# Approximate Controllability of Fractional Nonlinear Hybrid Differential Systems via Resolvent Operators

Authors: Mohammed M. Matar
Journal: Journal of Mathematics
 (2019)

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## Abstract

We obtain sufficient conditions for the approximate controllability of a fractional nonlinear hybrid differential system. The results are obtained by using resolvent and sectorial operators technique via Dhage fixed point theorem.


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## Body

## 1. Introduction

Fractional differential systems have described many practical dynamical phenomena more efficiently than the corresponding integer-order systems; hence they have attracted the attention of many researchers in such fields (see [1–10] and references cited therein).One of these systems is the fractional control system with all its branches such as stability, controllability, and observability. In the recent years, many investigations on the controllability problems of fractional behaviour have extensively appeared with various applications on linear and nonlinear systems. Particularly, the researches have focused on exact (complete) and approximate controllability (see the articles [11–17] and the references therein).The fractional control systems involving a linear closed (unbounded) operator which generates resolvent operators were considered recently by many authors [15, 18–20]. The lack of the semigroup property of the generated resolvent operator was the most popular difficulty that has been faced by the interested researchers. However, some authors used the idea of analytic sectorial operators to overcome this problem. For more details, we refer the reader to the papers [20–23] and references therein. To the best of our knowledge, there is not any investigation in the controllability problem via resolvent operators applied on hybrid systems such as the system that has been discussed in the article [24].In this article, we study the approximate controllability for a fractional hybrid differential system of the form(1)Dc0α+Axt=gt,xtI01-αBut+ft,xt,0<t≤b,x0=x0,where Dc0α is the Caputo fractional derivative of order α such that 0<α<1, X and U are two real Hilbert spaces, A:DA⊆X→X is the infinitesimal generator of a resolvent operator Sαt,t∈J=0,b, B:U→X is a bounded linear operator, u∈L2J,U, I01-α denotes the 1-α-order fractional integral, and f,g:J×X→X are given functions such that g does not vanish on J×X.


## 2. Preliminaries

LetC(J,X) be the space of all X-valued continuous functions defined on J with the norm x=supxt,t∈J, and let Lp(J,X) be the space of X-valued Bochner integrable functions defined on J with the norm fLp=∫0b‍ftpdt1/p, where 1≤p≤∞.Now, let us recall some basic preliminaries on fractional calculus [25] and operator theory [26].Definition 1.
The fractional-order integral of a functionf∈C(J,X)(or L1(J,X)) of order α>0 is defined by (2)I0αft=1Γα∫0tt-sα-1fsds.Definition 2.
The Caputo fractional derivative of orderα∈0,1 of a function f∈C1(J,X)(or Lp(J,X)) is defined by (3)Dc0αft=I01-αf′t=1Γ1-α∫0tt-s-αf′sds.Definition 3.
The Laplace transform of a functionf is given by(4)Lfλ=∫0∞e-λtftdt,λ∈C.The Laplace transform of the Caputo fractional derivative is given by(5)LDc0αftλ=λαLft-λα-1f0,0<α<1.The Laplace transform of the fractional integral I01-α is given by (6)LI01-αft=λα-1Lft,α>0.The inverse Laplace transform of a functionF=Lf is given by(7)L-1Fλ=12πi∫ceλtFλdλ,for some suitable path c to ensure the existence of the integral.The resolvent operator of an operatorA is defined as(8)Rλ,A=λαI+A-1,λ∈C.The resolvent setρA is the set of all regular values of λ∈C such that Rλ,A is injective, bounded linear operator.The following fixed point theorem, which is due to Dhage [27], is essential tool for the proof of the main result.Theorem 4.
LetΩ be a nonempty bounded closed convex subset of a Banach algebra X. Let Φ:Ω→X and Θ:X→X be continuous operators satisfying the following:(a)
Φ is completely continuous,(b)
Θ is Lipschitzian with a Lipschitz constant kΘ,(c)
x=ΦyΘx implies x∈Ω for all y∈Ω, and(d)
MkΘ<1, where M=supΦx:x∈Ω.
Then the operator equationx=ΦxΘx has a solution in Ω.


## 3. Fractional Control Systems via Resolvent Operators

LetA:DA→X be a linear operator defined on the subspace DA⊆X, the domain of A to the space X. An operator A is said to be closed if and only if its domain DA is a complete space with respect to the norm xDA=x+Ax. An operator A is said to be densely defined if its domain is dense in X. The denseness of the domain is necessary and sufficient for the existence of the adjoint. The adjoint operator of unbounded operators can be defined as bounded operators. For more details on these topics, the reader may refer to [26, 28].Next, we introduce some information about solution operators [29].Consider system(9)Dc0αxt=-Axt,x0=x∈DA,which has an integral solution given by(10)xt=x-∫0tt-sα-1ΓaAxsds∈DA,t∈J.Definition 5.
LetA be a closed and densely defined operator on X. A family Sαtt≥0 of bounded linear operators in X is called a solution operator (or α-resolvent) generated by A if the following conditions are satisfied:(S1)
Sα(t) is strong continuous on R+ and Sα0=I, where I is the identity operator.(S2)
SαtDA⊆DA and ASαtx=SαtAx for all x∈DA and t≥0.(S3)
Sαtx is a solution of the integral equation (10).Moreover, a solution operatorSαt is called compactif for every t>0, Sαt is a compact operator. If Sαt is a solution operator of system (9), then by (S3), we deduce that(11)Ax=Γα+1limt→0+x-Sαtxtα,whereDA consists of all x for which the limit exists. We call A as the infinitesimal generator of Sαt or simply we say that A generates the solution operator Sαt.Definition 6.
LetA:DA⊆X→X be a closed linear operator. A is said to be a sectorial operator of type M,θ,α,μ if there exist μ≤0,0<θ<π/2, and M>0 such that the solution operator Sα of A exists throughout the sector (12)γ=λ∈C:Re⁡λ>-μ1/α;arg⁡λ<θ⊆ρA,and Rλα,A≤M/λα+μ,λ∈γ.Hereafter, we assume thatA is a sectorial operator of type M,θ,α,μ that generates the solution operator Sαt. In this case, we can write the solution operator Sαt of system (9) as(13)Sαt=L-1λα-1Rλα,A=12πi∫ceλtλα-1Rλα,Adλ,withc being a suitable path in a sector γ.Lemma 7.
The linear fractional system(14)Dc0α+Axt=gtI01-αBut+ft,x0=x0,has an integral solution given by(15)xt=gtSαtx0g0-1+∫0tSαt-sBus+fsds.Proof.
Lettingg(t)≠0, for any t∈J, then system (14) is equivalent to system(16)Dc0αxtgt+Axtgt=I01-αBut+ft,x0=x0,Applying the Laplace transform to system (16), we have (17)LDc0αxtgt+ALxtgt=LI01-αBut+ft,and that implies(18)λαLxtgt-λα-1x0g0+ALxtgt=λα-1LBut+ft.Therefore, (19)Lxtgt=λα-1λαI+A-1x0g0-1+λα-1λαI+A-1LBut+ft.Now, taking the inverse Laplace transform, we get the solution (15). This finishes the proof.We define a mild solution for system (1).Definition 8.
A functionxt∈CJ,X is called a mild solution of system (1) if it satisfies (20)xt=gt,xtSαtx0g0,x0-1+∫0tSαt-sBus+fs,xsds,for any t∈J.We introduce some preliminaries about controllability (see [3, 11–13, 18–20, 27]). We assume that x is a mild solution (we call it now as state function) of the fractional differential system (1) corresponding to a control u.Definition 9.
System (1) is said to be approximately controllable on J if for every desired final state xb∈X and ε>0, there exists a control u∈L2(J,U) such thatx satisfies xb;u-xb<ε.The set(21)R=xb;u∈X:u∈L2J,U,xis  the  mild  solution  of1with  controlu,is called the reachability set of system (1). Therefore, the fractional system (1) is said to be approximately controllable on J if R¯=X, where R¯ denotes the closure of R. If the used control function is fixed, the symbol x(b) is used instead of x(b;u).We define the controllability operatorLt:L2J,U→X as(22)Ltu=∫0tSαt-sBusds,t∈J.ThenLt is a bounded linear operator defined on L2J,U. The adjoint operator (23)Lb∗:X→L2J,Uof Lb is given by(24)Lb∗=B∗Sα∗b-·.The controllability GramnianW:X→X is defined by(25)W=LbLb∗=∫0bSαb-sBB∗Sα∗b-sds.Following the idea, as in [20], the suggested control function u for system (1) can be written in the form. (26)ut=B∗Sα∗b-tλI+W-1Λx,t∈J,where(27)Λx=xbgb,xb-1-Sαbx0g0,x0-1-∫0bSαb-sfs,xsds.


## 4. Approximate Controllability

We prove the approximate controllability of the fractional control system (1) by using the mild solution (20) and the control defined by (26). More precisely, we prove the existence of at least one state x∈CJ,X satisfying (20) and (26) following the same arguments presented in [20], but using Dhage fixed point theorem. For this lets (28)Θxt=gt,xt,and (29)Φxt=Sαtx0g0,x0-1+∫0tSαt-sBus+fs,xsds,where u is given by (26).IfSα is compact C0-semigroup, then the Cauchy operator Ψ:CJ,X→CJ,X defined as (30)Ψht=∫0tSαt-shsds,t∈J,is also compact. Unfortunately, the resolvent operator does not have the property of semigroups which leads to the impossibility of obtaining the compactness of the Cauchy operator Ψ. However, we can prove the continuity of the solution operator in the case of analytic operators by which we can prove the compactness of the Cauchy operator Ψ.Letr be a fixed positive real number such that Br=x∈CJ,X:x≤r. Clearly, Br is a bounded closed and convex set. We need the following assumptions:(H1)
Sαt is compact analytic operator such that MS=supSαt:t∈J<∞.(H2)
f:J×X→X is continuous and there exists a positive constant Kf such that ft,x<Kf, for all (t,x)∈J×X.(H3)
g:J×X→X is continuous and there exists a function ρ∈L1J,R+ such that g(t,x)-gt,y<ρL1x-y, for all (t,x)∈J×X.(H4)
B:U→X is a linear bounded operator and there exists N>0 such that B=N.(H5)
LetM≔N2Ms2/λxbgb,xb-1+Msx0g0,x0-1+MsKfb+Msx0g0,x0-1+Ms2Kfb, such that MρL1<1.(H6)
(λI+W)-1≤1/λ, λ>0.Theorem 10.
Assume that conditions (H1)-(H6) are satisfied. Then system (1) has a mild solution on J.Proof.
We show the operatorsΘ and Φ satisfying the hypotheses of Dhage fixed point theorem. For the sake of clarity, we split the proof into two main steps.
Step 1. Firstly, we prove the continuity of Θ and Φ. Let xnn≥1 be a sequence in C(J,X) with limn→∞xn=x in C(J,X). By the hypotheses (H2) and (H3), we obtain the convergence of f(t,xn(t)) and gt,xnt to ft,xt and g(t,x(t)), respectively, for any t∈J. Hence(31)Λxn-Λx≤xbgb,xnb-1-gb,xb-1+MS∫0bfs,xns-fs,xsds.Thus, by Lebsegue Dominated Convergence Theorem and the fact that g(b,xn(b))-1≥δ, for some δ>0 and for any n∈N, we have limn→∞Λxn-Λx=0. Then(32)Φxnt-Φxt≤MS2N2λ∫0bΛxns-Λxsds+MS∫0bfs,xns-fs,xsds.As n→∞ and using again dominated convergence theorem we have Φxn(t)-Φx(t)→0. Thus Θ and Φ are continuous on C(J,X). Next, we show that Φ is bounded on C(J,X). In fact, for all x∈C(J,X), we have(33)Φx≤Msx0g0,x0-1+N2Ms2λxbgb,xb-1+Msx0g0,x0-1+MsKfb+Ms2Kfb.Then, the inequality Φx≤M holds for all x∈Br.
The last thing in this step, we show thatΦ:C(J,X)→C(J,X) is a compact operator. It is sufficient to prove that (34)∫0tSαt-sBus+fs,xsdsis compact. But this has been proved in many articles see, for example, ([20]: Theorem 3.3) by using the Ascoli-Arzela theorem. Hence we conclude that Φ is compact. Therefore, Φ is completely continuous. By following the same arguments presented in [13], we can prove.....
Step 2. The hypothesis (H3) shows that the operator Θ is Lipschitz with Lipschitzian constant kΘ=ρL1. Next, we show that x∈Br whenever x=ΦyΘx for all y∈Br. For this, letting y∈Br and k0=supt∈Jgt,0, we have(35)x≤gt,x-gt,0+gt,0M≤ρL1x+k0M.In consequence, this implies that(36)x≤k0M1-MρL1.Ifr is chosen large enough such that r>Mmax⁡1,k0/1-MρL1, then we ensure that x∈Br. Therefore, all hypotheses of Dhage Theorem are satisfied; then there exists a fixed point x∈Br satisfying the operator equation x=ΘxΦx, which is a solution of system (1).Next result, we investigate the approximate controllability of the fractional control system (1). We introduce the following extra conditions:(H7)
λ(λI+W)-1→0 as λ→0+ in the strong operator topology.(H8)
The sequenceg(·,xλ(·)):λ>0 is bounded in L2(J,X).Theorem 11.
Assume that conditions (H1)-(H8) are satisfied. Then, the fractional system (1) is approximately controllable on J.Proof.
In virtue of Theorem10, there exists a mild solution xλ∈C(J,X) such that(37)xλt=gt,xλtSαtx0g0,x0-1+∫0tSαt-sBus+fs,xλsds,
where(38)ut=B∗Sα∗b-tλI+W-1xbgb,xλb-1-Sαbx0g0,x0-1-∫0bSαb-sfs,xsds.
Therefore,(39)xλb=gb,xλbSαbx0g0,x0-1+∫0bSαt-sBus+fs,xλsds=gb,xλbSαbx0g0,x0-1+λI+W-λIλI+W-1×xbgb,xλb-1-Sαbx0g0,x0-1-∫0bSαb-sfs,xλsds+∫0bSαb-sfs,xλsds.Hence,(40)xλb-xb=-gb,xλbλλI+W-1xbgb,xλb-1-Sαbx0g0,x0-1-∫0bSαb-sfs,xλsds.
Now, by condition (H2), we have(41)∫0bfs,xλs2ds≤Kf2b,
which implies that the sequence{f(·,xλ(·)):λ>0} is bounded in the Hilbert space L2(J,X). Together with (H8), there exist subsequences of {f(·,xλ(·)):λ>0} and {g(·,xλ(·)):λ>0} in L2(J,X) converging weakly to some points ω,ν∈L2(J,X), respectively. Let(42)η=xbνb-1-Sαbx0g0,x0-1-∫0bSαb-sωsds.Thus, (43)xbgb,xλb-1-Sαbx0g0,x0-1-∫0bSαb-sfs,xλsds-η≤gb,xλb-1-νb-1xb+∫0bSαb-sfs,xλs-ωsds.
Using the compactness ofSα(t), we can deduce that the mapping(44)xt→∫0tSαt-sxsds,
fromL2(J,X) to C(J,X) is compact. So, we obtain that(45)gb,xλb-1-νb-1xb+∫0bSαb-sfs,xλs-ωsds→0,
asλ→0+. This implies that (46)xbgb,xλb-1-Sαbx0g0,x0-1-∫0bSαb-sfs,xλsds-η→0,
asλ→0+. In view of (40) and condition (H5), we obtain that(47)xλb-xb≤gb,xλbλRλ,Wxbgb,xλb-1-Sαbx0g0,x0-1-∫0bSαb-sfs,xλsds≤gb,xλbλRλ,Wxbgb,xλb-1-Sαbx0g0,x0-1-∫0bSαb-sfs,xλsds-η+gb,xλbλRλ,Wη.
Hencexλ(b)-xb→0 as λ→0+, which implies that the fractional system (1) is approximately controllable on J. This finishes the proof.Example 12.
Consider the fractional control system(48)Dc00.8xt,y1+Lt2sin⁡xt,y=-∂2∂y2xt,y1+Lt2sin⁡xt,y+I01-αut,xt,y+xt,y1+xt,y,y∈0,π,t∈0,1,xt,0=xt,π=0,t∈0,1,x0,y=x0y,y∈0,π.
LetX=U=L20,π and A=∂2/∂y2, where DA={φ∈X:φ and φ′ are absolutely continuous, φ′′∈X, φ(0)=φ(π)=0}. Then A is an infinitesimal generator of an analytic semigroup (t), t>0, which can be written in the form(49)Stφ=∑n=1∞e-n2π2tφ,enen,t>0,x∈X,where en(y)=2/πsin⁡ny,n=1,2,... is the orthonormal basis for X. It can be shown that A is also a generator of a compact analytic operator Sα(t), t>0, given by(50)Sαt=∫0∞Sstαψαsds,t>0,where (51)ψαs=∑n=0∞-snn!Γ1-α-nα,0<α<1is the Wright function (see [19, 21]). Setting x(t)y=x(t,y), (Bu)(t)(y)=u(t,y), gt,xty=1+Lt2sin⁡xt,y, and ft,xty=x(t,y)/1+x(t,y), then system (48) is equivalent to system (1) for any t∈0,1. The operator Sα(t) satisfies the hypothesis (H1) such that MS=1. Simple calculations lead to Kf=1, ρL1=L/3, and N=1. Therefore, if we choose L such that ML<3 and that (H6) and (H7) are both satisfied, then, using Theorems 10, and 11, we ensure that system (48) is approximately controllable on [0,1].



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