# Approximate Controllability for Impulsive Riemann-Liouville Fractional Differential Inclusions

Authors: Zhenhai Liu; Maojun Bin
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We study the control systems governed by impulsive Riemann-Liouville fractional differential inclusions and their approximate controllability in Banach space. Firstly, we introduce thePC1-α-mild solutions for the impulsive Riemann-Liouville fractional differential inclusions in Banach spaces. Secondly, by using the fractional power of operators and a fixed point theorem for multivalued maps, we establish sufficient conditions for the approximate controllability for a class of Riemann-Liouville fractional impulsive differential inclusions, which is a generalization and continuation of the recent results on this issue. At the end, we give an example to illustrate the application of the abstract results.


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## Body

## 1. Introduction

The concept of controllability plays an important part in the analysis and design of control systems. Since Kalman [1] first introduced its definition in 1963, controllability of the deterministic and stochastic dynamical control systems in finite-dimensional and infinite-dimensional spaces is well developed in different classes of approaches, and more details can be found in papers [2–4]. Some authors [5–7] have studied the exact controllability for nonlinear evolution systems by using the fixed point theorems. In [5–7], to prove the controllability results for fractional-order semilinear systems, the authors made an assumption that the semigroup associated with the linear part is compact. But if C0-semigroup T(t) is compact or the operator B is compact, then the controllability operator is also compact and hence the inverse of it does not exist if the state space V is infinite dimensional [8]. Thus, it is shown that the concept of exact controllability is difficult to be satisfied in infinite-dimensional space. Therefore, it is important to study the weaker concept of controllability, namely, approximate controllability for differential equations. In these years, several researchers [9–17] have studied it for control systems.In [13], Sakthivel et al. studied on the approximate controllability of semilinear fractional differential systems:
(1)cDαx(t)=Ax(t)+Bu(t)+f(t,x(t)),t∈J=[0,T],x(0)=x0,
where cDtα is Caputo’s fractional derivative of 0<α<1 and A is the infinitesimal generator of a C0-semigroup T(t) of bounded operators on the Hilbert space X; the control function u(·) is given in L2(J,U); U is a Hilbert space; B is a bounded linear operator from U to X; f:J×X→X is a given function satisfying some assumptions and x0 is an element of the Hilbert space X.In [16], Sukavanam and Kumar researched approximate controllability of fractional-order semilinear delay systems:
(2)dαx(t)tα=Ax(t)+Bu(t)+f(t,xt,u(t)),t∈[0,τ],x0(θ)=ϕ(θ),θ∈[-h,0],
where 1/2<α<1; A:D(A)⊆V→V is a closed linear operator with dense domain D(A) generating a C0-semigroup S(t); the state x(·) takes values in the Banach space V; the control function u(·) takes values in V^; B is a bounded linear operator from L2([0,τ];V^) to L2([0,τ];V); the operator f:[0,τ]×C([-h,0];V)×V^→V is nonlinear. If x:[-h,τ]→V is a continuous function, then xt:[-h,0]→V is defined as xt(θ)=x(t+θ) for θ∈[-h,0] and ϕ∈C([-h,0];V).In [18], Rykaczewski studied the approximate controllability of an inclusion of the form
(3)x˙(t)∈Ax(t)+F(t,x(t))+Bu(t),t∈J=[0,b],x(0)=x0,
where A is a linear operator which generates a compact semigroup, F is u.h.c. multivalued perturbation with weakly compact values, and the state x(·) takes values in the Hilbert space H. U is a Hilbert space of all admissible controls. B:U→H is a continuous linear operator.Fractional differential equations have recently proved to be valuable tools in the modeling of many phenomena in various fields of engineering, physics, and economics. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetic, and so forth; see [19–27] for example. As a consequence there was an intensive development of the theory of differential equations of fractional order. One can see the monographs of Kilbas et al. [28] and Podlubny [29] and the references therein. The definitions of Riemann-Liouville fractional derivatives or integrals with initial conditions are strong tools to resolve some fractional differential problems in the real world. Heymans and Podlubny [30] have verified that it was possible to attribute physical meaning to initial conditions expressed in terms of Riemann-Liouville fractional derivatives or integrals, and such initial conditions are more appropriate than physically interpretable initial conditions. Furthermore, they have investigated that the impulse response with Riemann-Liouville fractional derivatives was seldom used in the fields of physics, such as viscoelasticity. In recent years, many authors [18, 27, 31] were devoted to mild solutions to fractional evolution equations with Caputo fractional derivative, and there have been a lot of interesting works. As for the study of the fractional differential systems with Caputo fractional derivative, we can refer to [27, 31, 32] for the existence results. Its approximate controllability was considered in [9, 13–16]. The approximate controllability of Caputo fractional inclusion systems has been investigated by [10]. We know that differential inclusions are strong tools to solve some problems in various fields of engineering, physics, and optimal control; see [10, 32–35]. However, the approximate controllability for the impulsive fractional differential evolution inclusion with Riemann-Liouville fractional derivatives is still an untreated topic in the literature.Motivated by the above work, in this paper, we consider the following system:(4)Dtαx(t)∈Ax(t)+F(t,x(t))+Bu(t),t∈J′=(0,b]∖{t1,t2,…,tm},ΔI0+1-αx(t)|t=tk=Ik(x(tk-)),k=1,2,…,m,It1-αx(t)|t=0=x0∈X,
where 1/2<α≤1 and Dtα denotes the Riemann-Liouville fractional derivative of order α with the lower limit zero. F:J×X→𝒫(X):=2X∖{∅} is a nonempty, bounded, closed, and convex multivalued map. A:D(A)⊆X→X is the infinitesimal generator of a C0-semigroup T(t) (t≥0) on a Banach space X. 0=t0<t1<⋯<tm<tm+1=b, Ik:X↦X, ΔI0+1-αx(tk)=I0+1-αx(tk+)-I0+1-αx(tk-), I0+1-αx(tk+) and I0+1-αx(tk-) denote the right and the left limits of I0+1-αx(t) at t=tk, k=1,2,…,m. The control function u(t) takes value in V=Lp([0,b];U), p>1/α, and U is a Banach space; B is a linear operator from V to Lp([0,b];X).The purpose of this paper is to provide some suitable sufficient conditions for the existence of mild solutions and approximate controllability results for the impulsive fractional abstract Cauchy problems with Riemann-Liouville fractional derivatives. The main tools used in our study are fixed point theorem, semigroup theory for multivalued maps, and the theory from fractional differential equations. The rest of this paper is organized as follows. In Section2, we present some preliminaries to prove our main results. In Section 3, by applying some standard fixed point principles, we prove the existence of the mild solutions for semilinear fractional differential equations, and the approximate controllability of the system (4) is proved. In Section 4, we give an example to illustrate our main results.


## 2. Preliminaries

In this section, we introduce some basic definitions and preliminaries which are used throughout this paper. The norm of a Banach spaceX will be denoted by ∥·∥X. Lb(X,Y) denotes the space of bounded linear operators from X to Y. For the uniformly bounded C0-semigroup T(t) (t≥0), we set M:=supt∈[0,∞)∥T(t)∥Lb(X)<∞. Let C(J,X) denote the Banach space of all X value continuous functions from J=[0,b] to X with the norm ∥x∥C=supt∈J∥x(t)∥X. Let C1-α(J,X)={x:t1-αx(t)∈C(J,X)} with the norm
(5)∥x∥C1-α=sup{t1-α∥x(t)∥X:t∈J}.
Obviously, the space C1-α(J,X) is a Banach space.To define the mild solutions of (4), we also consider the Banach space PC1-α(J,X)={x:(t-tk)1-αx(t)∈C((tk,tk+1],X), limt→tk+(t-tk)1-αx(t) exist, and x(tk-) with x(tk-)=x(tk), k=0,1,2,…,m} with the norm
(6)∥x∥PC1-α=max{sup(t-tk)1-α∥x(t)∥X:t∈(tk,tk+1],k=0,1,2,…,m∥x(t)∥X}.
It is easily known that the space PC1-α(J,X) is a Banach space.Let(X,d) be a metric space. We use the notations
(7)Pcl(X)={Y∈𝒫(X):Yisclosed},Pb(X)={Y∈𝒫(X):Yisbounded},Pcv(X)={Y∈𝒫(X):Yisconvex},Pcp(X)={Y∈𝒫(X):Yiscompact}.Firstly, let us recall the following basic definitions from fractional calculus. For more details, one can see [28, 29].Definition 1.
The integral(8)Itαf(t)=1Γ(α)∫0t‍(t-s)α-1f(s)ds,α>0,
is called Riemann-Liouville fractional integral of order α, where Γ is the gamma function.Definition 2.
For a functionf(t) given in the interval [0,∞), the expression
(9)Dtαf(t)=1Γ(n-α)(ddt)n∫0t‍(t-s)n-α-1f(s)dt,
where n=[α]+1 and [α] denotes the integer part of number α, is called the Riemann-Liouville fractional derivative of order α.Lemma 3 (see [28]).
Letα>0, m=[α]+1, and let xm-α(t)=Itm-αx(t) be the fractional integral of order m-α. If x(t)∈L1(J,X) and xm-α(t)∈ACm(J,X), then one has the following equality:
(10)ItαDtαx(t)=x(t)-∑k=1m‍xm-α(m-k)(0)Γ(α-k+1)tα-k.In order to study thePC-mild solutions of (4) in Banach space PC1-α(J,X), we give the following results which will be used throughout this paper.Lemma 4.
Let0<α≤1, and let x1-α(t)=It1-αx(t) be the fractional integral of order 1-α. If x(t)∈PC1-α(J,X) and x1-α(t)∈PC(J,X), then one has the following equality:
(11)ItαDtαx(t)={x(t)-x1-α(t)|t=0tα-1Γ(α),t∈[0,t1],x(t)-∑k=1m‍Δx1-α(ti)Γ(α)(t-ti)α-1-x1-α(t)|t=0tα-1Γ(α),t∈(tk,tk+1],
where Δx1-α(tk)=x1-α(tk+)-x1-α(tk-), k=1,2,…,m.Proof.
Ift∈[0,t1], then, by Lemma 3, we easily get
(12)I0+αD0+αx(t)=x(t)-x1-α(t)|t=0tα-1Γ(α).
If t∈(t1,t2], since
(13)ItαDtαx(t)=1Γ(α)∫0t‍(t-s)α-1LDtαx(s)ds=ddt{1Γ(α+1)∫0t‍(t-s)αLDtαx(s)ds},
then, by (13), we have
(14)1Γ(α+1)∫0t‍(t-s)αLDtαx(s)ds=1Γ(α+1)∫0t‍(t-s)αdds{I0+1-αx(s)}ds=1Γ(α+1)Γ(1-α)×∫0t‍(t-s)αdds∫0s‍(s-τ)1-α-1x(τ)dτds=1Γ(α+1)Γ(1-α)×∫0t1‍(t-s)αdds∫0s‍(s-τ)1-α-1x(τ)dτds+1Γ(α+1)Γ(1-α)×∫t1t‍(t-s)αdds∫0t1‍(s-τ)1-α-1x(τ)dτds+1Γ(α+1)Γ(1-α)×∫t1t‍(t-s)αdds∫t1s‍(s-τ)1-α-1x(τ)dτds=1Γ(α)Γ(1-α)×∫0t1‍(t-s)α-1∫0s‍(s-τ)1-α-1x(τ)dτds+1Γ(α+1)Γ(1-α)×[(t-s)α∫0s‍(s-τ)1-α-1x(τ)dτ]|s=0s=t1+1Γ(α)Γ(1-α)×∫t1t‍(t-s)α-1∫0t1‍(s-τ)1-α-1x(τ)dτds+1Γ(α+1)Γ(1-α)×[(t-s)α∫0t1‍(s-τ)1-α-1x(τ)dτ]|s=t1s=t+1Γ(α)Γ(1-α)×∫t1t‍(t-s)α-1∫t1s‍(s-τ)1-α-1x(τ)dτds+1Γ(α+1)Γ(1-α)×[(t-s)α∫t1s‍(s-τ)1-α-1x(τ)dτ]|s=t1s=t=1Γ(α)Γ(1-α)×∫0t1‍x(τ)dτ∫τt1‍(t-s)α-1(s-τ)1-α-1ds+1Γ(α)Γ(1-α)×∫0t1‍x(τ)dτ∫t1t‍(t-s)α-1(s-τ)1-α-1ds+1Γ(α)Γ(1-α)×∫t1t‍x(τ)dτ∫τt‍(t-s)α-1(s-τ)1-α-1ds-x1-α(0)Γ(α+1)tα-Δx1-α(t1)Γ(α+1)(t-t1)α=∫0t‍x(τ)dτ-x1-α(0)Γ(α+1)tα-Δx1-α(t1)Γ(α+1)(t-t1)α.
Thus, by (13) and (14), we get
(15)ItαDtαx(t)=x(t)-x1-α(0)Γ(α)tα-1-Δx1-α(t1)Γ(α)(t-t1)α-1.
Similarly, if t∈(tk,tk+1], k=2,…,m, we can get
(16)ItαDtαx(t)=x(t)-∑k=1m‍Δx1-α(tk)Γ(α)(t-tk)α-1-x1-α(t)|t=0tα-1Γ(α).
The proof is completed.The Laplace transform formula for the Riemann-Liouville fractional integral is defined by(17)L{Itαx(t);λ}=1λαx^(λ),
where x^(λ) is the Laplace of x defined by
(18)x^(λ)=∫0∞‍e-λtx(t)dt,Reλ>ω,|x(t)|≤ceωt,cisaconstant.Lemma 5.
Letα∈(0,1] and h∈Lp(J,X), p>1/α; if x(t)∈PC1-α(J,X), x1-α(t)∈PC(J,X), and x is a solution of the following problem:
(19)Dtαx(t)=Ax(t)+h(t),t∈(0,b],t≠tk,k=1,2,…,m,ΔIt1-αx(t)|t=tk=Ik(x(tk-)),k=1,2,…,m,It1-αx(t)|t=0=x0∈X,
then, x satisfies the following equation:
(20)x(t)={tα-1Tα(t)x0+∫0t‍(t-s)α-1Tα(t-s)h(s)ds,t∈[0,t1],tα-1Tα(t)x0+∑k=1m‍Tα(t-tk)(t-tk)α-1ΔI0+1-αx(tk)+∫0t‍(t-s)α-1Tα(t-s)h(s)ds,t∈(tk,tk+1],
where k=1,2,…,m,
(21)Tα(t)=α∫0∞‍θξα(θ)T(tαθ)dθ,ξα(θ)=1αθ-1-(1/α)ϖα(θ-1/α),ϖα(θ)=1π∑n=1∞‍(-1)n-1θ-nα-1Γ(nα+1)n!sin(nπα),1111111111111111111111111θ∈(0,∞),
where ξα is a probability density function defined on (0,∞); that is,
(22)ξα(θ)≥0,θ∈(0,∞),∫0∞‍ξα(θ)dθ=1.Proof.
We observe thatx(·) can be decomposed to p(·)+q(·), where p is the continuous mild solution for
(23)Dtαp(t)=Ap(t)+h(t),t∈(0,b],It1-αp(t)|t=0=x0∈X,
and q is the PC1-α-mild solution for
(24)Dtαq(t)=Aq(t),t∈(0,b],t≠tk,k=1,2,…,m,ΔIt1-αq(t)|t=tk=yk,k=1,2,…,m,It1-αq(t)|t=0=0∈X.
Indeed, by adding together (23) and (24), it follows by (19). Since p is continuous, then p(tk+)=p(tk-), k=1,2,…,m. On the other hand, any solution of (19) can be decomposed to (23) and (24). So we show the results by the following.
At first, we calculate the mild solution of (23).
Apply Riemann-Liouville fractional integral operator on both sides of (23); then, by Lemma 3, we get
(25)p(t)=It1-αp(t)|t=0Γ(α)tα-1+ItαAp(t)+Itαh(t)=tα-1Γ(α)x0+ItαAp(t)+Itαh(t).
That is,
(26)p(t)=tα-1Γ(α)x0+1Γ(α)∫0t‍(t-s)α-1[Ap(s)+h(s)]ds.
Let λ>0; taking the Laplace transformations
(27)p^(λ)=∫0∞‍e-λtp(t)dt,h^(λ)=∫0∞‍e-λth(t)dt
to (26), we obtain
(28)p^(λ)=1λαx0+1λαAp^(λ)+1λαh^(λ)=(λαI-A)-1x0+(λαI-A)-1h^(λ)=∫0∞‍e-λαtT(t)x0dt+∫0∞‍e-λαtT(t)h^(λ)dt.
Consider the one-sided stable probability density
(29)ϖα(θ)=1π∑n=1∞‍(-1)n-1θ-nα-1Γ(nα+1)n!sin(nπα),11111111111111111111111111θ∈(0,∞),
whose Laplace transformation is given by
(30)∫0∞‍e-λθϖα(θ)dθ=e-λα,α∈(0,1).
Hence, it follows from (28) and (30) that
(31)∫0∞‍e-λαtT(t)x0dt=∫0∞‍e-(λs)αT(sα)x0dsα(t=sα)=α∫0∞‍e-(λs)αT(sα)sα-1x0ds=α∬0∞ϖα(θ)e-λsθT(sα)sα-1x0dθds=α∬0∞ϖα(θ)e-λuT(uαθα)sα-1θαx0dudθ(u=θs)=∫0∞‍e-λu[α∫0∞‍ϖα(θ)T(uαθα)sα-1θαx0dθ]du,∫0∞‍e-λαtT(t)h^(λ)dt=∫0∞‍e-λαtT(t)[∫0∞‍e-λsh(s)ds]dt=∬0∞e-λαtT(t)e-λsh(s)dsdt=∬0∞e-(λμ)αT(μα)qμα-1e-λsh(s)dsdμ(t=μα)=∭0∞αϖα(θ)e-λμθT(μα)μα-1e-λsh(s)dθdsdμ=∭0∞αϖα(θ)e-λνT(ναθα)×να-1θαe-λsh(s)dνdθds(ν=μθ)=∭0∞αϖα(θ)e-λ(ν+s)T(ναθα)να-1θαh(s)dνdθds=∬0∞∫s∞‍αϖα(θ)e-λτT((τ-s)αθα)×(τ-s)α-1θαh(s)dτdθds(τ=ν+s)=∫0∞‍e-λτ[((τ-s)αθα)α∫0τ‍∫0∞‍ϖα(θ)T((τ-s)αθα)×(τ-s)α-1θαh(s)dθds((τ-s)αθα)]dτ.
According to the above work, we get(32)p^(λ)=∫0∞‍e-λt[α∫0∞‍ϖα(θ)T(tαθα)tα-1θαx0dθ+α∫0t‍∫0∞‍ϖα(θ)T((t-s)αθα)×(t-s)α-1θαh(s)dθdsα∫0∞‍ϖα(θ)T(tαθα)tα-1θαx0dθ]dt.
Now, we can invert the Laplace transform to (20) and obtain
(33)p(t)=α∫0∞‍ϖα(θ)T(tαθα)tα-1θαx0dθ+α∫0t‍∫0∞‍ϖα(θ)T((t-s)αθα)(t-s)α-1θαh(s)dθdt=α∫0∞‍1αθ-1-(1/α)ϖα(θ-1/α)θT(tαθ)tα-1x0dθ+α∫0t‍∫0∞‍(t-s)α-11αθ-1-(1/α)ϖα(θ-1/α)θT×((t-s)αθ)h(s)dθdt.
Let(34)ξα(θ)=1αθ-1-(1/α)ϖα(θ-1/α),Tα(t)=α∫0∞‍θξα(θ)T(tαθ)dθ.
Then, we get(35)p(t)=tα-1Tα(t)x0+∫0t‍(t-s)α-1Tα(t-s)h(s)ds.
Now we calculate thePC1-α-mild solution of (24).
Applying Riemann-Liouville fractional integral operator on both sides of (24), then by Lemma 4, we get
(36)q(t)={1Γ(α)∫0t‍(t-s)α-1Aq(s)ds,t∈[0,t1],∑k=1m‍Ik(x(tk-))Γ(α)(t-ti)α-1+1Γ(α)∫0t‍(t-s)α-1Aq(s)ds,t∈(tk,tk+1].
The above equation (36) can be rewritten as
(37)q(t)=∑k=1m‍Ik(x(tk-))Γ(α)(t-tk)α-1χk(t)+1Γ(α)∫0t‍(t-s)α-1Aq(s)ds,
where
(38)χk(t)={0,t≤tk,1,t>tk.
Letλ>0; taking the Laplace transformation to (37), we obtain
(39)q^(λ)=∑k=1m‍Ik(x(tk-))e-λtkλα+1λαAq^(λ).
That is,
(40)q^(λ)=∑k=1m‍(λαI-A)-1Ik(x(tk-))e-λtk.
Notice that the Laplace transform of tα-1Tα(t)yk is (λαI-A)-1yk. Thus one can calculate the mild solution of (24) as
(41)q(t)=∑k=1m‍χk(t)(t-ti)α-1Tα(t-ti)Ik(x(tk-)).
By the above work, thePC1-α-mild solution of (19) is given by
(42)x(t)=tα-1Tα(t)x0+∑k=1m‍χk(t)(t-tk)α-1Tα(t-tk)Ik(x(tk-))+∫0t‍(t-s)α-1Tα(t-s)h(s)ds.
That is,
(43)x(t)={tα-1Tα(t)x0+∫0t‍(t-s)α-1Tα(t-s)h(s)ds,t∈[0,t1],tα-1Tα(t)x0+∑k=1m‍Tα(t-tk)(t-tk)α-1Ik(x(tk-))+∫0t‍(t-s)α-1Tα(t-s)h(s)ds,t∈(tk,tk+1],
where k=1,2,…,m,
(44)Tα(t)=α∫0∞‍θξα(θ)T(tαθ)dθ,ξα(θ)=1αθ-1-(1/α)ϖα(θ-1/α),ϖα(θ)=1π∑n=1∞‍(-1)n-1θ-nα-1Γ(nα+1)n!sin(nπα),1111111111111111111111111θ∈(0,∞),
where ξα is a probability density function defined on (0,∞); that is,
(45)ξα(θ)≥0,θ∈(0,∞),∫0∞‍ξα(θ)dθ=1.
This completes the proof of the lemma.According to Lemma5, we give the following definition.Definition 6.
A functionx∈PC1-α(J,X) is called a mild solution of (4) if  It1-αx(t)|t=0=x0 and there exits f∈L1(J<X) such that f(t)∈F(t,x(t)) a.e. on t∈J and
(46)x(t)={tα-1Tα(t)x0+∫0t‍(t-s)α-1Tα(t-s)[Bu(s)+f(s,x(s))]ds,t∈[0,t1],tα-1Tα(t)x0+∑k=1m‍Tα(t-tk)(t-tk)α-1Ik(x(tk-))+∫0t‍(t-s)α-1Tα(t-s)[Bu(s)+f(s,x(s))]ds,t∈(tk,tk+1],
where k=1,2,…,m,
(47)Tα(t)=α∫0∞‍θξα(θ)T(tαθ)dθ,ξα(θ)=1αθ-1-(1/α)ϖα(θ-1/α),ϖα(θ)=1π∑n=1∞‍(-1)n-1θ-nα-1Γ(nα+1)n!sin(nπα),1111111111111111111111111θ∈(0,∞),
where ξα is a probability density function defined on (0,∞); that is,
(48)ξα(θ)≥0,θ∈(0,∞),∫0∞‍ξα(θ)dθ=1.
Due to the work of the paper [31], we have the following result.Lemma 7.
The operatorTα(t) has the following properties.(i)
For any fixedt≥0, Tα(t) is linear and bounded operator; that is, for any x∈X,
(49)∥Tα(t)x∥≤MΓ(α)∥x∥.(ii)
Tα(t) (t≥0) is strongly continuous.Now, we also introduce some basic definitions on multivalued maps. For more details, see [36–38].A multivalued mapG:X→𝒫(X) is convex (closed) valued if G(x) is convex (closed) for all x∈X. G is bounded on bounded sets if G(B)=⋃x∈B‍. G(x) is bounded on X for any bounded set B of X; that is, supx∈B{sup{∥y∥:y∈G(x)}}<∞.G is called upper semicontinuous (u.s.c.) on X if, for each x0∈X, the set G(x0) is a nonempty closed subset of X, and if, for each open set U of X containing G(x0), there exists an open neighborhood V of x0 such that G(V)⊆U.G is said to be completely continuous if G(B) is relatively compact for every B∈Pb(X).If the multivalued mapG is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph (i.e., xn→x*, yn→y*, yn∈G(xn) imply y*∈G(x*)).We say thatG has a fixed point if there is a x∈X such that x∈G(x).A multivalued mapG:J→Pcl(X) is said to be measurable if for each x∈X the function Y:J→R+ defined by Y(t)=d(x,G(t))=inf{∥x-z∥:z∈G(t)} is measurable.Definition 8.
The system (4) is said to be exactly controllable on J, if, for all x0,x1∈X, there exists a control u∈Lp(J,U) (p>1/α) such that the mild solution of (4) satisfies x(0;u)=x0 and x(b;u)=x1.Definition 9.
The system (4) is said to be approximately controllable on the interval J, if, for all x0∈X, one has ℛ(b,x0)¯=X, where ℛ(b,x0)¯={x(b;u):u∈Lp(J,U)(p>1/α),x(0;u)=x0} is the reachable set of system (4) with the initial values x0,x1 at the terminal time b.
It is convenient at this point to introduce two relevant operators:(50)Γ0b=∫0b‍(b-s)2α-2Tα(b-s)BB*Tα*(b-s)ds,1111111111111111111111111112<α≤1,R(a,Γ0b)=(aI+Γ0b)-1,a>0,
where B* denotes the adjoint of B and Tα*(t) is the adjoint of Tα(t). It is straightforward that the operator Γ0b is a linear bounded operator.
We consider the following linear fractional differential system:(51)cDαx(t)=Ax(t)+Bu(t),12<α≤1,t∈J=(0,b],It1-αx(t)|t=0=x0∈X.Lemma 10.
The linear fractional differential system (51) is approximately controllable on J if and only if aR(a,Γ0b)→0 as a→0+ in the strong operator topology.The proof of this lemma is a straightforward adaptation of the proof of [3].Lemma 11.
LetE be a Banach space and let 𝒲⊂L1(J,E) be integrably bounded. If, for all t∈J, there is a relatively weakly compact set C(t)⊂E such that w(t)∈C(t) for every w∈𝒲, then 𝒲 is relatively weakly compact in L1(J,E).Lemma 12 (Lasota and Opial [39]).
LetJ be a compact real interval and let X be a Banach space. The multivalued map F:J×X→𝒫b,cl,cv(X) is measurable to t for each fixed x∈X, u.s.c. to x for each t∈J, and for each x∈C(J,X) the set SF,x={f∈L1(J,X):f(t)∈F(t,x(t)),t∈J} is nonempty. Let Γ be a linear continuous mapping from L1(J,X) to C(J,X); then, the operator
(52)Γ∘SF:C(J,X)⟶𝒫b,cl,cv(C(J,X)),x⟼(Γ∘SF)(x)=Γ(SF,x)
is a closed graph operator in C(J,X)×C(J,X).Lemma 13 (see [37]).
LetD be a bounded, convex, and closed subset in the Banach space E and let G:D→2X∖{∅} be a u.s.c. condensing multivalued map. If, for every x∈D, G(x) is a closed and convex set in D, then G has a fixed point.


## 3. Main Results

In this section, we present our main result on approximate controllability of system (4). To do this, we first prove the existence of solutions for fractional control system. Secondly, we show that, under certain assumptions, the approximate controllability of (4) is implied by the approximate controllability of the corresponding linear system.For convenience, let us introduce some notations:(53)Mb=∥B∥,β=(1-γα-γb(α-γ)/(1-γ))1-γ,Λ=MβΓ(α)+M3MB2b2α-1βa(2α-1)(Γ(α))3.Before stating and proving our main results, we introduce the following assumptions.H(1): Tα(t) is compact, ∥aR(a,Γ0b)∥≤1.H(2): F is a multivalued map satisfying F:J×X→𝒫cp,cv(X) which is measurable to t for each fixed x∈D, u.s.c. to x for each t∈J, and for each x∈PC1-α(J,X) the set
(54)SF,x={f∈L1(J,X):f(t)∈F(t,x(t))}is nonempty.H(3): There exist a function P(t)∈L1/γ(J,R+), γ∈(0,α), and a nondecreasing continuous function ψ:→R+, such that
(55)∥F(t,x(t))∥X=sup{∥f(t)∥X:f(t)∈F(t,x(t))}≤P(t)ψ(∥x∥D),for a.e.t∈J, for all x∈D, and for each r>0, there exists 0<ρ<1, such that
(56)limr→∞infψ(r)r∥P∥L2=ρ<1.Theorem 14.
If the conditionsH(1)–H(4) are held, then the system (4) has a mild solution.Proof.
We consider a set(57)Br={x∈PC1-α(J,X):∥x∥≤r,r>0}
on the space PC1-α(J,X). We easily know that Br is a bounded, closed, and convex set in PC1-α(J,X). For a>0, for all x(·)∈PC1-α(J,X), x1∈X, we take the control function as
(58)u(t)=(b-t)α-1B*Tα*(b-t)R(a,Γ0b)P(x(·)),
where
(59)P(x(·))=x1-tα-1Tα(b)x0-∑k=1m‍Tα(t-tk)(t-tk)α-1Ik(x(tk-))-∫0b‍(b-s)α-1Tα(b-s)f(s)ds,f∈SF,x.
By this control, we define the operator Φa:PC1-α(J,X)→𝒫(PC1-α(J,X)) as follows:
(60)Φa(x)={∑k=1m‍τ∈PC1-α(J,X):τ(t)=tα-1Tα(t)x0+∫0t‍(t-s)α-1Tα(t-s)f(s)ds+∫0t‍(t-s)α-1Tα(t-s)Bu(s)ds,f∈SF,x,t∈(0,t1],tα-1Tα(t)x0+∑k=1m‍Tα(t-tk)(t-tk)α-1Ik(x(tk-))+∫0t‍(t-s)α-1Tα(t-s)f(s)ds+∫0t‍(t-s)α-1×Tα(t-s)Bu(s)ds,f∈SF,x,t∈(tk,tk+1],k=1,2,…,m∑k=1m‍}.
We will show that, for all a>0, the operator Φa:PC1-α(J,X)→𝒫(PC1-α(J,X)) has a fixed point. For the sake of convenience, we subdivide the proof into several steps.
Step  1. For each a>0, the operator Φa(x) is convex for each x∈PC1-α(J,X).
In fact, ifτ1,τ2∈Φa(x), then, for each t∈(tk,tk+1], k=1,2,…,m, there exist f1,f2∈SF,x such that
(61)τi(t)=tα-1Tα(t)x0+∑k=1m‍Tα(t-tk)(t-tk)α-1Ik(x(tk-))+∫0t‍(t-s)α-1Tα(t-s)fi(s)ds+∫0t‍(t-s)α-1Tα(t-s)BB*Tα*(b-t)R(a,Γ0b)×[∫0b‍(b-μ)α-1Tα(b-μ)fi(μ)x1-tα-1Tα(t)x0-∑k=1m‍Tα(t-tk)(t-tk)α-1Ik(x(tk-))-∫0b‍(b-μ)α-1Tα(b-μ)fi(μ)dμ]ds,1111111111111111111111111111111111i=1,2.
Let 0≤λ≤1; then, for each t∈(tk,tk+1], k=1,2,…,m, we have
(62)λτ1(t)+(1-λ)τ2(t)=tα-1Tα(t)x0+∑k=1m‍Tα(t-tk)(t-tk)α-1Ik(x(tk-))+∫0t‍(t-s)α-1Tα(t-s)[λf1(s)+(1-λ)f2(s)]ds+∫0t‍(t-s)α-1Tα(t-s)BB*Tα*(b-t)R(a,Γ0b)×[∑k=1m‍x1-tα-1Tα(t)x0-∑k=1m‍Tα(t-tk)(t-tk)α-1Ik(x(tk-))-∫0b‍(b-μ)α-1Tα(b-μ)×[λf1(μ)+(1-λ)f2(μ)]dμ∑k=1m‍]ds.
Since SF,x is convex (because F has convex values), λf1+(1-λ)f2∈SF,x; thus, λτ1(t)+(1-λ)τ2∈Φa(x).
Step  2. For each a>0, there is a positive constant r0=r(a), such that Φa(Br0)⊂Br0.
If this is not true, then there existsa>0 such that, for every r>0, there exists a x¯∈Br such that Φa(x¯)⊈Br; that is,
(63)∥Φa(x¯)∥≡sup{∥τ∥PC1-α(J,X):τ∈Φa(x¯)}>r,τ(t)=tα-1Tα(t)x0+∑k=1m‍Tα(t-tk)(t-tk)α-1Ik(x(tk-))+∫0t‍(t-s)α-1Tα(t-s)f(s)ds+∫0t‍(t-s)α-1Tα(t-s)BB*Tα*(b-t)R(a,Γ0b)×[∫0b‍x1-bα-1Tα(b)x0-∑k=1m‍Tα(b-tk)(b-tk)α-1Ik(x(tk-))-∫0b‍(b-μ)α-1Tα(b-μ)f(μ)dμ∫0b‍]ds,
for some SF,x¯.
By using Holder’s inequality andH(3), we have
(64)∫0t‍(t-s)α-1∥u(s)∥ds≤MMBb2α-1aΓ(α)(2α-1)×[∥x1∥+M∥x0∥Γ(α)+∑k=1m‍MΓ(α)(dk∥x∥+m(b-tk)α-1∥Ik(0)∥)+Mψ(r)βΓ(α)∥P∥L1/γ].
Then, we obtain
(65)(t-tk)1-α∥τ(t)∥≤(t-tk)1-α∥tα-1Tα(t)x0∥+(t-tk)1-α×∑k=1m‍∥Tα(t-tk)∥(t-tk)α-1∥Ik(x(tk-))∥+(t-tk)1-α×∫0t‍(t-s)α-1∥Tα(t-s)∥∥f(s)∥ds+(t-tk)1-α×∫0t‍(t-s)α-1∥Tα(t-s)∥∥B∥∥u(s)∥ds≤Mbα-1(b-tk)1-αΓ(α)∥x0∥+(b-tk)1-α×∑k=1m‍MΓ(α)(dk∥x∥+m(b-tk)α-1∥Ik(0)∥)+M(b-tk)1-αΓ(α)∫0t‍(t-s)α-1P(s)ψ(r)ds+MMB(b-tk)1-αΓ(α)∫0t‍(t-s)α-1u(s)ds≤Mbα-1(b-tk)1-αΓ(α)∥x0∥+(b-tk)1-α×∑k=1m‍MΓ(α)(dk∥x∥+m(b-tk)α-1∥Ik(0)∥)+b2α-1M2MB2(b-tk)1-αa(2α-1)(Γ(α))2×[∑k=1m‍∥x1∥+M∥x0∥Γ(α)+∑k=1m‍MΓ(α)(dk∥x∥+m(b-tk)α-1∥Ik(0)∥)]+ψ(r)Λ∥P∥L1/γ.
Thus,
(66)r≤Mbα-1(b-tk)1-αΓ(α)∥x0∥+(b-tk)1-α×∑k=1m‍MΓ(α)(m(b-tk)α-1dk∥x∥+m(b-tk)α-1∥Ik(0)∥)+b2α-1M2MB2(b-tk)1-αa(2α-1)(Γ(α))2×[∑k=1m‍MΓ(α)(dk∥x∥+m(b-tk)α-1∥Ik(0)∥)∥x1∥+M∥x0∥Γ(α)+∑k=1m‍MΓ(α)(dk∥x∥+m(b-tk)α-1∥Ik(0)∥)]+ψ(r)Λ∥P∥L1/γ.
Dividing both sides by r and taking the low limit as r→∞, we get
(67)1≤limr→∞infψ(r)r∥P∥L1/γ,
which is a contradiction to H(3). Thus, for each a>0, there exists r0 such that Φa(x) maps Br0 into itself.
Step  3.
Φ
α
(
x
) is closed for each x∈PC1-α(J,X).
Indeed, for each givenx∈PC1-α(J,X), let {τn}n≥0⊂Φa(x) such that τn→τ in PC1-α(J,X). Then, there exists fn∈SF,x such that, for each t∈J,
(68)τn(t)=tα-1Tα(t)x0+∑k=1m‍Tα(t-tk)(t-tk)α-1Ik(x(tk))+∫0t‍(t-s)α-1Tα(t-s)fn(s)ds+∫0t‍(t-s)α-1Tα(t-s)Bun(s)ds,
where
(69)un(t)=B*Tα*(b-t)R(a,Γ0b)×[∫0b‍x1-bα-1Tα(b)x0-∑k=1m‍Tα(b-tk)(b-tk)α-1Ik(x(tk-))-∫0b‍(b-s)α-1Tα(b-s)fn(s)ds].
Because of [40, Proposition 3.1], SF,x is weakly compact in L1(J,X) which implies that fn converges weakly to some f∈SF,x in L1(J,X). Thus, un⇀u, and
(70)u(t)=B*Tα*(b-t)R(a,Γ0b)×[∫0b‍x1-bα-1Tα(b)x0-∑k=1m‍Tα(b-tk)(b-tk)α-1Ik(x(tk-))-∫0b‍(b-s)α-1Tα(b-s)f(s)ds].
Then, for each t∈J,
(71)τn(t)⟶τ(t)=tα-1Tα(t)x0+∑k=1m‍Tα(t-tk)(t-tk)α-1Ik(x(tk-))+∫0t‍(t-s)α-1Tα(t-s)f(s)ds+∫0t‍(t-s)α-1Tα(t-s)BB*Tα*(b-t)R(a,Γ0b)×[∫0b‍x1-bα-1Tα(b)x0-∑k=1m‍Tα(b-tk)(b-tk)α-1Ik(x(tk-))-∫0b‍(b-μ)α-1Tα(b-μ)f(μ)dμ]ds.
Thus, we show that τ∈Φa(x).
Step  4.
Φ
a is u.s.c and condensing.
We decomposeΦa as Φa=Φa1+Φa2, where the operators Φa1 and Φa2 are defined by
(72)(Φa1x)(t)=∑k=1m‍Tα(t-tk)(t-tk)α-1Ik(x(tk-)),1111111111111t∈J,k=1,2,…,m,Φa2(x)={∑k=1m‍τ∈PC1-α(J,X):τ(t)=tα-1Tα(t)x0+∫0t‍(t-s)α-1Tα(t-s)f(s)ds+∫0t‍(t-s)α-1Tα(t-s)×BB*Tα*(b-t)R(a,Γ0b)×[∫0b‍x1-bα-1Tα(b)x0-∑k=1m‍Tα(b-tk)(b-tk)α-1Ik(x(tk-))-∫0b‍(b-μ)α-1Tα(b-μ)f(μ)dμ]ds,f∈SF,x,t∈J∑k=1m‍}.
According to [41, Corollary 2.2.1], we will prove that Φa1 is a contraction operator, while Φa2 is a completely continuous operator.
Let us begin proving thatΦa1 is a contraction operator. For any x,y∈X, we obtain
(73)∥(Φa1x)(t)-(Φa1y)(t)∥≤∥∑k=1m‍Tα(t-tk)(t-tk)α-1Ik(x(tk-))-∑k=1m‍Tα(t-tk)(t-tk)α-1Ik(y(tk-))∥≤MΓ(α)∑k=1m‍dk∥x-y∥.
Then, Φa1 is a contraction operator, since (M/Γ(α))∑k=1m‍dk<1.
Next, we prove thatΦa2 is u.s.c and completely continuous. We subdivide the proof into several claims.
Claim  1. There exists a positive constant r such that Φa2(Br)⊆Br.
By employing the technique used in Step 2, one can easily show that there existsr>0 such that Φa2(Br)⊆Br.
Claim  2.
Φ
a
2
(
B
r
) is a family of equicontinuous functions.
Let0≤s≤t1≤t2≤b. For each x∈Br, φ∈Φa2(x), there exists f∈SF,x such that
(74)τ(t)=tα-1Tα(t)x0+∫0t‍(t-s)α-1Tα(t-s)f(s)ds+∫0t‍(t-s)α-1Tα(t-s)Bu(s)ds,t∈J.
Then, we have
(75)∥τ(t2)-τ(t1)∥≤∥∫0t1‍[(t2-s)α-1-(t1-s)α-1]Tα(t2-s)f(s)ds∥×∥∫0t1‍(t1-s)α-1[Tα(t2-s)-Tα(t1-s)]f(s)ds∥+∥∫t1t2‍(t2-s)α-1Tα(t2-s)f(s)ds∥×∥∫0t1‍[(t2-s)α-1-(t1-s)α-1]Tα(t2-s)u(s)ds∥×∥∫0t1‍(t1-s)α-1[Tα(t2-s)-Tα(t1-s)]u(s)ds∥+∥∫t1t2‍(t2-s)α-1Tα(t2-s)u(s)ds∥≤I1+I2+I3+I4+I5+I6.
By using Hölder’s inequality and assumption H(3), we get
(76)I1=Mψ(r)∥P∥L1/γΓ(α)×[(1-γα-γt1(α-γ)/(1-γ))1-γ+(1-γα-γ(t2-t1)(α-γ)/(1-γ))1-γ-(1-γα-γt2(α-γ)/(1-γ))1-γ],I2=ψ(r)∥P∥L1/γ(1-γα-γt1(α-γ)/(1-γ))1-γ×sups∈[0,t1]∥Tα(t2-s)-Tα(t1-s)∥,I3=Mψ(r)∥P∥L1/γΓ(α)[1-γα-γ(t2-t1)(α-γ)/(1-γ)]1-γ,I4=MMBΓ(α)∫t1t2‍[(t2-s)α-1-(t1-s)α-1]∥u(s)∥ds,I5=MBsups∈[0,t1]∥Tα(t2-s)-Tα(t1-s)∥×∫0t1‍(t1-s)α-1∥u(s)∥ds,I6=MMBΓ(α)∫t1t2‍(t2-s)α-1∥u(s)∥ds.
It is easy to see that I2 tends to zero independently of x∈Br as t2→t1. Note that, from Lemma 10, Tα(t) is continuous in the uniform operator topology for t>0; we can directly obtain I1 and I3 tending to zero independently of x∈Br as t2→t1. Applying the absolute continuity of the Lebesgue integral, we have I4, I5, and I6 tending to zero independently of x∈Br as t2→t1.
Therefore,Φa2(Br)⊂PC1-α(J,X) is equicontinuous.
Claim  3. The set Π(t)={τ(t):τ∈Φa2(Br)}⊂X is relatively compact for each t∈J.
Let0<t≤b be fixed. For x∈Br and τ∈Φa2(x), there exists f∈SF,x such that, for each t∈J,
(77)τ(t)=tα-1Tα(t)x0+∫0t‍(t-s)α-1Tα(t-s)f(s)ds+∫0t‍(t-s)α-1Tα(t-s)BB*Tα*(b-t)R(a,Γ0b)×[∫0b‍x1-bα-1Tα(b)x0-∑k=1m‍Tα(b-tk)(b-tk)α-1Ik(x(tk-))-∫0b‍(b-μ)α-1Tα(b-μ)f(μ)dμ]ds,t∈J.
For allε∈(0,t) and for all δ>0, define
(78)τε,δ(t)=tα-1Tα(t)x0+α∫0t-ε‍∫δ∞‍θ(t-s)α-1ξα(θ)T((t-s)αθ)f(s)dθds+α∫0t-ε‍∫δ∞‍θ(t-s)α-1ξα(θ)T((t-s)αθ)×BB*Tα*(b-s)R(a,Γ0b)×[∑k=1m‍x1-bα-1Tα(b)x0-∑k=1m‍Tα(b-tk)(b-tk)α-1Ik(x(tk-))-∫0b‍(b-μ)α-1Tα(b-μ)f(μ)dμ]dθds=tα-1Tα(t)x0+T(εαδ)×{∑k=1m‍α∫0t-ε‍∫δ∞‍θ(t-s)α-1ξα(θ)T×((t-s)αθ-εαδ)f(s)dθds+α∫0t-ε‍∫δ∞‍θ(t-s)α-1ξα(θ)T((t-s)αθ-εαδ)×BB*Tα*(b-s)R(a,Γ0b)×[∑k=1m‍x1-bα-1Tα(b)x0-∑k=1m‍Tα(b-tk)(b-tk)α-1Ik(x(tk-))-∫0b‍(b-μ)α-1Tα(b-μ)f(μ)dμ∑k=1m‍]dθds}.
By the compactness ofT(εαδ) (εαδ>0), we obtain the set
(79)Πε,δ(t)={φε,δ(t):τ∈Φa2(Br)}
which is relatively compact in X for all ε∈(0,t) and δ>0. Moreover, we have
(80)∥τ(t)-τε,δ(t)∥≤{αMψ(r)βΓ(α)∥P∥L1/γ+αM2MB2b2α-1a(2α-1)Γ(α)×[∥x1∥+M∥x0∥Γ(α)+∑k=1m‍MΓ(α)(dk∥x∥+m(b-tk)α-1∥Ik(0)∥)+Mψ(r)βΓ(α)∥P∥L1/γ∥x1∥+M∥x0∥Γ(α)]αM2MB2b2α-1a(2α-1)Γ(α)}∫0δ‍θξα(θ)dθ+Mψ(r)Γ(α)∥P∥L1/γ(1-γα-γε(α-γ)/(1-γ))1-γ+M2MB2ε2α-1a(2α-1)(Γ(α))2×[∥x1∥+M∥x0∥Γ(α)+∑k=1m‍MΓ(α)(dk∥x∥+m(b-tk)α-1∥Ik(0)∥)+Mψ(r)βΓ(α)∥P∥L1/γM∥x0∥Γ(α)].
The right-hand side of the above inequality tends to zero as ε→0. Therefore, there are relatively compact sets arbitrarily close to the set Π(t), t>0. Hence the set Π(t), t>0 is also relatively compact in X. As a consequence of Claims 1–3 together with the Arzola-Ascoli theorem, we can conclude that Φa2 is completely continuous.
Claim  4.
Φ
α
2 has a closed graph.
Letxn→x* (n→∞), τn∈Φa2(xn), τn→τ* (n→∞). We will prove that τ*∈Φa2(x*). Since τn∈Φa2(xn), there exists fn∈SF,xn, such that, for each t∈J,
(81)τn(t)=tα-1Tα(t)x0+∫0t‍(t-s)α-1Tα(t-s)fn(s)ds+∫0t‍(t-s)α-1Tα(t-s)BB*Tα*(b-t)R(a,Γ0b)×[∫0b‍x1-bα-1Tα(b)x0-∑k=1m‍Tα(b-tk)(b-tk)α-1Ik(x(tk-))-∫0b‍(b-μ)α-1Tα(b-μ)fn(μ)dμ]ds.
We must prove that there exists f*∈SF,x*, such that, for each t∈J,
(82)τ*(t)=tα-1Tα(t)x0+∫0t‍(t-s)α-1Tα(t-s)f*(s)ds+∫0t‍(t-s)α-1Tα(t-s)BB*Tα*(b-t)R(a,Γ0b)×[∫0b‍x1-bα-1Tα(b)x0-∑k=1m‍Tα(b-tk)(b-tk)α-1Ik(x(tk-))-∫0b‍(b-μ)α-1Tα(b-μ)f*(μ)dμ]ds.
Since τn→τ* (n→∞), we can obtain
(83)∥∑k=1m‍(∑k=1m‍τn(t)-tα-1Tα(t)x0-∫0t‍(t-s)α-1Tα(t-s)BB*Tα*(b-t)R(a,Γ0b)×[∑k=1m‍x1-bα-1Tα(b)x0-∑k=1m‍Tα(b-tk)(b-tk)α-1Ik(x(tk-))]ds)-(∑k=1m‍τ*(t)-tα-1Tα(t)x0-∫0t‍(t-s)α-1Tα(t-s)BB*Tα*(b-t)R(a,Γ0b)×[∑k=1m‍x1-bα-1Tα(b)x0-∑k=1m‍Tα(b-tk)(b-tk)α-1×Ik(x(tk-))∑k=1m‍]ds)∑k=1m‍∥⟶0,asn⟶∞.
Consider the linear continuous operator
(84)Γ:L1/γ(J,X)⟶C1-α(J,X),(Γf)(t)=∫0t‍(t-s)α-1Tα(t-s)×[∫0b‍f(s)-B(b-s)α-1B*Tα*(b-s)R(a,Γ0b)×∫0b‍Tα(b-η)f(η)dη]ds.
Clearly it follows from Lemma 13 that Γ∘SF is a closed graph operator. Moreover, we have
(85)τn(t)-tα-1Tα(t)x0-∫0t‍(t-s)α-1Tα(t-s)BB*Tα*(b-s)R(a,Γ0b)×[∑k=1m‍x1-bα-1Tα(b)x0-∑k=1m‍Tα(b-tk)(b-tk)α-1Ik(x(tk-))]ds∈Γ(SF,xn).
Since xn→x*, it follows from Lemma 13 that
(86)τ*(t)-tα-1Tα(t)x0-∫0t‍(t-s)α-1Tα(t-s)BB*Tα*(b-s)R(a,Γ0b)×[∑k=1m‍x1-bα-1Tα(b)x0-∑k=1m‍Tα(b-tk)(b-tk)α-1Ik(x(tk-))]ds∈Γ(SF,x*).
Therefore, Φa2 has a closed graph. Since Φa2 is a completely continuous multivalued map with compact value, we have that Φa2 is u.s.c.
ThusΦa=Φa1+Φa2 is u.s.c and condensing. Therefore, applying Lemma 13, we conclude that Φa has a fixed point x(·) on Br0. Thus, the fractional control system (4) has a mild solution on J.
The proof is complete.The following result concerns the approximate controllability of that problem (4). We assume that the following assumption be held.H′(3): There exists a positive constant L such that ∥F(t,x(t))∥≤L for all (t,x)∈J×X.Theorem 15.
Assume that assumptionsH(1), H(2), H′(3), and H(4) are satisfied and the linear system (24) is approximately controllable on J. Then system (4) is approximately controllable on J.Proof.
By employing the technique used in Theorem14, we can easily show that, for all 0<a<1, the operator Φa has a fixed point in Br0, where r0=r(a). Let xa(·) be a fixed point of Φa in Br0. Any fixed point of Φa is a mild solution of (4); this means that there exists fa∈SF,xa such that, for each t∈J′,
(87)xa(t)=tα-1Tα(t)x0+∑k=1m‍Tα(t-tk)(t-tk)α-1Ik(x(tk-))+∫0t‍(t-s)α-1Tα(t-s)fa(s)ds+∫0t‍(t-s)α-1Tα(t-s)BB*Tα*(b-s)R(a,Γ0b)×[∫0b‍x1-bα-1Tα(b)x0-∑k=1m‍Tα(b-tk)(b-tk)α-1Ik(x(tk-))-∫0b‍(b-μ)α-1Tα(b-μ)fa(μ)dμ]ds.
Define
(88)P(fa)=x1-bα-1Tα(b)x0-∑k=1m‍Tα(b-tk)(b-tk)α-1Ik(x(tk-))-∫0b‍(b-s)α-1Tα(b-s)fa(s)ds,forsomefa∈SF,xa.
Noting that I-Γ0bR(a,Γ0b)=aR(a,Γ0b), we get
(89)xa(b)=x1-aR(a,Γ0b)P(fa).
By assumption H′(3),
(90)∫0b‍∥fa(s)∥2ds≤L2b.
Consequently the sequence {fa} is uniformly bounded in L1/γ(J,X). Thus, there is a subsequence, still denoted by {fa}, that converges weakly to, say, f in L1/γ(J,X). Denoting
(91)h=x1-bα-1Tα(b)x0-∑k=1m‍Tα(b-tk)(b-tk)α-1Ik(x(tk-))-∫0b‍(b-s)α-1Tα(b-s)f(s)ds,
we see that
(92)∥P(fα)-h∥=∥∫0b‍(b-s)α-1Tα(b-s)[fa(s)-f(s)]ds∥≤sup0≤t≤b∥∫0b‍(b-s)α-1Tα(b-s)[fα(s)-f(s)]ds∥.
By the Ascoli-Arzela theorem, we can show that the linear operator g→∫0·‍(·-s)α-1Tα(·-s)g(s)ds:L1/γ(J,X)→PC1-α(J,X) is compact; consequently the right-hand side of (92) tends to zero as a→0+.
This implies that(93)∥xa(b)-x1∥=∥aR(a,Γ0b)P(fa)∥≤∥aR(a,Γ0b)(h)∥+∥aR(α,Γ0b)(P(fa)-h)∥≤∥aR(a,Γ0b)(h)∥+∥P(fa)-h∥⟶0,asa⟶0+.
This proves the approximate controllability of system (4).Remark 16.
In [9], Ganesh et al. have studied the approximate controllability of fractional integrodifferential evolution equations. If our problem (4) can be changed as
(94)Dtαx(t)∈Ax(t)+Bu(t)+F(t,x(t),(Hx)(t)),t∈(0,b],12<α≤1,I0+1-αx(t)|t=0=x0+g(x)∈X,
where Dtα denotes the Riemann-Liouville fractional derivative, the operator H is defined by (Hx)(t)=∫0b‍h(t,s,x(s))ds. By a similar way, we can get the approximate controllability of the system (94).Remark 17.
In [15], Rathinasamy and Yong have researched the approximate controllability of fractional differential equations with state-dependent delay. Our problem (4) can be adopted to the impulsive fractional differential inclusion system with state-dependent delay:
(95)Dtαx(t)∈Ax(t)+Bu(t)+F(t,xρ(t,xt)),t∈(0,b],12<α≤1,ΔI0+1-αx(t)|t=tk=Ik(x(tk-)),k=1,2,…,m,I0+1-αx(t)|t=0=ϕ∈X,
where Dtα is the Riemann-Liouville fractional derivative, A is the infinitesimal generator of a C0-semigroup {T(t),t≥0} on a Banach space X, the state x(·) takes values in X, the control function u(·) is given in Lp(J,U) (p>1/α), U is a Banach space of admissible control, and B:U→X is a bounded linear operator. F:J×D→𝒫(X):=2X∖{∅} with D:=C((-∞,0],X) is a multivalued map. For a continuous function x:J*:=(-∞,b]→X, xt is the element of D defined by xt(s)=x(t+s), -∞≤s≤0, and ϕ∈D. Further, ρ:[0,b]×𝒫(X)→(-∞,b] are appropriate nonlinear functions.


## 4. An Example

In this final section, we give an example to illustrate our abstract results.Example 1.
Consider the following fractional partial differential inclusion with control(96)Dtαx(t,y)∈∂2∂y2x(t,y)+F(t,x(t,y))+Bu(t,y),t∈J′=[0,1]∖{12},y∈[0,π],ΔI0+1-αx(12,y)=|x(y)|2π+|x(y)|,y∈[0,π],x(t,0)=x(t,π)=0,t∈J=[0,1],I0+1-αx(t,y)|t=0=x0(y),t∈[0,1],y∈[0,π],
where Dα is the Riemann-Liouville fractional partial derivative of order 3/5, J=[0,1].
LetX=U=L2[0,π], define; the operator A by Ax=xyy with the domain
(97)D(A)={t,0x∈X:x,x′areabsolutelycontinuous,xyy∈X,x(t,0)=x(t,π)=0x′areabsolutelycontinuous}.
Then,
(98)Ax=-∑n=1∞‍n2〈x,en〉en,x∈D(A),
where en(y)=2/πsinny, 0≤y≤π, n=1,2,…, is the orthogonal set of eigenvectors of A. It is easily shown that operator A generates a strongly continuous semigroup {T(t):t≥0} on X, which are compact, analytic, and self-adjoint in X.
It can be easily seen that the linear system corresponding to (96) is approximately controllable on [0,1] [42].
Definex(t)(y)=x(t,y); let B:U→X be defined by Bu(t)(y)=Bu(t,y). Then system (96) can be written in the abstract form given by (4). We assume that F:J×D→𝒫(X) satisfy the following conditions.(F1): F:J×D→𝒫cp,cv(X) is measurable to t for each fixed x∈D, u.s.c. to x for a.e. t∈J, and for each x∈PC1-α(J,X) the set
(99)SF,x={f∈L1(J,X):f(t)∈F(t,x)}is nonempty.(F2): There exists a positive constant L such that ∥F(t,x)∥≤L for all (t,x)∈J×D, since, for any x,y∈AC(J,X), we have
(100)∥ΔIt1-αx(tk,z)-ΔIt1-αy(tk,z)∥≤∥x-y∥2.
Obviously, the hypotheses of Theorem15 are fulfilled. Thus, the system (96) is approximately controllable on [0,1].


## 5. Conclusions

In this paper, we study the approximate controllability for impulsive fractional differential inclusions with Riemann-Liouville fractional derivatives in Banach spaces. We mainly use fixed point theorem and semigroup theory; we obtain some new mild solutions for this kind of problems (4). An illustrative example is also discussed to show the effectiveness of the results in this paper. In the near future, we will consider Riemann-Liouville fractional partial differential problems, which will be more complicated.



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