# Regional Controllability of Riemann–Liouville Time-Fractional Semilinear Evolution Equations

Authors: Asmae Tajani; Fatima Zahrae El Alaoui; Ali Boutoulout
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, we discuss the exact regional controllability of fractional evolution equations involving Riemann–Liouville fractional derivative of orderq∈0,1. The result is obtained with the help of the theory of fractional calculus, semigroup theory, and Banach fixed-point theorem under several assumptions on the corresponding linear system and the nonlinear term. Finally, some numerical simulations are given to illustrate the obtained result.


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## Body

## 1. Introduction

Time-fractional systems have been proved, with the development of science and technology, to be one of the most effective tools in modeling many phenomena arising in physics, engineering, and real world problems [1–6]. Therefore, the research studies of fractional-order calculus attract lots of attention for these kinds of systems with several fractional derivatives (for more details, see [7–10] and the references therein). Zhou and Jiao [11] introduced a concept of a mild solution based on Laplace transform and probability density functions; several authors presented a tremendous amount of valuable results on controllability and observability, stability analysis, and so on [12–15].Similar to the integer-order control systems [16–21], the regional controllability problem of fractional systems is a class of control problems presented in many applications in real world. Regional controllability of linear and some nonlinear fractional systems is referred to in literature [22–24] and the references therein. However, regional controllability of Riemann–Liouville fractional semilinear evolution equations with analytic semigroup problem is still open. Then, this paper focuses on the existence of a bounded control steering the system into a bounded desired state defined only in a subregion of the whole evolution domain. Based on Banach fixed-point theorem and some properties of fractional operators, the main result is deduced.The rest of this paper is organized as follows. In Section2, some definitions and preparation results are introduced. In Section 3, the regional controllability of the considered system, using theory of analytical semigroup, is established under some conditions. At last, two examples are given to illustrate our given algorithm.


## 2. Preliminaries and Problem Formulation

In this section, we introduce some basic definitions of fractional operators present in the considered system which will be specified later and some properties which are used further in this paper.Definition 1.
(see [7]). The left sided Riemann–Liouville fractional integral (resp. derivative) of a function y at a point t of order q∈0,1 can be written as(1)I0+qyt=1Γq∫0tt−sq−1ysds,0<t≤T,(2)DRL0+qyt=ddtI0+1−qyt,0<t≤T.
Let us considerX and Y to be two Banach spaces; we have the following two propositions.Proposition 1 (see [25]).
Let us considerf∈Lloc10,T;X and T:0,T⟶ℒX,Y to be strongly continuous.
Then, the convolution(3)T∗ft≔∫0tTt−sfsds,exists in Bochner sense and defines a continuous function T∗f from 0,T into Y.Proposition 2 (see [25]).
(Young’s inequality).
Let us considerp,r,s≥1 such that 1/r+1/p=1+1/s. If T∈Lp0,T;ℒX,Y and f∈Lr0,T;X; then, T∗f∈Ls0,T;Y and(4)T∗fLs0,T;Y≤fLr0,T;X.TLp0,T;ℒX,Y.
Now, we present the considered system. For that, letΩ be a bounded subset of ℝn with a smooth boundary ∂Ω . Let us consider T>0 and denote(5)Q≔Ω×0,T,Σ≔∂Ω×0,T.We consider the following semilinear fractional system involving Riemann–Liouville derivative of orderq∈0,1:(6)D0+qRLyx,t+Ayx,t=Nyx,t+ℬut,inQ,yξ,t=0,onΣ,limt⟶0+I0+1−qyx,t=y0,inΩ,where −A is the infinitesimal generator of an analytic semigroup of uniformly bounded operator Gtt≥0 on the Hilbert space X=L2Ω. Without loss of generality, let 0∈ρA where ρA is the resolvent set of A. Then we define the fractional power Aα for 0<α<1, which is a closed linear operator. Its domain is DAα=Xα, which is a Banach space equipped with the norm .Xα=Aα.X, N:L20,T;Xα⟶L20,T;X is a nonlinear operator, and ℬ is the control operator which is linear (bounded or unbounded) from ℝp into X where p is the number of actuators, u is given in U≔L20,T,ℝp, and the initial state y0 is in Xα.We use the following definition of mild solution for the previous problem.Definition 2 (see [26]).
Fort∈0,T and any given u∈U, we say that a function yu∈C0,T;X is a mild solution of system (6) if it satisfies the following formula:(7)yu.,t=tq−1Kqty0+∫0tt−τq−1Kqt−τNyuτ+Buτdτ,where(8)Kqt=α∫0∞θϕqθGtqθdθ,in which ϕq is a probability density function defined in 0,∞.
MoreoverAα and Kq have the following properties.Proposition 3 (see [27]).
For anyt>0, we have(i)
∃Mα>0 such that(9)AαKqtℒX,X≤qMαtαq×Γ2−αΓ1+q1−α.(i)
∃M>0 such that(10)KqtℒXα,Xα≤MΓq.Corollary 1.
Let us consider(11)Hqt=tq−1Kqt.
Then, we have(12)Hq∈L10,T;ℒX,Xα.Proof.
We have0<α,q<1; then, q1−α>0. Therefore, t−αq+q−1∈L10,T, and by the previous proposition, we have the result.
For the rest of this paper, we denote(13)β≔HqL10,T;ℒX,XαℬℒX,ℝp.


## 3. Regional Controllability

In this section, we formulate and prove conditions for the regional controllability of semilinear Riemann–Liouville fractional control systems. To do this, letω be a subregion of Ω, and we define the restriction operator in ω by(14)χω:L2Ω⟶L2ω,y⟶y|ω,and we denote by χω∗ its adjoint.We have the following definition.Definition 3.
The system (6) is said to be exactly (respectively, approximately) ω-controllable if for all yd∈L2ω (respectively, for all ε>0 and for all yd∈L2ω), there exists a control u∈U such that χωyuT=yd (respectively, χωyuT−ydL2ω≤ε).
For the rest of this paper, we can write the mild solution as follows:(15)yu.,t=Hqty0+Hq∗Nyu.,t+Hq⋆Bu.,t,and we define the restriction of the controllability operator in ω by(16)HTωq:U⟶L2ω,u⟶χωHq∗Bu.,T.
Consider now the following associate linear system of equation (6):(17)D0+qRLyux,t=Ayux,t+ℬut,inQ,yuξ,t=0,onΣ,limt⟶0+I0+1−qyux,t=y0,inΩ,which we assume to be approximately ω−controllable.
Next, we will study the regional controllability of the system (6) in ImHTωq endowed with the norm(18)ydImHTωq=HTωq†ydU,where(19)HTωq†≔HTωq∗HTωqHTωq∗−1Pseudo−inverse operator ofHTωq.
Now, we shall present the main result; we first define the operator(20)ψyd,u=HTωq†yd−χωHqTy0−χωHq∗Nyu.,T,and we make the following assumptions:(i)
(H1) For arbitrary x,y∈L20,T;Xα,(21)N0=0,Nx−NyL20,T;X≤LNx,yx−yL20,T;Xα,withLN:ℝ+×ℝ+⟶ℝ+ satisfying lima,ν⟶0,0LNa,ν=0.(ii)
H2χωHq.ℒX,ImHTωq≔gq∈L20,T.
We obtain the following theorem.Theorem 1.
If the hypothesesH1 and H2 are satisfied, then the following assertions hold:(1)
There existρ>0, μ=μρ>0, and m=mρ>0 such that, under the assumption(22)ρ>1+βMTΓqy0Xα,for any stateyd in B0,μ⊂ImHTωq, the operator Ψyd,. has a unique fixed point u∗ in B0,m that steers system (6) to yd in ω.(2)
The mapping(23)F:B0,μ⟶U,yd⟶u∗is a Lipschitz mapping.Proof.
(1)
Let us consider(24)A1=βgqL20,Tsupa,ν≤ρLNa,ν,A2=HqL10,T;ℒX,Xαsupa,ν≤ρLNa,ν+MTρΓqy0Xα,A3=HqL10,T;ℒX,Xαsupa,ν≤ρLNa,ν.Using the limit ofLN.,. near 0,0, we can see that for ρ>0, there exists l>0 such that(25)LNa,ν≤l<1−1+βMT/ρΓqy0XαHqL10,T;ℒX,Xα+βgqL20,T∀a,ν≤ρ,which givesA1,A2,A3<1.Let us consider(26)m=ρβ1−HqL10,T;ℒX,Xαsupa≤ρLNa,0−MTρΓqy0Xα.We havem>0 and the mapping f:B0,m⟶B0,ρ such thatfu=yu is a Lipschitz mapping with constant β/1−A3. In fact,(27)HqL10,T;ℒX,Xαsupa≤ρLNa,0+MTρΓqy0Xα≤A2<1,and hencem>0.To show the Lipschitz condition of the functionf, we use equation (15), and Corollary 1, we have for all u,v∈B0,m(28)yu−yvL20,T;Xα=Hq∗Nyu−yv.+Hq⋆Bu−v.L20,T;Xα≤HqL10,T;ℒX,XαNyu−yvL20,T;X+HqL10,T;ℒX,XαℬℒX,ℝpu−vU.Therefore, by hypothesis (H1), we obtain(29)yu−yvL20,T;Xα≤A3yu−yvL20,T;Xα+βu−vU,and hencef is a Lipschitz mapping with constant β/1−A3.Now, letμ=ρ/β1−1+βMT/ρΓqy0Xα+A1+A3>0 and consider yd∈B0,μ; we show that Ψyd,. has a unique fixed point in B0,m.Let us consideru,v∈B0,m; we have(30)Ψyd,u−Ψyd,vU=∫0TχωHqT−sNyus−NyvsdsImHTωq≤gqL20,TNyu−NyvL20,T;X≤gqL20,Tsupa≤ρLNa,0yu−yvL20,T;Xα≤A1βyu−yvL20,T;Xα.Sincef is Lipschitz, then(31)Ψyd,u−Ψyd,vU≤A11−A3u−vU.ForA4≔A1/1−A3, using inequality (15), we have A4<1, and thus Ψyd,. is a strict contraction mapping.Let us show that(32)ΨB0,μ,B0,m⊂B0,m.For that, letu∈B0,m; then, yu∈B0,ρ and(33)Ψyd,uU=yd−χωHqTy0−∫0TχωHqT−sNyusdsImHTωq≤ydImHTωq+MTΓqy0+gqL20,TNyuL20,T;X≤ydImHTωq+MTΓqy0+gqL20,Tρsupa≤ρLNa,0.Therefore, foryd∈B0,μ, we have(34)Ψyd,uU≤ρβ1−HqL10,T;ℒX,Xαsupa≤ρLNa,0−MTρΓqy0Xα=m.Then,ΨB0,μ,B0,m⊂B0,m; finally, we deduce from Banach fixed-point theorem that Ψyd,. admits a unique fixed point u∗∈B0,m.A direct calculation shows us thatu∗ obtained is the solution of the regional controllability problem of system (3).(2)
Let us considerzd and yd in B0,μ; we have(35)Fzd−Fyd=Ψzd,Fzd−Ψzd,Fyd+Ψzd,Fyd−Ψyd,Fyd.On the other hand,(36)Ψzd,Fzd−Ψzd,FydU≤A4Fzd−FydU,Ψzd,Fyd−Ψyd,FydU=zd−ydImHTωq.Then,(37)Fzd−FydU≤11−A4zd−ydImHTωq.Therefore,F satisfies the Lipschitz condition.Remark 1.
In the case whereℬ is unbounded, we suppose that ℬ is an admissible control operator for Hq (see [28]), and consequently we can demonstrate the same result with a suitable β.
We give the following proposition.Proposition 4.
The sequence(38)u0=0,un+1=HTωq†yd−χωHqTy0−χωHq∗NyunT,converges in B0,m⊂U to u∗.Proof.
Let us considern,k∈ℕ∗; we have(39)un+k−unU≤∑l=nn+k−1ul+1−ulU.
From inequality (31), we obtain(40)ul+1−ulU=Ψyd,ul−Ψyd,ul−1U≤A4ul−ul−1U≤A4lu1U.
SinceA4n⟶0n⟶+∞, limn⟶+∞un+k−unU=0.
Then,unn is a Cauchy sequence on B0,m, and we conclude that unn converges to u∗ in B0,m.
Passing to the limit in (8), we have u∗=Ψyd,u∗; therefore, u=u∗ because Ψyd,. has a unique fixed point in B0,m.
Accordingly, we implement the algorithm as follows.(i)
Step 1q, y0, the actuator, the subregion ω, yd and ε small enough.Chooser1=yd.Calculateu1=HTωq†r1 and obtain yu1.,T.(ii)
Step 2Repeatrn=rn−1+yd−χωyun−1.,T,n≥2.Calculateun=HTωq†rn and obtain yun.,T.Untilχωyun.,T−ydImHTωq<ε.


## 4. Numerical Simulations

In this section, we present two numerical simulations illustrating our theoretical result where the first one is given by using zonal actuator and the second example is given by using a pointwise actuator.


### 4.1. Zonal Actuator

Let us consider the following one-dimensional fractional system withq=0.3.(41)D0+0.3RLyux,t=∂2yux,t∂x2+∑j=1∞yu,φj2φjx+χDutin0,1×0,2,yuξ,t=0on0,1×0,2,limt⟶0+I0+0.7yux,t=sinπxin0,1,where φj=2sinjπx and D=0.4,0.6.The subregion where we search the controllability of the considered system isω=0.2,0.5 and let ydx=12x2x−1x−0.4. By using the previous algorithm, we have the following figures.In Figure1, we remark that with the given zonal sensor, we obtain successful results which validate the used method and the previous algorithm; indeed, the desired and estimated final states are very close in the subregion ω=0.2,0.5 with the error ϵ=3×10−6 which is very small. Figure 2 presents the evolution of control function which has a transfer cost equal to u∗2=0.21.Figure 1
Desired state and estimated final state inω.Figure 2
Control input function.


### 4.2. Pointwise Actuator

We consider the following system:(42)D0+0.5RLyux,t=∂2yux,t∂x2+∑j=1∞yu,φjyu,φjφjx+δbutin0,1×0,2,yuξ,t=0on0,1×0,2,limt⟶0+I0+0.5yux,t=sinπxin0,1.By applying the above algorithm forb=0.2, ω=0.35,0.55, and(43)ydx=−52xx−1x−0.33x−0.54,we have the following result.In Figure3, like the previous part, the given algorithm leads to good results; we remark that the desired state and the reached one are close in w=0.35,0.55 using the given pointwise actuator. In this case, the reconstruction error is very small, and it is of order ε=10−5. Figure 4 shows the evolution of control function depending on the time t with the transfer cost u∗2=4×10−3.Figure 3
Estimated state and desired state inω.Figure 4
Control function.


## 4.1. Zonal Actuator

Let us consider the following one-dimensional fractional system withq=0.3.(41)D0+0.3RLyux,t=∂2yux,t∂x2+∑j=1∞yu,φj2φjx+χDutin0,1×0,2,yuξ,t=0on0,1×0,2,limt⟶0+I0+0.7yux,t=sinπxin0,1,where φj=2sinjπx and D=0.4,0.6.The subregion where we search the controllability of the considered system isω=0.2,0.5 and let ydx=12x2x−1x−0.4. By using the previous algorithm, we have the following figures.In Figure1, we remark that with the given zonal sensor, we obtain successful results which validate the used method and the previous algorithm; indeed, the desired and estimated final states are very close in the subregion ω=0.2,0.5 with the error ϵ=3×10−6 which is very small. Figure 2 presents the evolution of control function which has a transfer cost equal to u∗2=0.21.Figure 1
Desired state and estimated final state inω.Figure 2
Control input function.


## 4.2. Pointwise Actuator

We consider the following system:(42)D0+0.5RLyux,t=∂2yux,t∂x2+∑j=1∞yu,φjyu,φjφjx+δbutin0,1×0,2,yuξ,t=0on0,1×0,2,limt⟶0+I0+0.5yux,t=sinπxin0,1.By applying the above algorithm forb=0.2, ω=0.35,0.55, and(43)ydx=−52xx−1x−0.33x−0.54,we have the following result.In Figure3, like the previous part, the given algorithm leads to good results; we remark that the desired state and the reached one are close in w=0.35,0.55 using the given pointwise actuator. In this case, the reconstruction error is very small, and it is of order ε=10−5. Figure 4 shows the evolution of control function depending on the time t with the transfer cost u∗2=4×10−3.Figure 3
Estimated state and desired state inω.Figure 4
Control function.


## 5. Conclusion

In this paper, we have established the regional controllability for a class of Riemann–Liouville fractional semilinear control systems. The idea of applying control theory for this kind of systems is very interesting and constitutes a new issue in the applications. The presented method in this paper covers a large class of this kind of systems. We have also given an algorithm which has been implemented numerically and has very satisfactory results. In addition, the problem of regional controllability remains open for other types of fractional systems that will be the subject of future research.



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