# Complete Controllability of Fractional Neutral Differential Systems in Abstract Space

Authors: Fang Wang; Zhen-hai Liu; Jing Li
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

By using fractional power of operators and Sadovskii fixed point theorem, we study the complete controllability of fractional neutral differential systems in abstract space without involving the compactness of characteristic solution operators introduced by us.


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## Body

## 1. Introduction

Recently, fractional differential systems have been proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetic, and so forth (see [1–5]). There has been a great deal of interest in the solutions of fractional differential systems in analytic and numerical sense. One can see the monographs of Kilbas et al. [6], Miller and Ross [7], Podlubny [8], Lakshmikantham et al. [9], Tarasov [10], Wang et al. [11–13] and the survey of Agarwal et al. [14] and the reference therein. In order to study the fractional systems in the infinite dimensional space, the first important step is how to introduce a new concept of mild solutions. A pioneering work has been reported by EI-Borai [15] and Zhou and Jiao [16].In recent years, controllability problems for various types of nonlinear fractional dynamical systems in infinite dimensional spaces have been considered in many publications. An extensive list of these publications focused on the complete and approximate controllability of the fractional dynamical systems can be found (see [17–34]). Although the controllability of fractional differential systems in abstract space has been discussed, Hernández et al. [35] point out that some papers on controllability of abstract control systems contain a similar technical error when the compactness of semigroup and other hypotheses is satisfied, more precisely, in this case the application of controllability results are restricted to the finite dimensional space. Ji et al. [32] find some conditions guaranteeing the controllability of impulsive differential system when the Banach space is nonseparable and evolution systems are not compact, by means of Möch fixed point theorem and the measure of noncompactness. Meanwhile, Wang et al. [19, 20] have researched the complete controllability of fractional evolution systems without involving the compactness of characteristic solution operators. Neutral differential equations arise in many areas of applied mathematics and for this reason these equations have received much attention in the last decades. Sakthivel and Ren [29] have established a new set of sufficient conditions for the complete controllability for a class of fractional order neutral systems with bounded delay under the natural assumption that the associated linear control is completely controllable. To the author’s knowledge, there are few papers on the complete controllability of the abstract neutral fractional differential systems with unbounded delay.In the present paper, we introduce a suitable concept of the mild solutions including characteristic solution operatorsφ(·) and S(·) which are associated with operators semigroup {T(t);t≥0} and some probability density functions ξq. Then also without involving the compactness of characteristic solution operators, we obtain the controllability of the following abstract neutral fractional differential systems with unbounded delay:
(1)Dtqc(x(t)+F(t,xt))+Ax(t)=Cu(t)+G(t,xt),t∈(0,a],x0(ϑ)=ϕ(ϑ)∈B,ϑ∈(-∞,0],
where the state variable x(·) takes values in Banach space X,  xt:(-∞,0]→X,xt(ϑ)=x(t+ϑ) belongs to some abstract phase space B, and B is the phase space to be specified later. The control function u(·) is given in L2([0,a];U), with U as a Banach spaces. C is a bounded linear operator from U to X. The operator -A is a generator of a uniformly bounded analytic semigroup {T(t),t≥0} in which X, F,G:[0,a]×B→X are appropriate functions.


## 2. Preliminaries

Throughout this paperX will be a Banach space with norm ∥·∥ and Y is another Banach space, Lb(X,Y) denote the space of bounded linear operators from X to Y. We also use ∥f∥Lp([0,a],R+) to denote the Lp([0,a],R+) of norm of f whenever f∈Lp([0,a],R+) for some p with 1≤p<∞. Let Lp([0,a],R+) denote the Banach space of functions f:[0,a]→X which are Bochner integrable normed by ∥f∥Lp([0,a],R+). Let -A:D(A)→X be the infinitesimal generator of a uniformly bounded analytic semigroup T(t). Let 0∈ρ(A), then it is possible to define the fractional power Aα, for 0<α≤1, as a closed linear operator on its domain D(Aα). Furthermore, the subspace D(Aα) is dense in X and the expression
(2)∥x∥α=∥Aαx∥,x∈D(Aα)
defines a norm on D(Aα). Hereafter we denote by Xα the Banach space D(Aα) normed with ∥x∥α. Then for each 0<α≤1, Xα the Banach space, and ∥x∥α↪∥x∥β for 0<β<α≤1 and the imbedding is compact whenever the resolvent operator of A is compact. For a uniformly bounded analytic semigroup {T(t);t≥0} the following properties will be used:(a)
there is aM≥0 such that ∥T(t)∥≤M for all t≥0.(b)
for anyα≥0, there exists a positive constant Cα such that
(3)∥AαT(t)∥≤Cαtα,0<t≤a.For more details about the above preliminaries, we can refer to [16].Although the semigroup{T(t);t≥0} is only the uniformly bounded analytic semigroup but not compact, we can also give the definition of mild solution for our problem by using the similar method introduced in [36].Definition 1.
We say that a functionx(·):(-∞,a]→X is a mild solution of the system (1) if x0=ϕ, the restriction of x(·) to the interval [0,a] is continuous and for each 0≤t≤a, the function AS(t-s)F(s,xs),s∈[0,t] is integrable and satisfies the following integral equation:
(4)x(t)=φ(t)[ϕ(0)+F(0,ϕ)]-F(t,xt)-∫0t‍(t-s)q-1AS(t-s)F(s,xs)ds+∫0t‍(t-s)q-1S(t-s)[Cu(s)+G(s,xs)]ds,
where φ(t) and S(t) are called characteristic solution operators and are given by
(5)φ(t)=∫0∞‍ξq(θ)T(tqθ)dθ,S(t)=q∫0∞‍θξq(θ)T(tqθ)dθ,
and for θ∈(0,∞), ξq(θ)=(1/q)θ-1-1/qw-q(θ-1/q)≥0,
(6)w-q(θ)=1π∑n=1∞‍(-1)n-1ϑ-qn-1Γ(nq+1)n!sin(nπq).
Here, ξq is a probability density function defined on (0,∞), that is, ξq(θ)≥0, θ∈(0,∞), and ∫0∞‍ξq(θ)dθ=1.Definition 2 (complete controllability).
The fractional system  (1) is said to be completely controllable on the interval [0,a] if, for every initial function ϕ∈B and x1∈X there exists a control u∈L2([0,a],U) such that the mild solution x(·) of  (1) satisfies x(a)=x1.The following results ofφ(t) and S(t) will be used throughout this paper.Lemma 3.
The operatorsφ(t) and S(t) have the following properties:(i)
for any fixedt≥0, φ(t) and S(t) are linear and bounded operators, that is, for any x∈X,
(7)∥φ(t)x∥≤M0∥x∥,∥S(t)x∥≤qM0Γ(1+q)∥x∥;(ii)
{φ(t),t≥0} and {S(t),t≥0} are strongly continuous and there exists M1,M2 such that ∥φ(t)∥≤M1, ∥S(t)∥≤M2 for any t∈[0,a];(iii)
fort∈[0,a] and any bounded subsets D⊂X, t→{φ(t)x:x∈D} and t→{S(t)x:x∈D} are equicontinuous if  ∥T(t2qθ)x-T(t1qθ)x∥→0 with respect to x∈D as t2→t1 for each fixed θ∈[0,∞].The proof of Lemma3 we can see in [33].To end this section, we recall Kuratowski’s measure of noncompactness, which will be used in the next section to study the complete controllability via the fixed points of condensing operator.Definition 4.
LetX be a Banach space and ΩX the bounded set of X. The Kuratowski’s measure of noncompactness is the map α:ΩX→[0,∞) defined by
(8)α(D)=inf{d>0:D⊆⋃i=1n‍Di,diam(Di)≤d},
here D∈ΩX.One will use the following basic properties of theα measure and Sadovskii's fixed point theorem here (see [37–39]).Lemma 5.
LetD1 and D2 be two bounded sets of a Banach space X. Then(i)
α(D1)=0 if and only if D1 is relatively compact;(ii)
α(D1)≤α(D2) if D1⊆D2;(iii)
α(D1+D2)≤α(D1)+α(D2).Lemma 6 (sadovskii’s fixed point theorem).
LetN be a condensing operator on a Banach space X, that is, N is continuous and takes bounded sets into bounded sets, and α(N(D))<α(D) for every bounded set D of X with α(D)>0. If N(S)⊂S for a convex closed and bounded set S of X, then N has a fixed point in S.


## 3. Complete Controllability Result

To study the system (1), we assume the function xt represents the history of the state from -∞ up to the present time t and xt:(-∞,0]→X,xt(ϑ)=x(t+ϑ) belongs to some abstract phase space B, which is defined axiomatically. In this article, we will employ an axiomatic definition of the phase space B introduced by Hale and Kato [40] and follow the terminology used in [41]. Thus, B will be a linear space of functions mapping (-∞,0] into X endowed with a seminorm ∥·∥B. We will assume that B satisfies the following axioms:(A)
Ifx∈(-∞,σ+a)→X, a>0, is continuous on [σ,σ+a] and xσ∈B, then for every t∈[σ,σ+a] the following conditions hold:(i)
xt is in B;(ii)
∥x(t)∥≤H∥xt∥B;(iii)
∥xt∥B≤K(t-σ)sup{∥x(t)∥:σ≤s≤t}+M(t-σ)∥xσ∥B.HereH≥0 is a constant, K,M:[0,+∞)→[0,+∞), K is continuous and M is locally bounded, and H,K,M are independent of x(t).(B)
For the functionx(·) in (A), xt is a B-valued continuous function on [σ,σ+a].(C)
The spaceB is complete.Now we give the basic assumptions on the system (1).(
H
0
)
(i)A generates a uniformly bounded analytic semigroup {T(t),t≥0} in X; (ii) for all bounded subsets D⊂X and x∈D, ∥T(t2qθ)x-T(t1qθ)x∥→0 as t2→t1 for each fixed θ∈[0,∞].
(
H
1
)
F
:
[
0
,
a
]
×
B
→
X is continuous function, and there exists a constant β∈(0,1) and L,L1>0 such that the function F is Xβ-valued and satisfies the Lipschitz condition:
(9)∥AβF(s1,ϕ1)-AβF(s2,ϕ2)∥≤L(|s1-s2|+∥ϕ1-ϕ2∥B),
for0≤s1, s2≤a, ϕ1,ϕ2∈B, and the inequality
(10)∥AβF(t,ϕ)∥≤L1(∥ϕ∥B+1)
holds fort∈[0,a],ϕ∈B.
(
H
2
)
The functionG:[0,a]×B→X satisfies the following conditions:
(i)
for eacht∈[0,a], the function G(t,·):B→X is continuous and for each ϕ∈B the function G(·,ϕ):[0,a]→X is strongly measureable;
(ii)
for each positive numberk, there is a positive function gk∈L1/q1([0,a]),0<q1<q such that
(11)sup∥ϕ∥B≤k∥G(t,ϕ)∥≤gk(t),liminf1k∥gk∥L1/q1[0,a]=γ<∞.(
H
3
)
The linear operatorC is bounded, W from U into X is defined by
(12)Wu=∫0a‍(a-s)q-1S(a-s)Cu(s)ds
and there exists a bounded invertible operatorW-1 defined on L2([0,a];U)/kerW and there exist two positive constants M3,M4>0 such that ∥B∥Lb(U,X)≤M3, ∥W-1∥Lb(X,L2([0,a],U)/kerW)≤M4.
(
H
4
)
For all bounded subsetsD⊆X, the set
(13)Πh,δ(t)={Q2,h,δz(t)∣z∈D},
where(14)Q2,h,δz(t)=∫0t-h‍∫δ∞‍(t-s)q-1θξq(θ)qT((t-s)qθ)×[Cu(s)+G(s,z-s+ys)dθds
is relatively compact inX for arbitrary h∈(0,t) and δ>0.Theorem 7.
Letϕ∈B. If the assumptions (H0)–(H4) are satisfied, then the system (1) is controllable on interval [0,a] provided that
(15)M5LKa+C1-βΓ(1+β)aqββΓ(1+qβ)LKa<1,(16)(1+aM2M3M4)×(L1M5Ka+M2((1-q1q-q1)a(q-q1)/(1-q1))1-q1Kaγ+C1-βΓ(1+β)aqββΓ(1+qβ)L1Ka)<1,
where M5=∥A-β∥, Ka=sup{K(t):0≤t≤a} and C1-β is from (3).Proof.
Using the assumption(H3), for arbitrary function x(·) define the control
(17)u(t)=W-1[∫0a‍(a-s)q-1AS(a-s)F(s,xs)dsx1-φ(a)(ϕ(0)+F(0,ϕ))+F(a,xa)+∫0a‍(a-s)q-1AS(a-s)F(s,xs)ds-∫0a‍(a-s)q-1S(a-s)G(s,xs)ds](t).
It will be shown that when using this control the operator P defined by
(18)Px(t)=φ(t)[ϕ(0)+F(0,ϕ)]-F(t,xt)-∫0t‍(t-s)q-1AS(t-s)F(s,xs)ds+∫0t‍(t-s)q-1S(t-s)[Cu(s)+G(s,xs)]ds
has a fixed point x(·). Then x(·) is a mild solution of system (1), and it is easy to verify that x(a)=Px(a)=x1, which implies that the system is controllable.
Next we will prove thatP has a fixed point using the fixed point theorem of Sadovskii [38].
Lety(·):(-∞,a]→X be the function defined by
(19)y(t)={φ(t)ϕ(0),t∈[0,a],ϕ(t),-∞<t<0,
then y0=ϕ and the map t→yt is continuous. We can assume N=sup{∥yt∥:0≤t≤a}. For each z∈C([0,a]:X),z(0)=0. We can denote by z- the function defined by
(20)z-(t)={z(t),0≤t≤a,0,-∞<t<0.
If x(·) satisfies (18), we can decompose it as x(t)=z-(t)+y(t), 0≤t≤a, which implies xt=z-t+yt for every 0≤t≤a and the function z(·) satisfies
(21)z(t)=φ(t)F(0,ϕ)-F(t,z-t+yt)-∫0t‍(t-s)q-1AS(t-s)F(s,z-s+ys)ds+∫0t‍(t-s)q-1S(t-s)[Cu(s)+G(s,z-s+ys)]ds.
Moreover z-0=0. Let Q be the operator on C([0,a],X) defined by
(22)Qz(t)=φ(t)F(0,ϕ)-F(t,z-t+yt)-∫0t‍(t-s)q-1AS(t-s)F(s,z-s+ys)ds+∫0t‍(t-s)q-1S(t-s)[Cu(s)+G(s,z-s+ys)]ds.
Obviously the operator P has a fixed point is equivalent to Q has a fixed point, so it turns out to prove that Q has a fixed point. For each positive number k, let
(23)Bk={z∈C([0,a]:X):z(0)=0,∥z(t)∥≤k,0≤t≤a},
then for each k, Bk is clearly a bounded closed convex set in C([0,a]:X). Since by (3) and (10) the following relation holds:
(24)∥AS(t-s)F(s,z-s+ys)∥≤∥A1-βS(t-s)AβF(s,z-s+ys)∥≤C1-βqΓ(1+β)Γ(1+qβ)(t-s)-(1-β)q×L1(∥z-s+ys∥B+1)≤C1-βqΓ(1+β)Γ(1+qβ)(t-s)-(1-β)q×L1(∥z-s∥B+∥ys∥B+1)≤C1-βqΓ(1+β)Γ(1+qβ)(t-s)-(1-β)q×L1(kKa+N+1),
then from Bocher’s theorem [42] it follows that AS(t-s)F(s,z-s+ys) is integrable on [0,a], so Q is well defined on Bk.
In order to make the following process clear we divide it into several steps.
Step  1. We claim that there exists a positive number k such that Q(Bk)⊆Bk.
If it is not true, then for each positive numberk, there is a function zk(·)∈Bk, but Qzk∉Bk, that is, ∥Qzk(t)∥>k for some t∈[0,a]. However, on the other hand, we have
(25)k<∥Qzk(t)∥=∥∫0t‍φ(t)F(0,ϕ)-F(t,z-k,t+yt)-∫0t‍(t-s)q-1×AS(t-s)F(s,z-k,s+ys)ds+∫0t‍(t-s)q-1S(t-s)×[Cuk(s)+G(s,z-k,s+ys)]ds∫0t‍∥=∥∫0t‍φ(t)F(0,ϕ)-F(t,z-k,t+yt)-∫0t‍(t-s)q-1AS(t-s)×F(s,z-k,s+ys)ds+∫0t‍(t-s)q-1S(t-s)×[∫0a‍(a-τ)q-1CW-1{∫0a‍(a-τ)q-1x1-φ(a)[ϕ(0)+F(0,ϕ)]+F(a,z-k,a+ya)+∫0a‍(a-τ)q-1AS(a-τ)×F(τ,z-k,τ+yτ)dτ-∫0a‍(a-τ)q-1S(a-τ)×G(τ,z-k,τ+yτ)dτ∫0a‍}(s)ds+∫0t‍(t-s)q-1S(t-s)×G(s,z-k,s+ys)∫0a‍(a-τ)q-1]ds∫0a‍∥≤M1∥F(0,ϕ)∥+∥F(t,z-k,t+yt)∥+∫0t‍(t-s)q-1∥AS(t-s)F(s,z-k,s+ys)∥ds+∫0t‍M2M3M4×{∫0a‍M2∫0a‍∥x1∥+M1∥ϕ(0)is+F(0,ϕ)∥+∥F(a,z-k,a+ya)∥+∫0a‍(a-τ)q-1×∥AS(a-τ)F(τ,z-k,τ+yτ)∥dτ+∫0a‍M2(a-τ)q-1×∥G(τ,z-k,τ+yτ)∥dτ∫0a‍M2∫0a‍}(s)ds+∫0t‍M2(t-s)q-1∥G(s,z-k,s+ys)∥ds,
where uk is the corresponding control of xk, xk=z-k+y. Since
(26)∥∫0t‍(t-s)q-1AS(t-s)F(s,z-k,s+ys)ds∥≤∫0t‍(t-s)q-1∥A1-βS(t-s)AβF(s,z-k,s+ys)∥ds≤C1-βqΓ(1+β)Γ(1+qβ)×∫0t‍(t-s)q-1(t-s)-(1-β)qL1(kKa+N+1)≤C1-βΓ(1+β)aqββΓ(1+qβ)L1(kKa+N+1),∥F(t,z-k,t+yt)∥=∥A-βAβF(t,z-k,t+yt)∥≤M5L1(kKa+N+1),∫0t‍∥(t-s)q-1G(s,z-k,s+ys)∥ds≤∫0t‍(t-s)q-1gkKa+N(s)ds,
there holds
(27)k<M1∥F(0,ϕ)∥+M5L1(kKa+N+1)+C1-βΓ(1+β)aqββΓ(1+qβ)L1(kKa+N+1)+aM2M3M4{∫0a‍∥x1∥+M1∥ϕ(0)+F(0,ϕ)∥+M5L1(kKa+N+1)+C1-βΓ(1+β)aqββΓ(1+qβ)L1(kKa+N+1)+M2∫0a‍(a-τ)q-1gkKa+N(τ)dτ}+M2∫0a‍(a-s)q-1gkKa+N(s)ds=M5L1kKa(1+aM2M3M4)+M2(1+aM2M3M4)∫0a‍(a-s)q-1gkKa+N(s)ds+C1-βΓ(1+β)aqββΓ(1+qβ)L1(kKa+N+1)(1+aM2M3M4)+M1∥F(0,ϕ)∥+M5L1N+M5L1+aM2M3M4∥x1∥+aM2M3M4∥ϕ(0)+F(0,ϕ)∥+M5L1NaM2M3M4+M5L1aM2M3M4=M*+(1+aM2M3M4)×[M5L1Kak+M2(∫0a‍(a-s)(q-1)/(q-q1)ds)1-q1×(∫0a‍(gkKa+N(s))1/q1ds)q1+C1-βΓ(1+β)aqββΓ(1+qβ)L1(kKa+N+1)]=M*+(1+aM2M3M4)×[M5L1Kak+M2(1-q1q-q1a(q-q1)/(1-q1))1-q1×∥gkKa+N∥L1/q1[0,a]+C1-βΓ(1+β)aqββΓ(1+qβ)L1(kKa+N+1)],
where
(28)M*=M1∥F(0,ϕ)∥+M5L1N+M5L1+aM2M3M4∥x1∥+aM2M3M4∥ϕ(0)+F(0,ϕ)∥+M5L1NaM2M3M4+M5L1aM2M3M4.
Dividing on both sides by k and taking the low limit, we get
(29)(1+aM2M3M4)×(L1M0Ka+M2((1-q1q-q1)a(q-q1)/(1-q1))1-q1Kaγ+C1-βΓ(1+β)aqββΓ(1+qβ)L1Ka)≥1.
This contradicts (16). Hence for some positive number k, QBk⊆Bk.
Now, we define operatorQ1 and Q2 on Bk as
(30)(Q1z)(t)=φ(t)F(0,ϕ)-F(t,z-t+yt)-∫0t‍(t-s)q-1AS(t-s)F(s,z-s+ys)ds,(Q2z)(t)=∫0t‍(t-s)q-1S(t-s)×[Cu(s)+G(s,z-s+ys)]ds,
for all t∈[0,a], respectively.
We prove thatQ1 is contraction, while Q2 is completely continuous.
Step  2. Q1 is contraction.
Letz1,z2∈Bk. Then, for each t∈[0,a], and by axiom (A)-(iii) and (15), we have
(31)∥Q1z1(t)-Q1z2(t)∥=∥∫0t‍(t-s)q-1φ(t)F(0,ϕ)-F(t,z-1,t+yt)-∫0t‍(t-s)q-1AS(t-s)F(s,z-1,s+ys)ds-φ(t)F(0,ϕ)+F(t,z-2,t+yt)+∫0t‍(t-s)q-1AS(t-s)F(s,z-2,s+ys)ds∥≤∥F(t,z-1,t+yt)-F(t,z-2,t+yt)∥+∥∫0t‍(t-s)q-1AS(t-s)×(F(s,z-1,s+ys)-F(s,z-2,s+ys))ds∫0t‍(t-s)q-1∥≤∥A-βAβF(t,z-1,t+yt)-A-βAβF(t,z-2,t+yt)∥+∫0t‍(t-s)q-1×∥A1-βS(t-s)Aβ(F(s,z-1,s+ys)-F(s,z-2,s+ys))∥ds≤M5LKa∥z-1,t-z-2,t∥B+∫0a‍(t-s)q-1C1-βqΓ(1+β)Γ(1+qβ)(t-s)-(1-β)q×L∥z-1,s-z-2,s∥Bds≤M5LKasup0≤s≤a∥z1(s)-z2(s)∥+C1-βΓ(1+β)aqββΓ(1+qβ)L1Kasup0≤s≤a∥z1(s)-z2(s)∥≤(M5LKa+C1-βΓ(1+β)aqββΓ(1+qβ)L1Ka)×sup0≤s≤a∥z1(s)-z2(s)∥.
Thus
(32)∥Q1z1(t)-Q1z2(t)∥<∥z1-z2∥,
and Q1 is contraction.
Step  3. Q2 is completely continuous.
Let{zn}⊆Bk with zn→z in Bk, then for each s∈[0,a],z-n,s→z-s, and by (H1) and (H2)-(i), we have
(33)G(s,z-n,s+ys)-G(s,z-s+ys)→0,F(s,z-n,s+ys)-F(s,z-s+ys)→0,un(s)-u(s)→0,
as n→∞.
Since∥G(s,z-n,s+ys)-G(s,z-s+ys)∥≤2gkKa+N(s), then by the dominated convergence theorem we have
(34)∥Q2zn(t)-Q2z(t)∥=sup0≤t≤a∥∫0t‍(t-s)q-1S(t-s)Cun(s)ds+∫0t‍(t-s)q-1S(t-s)G(s,z-n,s+ys)ds-∫0t‍(t-s)q-1S(t-s)Cu(s)ds-∫0t‍(t-s)q-1S(t-s)G(s,z-s+ys)ds∥≤∥∫0t‍(t-s)q-1S(t-s)(Cun(s)-Cu(s))ds∥+∥∫0t‍(t-s)q-1S(t-s)×(G(s,z-n,s+ys)-G(s,z-s+ys))ds∫0t‍∥→0,
as n→∞, that is, Q2 is continuous.
Next we prove that the family{Q2z:z∈Bk} is an equicontinuous family of functions. To do this, let 0≤t1<t2≤a, then
(35)∥Q2z(t2)-Q2z(t1)∥=∥∫0t2‍(t2-s)q-1S(t2-s)Cu(s)ds+∫0t2‍(t2-s)q-1S(t2-s)G(s,z-s+ys)ds-∫0t1‍(t1-s)q-1S(t1-s)Cu(s)ds-∫0t1‍(t1-s)q-1S(t1-s)G(s,z-s+ys)ds∥≤M3∫0t1‍(t2-s)q-1∥(S(t2-s)-S(t1-s))u(s)∥ds+M3∫0t1‍((t2-s)q-1-(t1-s)q-1)×∥S(t1-s)∥∥u(s)∥ds+∫0t1‍(t2-s)q-1×∥(S(t2-s)-S(t1-s))G(s,z-s+ys)∥ds+∫0t1‍((t2-s)q-1-(t1-s)q-1)×∥S(t1-s)∥∥G(s,z-s+ys)∥ds+M3∫t1t2‍(t2-s)q-1∥S(t2-s)∥∥u(s)∥ds+∫t1t2‍(t2-s)q-1∥S(t2-s)∥∥G(s,z-s+ys)∥ds.
Noting that
(36)∥u(s)∥≤M4[∫0a‍∥x1∥+M1∥ϕ(0)+F(0,ϕ)∥+∥F(a,xa)∥+∫0a‍(a-s)q-1∥S(a-s)F(s,xs)∥ds+∫0a‍(a-s)q-1∥S(a-s)G(s,xs)∥ds]≤M4[C1-βΓ(1+β)aqββΓ((1+qβ))∥x1∥+M1∥ϕ(0)+F(0,ϕ)∥+∥F(a,xa)∥+M5L1(kKa+N+1)+C1-βΓ(1+β)aqββΓ(1+qβ)L1(kKa+N+1)+M2(1-q1q-q1a(q-q1)/(1-q1))1-q1×∥gkKa+N∥L1/q1[0,a]C1-βΓ(1+β)aqββΓ(1+qβ)],∫0t‍∥(t-s)q-1G(s,z-k,s+ys)∥ds≤∫0t‍(t-s)q-1gkKa+N(s)ds≤∫0a‍(a-s)q-1gkKa+N(s)ds≤((1-q1q-q1)a(q-q1)/(1-q1))1-q1∥gkKa+N∥L1/q1[0,a].
We see that ∥Q2z(t2)-Q2z(t1)∥ tends to zero independently of z∈Bk as t2→t1 since for t∈[0,a] and any bounded subsets D⊂X, t→{S(t)x:x∈D} is equicontinuous.
Hence,Q2 maps Bk into an equicontinuous family functions.
It remains to prove thatV(t)={(Q2z)(t):z∈Bk} is relatively compact in X. let 0≤t≤a be fixed, 0<ϵ<t, for z∈Bk, we define Π=Q2Bk and Π(t)={Q2z(t)∣z∈Bk}, for t∈[0,a].
Clearly,Π(0)={Q2z(0)∣z∈Bk}={0} is compact, and hence, it is only to consider 0<t≤a. For each h∈(0,t),t∈(0,a], arbitrary δ>0, define
(37)Πh,δ(t)={Q2,h,δz(t)∣z∈Bk},
where
(38)Q2,h,δz(t)=∫0t-h‍∫δ∞‍(t-s)q-1θξq(θ)qT((t-s)qθ)×[Cu(s)+G(s,z-s+ys)]dθds.
Then the sets {Q2,h,δz(t)∣z∈Bk} are relatively compact in X since the condition (H4). It comes from the following inequalities:
(39)∥Q2z(t)-Q2,h,δz(t)∥=∥∫0t‍∫0∞‍(t-s)q-1θξq(θ)qT((t-s)qθ)×[Cu(s)+G(s,z-s+ys)]dθds-∫0t-h‍∫δ∞‍(t-s)q-1θξq(θ)qT((t-s)qθ)×[Cu(s)+G(s,z-s+ys)]dθds∫0t-h‍∫δ∞‍∥=∥∫0t‍∫0δ‍(t-s)q-1θξq(θ)qT((t-s)qθ)×[Cu(s)+G(s,z-s+ys)]dθds+∫0t‍∫δ∞‍(t-s)q-1θξq(θ)qT((t-s)qθ)×[Cu(s)+G(s,z-s+ys)]dθds-∫0t-h‍∫δ∞‍(t-s)q-1θξq(θ)qT((t-s)qθ)-∫0t-h‍∫δ∞‍(t-s)q-1θξq(θ)qT((t-s)qθ)×[Cu(s)+G(s,z-s+ys)]dθds∫δ∞‍∥≤MM3∥u(s)∥q∫0t‍(t-s)q-1ds∫0δ‍θξq(θ)dθ+Mq∫0t‍(t-s)q-1gkKa+N(s)ds∫0δ‍θξq(θ)dθ+MM3∥u(s)∥∫t-ht‍(t-s)q-1ds·q∫δ∞‍θξq(θ)dθ+M∫t-ht‍(t-s)q-1gkKa+N(s)ds·q∫δ∞‍θξq(θ)dθ≤{C1-βΓ(1+β)aqββΓ(1+qβ)((1-q1q-q1)a(q-q1)/(1-q1))1-q1MM3M4[C1-βΓ(1+β)aqββΓ(1+qβ)∥x1∥+M1∥ϕ(0)+F(0,ϕ)∥+∥F(a,xa)∥+M5L1(kKa+N+1)+C1-βΓ(1+β)aqββΓ(1+qβ)L1(kKa+N+1)+M((1-q1q-q1)a(q-q1)/(1-q1))1-q1×∥gkKa+N∥L1/q1[0,a]((1-q1q-q1)a(q-q1)/(1-q1))1-q1]aq+M((1-q1q-q1)a(q-q1)/(1-q1))1-q1∥gkKa+N∥L1/q1[0,a]}q×∫0δ‍θξq(θ)dθ+M((1-q1q-q1)h(q-q1)/(1-q1))1-q1×∥gkKa+N∥L1/q1[0,a]q∫0∞‍θξq(θ)dθ.
Therefore,Π(t)={Q2z(t)∣z∈Bk} is relatively compact in X for all t∈[0,a].
Thus, the continuity ofQ2 and relatively compact of {Q2z(t)∣z∈Bk} imply that Q2 is a completely continuous operator.
These arguments enable us to conclude thatQ=Q1+Q2 is a condense mapping on Bk, and by the fixed point theorem of Sadovskii there exists a fixed point z(·) for Q on Bk. In fact, by Step  1–Step  3 and Lemma 3, we can conclude that Q=Q1+Q2 is continuous and takes bounded sets into bounded sets. Meanwhile, it is easy to see α(Q2(Bk))=0 since Q2(Bk) is relatively compact. Since Q1(Bk))⊆Bk and α(Q2(Bk))=0, we can obtain α(Q(Bk))≤α(Q1(Bk))+α(Q2(Bk))≤α(Bk) for every bounded set Bk of X with α(Bk)>0, that is, Q=Q1+Q2 is a condense mapping on Bk. If we define x(t)=z(t)+y(t),-∞<t≤a, it is easy to see that x(·) is a mild solution of (1) satisfying x0=ϕ,x(a)=x1. Then the proof is completed.Remark 8.
In order to describe various real-world problems in physical and engineering sciences subject to abrupt changes at certain instants during the evolution process, impulsive fractional differential equations always have been used in the system model. So we can also consider the complete controllability for (1) with impulses.Remark 9.
Since the complete controllability steers the systems to arbitrary final state while approximate controllability steers the system to arbitrary small neighborhood of final state. In view of the definition of approximate controllability in [28], we can deduce that the considered systems (1) is also approximate controllable on the interval [0,a].


## 4. An Example

As an application of Theorem7, we consider the following system:
(40)∂2/3∂t2/3[z(t,x)+∫-∞t‍∫0π‍b(s-t,y,x)z(s,y)dyds]-∂2∂x2z(t,x)=Cu(t)+a0(x)z(t,x)+∫-∞t‍a1(s,t)z(s,x)ds+a2(t,x),0≤t≤a,0≤x≤π,z(t,0)=z(t,π)=0,z(ϑ,x)=ϕ(ϑ,x),ϑ≤0.To write system (40) to the form of (1), let X=L2([0,π]) and A defined by Af=-f′′ with domain D(A)={f(·)∈X:f,f′absolutelycontinuous,f′′∈X,f(0)=f(π)=0}, the -A generates a uniformly bounded analytic semigroup which satisfies the condition (H0). Furthermore, A has a discrete spectrum, the eigenvalues are -n2, n∈N, with the corresponding normalized eigenvectors zn(x)=(2/π)1/2sin(nx). Then the following properties hold.(i) IfA∈D(A), then
(41)Af=∑n=1∞‍n2〈f,zn〉zn.(ii) For eachf∈X,
(42)A-1/2f=∑n=1∞‍1n〈f,zn〉zn.In particular,∥A-1/2∥=1.(iii) The operatorA1/2 is given by
(43)A1/2f=∑n=1∞‍n〈f,zn〉znon the spaceD(A1/2)={f(·)∈X,A1/2f∈X}.Here we take the phase spaceB=C0×L2(g,X), which contains all classes of functions ϕ:(-∞,0]→X such that ϕ is Lebesgue measurable and g(·)∥ϕ(·)∥2 is Lebesgue integrable on (-∞,0) where g:(-∞,0)→R is a positive integrable function. The seminorm in B is defined by
(44)∥ϕ∥B=∥ϕ(0)∥+(∫-∞0‍g(ϑ)∥ϕ(ϑ)∥2dϑ)1/2.
From [41], under some conditions B is a phase space verifying (A), (B), (C), and in this case K(t)=1+(∫-t0‍g(ϑ)dϑ)1/2 (see [41] for the details).We assume the following conditions hold.(a)
The functionb is measurable and ∫0π‍∫-∞0‍∫0π‍(b2(ϑ,y,x)/g(ϑ))dydϑdx<∞.(b)
The function(∂/∂x)b(ϑ,y,x) is measurable, b(ϑ,y,0)=b(ϑ,y,π)=0 and let N1=∫0π‍∫-∞0‍∫0π‍(1/g(ϑ))((∂/∂x)b(ϑ,y,x))2dydϑdx<∞.(c)
The functiona0(·)∈L∞([0,π]), a(·) is measurable, with ∫-∞0‍(a12(ϑ))/g(ϑ)dϑ<∞, the function a2(t,·)∈L2([0,π]) for each t≥0 is measurable in t.(d)
The functionϕ defined by ϕ(ϑ)(x)=ϕ(ϑ,x) belongs to B.(e)
The linear operatorW:U→X is defined by
(45)Wu=∫0a‍(a-s)-1/3S(a-s)Cu(s)dsand has a bounded invertible operatorW-1 defined L2([0,a]);U)/kerW.We defineF,G:[0,a]×B→X by F(t,ϕ)=Z1(ϕ) and G(t,ϕ)=Z2(ϕ)+h(t), where
(46)Z1(ϕ)=∫-∞0‍∫0π‍b(ϑ,y,x)ϕ(ϑ,x)dydϑ,Z2(ϕ)=a0(x)ϕ(0,x)+∫-∞0‍a1(ϑ)ϕ(ϑ,x)dϑ,h(t)=a2(t,·).
From (a) and (c) it is clear that Z1 and Z2 are bounded linear operators on B. Furthermore, Z1(ϕ)∈D(A1/2), and ∥A1/2Z1∥≤N1. In fact, from the definition of Z1 and (b) it follows that 〈Z1(ϕ),zn〉=(1/n)(2/π)1/2〈Z(ϕ),con(nx)〉, where Z(ϕ)=∫-∞0‍∫0π‍(∂/∂x)b(ϑ,y,x)ϕ(ϑ,x)dϑ. From (b) we know that Z:B→X is a bounded linear operator with ∥Z∥≤N1. Hence ∥A1/2Z1(ϕ)∥=∥Z(ϕ)∥, which implies the assertion. Therefore, from Theorem 7, the system (40) is completely controllable on [0,a] under the above assumptions.


## 5. Conclusion

In this paper, by using the uniformly boundedness, analyticity, and equicontinuity of characteristic solution operators and the Sadovskii fixed point theorem, we obtained the complete controllability of the abstract neutral fractional differential systems with unbounded delay in a Banach space. It shows that the compactness of the characteristic solution operators can be weakened to equicontinuity. Our theorem guarantees the effectiveness of complete controllability results under some weakly compactness conditions.



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