H3B.
From (35), we deduce that P1 is a contraction.Step 3.
We show thatP2 is a continuous and compact operator.
It is easy to show that the operatorP2 is continuous since f is continuous. Next, we prove that P2 is compact to this end, and we show that P2ℬr⊂CJ,ℝn is bounded and equicontinuous.
For eachx∈ℬr and t1,t2∈J such that t2>t1, we have(42)P2xt2−P2xt1=∫t0t21ψt0s,αEαψt0A,t2,t0Eαψt0−A,s,t0fs,xsds−∫t0t11ψt0s,αEαψt0A,t1,t0Eαψt0−A,s,t0fs,xsds≤J1+J2,where(43)J1=Eαψt0A,t2,t0−Eαψt0A,t1,t0∫t1t21ψt0s,αEαψt0NA,s,t0fs,xsds,J2=Eαψt0NA,t2,t0∫t1t21ψt0s,αEαψt0NA,s,t0fs,xsds.
Now, we prove thatJ1 and J2 tend to zero as t1 tends to t2. According to (A1), (A2), and Remark 3, we get(44)J1≤Eαψt0A,t2,t0−Eαψt0A,t1,t0∫t0t11ψt0s,αEαψt0NA,s,t0Lfsxsds+Eαψt0A,t2,t0−Eαψt0A,t1,t0∫t0t11ψt0s,αEαψt0NA,s,t0fs,0ds≤rNAEαψt0A,t2,t0−Eαψt0A,t1,t0Eαψt0NA,t1,t0−1LfCJ,ℝ++ℛfNAEαψt0A,t2,t0−Eαψt0A,t1,t0Eαψt0NA,t1,t0−1,J2≤Eαψt0NA,t2,t0∫t1t21ψt0s,αEαψt0NA,s,t0Lfsxsds+Eαψt0NA,t2,t0∫t1t21ψt0s,αEαψt0NA,s,t0fs,0ds≤rNAEαψt0NA,t2,t0Eαψt0NA,t2,t0−Eαψt0NA,t1,t0LfCJ,ℝ++ℛfNAEαψt0NA,t2,t0Eαψt0NA,t2,t0−Eαψt0NA,t1,t0.
Ast1 tends to t2, the right side of the above two inequalities tends to zero. Hence, P2ℬr is equicontinuous. It remains to show that P2ℬr is bounded.
For eachx∈ℬr and t∈J, according to (A1), (A2), and Remark 3, we get(45)P2xt≤Eαψt0NA,t,t0∫t0t1ψt0s,αEαψt0NA,s,t0fs,xsds≤Eαψt0NA,t,t0∫t0t1ψt0s,αEαψt0NA,s,t0Lfsxsds+Eαψt0NA,t,t0∫t0t1ψt0s,αEαψt0NA,s,t0fs,0ds≤rNAEαψt0NA,b,t0Eαψt0NA,b,t0−1LfCJ,ℝ++ℛfNAEαψt0NA,b,t0Eαψt0NA,b,t0−1.
So,P2ℬr is bounded. As a consequence of Krasnoselskii’s fixed point theorem, we conclude that P has a fixed point x∈ℬr which is a solution of problem (28) satisfying xb=x1.


## 5. Examples

Example 1.
Let us consider system (7) with ψt0α,t=1+1−αt−t0, t0=0,b=1,α=1/2,(46)A=2001,B=1001.
Then, the conformable Gram matrix is given by(47)W=∫t0b1ψt0s,αEαψt0A,b,t0Eαψt0−A,s,t0BEαψt0A,b,t0Eαψt0−A,s,t0B∗ds=∫0111+1/2s32+s40032+s232+s40032+s2∗ds=2∫01382+s900342+s5ds=6.1572002.0313.
So,(48)W−1=0.1624000.4923,and it implies that W is nonsingular; therefore, the system is controllable on 0,1.Example 2.
Let us consider system (28) with n=r=2, ψt0α,t=1+1−αt−t0, t0=0,b=1,α=1/2,(49)A=0.1000.3,B=1001,ft,xt=110t+1arctanx1t110t+1arctanx2t.
Clearly, the function1/ψt0 is square integrable. The conformable Gram matrix is given by(50)W=∫t0b1ψt0s,αEαψt0A,b,t0Eαψt0−A,s,t0BEαψt0A,b,t0Eαψt0−A,s,t0B∗ds=∫0111+1/2s32+s0.20032+s0.632+s0.20032+s0.6∗ds=2∫0130.42+s1.40031.22+s2.2ds=0.8804001.0445.
So,(51)W−1=1.1358000.9574.
We haveW−1ℒℝn,L2J,ℝr\kerW=W−11/2 as in the case of the conformable derivative given in [19]. Thus, R=1.0657.
In addition, we have for allx,y∈ℝ2 and t∈J,(52)ft,x−ft,y=110t+1arctanx1t−arctany1t2+arctanx2t−arctany2t21/2≤110t+1x1t−y1t2+x2t−y2t21/2≤110t+1x−y.
Then,Lft=1/10t+1∈CJ,ℝ+ and LfCJ,ℝ+=1/5. Moreover, we get(53)NA=A=0.3,H2=1NAEαψt0NA,b,t0Eαψt0NA,b,t0−1LfCJ,ℝ+=0.23418,H3=REαψt0NA,b,t0∫t0b1ψt02s,αEαψt02NA,s,t0ds1/2=1.2492.
Therefore,(54)H21+H3B=0.52672<1.
All the conditions of Theorem3 are satisfied. Then, the system is completely controllable on 0,1.


## 6. Conclusion

In this paper, the controllability of general conformable linear and semilinear systems is introduced and described. Sufficient and necessary conditions to investigate that a linear general conformable system is null completely controllable are given. In addition, Krasnoselskii’s fixed point theorem to obtain a complete controllability result for a semilinear general conformable system is applied. Note that we can extend this work to stochastic general conformable linear and semilinear systems in the future.



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