# Approximate Controllability of Fractional Neutral Evolution Equations in Banach Spaces

Authors: N. I. Mahmudov
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We discuss the approximate controllability of semilinear fractional neutral differential systems with infinite delay under the assumptions that the corresponding linear system is approximately controllable. Using Krasnoselkii's fixed-point theorem, fractional calculus, and methods of controllability theory, a new set of sufficient conditions for approximate controllability of fractional neutral differential equations with infinite delay are formulated and proved. The results of the paper are generalization and continuation of the recent results on this issue.


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## Body

## 1. Introduction and Preliminaries

Many social, physical, biological and engineering problems can be described by fractional partial differential equations. In fact, fractional differential equations are considered as an alternative model to nonlinear differential equations. In the last two decades, fractional differential equations (see, e.g., Samko et al. [1] and references therein) have attracted many scientists, and notable contributions have been made to both theory and applications of fractional differential equations.Nowadays, controllability theory for linear systems has already been well established, for finite and infinite dimensional systems; see, for instance, [2]. Several authors have extended these concepts to infinite-dimensional systems represented by nonlinear evolution equations in infinite-dimensional spaces, see [3–30]. On the other hand, approximate controllability problems for fractional evolution equations in Hilbert spaces are not yet sufficiently investigated, and there are only few works on it [13–21, 28–30]. So far, the overwhelming majority of the approximate controllability results have only been available for semilinear evolution differential systems in Hilbert spaces, with the exception of the case of [11]. Motivated by the fact that many partial fractional differential equations can be converted into fractional PDE in some Banach spaces, we consider that there is a realistic need to discuss the approximate controllability problem of fractional-order differential systems in Banach spaces. Note that our results are new even for the approximate controllability of fractional neutral differential equations with infinite delay in Hilbert spaces.Consider the following fractional neutral evolution differential system with infinite delay:(1)dαdtα[x(t)-h(t,xt)]=-Ax(t)+Bu(t)+f(t,xt),t∈[0,T],x(t)=ϕ(t)∈Bl,
where the state x takes values in a Banach space X and the control function takes values in a Hilbert space U. The functions h,f will be specified in the sequel. Let xt(·) denote xt(θ)=x(t+θ), θ∈(-∞,0]. Assume that l:(-∞,0]→(-∞,0) is a continuous function satisfying l=∫-∞0l(t)dt<∞. The Banach space (Bl,∥·∥l) induced by function h is defined as follows:
(2)Bl∶={φ:(-∞,0]⟶X:foranyc>0,φ(θ)isaboundedandmeasurablefunctionon[-c,0],and∫-∞0l(t)supt≤θ≤0∥φ(θ)∥dt<∞}
endowed with the norm ∥φ∥Bl∶=∫-∞0h(t)supt≤θ≤0∥φ(θ)∥dt. It should be mentioned that (approximate) controllability results for first- and second-order partial neutral functional differential equations with infinite delay were considered by Sakthivel et al. [18], Chalishajar [8], and Chalishajar and Acharya [9].Throughout this paper, unless otherwise specified, the following notations will be used. Let(X,∥·∥) be a separable reflexive Banach space, and let (X*,∥·∥*) stand for its dual space with respect to the continuous pairing 〈·,·〉. We may assume, without loss of generality, that X and X* are smooth and strictly convex, by virtue of renorming theorem (see, e.g., [10]). In particular, this implies that the duality mapping J of X into X* given by the following relations:
(3)∥J(z)∥*=∥z∥,〈J(z),z〉=∥z∥2,∀z∈X
is bijective, homogeneous, and demicontinuous, that is, continuous from X with a strong topology into X* with weak topology and strictly monotonic. Moreover, J-1:X*→X is also duality mapping.In this paper, we also assume that-A:D(A)⊂X→X is the infinitesimal generator of a compact analytic semigroup S(t), t>0, of uniformly bounded linear operator in X, that is, there exists M>1 such that ∥S(t)∥L(X)≤M for all t≥0. Without loss of generality, let 0∈ρ(A), where ρ(A) is the resolvent set of A. Then, for any β>0, we can define A-β by
(4)A-β∶=1Γ(β)∫0∞tβ-1S(t)dt.
It follows that each A-β is an injective continuous endomorphism of X. Hence, we can define Aβ∶=(A-β)-1, which is a closed bijective linear operator in X. It can be shown that each Aβ has dense domain and that D(Aγ)⊂D(Aβ) for 0≤β≤γ. Moreover, Aβ+γx=AβAγx=AγAβx for every β, γ∈R and x∈D(Aμ) with μ∶=max(β,γ,β+γ), where A0=I and I is the identity in X.We denote byXβ the Banach space of D(Aβ) equipped with norm ∥x∥β∶=∥Aβx∥ for x∈D(Aβ), which is equivalent to the graph norm of Aβ. Then, we have Xγ↪Xβ, for 0≤β≤γ (with X0=X ), and the embedding is continuous. Moreover, Aβ has the following basic properties.Lemma 1 (see [31]).
A
β has the following properties:(i)
S(t):X→Xβ for each t>0 and β≥0.(ii)
AβS(t)x=S(t)Aβx for each x∈D(Aβ) and t≥0.(iii)
For everyt>0, AβS(t) is bounded in X, and there exists Mβ>0 such that
(5)∥AβS(t)∥≤Mβt-β.(iv)
A-β is a bounded linear operator for 0≤β≤1 in X.From Lemma1(iv), since A-β is a bounded linear operator for 0≤β≤1, there exists a constant Cβ such that ∥A-β∥≤Cβ for 0≤β≤1.Let us recall the following known definitions in fractional calculus. For more details, see [1].Definition 2.
The fractional integral of orderα>0 with the lower limit 0 for a function f is defined as
(6)Iαf(t)=1Γ(α)∫0tf(s)(t-s)1-αds,t>0,α>0,
provided that the right-hand side is pointwise defined on [0,∞), where Γ is the gamma function.Definition 3.
Riemann-Liouville derivative of orderα with the lower limit 0 for a function f:[0,∞)→R can be written as
(7)LDαf(t)=1Γ(n-α)dndtn∫0tf(s)(t-s)α+1-nds,t>0,n-1<α<n,Definition 4.
The Caputo derivative of orderα for a function f:[0,∞)→R can be written as
(8)cDαf(t)=LDα(f(t)-∑k=0n-1tkk!f(k)(0)),t>0,n-1<α<n.Forx∈X, we define two families {𝔖α(t):t≥0} and {𝔄α(t):t≥0} of operators by
(9)𝔖α(t)=∫0∞Ψα(θ)S(tαθ)dθ,𝔄α(t)=α∫0∞θΨα(θ)S(tαθ)dθ,
where
(10)Ψα(θ)=1πα∑n=1∞(-1)n-1Γ(nα+1)n!sin(nπα),θ∈(0,∞),
is the function of Wright type defined on (0,∞) which satisfies
(11)Ψα(θ)≥0,∫0∞Ψα(θ)dθ=1,∫0∞θζΨα(θ)dθ=Γ(1+ζ)Γ(1+αζ),ζ∈(-1,∞).The following lemma follows from the results in [32–34].Lemma 5.
The operators𝔖α and 𝔄α have the following properties:(i)
for any fixedt≥0 and any x∈Xβ, one has the operators 𝔖α(t) and 𝔄α(t) which are linear and bounded operators; that is, for any x∈Xβ,(12)∥𝔖α(t)x∥β≤M∥x∥β,∥𝔄α(t)x∥β≤MΓ(α)∥x∥β;(ii)
the operators𝔖α(t) and 𝔄α(t) are strongly continuous for all t≥0;(iii)
𝔖α(t) and 𝔄α(t) are norm continuous in X for t>0;(iv)
𝔖α(t) and 𝔄α(t) are compact operators in X for t>0;(v)
for everyt>0, the restriction of 𝔖α(t) to Xβ and the restriction of 𝔄α(t) to Xβ are norm continuous;(vi)
for everyt>0, the restriction of 𝔖α(t) to Xβ and the restriction of 𝔄α(t) to Xβ are compact operators in Xβ;(vii)
for allx∈X and t∈[0,T],
(13)∥Aβ𝔄α(t)x∥≤Cβt-αβ∥x∥,Cβ∶=MβαΓ(2-β)Γ(1+α(1-β)).In this paper, we adopt the following definition of mild solution of (1).Definition 6.
A solutionx(·;u)∈C([0,T],X) is said to be a mild solution of (1) if for any u∈L2([0,T],U) the integral equation
(14)x(t)=𝔖α(t)(ϕ(0)+g(0,ϕ))-g(t,xt)=-∫0t(t-s)α-1A𝔄α(t-s)g(s,xs)ds=+∫0t(t-s)α-1𝔄α(t-s)[Bu(s)+f(s,xs)]ds
is satisfied.Letx(T;u) be the state value of (14) at terminal time T corresponding to the control u. Introduce the set ℜ(T)={x(T;u):u∈L2([0,T],U)}, which is called the reachable set of the system (14) at terminal time T, and its closure in X is denoted by ℜ(T)¯.Definition 7.
The system (1) is said to be approximately controllable on [0,T] if ℜ(T)¯=X; that is, given an arbitrary ε>0, it is possible to steer from the point x0 to within a distance ε from all points in the state space X at time T.To investigate the approximate controllability of the system (14), we assume the following conditions.(H1)
A is the infinitesimal generator of an analytic semigroup of bounded linear operators S(t) in X, 0∈ρ(A), S(t) is compact for t>0, and there exists a positive constant M such that ∥S(t)∥≤M.(H2)
The functiong:[0,T]×Bl→X is continuous, and there exists some constant Mg>0,0<β<1, such that g is Xβ-valued and
(15)∥Aβg(t,x)-Aβg(t,y)∥≤Mg∥x-y∥Bl,x,y∈Bl,t∈[0,T],∥Aβg(t,x)∥≤Mg(1+∥x∥Bl).(H3)
The functionf:[0,T]×Bl→X satisfies the following:
(a)
f(t,·):Bl→X is continuous for each t∈[0,T] and for each x∈Bl,f(·,x):[0,T]→X is strongly measurable;(b)
there is a positive integrable functionn∈L∞([0,T],[0,+∞)) and a continuous nondecreasing function Λf:[0,∞)→(0,∞) such that for every (t,x)∈[0,T]×Bl, we have
(16)∥f(t,x)∥≤n(t)Λf(∥x∥Bl),liminfr→∞Λf(r)r=σf<∞.(H4)
The following relationship holds:(17)(1+1εMB2M𝔄2T2α-12α-1)×(Mg∥A-β∥l+C1-βMgTαβαβl+MΓ(α)Tαασfsups∈Jn(s))<1.
Here, MB∶=∥B∥,M𝔄∶=∥𝔄α∥,andC1-β=αM1-βΓ(1+β)/Γ(1+αβ).(Hac)
For everyh∈Xzα(h)=ε(εI+Γ0TJ)-1(h) converges to zero as ε→0+ in strong topology, where
(18)Γ0T∶=∫0T‍(T-s)2(α-1)𝔄α(T-s)BB*𝔄α*(T-s)ds
and zε(h) is a solution of the equation
(19)εzε+Γ0TJ(zε)=εh.Let(20)CT={x:x∈C((-∞,T],X),x0=ϕ∈Bl}.
set ∥·∥T be a seminorm defined by
(21)∥x∥T=∥x0∥Bl+sup0≤s≤T∥x(s)∥,x∈CT.Lemma 8 (see [8]).
Assume thatx∈CT, then for all t∈[0,T],xt∈Bl and
(22)l∥x(t)∥≤∥xt∥Bl≤lsup0≤s≤t∥x(s)∥+∥x0∥Bl.


## 2. Existence Theorem

In order to formulate the controllability problem in the form suitable for application of fixed-point theorem, it is assumed that the corresponding linear system is approximately controllable. Then, it will be shown that the system (1) is approximately controllable if for all ε>0 there exists a continuous function x∈C([0,T],X) such that
(23)uε(t,x)=(T-t)α-1B*𝔄α*(T-t)J((εI+Γ0TJ)-1p(x)),x(t)=𝔖α(t)(ϕ(0)+g(0,ϕ))-g(t,xt)=-∫0t(t-s)α-1A𝔄α(t-s)g(s,xs)ds=+∫0t(t-s)α-1𝔄α(t-s)[Buε(s,x)+f(s,xs)]ds,
where
(24)p(x)=h-𝔖α(T)(ϕ(0)+g(0,ϕ))+g(T,xT)=+∫0T‍(T-s)α-1A𝔄α(T-s)g(s,xs)ds=-∫0T‍(T-s)α-1𝔄α(T-s)f(s,xs)ds,h∈X.
Having noticed this fact, our goal in this section is to find conditions for solvability of (23). Note that it will be shown that the control in (23) drives the system (1) from ϕ(0) to
(25)h-εJ((εI+Γ0TJ)-1p(x))
provided that the system (23) has a solution.Theorem 9.
Assume that assumptions (H1)–(H4) hold and 1/2<α<1. Then, there exists a solution to (23).Proof.
The proof of Theorem9 follows from Lemmas 10–14 and infinite dimensional analogue of Arzela-Ascoli theorem.For allε>0, consider the operator Φε:CT→CT defined as follows:
(26)(Φεx)(t)∶={ϕ(t)t∈(-∞,0]𝔖α(t)(ϕ(0)+g(0,ϕ))-g(t,xt)-∫0t‍(t-s)α-1A𝔄α(t-s)g(s,xs)ds+∫0t‍(t-s)α-1𝔄α(t-s)+∫0t‍×[Buε(s,x)+f(s,xs)]ds,t∈[0,T].
It will be shown that for all ε>0, the operator Φε:CT→CT has a fixed point.Suppose thatx(t)=ϕ~(t)+z(t),t∈(-∞,T], where ϕ~(t) is taken as ϕ(t) for t∈(-∞,0], while for t∈[0,T], it is defined as 𝔖α(t)ϕ(0). Set
(27)CT0={z∈CT:z0=0∈Bl}.
For any z∈CT0, we have
(28)∥z∥T=∥z0∥Bl+sup0≤s≤T∥z(s)∥=sup0≤s≤T∥z(s)∥.
Thus, (CT0,∥·∥T) is a Banach space. For each positive number r>0, set
(29)Br∶={z∈CT0:∥z∥T≤r}.
It is clear that Br is bounded closed convex set in CT0. For any z∈Br, we see that
(30)∥ϕ~t+zt∥Bl≤∥ϕ~t∥Bl+∥zt∥Bl≤lsup0≤s≤t∥ϕ~(s)∥+∥ϕ~0∥Bl≤+lsup0≤s≤t∥z(s)∥+∥z0∥Bl≤l(M∥ϕ(0)∥+r)+∥ϕ∥Bl∶=R(r).
Consider the maps Πε,Θε:CT0→CT0 defined by
(31)(Πεz)(t)∶={0t∈(-∞,0],𝔖α(t)g(0,ϕ)-g(t,ϕ~t+zt)-∫0t‍(t-s)α-1A𝔄α(t-s)g(s,ϕ~s+zs)dst∈[0,T],(Θεz)(t)∶={0t∈(-∞,0],∫0t‍(t-s)α-1𝔄α(t-s)∫0t‍×[Buε(s,ϕ~+z)+f(s,ϕ~s+zs)]ds,t∈[0,T].
Obviously, the operator Φε has a fixed point if and only if operator Πε+Θε has a fixed point. In order to prove that Πε+Θε has a fixed point we will employ the Krasnoselskii’s fixed-point theorem.Lemma 10.
Under assumptions (H1)–(H4), for any ε>0, there exists a positive number r∶=r(ε) such that Πεzr+Θεyr⊂Br for all z,y∈Brr.Proof.
Letε>0 be fixed. If it is not true, then for each r>0, there are functions zr,yr∈Br, but Πε(zr)+Θε(yr)∉Br. So for some t=t(r)∈[0,T], one can show that
(32)r≤∥Πε(zr)(t)+Θε(yr)(t)∥≤∥𝔖α(t)g(0,ϕ)∥+∥g(t,ϕ~t+zt)∥≤+∥∫0t‍(t-s)α-1A𝔄α(t-s)g(s,ϕ~s+zs)ds∥≤+∥∫0t‍(t-s)α-1𝔄α(t-s)f(s,ϕ~s+ys)ds∥≤+∥∫0t‍(t-s)α-1𝔄α(t-s)Buε(s,ϕ+y)ds∥=:I1+I2+I3+I4+I5.
Let us estimate Ii,i=1,…,4. By the assumption (H2), we have
(33)I1≤M∥A-β∥∥Aβg(0,ϕ)∥≤MMg∥A-β∥(1+∥ϕ∥Bl),I2≤∥A-β∥∥Aβg(t,ϕ~t+zt)∥≤Mg∥A-β∥(1+∥ϕ~t+zt∥Bl)(34)≤Mg∥A-β∥(1+R(r)).
Using Lemma 5 and the Hölder inequality, one can deduce that
(35)I3≤∫0t‍∥(t-s)α-1A1-β𝔄α(t-s)Aβg(s,ϕ~s+zs)∥ds≤M1-βαΓ(1+β)Γ(1+αβ)∫0t‍(t-s)αβ-1∥Aβg(s,ϕ~s+zs)∥ds≤K(α,β)∫0t‍(t-s)αβ-1Mg(1+∥ϕ~s+zs∥Bl)ds≤K(α,β)MgTαβαβ(1+R(r)).
Using the assumption (H3), one has
(36)I4≤∫0t‍∥(t-s)α-1𝔄α(t-s)f(s,ϕ~s+ys)∥ds≤MΓ(α)∫0t‍(t-s)α-1∥f(s,ϕ~s+ys)∥ds≤MΓ(α)∫0t‍(t-s)α-1n(s)Λf(∥ϕ~s+ys∥)ds≤MΓ(α)TααΛf(R(r))sups∈Jn(s).
Combining the estimates (32)–(36) yields
(37)I1+I2+I3+I4<MMg∥A-β∥(1+∥ϕ∥Bl)<+Mg∥A-β∥(1+R(r))<+C1-βMgTαβαβ(1+R(r))<+MΓ(α)TααΛf(R(r))sups∈Jn(s)∶=Δ.
On the other hand,
(38)I5≤∫0t‍∥(t-s)α-1𝔄α(t-s)Buε(s,ϕ+y)∥ds=∫0t‍∥((εI+Γ0TJ)-1p(ϕ+y))(t-s)α-1(T-s)α-1𝔄α(t-s)BB*𝔄α*=∫0t‍d×(T-t)J((εI+Γ0TJ)-1p(ϕ+y))∥ds≤∫0t‍∥(t-s)α-1(T-s)α-1𝔄α(t-s)BB*𝔄α*(T-t)∥ds≤×∥J((εI+Γ0TJ)-1p(ϕ+y))∥≤MB2M𝔄2T2α-12α-1∥J((εI+Γ0TJ)-1p(ϕ+y))∥=MB2M𝔄2T2α-12α-1∥(εI+Γ0TJ)-1p(ϕ+y)∥≤1εMB2M𝔄2T2α-12α-1∥p(ϕ+y)∥≤1εMB2M𝔄2T2α-12α-1Δ.
Thus,
(39)r≤∥Πε(zr)(t)+Θε(yr)(t)∥≤Δ+1εMB2M𝔄2T2α-12α-1Δ=(1+1εMB2M𝔄2T2α-12α-1)Δ.
Dividing both sides of (39) by r and taking r→∞, we obtain that
(40)(1+1εMB2M𝔄2T2α-12α-1)×(Mg∥A-β∥l+MgC1-βTαβαβl+MΓ(α)Tαασfsups∈Jn(s))≥1,
which is a contradiction by assumption (H4). Thus, Πεzr+Θεyr⊂Br for some r>0.Lemma 11.
Let assumptions (H1)–(H4) hold. Then, Θ1 is contractive.Proof.
Letx,y∈Br. Then,
(41)∥(Πεx)(t)-(Πεy)(t)∥≤∥g(t,ϕ~t+xt)-g(t,ϕ~t+yt)∥≤+∥∫0t‍(t-s)α-1A𝔄α(t-s)≤+MM×(g(s,ϕ~s+xs)-g(s,ϕ~s+ys))ds∫0t‍(t-s)α-1∥≤∥A-β∥Mg∥xt-yt∥Bl+C1-β≤×∫0t‍(t-s)αβ-1∥Aβ(g(s,ϕ~s+xs)-g(s,ϕ~s+ys))∥ds≤∥A-β∥Mg∥xt-yt∥Bl+C1-βMg≤×∫0t‍(t-s)αβ-1∥xs-ys∥Blds.
Hence,
(42)∥(Πεx)(t)-(Πεy)(t)∥≤Mgl(∥A-β∥+C1-βTαβαβ)sup0≤s≤t∥x(s)-y(s)∥,
where we have used the fact that x0=y0=0. Thus,
(43)sup0≤t≤T∥(Πεx)(t)-(Πεy)(t)∥≤Mgl(∥A-β∥+C1-βTαβαβ)sup0≤s≤T∥x(s)-y(s)∥,
so Θ1 is a contraction by assumption (H4).Lemma 12.
Let assumptions (H1)–(H4) hold. Then, Θε maps bounded sets to bounded sets in Br.Proof.
By the similar argument as Lemma10, we obtain
(44)∥(Θεz)(t)∥<(1+1εMB2M𝔄2T2α-12α-1)<×MΓ(α)TααΛf(R(r))sups∈Jn(s)∶=r1(ε)
which implies that Θεz∈Br1(ε).Lemma 13.
Let assumptions (H1)–(H4) hold. Then, the set {Θεz:z∈Br} is an equicontinuous family of functions on [0,T].Proof.
Let0<η<t<T and δ>0 such that
(45)∥𝔄α(s1)-𝔄α(s2)∥<η
for every s1,s2∈[0,T] with |s1-s2|<δ. For z∈Br, 0<|h|<δ,t+h∈[0,T], we have
(46)∥(Θεz)(t+h)-(Θεz)(t)∥≤∥∫0t‍((t+h-s)α-1-(t-s)α-1)𝔄α(t+h-s)i∫0t‍×[(T-s)α-1Bvε(s,ϕ~+z)+f(s,ϕ~s+zs)]ds∫0t‍∥≤+∥∫tt+h‍(t+h-s)α-1𝔄α(t+h-s)≤+∫tt+h‍×[(T-s)α-1Bvε(s,ϕ~+z)+f(s,ϕ~s+zs)]ds∫tt+h‍∥≤+∥∫0t‍(t-s)α-1(𝔄α(t+h-s)-𝔄α(t-s))≤m×[(T-s)α-1Bvε(s,ϕ~+z)+f(s,ϕ~s+zs)]ds∫0t‍∥.
Applying (38) and the Hölder inequality, we obtain
(47)∥(Θεz)(t+h)-(Θεz)(t)∥≤MΓ(α)Λf(R(r))∫0t‍((t+h-s)α-1-(t-s)α-1)n(s)ds≤+MΓ(α)1εMBM𝔄Δ≤+×∫0t‍((t+h-s)α-1-(t-s)α-1)(T-s)α-1ds≤+MΓ(α)Λf(R(r))∫tt+h‍(t+h-s)α-1n(s)ds≤+MΓ(α)1εMBM𝔄Δ∫tt+h‍(t+h-s)α-1(T-s)α-1ds≤+ηTααΛf(R(r))∫0t‍(t-s)α-1n(s)ds≤+ηTαα1εMBM𝔄Δ∫0t‍(t-s)α-1(T-s)α-1ds.
Therefore, for ε sufficiently small, the right-hand side of (47) tends to zero as h→0. On the other hand, the compactness of 𝔄α(t), t>0 implies the continuity in the uniform operator topology. Thus, the set {Θεz:z∈Br} is equicontinuous.Lemma 14.
Let assumptions (H1)–(H4) hold. Then, Θε maps Br onto a precompact set in Br.Proof.
Let0<t≤T be fixed and ε be a real number satisfying 0<λ<t. For δ>0, define an operator Θελ,δ on Br by
(48)(Θελ,δz)(t)=α∫0t-λ‍∫δ∞‍θ(t-s)α-1ηα(θ)S((t-s)αθ)=α∫0t-λ‍∫δ∞‍×[Buε(s,ϕ~+z)+f(s,ϕ~s+zs)]ds=αS(λαδ)∫0t-λ‍∫δ∞‍θ(t-s)α-1ηα(θ)S((t-s)αθ-λαδ)=αS(λαδ)∫0t-λ‍∫δ∞‍M×[Buε(s,ϕ~+z)+f(s,ϕ~s+zs)]ds.
Since S(t), t>0 is a compact operator, the set {(Θελ,δz)(t):z∈Br} is precompact in H for every 0<λ<t, δ>0. Moreover, for each z∈Br, we have
(49)∥(Θεz)(t)-(Θελ,δz)(t)∥≤αE∥∫0t‍∫0δ‍θ(t-s)α-1ηα(θ)S((t-s)αθ)≤αEmm×[Buε(s,ϕ~+z)+f(s,ϕ~s+zs)]dθds∫0t‍∫0δ‍∥≤+αE∥∫t-λt‍∫δ∞‍θ(t-s)α-1ηα(θ)S((t-s)αθ)≤+αEmm×[Buε(s,ϕ~+z)+f(s,ϕ~s+zs)]dθds∫t-λt‍∥∶=J1+J2.
A similar argument as before
(50)J1≤αM∫0t‍(t-s)α-1(∥Buε(s,ϕ~+z)∥+∥f(s,ϕ~s+zs)∥)ds≤×(∫0δ‍θηα(θ)dθ)≤αM(1εMBM𝔄Δ∫0t‍(t-s)α-1(T-s)α-1ds≤αMB+Λf(R(r))∫0t‍(t-s)α-1n(s)ds)(∫0δ‍θηα(θ)dθ),J2≤αM∫t-λt‍(t-s)α-1(∥Buε(s,ϕ~+z)∥+∥f(s,ϕ~s+zs)∥)ds≤×(∫δ∞‍θηα(θ)dθ)≤αMΓ(1+α)(1εMBM𝔄Δ∫t-λt‍(t-s)α-1(T-s)α-1ds≤αMΓ(1+α)B+Λf(R(r))∫t-λt‍(t-s)α-1n(s)ds),
where we have used the equality
(51)∫0∞‍θβηα(θ)dθ=Γ(1+β)Γ(1+αβ).
Form (49) to (50), one can see that for each z∈Br,
(52)∥(Θεz)(t)-(Θελ,δz)(t)∥⟶0asλ⟶0+,δ⟶0+.
Therefore, there are relatively compact sets arbitrary close to the set {(Θεz)(t):z∈Br}. Hence, the set {(Θεz)(t):z∈Br} is also precompact in Br.


## 3. Main Results

Consider the following linear fractional differential system:(53)Dtαx(t)=Ax(t)+Bu(t),t∈[0,T],(54)x(0)=ϕ(0).
The approximate controllability for the linear fractional system (53) is a natural generalization of approximate controllability of linear first-order control system. It is convenient at this point to introduce the controllability operator associated with (53) as
(55)L0T=∫0T‍(T-s)α-1𝔄α(T-s)Bu(s)ds,Γ0T=L0T(L0T)*=∫0T‍(T-s)2(α-1)𝔄α(T-s)BB*𝔄α*(T-s)ds,
respectively, where B* denotes the adjoint of B and 𝔄α*(t) is the adjoint of 𝔄α(t). It is straightforward that the operator L0T is a linear-bounded operator for 1/2<α≤1.Theorem 15 (see [11]).
The following three conditions are equivalent:(i)
Γ0T is positive, that is, 〈z*,Γ0Tz*〉>0 for all nonzero z*∈X*.(ii)
For allh∈X,J(zε(h)) converges to the zero as ε→0+ in the weak topology, where zε(h)=ε(εI+Γ0TJ)-1(h) is a solution of the equation
(56)εzε+Γ0TJ(zε)=αh.(iii)
For allh∈X,zε(h)=ε(εI+Γ0TJ)-1(h) converges to the zero as ε→0+ in the strong topology.Remark 16.
It is known that Theorem15(i) holds if and only if ImL0T¯=X. In other words, Theorem 15(i) holds if and only if the corresponding linear system is approximately controllable on [0,T].Theorem 17 (see [11]).
Letp:X→X be a nonlinear operator. Assume that zε is a solution of the following equation:
(57)εzε+Γ0TJ(zε)=αp(zε),(58)∥p(zε)-p∥⟶0asε⟶0+,p∈X.
Then, there exists a subsequence of the sequence {zε} strongly converging to zero as ε→0+.We are now in a position to state and prove the main result of the paper.Theorem 18.
Let1/2<α<1. Suppose that conditions (H1)–(H4) and (Hac) are satisfied. Besides, assume additionally that(Hgc)
g:[0,T]×X→X and Aβg(T,·) is continuous from the weak topology of X to the strong topology of X.(Hub)
There existsN∈L∞([0,T],[0,+∞)) such that
(59)supx∈Bl∥f(t,x)∥+supy∈X∥Aβg(t,y)∥≤N(t),fora.e.t∈[0,T].
Then, the system (1) is approximately controllable on [0,T].Proof.
Letxε be a fixed point of Φε in Br(ε). Then, xε is a mild solution of (1) on [0,T] under the control
(60)uε(t,xε)=(T-t)α-1B*S*(T-t)J((εI+Γ0TJ)-1p(xε)),p(xε)=h-𝔖α(T)(ϕ(0)+g(0,ϕ(0)))+g(T,xε(T))=+∫0T‍(T-s)α-1A𝔄α(T-s)g(s,xsε)ds=-∫0T‍(T-s)α-1𝔄α(T-s)f(s,xsε)ds
and satisfies the following equality:
(61)xε(T)=𝔖α(T)(ϕ(0)+g(0,ϕ(0)))-g(T,xε(T))=-∫0T‍(T-s)α-1A𝔄α(T-s)g(s,xsε)ds=+∫0T‍(T-s)α-1𝔄α(t-s)[Buε(s,x)+f(s,xsε)]ds=𝔖α(T)(ϕ(0)+g(0,ϕ(0)))-g(T,xε(T))=-∫0T‍(T-s)α-1A𝔄α(T-s)h(s,xsε)ds=+(-εI+εI+Γ0TJ)((εI+Γ0TJ)-1p(xε))=+∫0T‍(T-s)α-1𝔄α(t-s)f(s,xsε)ds=h-ε(εI+Γ0TJ)-1p(xε).
In other words, zε=h-xε(T) is a solution of the equation
(62)(εI+Γ0TJ)(zε)=εp(xε).
It follows that
(63)ε〈J(zε),zε〉+〈J(zε),Γ0TJ(zε)〉=ε〈J(zε),p(xε)〉,ε∥zε∥2+〈J(zε),Γ0TJ(zε)〉=ε〈J(zε),p(xε)〉,ε∥zε∥2≤ε〈J(zε),p(xε)〉≤ε∥zε∥∥p(xε)∥,∥zε∥=∥J(zε)∥≤∥p(xε)∥.
On the other hand, by (Hub),
(64)∥p(xε)∥≤∥h∥+M∥ϕ(0)∥+N(T)≤+M1-βαΓ(1+β)Γ(1+αβ)∫0T‍(T-s)αβ-1N(s)ds≤+MΓ(α)∫0T‍(T-s)α-1N(s)ds.
From (63) and (64), it follows that xε(T)⇀x~ weakly as ε→0+ and by the assumption (Hgc)  Aβg(T,xε(T))→Aβg(T,x~) strongly as ε→0+. Moreover, because of assumption (Hub),
(65)∫0T‍∥f(s,xsε)∥2ds+∫0T‍∥Aβg(s,xsε)∥2ds≤∫0T‍N(s)ds.
Consequently, the sequences {f(·,x·ε)},{Aβg(·,x·ε)} are bounded. Then, there is a subsequence still denoted by {f(·,x·ε),Aβg(·,x·ε)} which weakly converges to, say, (f(·),g(·)) in L2([0,T],X). Then,
(66)∥p(xε)-p∥=∥g(T,xε(T))-g(T,x~)∥=+∥∫0T‍(T-s)α-1A1-β𝔄α(T-s)[Aβg(s,xsε)-g(s)]ds∥=+∥∫0T‍(T-s)α-1𝔄α(T-s)[f(s,xsε)-f(s)]ds∥≤∥A-β(Aβg(T,xε(T))-Aβg(T,x~))∥≤+sup0≤t≤T∥∫0t‍(t-s)α-1A1-β𝔄α(t-s)[Aβg(s,xsε)-g(s)]ds∥≤+sup0≤t≤T∥∫0t‍(t-s)α-1𝔄α(t-s)[f(s,xsε)-f(s)]ds∥⟶0,
where
(67)p=h-𝔖α(T)(ϕ(0)+g(0,ϕ(0)))+g(T,x~)=+∫0T‍(T-s)α-1A1-β𝔄α(T-s)g(s)ds=-∫0T‍(T-s)α-1𝔄α(T-s)f(s)ds
as ε→0+ because of compactness of an operator f(·)→∫0·‍(·-s)α-1𝔄α(·-s)f(s)ds:L2([0,T],X)→C([0,T],X). Then, by Theorem 17(68)∥xε(T)-h∥=∥zε∥⟶0
as ε→0+. This gives the approximate controllability. The theorem is proved.Remark 19.
Theorem18 assumes that the operator A generates a compact semigroup. If the compactness condition holds on the bounded operator that maps the control function or the generated C0-semigroup, then the controllability operator L0T is also compact, and its inverse does not exist if the state space is infinite dimensional, and, consequently, the associated linear control system (53) is not exactly controllable. Therefore, the concept of complete controllability is too strong in infinite dimensional spaces, and the approximate controllability notion is more appropriate. Thus, Theorem 18 has no analogue for the concept of complete controllability.


## 4. Applications

In this section, we illustrate the obtained result. LetX=L2[0,π], and let A be defined as follows:
(69)Az=-z′′
with domain
(70)D(A)={z∈X∣z,dzdξareabsolutelycontinuous,=md2zdξ2∈Xandz(0)=z(π)=0}.
Recall that A is the infinitesimal generator of a strongly continuous semigroup S(t),t>0, on X which is analytic compact and self-adjoint, and the eigenvalues are -n2, n∈N, with corresponding normalized eigenvectors en(ξ)∶=(2/π)1/2sin(nξ) and
(71)S(t)en=e-n2ten,n=1,2,….
Moreover, the following hold.(a)
{en:n∈N} is an orthonormal basis of X.(b)
Ifz∈D(A), then A(z)=∑n=1∞‍n2〈z,en〉en.(c)
Forz∈ℍ, A-1/2z=∑n=1∞‍(1/n)〈z,en〉en.(d)
The operatorA1/2 is given as A1/2z=∑n=1∞‍n〈z,en〉en on the space D[A1/2]={z∈X:∑n=1∞‍n〈z,en〉en∈X}.For1/2<α<1, consider the neutral system
(72)∂α∂tα[x(t,ξ)+∫0π‍b(θ,ξ)x(t,θ)dθ]=∂2∂ξ2x(t,ξ)+p(t,x(t,ξ))+Bu(t,ξ),x(t,0)=x(t,π)=0,t≥0,x(0,ξ)=φ(ξ),0≤ξ≤π,
where p:[0,T]×R→R is continuous functions. B is a linear continuous mapping from U={u=∑n=2∞‍unen∣∥u∥U2∶=∑n=2∞‍un2<∞} to X as follows:
(73)Bu=2u2+∑n=2∞‍unen.To write the initial-boundary value problem (72) in the abstract form, we assume the following.(A1)
The functionb is measurable and
(74)∫0π∫0πb2(θ,ξ)dθdξ<∞.(A2)
The function(∂/∂ξ)b(θ,ξ) is measurable, b(θ,0)=b(θ,π)=0, and let
(75)L1=[∫0π∫0π(∂∂ξb(θ,ξ))2dθdξ]1/2.Definef,g:[0,T]×X→X by
(76)g(x)(ξ)=∫0π‍b(θ,ξ)x(θ)dθ,f(t,x)(ξ)=p(t,x(ξ)).From (A1), it is clear thatg is bounded linear operator on X. Furthermore, g(x)∈D[A1/2], and ∥A1/2g∥≤L1. In fact, from the definition of g and (A2), it follows that
(77)〈g(x),en〉=∫0π‍[∫0π‍b(θ,ξ)x(θ)dθ]en(ξ)dξ=1n(2π)1/2〈∫0π‍∂∂ξb(θ,ξ)x(θ)dθ,cos(nξ)〉=1n(2π)1/2〈g1(x),cos(nξ)〉,
where g1(x)=∫0π(∂/∂ξ)b(θ,ξ)x(θ)dθ. From (A2), we know that g1:X→X is a bounded linear operator with ∥g1∥≤L1. Hence, ∥A1/2g(x)∥=∥g1(x)∥, which implies the assertion. Moreover, assume that f and g satisfy conditions of Theorem 18. Thus, the problem (72) can be written in the abstract form
(78)dαdtα(x(t)+g(t,x(t)))=Ax(t)+f(t,x(t))+Bu(t),x(0)=x0,t∈[0,T].
Now, consider the associated linear system
(79)dαdtαx(t)=Ax(t)+Bu(t),x(0)=x0,t∈[0,T].
Show that it is approximately controllable on [0,T] for 1/2<α<1. It is easy to see that if z=∑n=1∞‍〈z,en〉en, then
(80)B*v=(2v1+v2)e2+∑n=3∞‍vnen,B*𝔄α*(T-s)z=B*α∫0∞‍θΨα(θ)S*((T-s)αθ)zdθ=α∫0∞‍θΨα(θ)(∑n=3∞‍e-n2(T-s)αθ(2〈z,e1〉e-(T-s)αθe1+〈z,e2〉e-4(T-s)αθ)e2=α∫0∞‍θΨα(θ)M+∑n=3∞‍e-n2(T-s)αθ〈z,en〉en)dθ=(2〈z,e1〉α∫0∞‍θΨα(θ)e-(T-s)αθdθ=m+〈z,e2〉α∫0∞‍θΨα(θ)e-4(T-s)αθdθ)e2=+α∑n=3∞‍∫0∞‍θΨα(θ)e-n2(T-s)αθdθ〈z,en〉en∥(T-s)α-1B*𝔄α*(T-s)z∥2=(T-s)2(α-1)(2α∫0∞‍θΨα(θ)e-(T-s)αθdθ〈z,e1〉=(T-s)2(α-1)mi+α∫0∞‍θΨα(θ)e-4(T-s)αθdθ〈z,e2〉)2=+(T-s)2(α-1)∑n=3∞‍(α∑n=3∞‍∫0∞‍θΨα(θ)e-n2(T-s)αθdθ)2=+×〈z,en〉2=0.
It follows that 〈z,e1〉=〈z,e2〉=⋯=〈z,en〉=⋯=0, and consequently, z=0, which means that (79) is approximately controllable on [0,T]. Therefore, from Theorem 18, the system (72) is approximately controllable [0,T].


## 5. Conclusion

In this paper, abstract results concerning the approximate controllability of fractional semilinear evolution systems with infinite delay in a separable reflexive Banach space are obtained. Approximate controllability result for semilinear systems is obtained by means of the Krasnoselskii’s fixed-point theorem under the compactness assumption. It is also proven that the controllability of the semilinear system is implied by the approximate controllability of the associated linear system under some natural conditions. Upon making some appropriate assumptions, by employing the ideas and techniques as in this paper, one can establish the approximate controllability results for a wide class of fractional deterministic and stochastic evolution equations.



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