# Approximate Controllability for Functional Equations with Riemann-Liouville Derivative by Iterative and Approximate Method

Authors: Badawi Hamza Elbadawi Ibrahim; Zhenbin Fan; Gang Li
Journal: Journal of Function Spaces
 (2017)

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## Abstract

We discuss the functional control systems governed by differential equations with Riemann-Liouville fractional derivative in general Banach spaces in the present paper. First we derive the uniqueness and existence of mild solutions for functional differential equations by the approach of fixed point and fractional resolvent under more general settings. Then we present new sufficient conditions for approximate controllability of functional control system by means of the iterative and approximate method. Our results unify and generalize some previous works on this topic.


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## Body

## 1. Introduction

As we all know, the study of fractional calculus theory can be traced back to the end of the seventeenth century. In recent years, a considerable interest has been paid to functional evolution equations with fractional derivatives since they are of importance in describing the natural phenomenon including the models in stochastic processes, finance, and physics (see [1–9]). On the other hand, the notion of controllability plays a central role in the study of the theory of control and optimization. Therefore, there are a lot of works on the controllability, approximate controllability, and optimal control of linear and nonlinear differential and integral systems in various frameworks (see [10–28] and references therein).Recently, the theory of resolvent families was formulated rapidly for the application of differential and integral equations, including the concepts of integrated solution operators [29], fractional resolvent operators [30], and a,k-regularized resolvent operators [19]. Furthermore, there are extensive studies for the control systems governed by Caputo fractional evolution equations via resolvent theory (see, for instance, [10–12, 15, 27, 31]). However, for the controllability of functional differential systems governed by Riemann-Liouville fractional derivatives there are few results to be shown so far.In [32, 33], the authors established the general theory of fractional resolvent and studied its application to the well posed problem for the evolution equation below:(1)Dβyt=Ayt+gt,t∈0,c,limt↓0⁡Γβt1-βyt=y∈X,where Dβ is the standard Riemann-Liouville derivative with 0<β≤1 and operator A:DA⊂X→X generates a β-order fractional resolvent Sβt. On the other hand, in [17], the authors considered the following control system governed by functional differential equations with Riemann-Liouville derivative in an abstract space X:(2)Dβyt=Ayt+But+gt,yt,t∈0,c,limt↓0⁡Γβt1-βyt=y0∈X,where Dβ is the standard Riemann-Liouville derivative with 0<β≤1 and operator A:DA⊂X→X generates a semigroup St, t≥0, on an abstract space X. B:V≐Lp0,c,U→Lp0,c,X belongs to the space BV,Lp0,c,X, U is a Banach space, and the control function u belongs to V.Motivated by the above-mentioned papers, we try to solve the approximate controllability for functional differential equation (2) again but under the assumption that operator A:DA⊂X→X is the infinitesimal generator of a resolvent Sβt, t>0, on the general Banach space X. Under this general condition, the difficulty on the well-posedness is how to deal with the singularity of resolvent operators and solutions at zero. We deal with this problem by utilizing the new space C1-β0,c,X and the theory of fractional resolvent developed in [32, 33]. In the present article, we first obtain the uniqueness and existence of mild solutions for functional differential equation (2) via fractional resolvent and topological approach. Then we will give the sufficient conditions for approximate controllability to equation (2) by means of the iterative and approximate method. Our main result seems to be more general and extends some recent related theorems.In this paper, we first review some basic definitions and give some necessary lemmas. We establish the uniqueness and existence results of mild solutions for functional differential equation (2) in the third part. We will solve the approximately controllable problem of functional control system (2) in the last section.


## 2. Preliminaries

This section is devoted to introducing some necessary concepts and auxiliary results which are used in the remainder of this article.LetX,Y be two Banach spaces with norm · and R+ be the set of nonnegative real numbers. We denote by BX all the continuous linear operators from Banach space X into itself, by C0,c,X the set of all the continuous functions from the interval 0,c to Banach space X with yC=sup⁡yt,t∈0,c, and by LP0,c,X the set of all Bochner integrable functions from the interval 0,c to Banach space X with gLp=∫0cgτpdτ1/p, where 1≤p<∞.LetAC0,c,X be the set of all absolutely continuous functions from the interval 0,c to Banach space X, and ACm0,c,X=g:0,c→Xandgm-1∈AC0,c,X. Let 0<β≤1; we consider the Banach space (3)C1-β0,c,X=y:τ1-βyτis  continuous  on  the  interval0,cwith  values  inXwith yC1-β=sup⁡τ1-βyτ:τ∈0,c. This space is of importance in dealing with the well-posedness of the functional equation (2).As usual, letΓz≐∫0∞e-ττz-1dτ denote the Gamma function and β the integer part of the real number β>0. We give the following definitions.Definition 1 (see [3, 34]).
The Riemann-Liouville fractional integral of orderβ>0 for a function h∈L10,c,X is defined by (4)Itβht=1Γβ∫0thτt-τ1-βdτ,0≤t≤c,if the integral above exists point-wisely.Definition 2 (see [3, 34]).
Letm=β+1. The Riemann-Liouville fractional derivative of order β>0 for a function h∈L10,c,X is defined by (5)Dβht=1Γm-βdmdtm∫0thτt-τβ+1-mdτ,0≤t≤c,if the derivative of m-order above exists point-wisely.Lemma 3 (see [3, 34]).
Letm=β+1 and ym-βt=Itm-βyt. Suppose y·∈L10,c,X and ym-β·∈ACm0,c,X. Then, one has (6)ItβDβyt=yt-∑k=1mym-βm-k0Γβ-k+1tβ-k,0≤t≤c.Definition 4 ([32, Definition  3.1]).
Let0<β≤1. A family of operators Sβtt>0 belonging to the space BX is called a β-order fractional resolvent if(a)
Sβt is continuous in the strong operator topology on 0,∞, and limτ↓0⁡Sβτ/τβ-1y=1/Γβy for y∈X.(b)
SβμSβτ=SβτSβμ for any μ,τ>0.(c)
SβμIτβSβτ-IμβSβμSβτ=μβ-1/ΓβIτβSβτ-τβ-1/ΓβIμβSβμ for all μ,τ>0.Definition 5.
The infinitesimal generatorA of the fractional resolvent Sβtt>0 is defined by (7)Ay=Γ2βlimτ↓0⁡Sβτy-τβ-1/Γβyτ2β-1with (8)DA=y∈X:limτ↓0⁡Sβτy-τβ-1/Γβyτ2β-1exists.Lemma 6 ([32, Theorem  3.2]).
LetA be the infinitesimal generator of the fractional resolvent Sβtt>0 on Banach space X. Then A is closed with dense domain and the following hold. (a)
SβτDA⊂DA and ASβτy=SβτAy for y∈DA, τ>0.(b)
Forτ>0, one has (9)Sβτy=τβ-1Γβy+AIτβSβτy,y∈X,Sβτy=τβ-1Γβy+IτβSβτAy,y∈DA.Lemma 7.
Letc>0 be fixed and operator A be the generator of a β-order fractional resolvent Sβtt>0 in Banach space X and (10)τ1-βSβτyτ=0≐limτ↓0⁡τ1-βSβτy=1Γβy,y∈X.Then there is a positive number M>0 satisfying supτ∈0,c⁡τ1-βSβτ≤M.Proof.
According to the definition ofβ-order fractional resolvent, for every y∈X there is a positive number My>0 dependent on y satisfying (11)τ1-βSβτy≤My,∀τ∈0,c.Moreover, since τ1-βSβτ∈BX,τ∈0,c, it follows from the uniform boundedness principle that there is a positive number M>0 independent of y satisfying supτ∈0,c⁡τ1-βSβτ≤M.Now, we turn to the concept of mild solutions and approximate controllability of functional equation (2). In [33], Fan proved that the convolution Sβ∗g exists and defines a continuous function on 0,c when the resolvent is uniformly integrable and the function f is continuous on 0,c. In fact, by the Proposition  1.3.4 in [35] and the Young inequality, we can also prove that the above convolution exists and defines a continuous function on 0,c under the assumption that Sβt is just a fractional resolvent and g∈L10,c,X. Thus, we have the following definitions.Definition 8.
A mappingy·∈C1-β0,c,X is called a mild solution of functional differential equation (2) if for u∈V, the integral equation (12)yt=Sβty0+∫0tSβt-τBuτ+gτ,yτdτ,0<t≤cis satisfied.Lety·;0,y0,u be a mild solution of functional equation (2) with the control function u belonging to the space V and initial value y0 in Banach space X. Define the reachable set of functional equation (2) at time c by Kc,y0≐yc;0,y0,u:u∈V.Definition 9.
The functional equation (2) is called approximate controllability on 0,c if Kc,y0¯=X.


## 3. The Result of Uniqueness and Existence for Mild Solutions

In this part, we study the existence and uniqueness of solutions for functional equation (2). To prove our result, we suppose the following conditions.(H1)A is the infinitesimal generator of an analytic β-order fractional resolvent of continuous linear operators Sβtt>0 on Banach space X.(H2) There are nonnegative functionμ·∈Lp0,c,R+ and a real number k>0 satisfying (13)gτ,y≤μτ+kτ1-βyfor y∈X and τ∈0,c.(H3) There is a nonnegative real numberL satisfying (14)gτ,y-gτ,z≤Ly-zfor τ∈0,c and y,z∈X.Now, we can give the existence result of mild solutions for the functional equation (2).Theorem 10.
Under the conditions (H1)–(H3), the functional equation (2) has one and only one mild solution in the space C1-β0,c,X for every u belonging to the control space V.Proof.
Define the mappingQ:C1-β0,c,X→C1-β0,c,X by(15)Qyt=Sβty0+∫0tSβt-τBuτ+gτ,yτdτ,0<t≤c.Obviously, it is well-defined. To prove our result, it is enough to verify the mapping Q has a unique fixed point in space C1-β0,c,X. We next verify that Qn is a contraction map on C1-β0,c,X for sufficiently large integer number n.
According to Lemma3, there is a constant M>0 satisfying supτ∈0,c⁡τ1-βSβτ≤M. Thus, for any y,z∈C1-β0,c,X and 0≤t≤c, one has(16)t1-βQyt-Qzt≤t1-β∫0tt-τβ-1t-τ1-βSβt-τgτ,yτ-gτ,zτdτ≤t1-β∫0tt-τβ-1τβ-1t-τ1-βSβt-ττ1-βLyτ-zτdτ≤t1-βML∫0tt-τβ-1τβ-1y-zC1-βdτ=tβMLΓβ2Γ2βy-zC1-β.Further,(17)t1-βQ2yt-Q2zt≤t1-βML∫0tt-τβ-1τβ-1τ1-βQyτ-Qzτdτ≤t1-βML∫0tt-τβ-1τβ-1MLΓβ2Γ2βτβy-zC1-βdτ≤t1-βML∫0tt-τβ-1τ2β-1MLΓβ2Γ2βy-zC1-βdτ=tβML2Γβ3Γ3βy-zC1-β.By repeating the above process, one has (18)t1-βQnyt-Qnzt≤cβMLnΓβn+1Γn+1βy-zC1-β.Thus, (19)Qny-QnzC1-β≤ΓβcβMLΓβnΓn+1βy-zC1-β.Note that the Mittag-Leffler function Eβ,βcβMLΓβ=∑k=0∞cβMLΓβk/Γkβ+β‍ is uniformly convergent; one has sufficiently large integer n, (20)ΓβcβMLΓβnΓn+1β<1,which implies Qn is a contraction map on the space C1-β0,c,X. Hence, it follows from the generalized Banach contraction principle that Q has one and only one fixed point y in C1-β0,c,X. This completes the proof.


## 4. The Result of Approximate Controllability

In this part, new sufficient conditions for the approximate controllability of the functional equation (2) are derived and proved by means of the iterative and approximate approach. For this purpose, we define operator F:C1-β0,c,X→Lp0,c,X and a continuous linear mapping G:Lp0,c,X→X, respectively, by (21)Fyτ=gτ,yτ,τ∈0,c,Gψ=∫0cSβc-τψτdτ,ψ·∈Lp0,c,X.Let the pair y,u be the mild solution of (2) with the control function u∈V=Lp0,c,U. We also denote the pair by yt=yt;0,y0,u and write the terminal state yc by (22)yc=yc;0,y0,u=Sβcy0+GFy+GBu.So the reachable set Kc,y0 of the functional equation (2) at time c is (23)Kc,y0=yc:yc=yc;0,y0,uforu∈V.Thus the approximate controllability of functional equation (2) means the set Kc,y0 is dense on space X. In other words, we have the following expression of approximate controllability.Definition 11.
Lety0∈X. We called the functional equation (2) approximately controllable on 0,c if, for any ε>0 and yc∈X, there is a control function uε∈V satisfying (24)yc-Sβcy0-GFyε-GBuε<ε,where yεt=yt;0,y0,uε,0<t≤c.Now, we give the following assumptions.(H3′) There is a nonnegative numberL′ satisfying (25)gτ,y-gτ,z≤L′τ1-βy-zfor all τ∈0,c and y,z∈X.(H4) For givenε>0 and ψ∈Lp0,c,X, there is a function u∈V satisfying (26)Gψ-GBu≤ε,BuLp≤λψLp,where λ is a positive real number independent of ψ.(H5) The following inequality holds:(27)ML′λp-1pβ-1p-1/pcEβML′cΓβ<1,where Eβz=∑k=0∞zk/Γkβ+1 is the Mittage-Leffler function and M comes from Lemma 3.Because condition (H3′) implies (H3), the existence result is still true if condition (H3) is replaced by (H3′) in Theorem10. Next, to prove the result of approximate controllability of functional equation (2), we need two lemmas below.Lemma 12.
Lety1,u1, y2,u2 be two pairs associated with the control system (2). Then under the hypotheses (H1), (H2), and (H3′) the following results hold: (28)y·;0,y0,uC1-β≤CEβMkcΓβ,∀u∈V,y1-y2C1-β≤NEβML′cΓβBu1-Bu2Lp,where C=My0+p-1/pβ-1p-1/pc1-1/pBuLp+μLp, N=Mp-1/pβ-1p-1/pc1-1/p.Proof.
The mild solutionyt=yt;0,y0,u of control system (2) in C1-β0,c,X satisfies (29)yt=Sβty0+∫0tSβt-τBuτ+gτ,yτdτ,0<t≤c.Thus, for 0≤t≤c, one has(30)t1-βyt≤t1-βSβty0+t1-β∫0tt-τβ-1t-τ1-βSβt-τBuτdτ+t1-β∫0tt-τβ-1t-τ1-βSβt-τgτ,yτdτ≤My0+t1-β∫0tt-τβ-1Buτdτ+t1-β∫0tt-τβ-1μτ+kτ1-βyτdτ≤My0+p-1pβ-1p-1/pc1-1/pBuLp+μLp+kc1-β∫0tt-τβ-1τ1-βyτdτ.Set ωt=t1-βyt,t∈0,c. Thus, (31)ωt≤C+Mkc1-β∫0tt-τβ-1ωτdτ.By a generalized Gronwall inequality for convolution type integral equations (see Corollary  2 in [36]), we obtain (32)ωt≤CEβMkcΓβ.It follows that (33)yC1-β=supt∈0,c⁡t1-βyt≤CEβMkcΓβ.Now, let us take y1,y2∈C1-β0,c,X to be the mild solutions for the control system (2) with the control functions u1,u2∈V, respectively. Then, by the same way, we can get (34)y1-y2C1-β≤NEβML′cΓβBu1-Bu2Lp.This ends the proof.Lemma 13.
Under the condition (H1), for anyy∈DA there exists ψ∈Lp0,c,X such that Gψ=y.Proof.
Fory∈DA and τ∈0,c, let (35)ψ1τ=Γ2βcc-τ21-βSβc-τy,ψ2τ=Γ2βc2τc-τ1-βdc-τ1-βSβc-τydτ.Then, we have (36)Gψ1=Γ2βc∫0cSβc-τc-τ21-βSβc-τydτ=Γ2βc∫0cc-τ21-βSβ2c-τydτ=Γ2βcτc-τ21-βSβ2c-τy0c-Γ2βc∫0cτdc-τ21-βSβ2c-τy=y-Γ2βc∫0c2τc-τ1-βSβc-τdc-τ1-βSβc-τy=y-Gψ2,which means that Gψ1+ψ2=y.
Moreover, it follows from the analyticity and the strong continuity ofτ1-βSβτ that ψ1+ψ2∈Lp0,c,X. This completes the proof.Now, we can give the result about the approximate controllability of this section.Theorem 14.
Suppose conditions (H1), (H2), (H3′), (H4), and (H5) hold. Then the functional equation (2) is approximately controllable on the interval 0,c.Proof.
SinceA is the infinitesimal generator of analytic β-order fractional resolvent of continuous operator Sβt on space X, the domain of operator A is dense in Banach space X. Thus, by the definition of approximate controllability, it suffices to prove that DA⊂Kc,y0. Next, we should prove that for any ε>0 and yc∈DA, there is a control function uε∈V with(37)yc-Sβcy0-GFyε-GBuε<ε,where yεt=yt;0,y0,uε satisfies(38)yεt=Sβty0+∫0tSβt-τFyετdτ+∫0tSβt-τBuετdτ,0<t≤c.
It follows from the analyticity of fractional resolvent thatSβcy0∈DA for y0∈X, which implies that yc-Sβcy0∈DA for yc∈DA. Then, by Lemma 13 there is a function ψ∈Lp0,c,X satisfying (39)Gψ=yc-Sβcy0.Thus, for every ε>0 and u1∈V, by means of hypothesis (H4), we can find a function u2∈V satisfying (40)yc-Sβcy0-GFy1-GBu2<ε32,where y1t=yt;0,y0,u1,0<t≤c. Further, for u2∈V, we determine again v2∈V by condition (H4) and Lemma 12 with the following two properties: (41)GFy2-Fy1-GBv2<ε33,Bv2Lp≤λFy1-Fy2Lp≤λL′c1/py2-y1C1-β≤λL′Mp-1pβ-1p-1/pcEβML′cΓβBu1-Bu2Lp,where y2t=yt;0,y0,u2,0<t≤c.
Thus, we may defineu3=u2-v2 in V, which has the following property (42)yc-Sβcy0-GFy2-GBu3≤yc-Sβcy0-GFy1-GBu2+GBv2-GFy2-Fy1≤132+133ε.By the same way, we obtain the sequence un:n≥1⊆V satisfying(43)yc-Sβcy0-GFyn-GBun+1≤132+⋯+13n+1ε,(44)Bun+1-BunLp≤λL′Mp-1pβ-1p-1/pcEβML′cΓβBun-Bun-1Lp,where ynt=yt;0,y0,un,0<t≤c,n=1,2,…. Since the condition (H5) is satisfied, it follows that the sequence Bun:n≥1 is a Cauchy sequence in Lp0,c,X and thus we can obtain a function v∈Lp0,c,X with (45)limn→∞⁡Bun=vinLp0,c,X.Note that the mapping G:Lp0,c,X→X is a continuous linear operator. Therefore for every ε>0, we can find a real integer number Nε>0 satisfying(46)GBuNε+1-GBuNε<ε3.
Consequently, by inequalities (43) and (46) we derive (47)yc-Sβcy0-GFyNε-GBuNε≤yc-Sβcy0-GFyNε-GBuNε+1+GBuNε+1-GBuNε≤132+⋯+13Nε+1ε+ε3≤ε,where yNεt=yt;0,y0,uNε,0<t≤c. This ends the proof.Remark 15.
In this article, we consider the control system with Riemann-Liouville derivative under the general assumption that linear operatorA:DA⊂X→X generates a fractional resolvent Sβt, t>0 on the Banach space X. By the resolvent theory, fixed point theorem, and iterative and approximate approach, we obtain the results of existence and uniqueness for mild solutions and approximate controllability of functional control system. From this point of view, our results extend the theorems in [17, 18], where the authors assumed that the operator A generates C0-semigroups. As for examples, we refer the reader to [17, 18, 33] since the examples therein still hold in our framework.



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