Zhang et al. Advances in Difference Equations (2015) 2015:91
DOI 10.1186/s13662-015-0412-z
RESEARCH
Open Access
Approximate controllability of impulsive
fractional stochastic differential equations
with state-dependent delay
Xiaozhi Zhang1 , Chuanxi Zhu1* and Chenggui Yuan2
*
Correspondence:
[email protected]
1
Department of Mathematics,
Nanchang University, Nanchang,
330031, P.R. China
Full list of author information is
available at the end of the article
Abstract
This work is concerned with the approximate controllability of nonlinear fractional
impulsive stochastic differential system under the assumption that the corresponding
linear system is approximately controllable. Using fractional calculus, stochastic
analysis, and the technique of stochastic control theory, a new set of sufficient
conditions for the approximate controllability of a fractional impulsive stochastic
differential system is obtained. The results in this paper are generalizations and
continuations of the recent results on this issue. An example is given to illustrate the
efficiency of the main results.
MSC: 93B05; 34A08; 34A37; 34F05
Keywords: approximate controllability; stochastic differential system;
state-dependent delay; nonlocal conditions
1 Introduction
In the last few decades, fractional differential systems have provided an excellent tool in
electrochemistry, physics, porous media, control theory, engineering etc., due to the descriptions of memory and hereditary properties of various materials and processes. The
research of fractional systems has received more and more attention very recently, see the
monographs of Kilbas et al. [], Miller and Ross [], Podlubny [], the recent papers [–]
and so forth. Controllability is one of the important concepts both in mathematics and in
control theory. Controllability of deterministic control systems has been well developed
by using different kinds of methods, which can be found in [–]. The approximate controllability, the weaker concept of controllability, has received much attention recently.
In this case it is possible to steer the system to an arbitrary small neighborhood of the
final state (see, for example, [–]). However, the extension of the deterministic controllability concepts to stochastic control system has rarely been reported. As a matter of
fact, the accurate analysis or assessment subjected to a realistic environment has to take
into account the potential randomness in the system properties, such as fluctuations in
the stock market or noise in a communication network. All these problems in mathematics are modeled and described by stochastic differential equations or stochastic integrodifferential equations with delay and impulse. Stochastic control theory is a stochastic
generalization of classical control theory [, ]. The biggest difficulty is the analysis of
a stochastic control system and stochastic calculations induced by the stochastic process.
© 2015 Zhang et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly credited.
Zhang et al. Advances in Difference Equations (2015) 2015:91
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In the deterministic case, Mahmudov and Zorlu [] discussed the approximate controllability of fractional evolution equations with a compact analytic semigroup. Ganesh et al.
[] derived a set of sufficient conditions for the approximate controllability of a class of
fractional integro-differential evolution equations. For more work, see [, , ] and the
references therein. In the stochastic case, by using the stochastic analysis technique and
the methods directly from deterministic control problems, Sakthivel et al. [] considered the approximate controllability of fractional stochastic evolution equations. Ahmed
[] considered the approximate controllability of impulsive neutral stochastic differential
equations with finite delay and fractional Brownian motion in a Hilbert space, and a new
set of sufficient conditions for approximate controllability was formulated and proved.
On the other hand, the stochastic differential equation with delay is a special type of
stochastic functional differential equations. Delay differential equations arise in many biological and physical applications, and it often forces us to consider variable or statedependent delays. The stochastic functional differential equations with state-dependent
delay have many important applications in mathematical models of real phenomena, and
the study of this type of equations has received much attention in recent years. Guendouzi
and Benzatout [] studied the existence of mild solutions for a class of impulsive stochastic differential inclusions with state-dependent delay. Sakthivel and Ren [] studied the
approximate controllability of fractional differential equations with state-dependent delay.
However, approximate controllability of stochastic differential system with state-dependent delay and impulse has rarely been considered in the literature. Therefore, this work
aims to study the fractional impulsive stochastic control system with state-dependent delay:
⎧
dw(t)
c α
⎪
⎨ Dt x(t) = Ax(t) + f (t, xρ(t,xt ) ) + Bu(t) + g(t, xρ(t,xt ) ) dt ,
x(tk ) = Ik (x(tk– )), k = , , . . . , n;
⎪
⎩
x() + h(x) = x = φ ∈ B,
q
t ∈ J := [, T], t = tk ;
(.)
where c Dt is the Caputo fractional derivative of order  < α < , the state variable x takes
values in a Hilbert space H, x(tk ) := x(tk+ ) – x(tk– ) represents the jump in the state x
at time tk ,  < t < t < · · · < tn < T. The history xs represents the function defined by
xs : (–∞, ] → H, xs (θ ) = x(s + θ ) belongs to some abstract phase space B described axiomatically and ρ : J × B → (–∞, T] is a continuous function. A is the infinitesimal generator of a compact semigroup {S(t), t ≥ } on the Hilbert space H, the control function
u is given in L ([, T], U), U is a Hilbert space, B is a bounded linear operator from U
to H. Let K be another Hilbert space, suppose {W (t)}t≥ is a given K -valued Brownian
motion or Wiener process with a finite trace nuclear covariance operator Q ≥  defined
on a complete probability space (, F , P). Denote PC(J, L (, F , P; H)) = {x(t) is continuous everywhere except for some tk at which x(tk– ) and x(tk+ ) exist and x(tk– ) = x(tk )}
be the Banach space with the norm xPC = supt∈J |x(t)| < ∞. PC(J, L ) is the closed subspace of PC(J, L (, F , P; H)) consisting of a measurable and Ft -adapted H-valued process x(·) ∈ PC(J, L (, F , P; H)) with the norm x = sup{Ex(t) , t ∈ J}. The functions
f , g, Ik , h are appropriate functions to be specified later.
The outline of this paper is as follows. In Section , we recall some preliminary notations
and results which will be needed in the sequel. Section  is devoted to studying the approximate controllability of the system (.) provided that the corresponding linear system
Zhang et al. Advances in Difference Equations (2015) 2015:91
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is approximately controllable. Finally, an example is given to illustrate the effectiveness of
the main results.
2 Preliminaries
In this section, we will introduce some preliminary definitions, notations, and results, to
establish our main results.
Let (, F , P) be a complete probability space with a filtration {Ft }t≥ . An H-valued
random variable is an F measurable function x(t) : → H, and the collection of random
variable C = {x(t, ω) : → H|t∈J } is a stochastic process. Here we suppress the dependence on ω ∈ and write x(t) instead of x(t, ω). Let {βn }n≥ be a sequence of real valued
√
independent Brownian motions. Set W (t) = ∞
n= λn βn en , t ≥ , where {en }n≥ is complete orthonormal system in K . Assume Q ∈ L(K, K) be an operator satisfying Qen = λn en
with tr(Q) < ∞. Then the above K -valued stochastic process W (t) is a Q-Wiener process.
Denote Ft = σ (W (s) :  ≤ s ≤ t) be the σ -algebra generated by W and FT = Ft . Take
√

φ ∈ L(K, H) and define φQ = tr(φQφ ∗ ) = ∞
n= λn φen < ∞, then φ is called a QHilbert Schmidt operator and we denote its set as LQ (K, H) with the norm φQ = φ, φ.
Throughout this paper, we assume that the abstract phase space (B, ·B ) is a seminorm
linear space of F -measurable functions mapping J = (–∞, ] to H and satisfying the
following fundamental axioms []:
(i) If x : (–∞, T) → H is continuous on [, T) and x ∈ B, then for each t ∈ [, T) the
following conditions hold:
(a) xt ∈ B;
(b) x(t) ≤ K xt B ;
(c) xt B ≤ K (t)x B + K (t) sup{x(s);  ≤ s ≤ T},
where K >  is a constant, K , K : [, ∞) → [, ∞), K is locally bounded, K is
continuous. Moreover, Ki (i = , , ) are independent of x.
(ii) For the function x(·) in (i), xt is a B-valued continuous functions on [, T).
(iii) The space B is complete.
Let x : (–∞, T] → H be an Ft -adapted measurable process such that we have the F adapted process x = φ(t) ∈ L (, B), then
 Ext B ≤ K  EφB + K  sup Ex(s) ,
≤s≤T
where K  = supt∈J K (t), K  = supt∈J K (t).
The next lemma is proved using the phase spaces axioms.
Lemma . ([]) Let φ ∈ B and I = (–∞, ] be such that φt ∈ B for each t ∈ I. Assume that there exists a locally bounded function H φ : I → [, ∞) such that Eφt B ≤
H φ (t)EφB for t ∈ I. Let x : (–∞, T] → H be functions such that x = φ and x ∈ PC(J, L ),
then

Exs B ≤ (H  + η)EφB + H  sup Ex(θ ) ; θ ∈ , max{, s} ,
s ∈ (–∞, T],
where η = supt∈I H φ (t), H  = supt∈J K  (t), H  = supt∈J K  (t).
Definition . A stochastic process J × → H is called a mild solution of the system
(.) if
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(i) x(t) is measurable and Ft -adapted for each t ∈ J;
(ii) x(t) ∈ H satisfies the following integral equation:
x(t) = T φ() – h(x) +
t

t
+
(t – s)α– S (t – s)f (s, xρ(s,xs ) ) ds
(t – s)α– S (t – s)Bu(s) ds

t
+

(t – s)α– S (t – s)g(s, xρ(s,xs ) ) dw(s) +
S (t – tk )Ik x tk– ;
<tk <t
(iii) x (·) = φ ∈ B on (–∞, ] satisfying φB < ∞, where
T (t) =
∞
ξα (θ )S t α θ dθ ,

ξα (θ ) =
∞
S (t) = α
θ ξα (θ )S t α θ dθ ,

 ––  –  θ α ωα θ α ,
α
∞
 (nα + )
ωα (θ ) =
(–)n– θ –αn–
sin(nπα),
π n=
n!
θ ∈ (, ∞).
Here ξα is a probability density function on (, ∞), that is, ξα (θ ) ≥  and
∞

ξα (θ ) = .
Lemma . ([]) The operators T and S have the following properties:
(i) For any fixed t ≥ , the operators T (t) and S (t) are linear and bounded operators,
i.e., for any x ∈ H,
T (t)x ≤ Mx and
S (t)x ≤
αM
x.
( + α)
(ii) The operators T (t) and S (t) are strongly continuous for all t ≥ , it means that for
every x ∈ H and  ≤ t < t ≤ T, we have
T t x – T t x → 
and S t x – S t x →  as t → t .
(iii) For every t > , T (t) and S (t) are also compact operators if S(t) is compact for all
t > .
Assume that the linear fractional differential system
c
Dαt x(t) = Ax(t) + Bu(t),
x() = x
t ∈ J;
(.)
is approximately controllable. It is convenient at this position to introduce the controllability operator associated with (.), thus
T =
T
(T – s)α– S (T – s)BB∗ S ∗ (T – s) ds,

where B∗ and S ∗ are the adjoint of B and S , respectively. It is straightforward that the
operator T is a linear bounded operator.
Zhang et al. Advances in Difference Equations (2015) 2015:91
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Let x(T; x , u) be the state value of (.) at terminal time T corresponding to the control u
and the initial value x . Introduce the set R(T, x ) = {x(T; x , u) : u ∈ L ([, T], U)}, which
is called the reachable set of system (.) at terminal time T, its closure in H is denoted by
R(T, x ).
Definition . ([]) The system (.) is said to be approximately controllable on [, T] if
R(T, x ) = L (, H), that is, given an arbitrary > , it is possible to steer from the point
x to within a distance from all points in the state space H at time T.
Lemma . ([]) The linear fractional control system (.) is approximately controllable
on [, T] if and only if λ(λI + T ) →  as λ → + in the strong operator topology.
Lemma . ([]) For any x̃T ∈ L (, H), there exists φ̃ ∈ L (, L (J, LQ (K, H))) such that
T
x̃T = Ex̃T +  φ̃(s) dw(s).
Lemma . (Krasnoselskii fixed point theorem [, ]) Let M be a closed, convex, and
nonempty subset of a Banach space X. Let A, B be the operators such that
(i) Ax + By ∈ M, wherever x, y ∈ M;
(ii) A is compact and continuous;
(iii) B is a contraction mapping.
Then there exists z ∈ M such that z = Az + Bz.
3 Main results
In this section, we aim to establish the approximate controllability of the system (.). To
do this, we need the following assumptions.
(H ) The function t → φt is well defined from Z (ρ – ) = {ρ(s, ψ); (s, ψ) ∈ J × B, ρ(s, ψ) ≤
} into B and there exists a continuous and bounded function H φ : Z (ρ – ) → (, ∞) such
that Ext B ≤ H φ (t)EφB for every t ∈ Z (ρ – ).
(Hf ) The function f : J × B → H is continuous and there exist two constants Mf and Lf
such that

Ef (t, x) ≤ Mf  + xB ,
Ef (t, x) – f (t, y) ≤ Lf x – yB .
(Hg ) The function g : J × B → H is continuous and there exist two constants Mg and Lg
such that

Eg(t, x) ≤ Mg  + xB ,
Eg(t, x) – g(t, y) ≤ Lg x – yB .
(HI ) The function Ik : H → H are continuous and there exist nondecreasing continuous
functions MIK : R+ → R+ such that for each x ∈ H,

EIk (x) ≤ MIK Ex
and
lim inf
r→∞
MIK (r)
= γk < ∞.
r
(Hh ) h is continuous and there exists some constant Mh satisfying

Eh(t, x) ≤ Mh  + xB .
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Now, for any λ >  and x̃T ∈ L (F , H), we define the control function
t
uλ (t, x) = B∗ S ∗ (T – t)R λ, T Ex̃T +
φ̃(s) dw(s) – T (T) φ() – h(x)
– B∗ S ∗ (T – t)

t
(T – s)α– R λ, sT S (T – s)f (s, xρ(s,xs ) ) ds
t
(T – s)α– R λ, sT S (T – s)g(s, xρ(s,xs ) ) dw(s)

– B∗ S ∗ (T – t)

T (T – tk )Ik x tk– ,
– B S (T – t)R λ, T
∗
∗
(.)
<tk <T
where R(λ, T ) = (λI + T )– .
Theorem . If the hypotheses (H ), (Hf ), (Hg ), (Hh ), (HI ) are satisfied, then the fractional
Cauchy problem (.) with u = uλ (t, x) has at least one mild solution provided that
Lr∗ H  + Lr < ,
T α–
M T α

L
L
+
H  < ,
f
g


(α) α
α – 
(.)
where
Lr = M n
n
k=
M M T α
γk  +   B  ,
λ (α)α
M MB T α
T α M
T α– M

Lr∗ =  M Mh +   Mf +
Mg  +  
.
α (α)
(α – )  (α)
λ (α)α 
Proof It is convenient to divide the proof into several steps.
Step : We can claim that Br ⊂ Br . If this is not true, then for any r > , there exist
xr ∈ Br and t r ∈ J such that r < xr (t r ) , from Lemma ., it follows that Exrρ(t,xr ) ≤
tr
(H + η)φB + H  r := r∗ , then one can see that

Euλ t r , xr ≤
t


r 
MB M

  Ex
+
E
φ̃(s)
ds
+
M
E
φ()
+
M
E
h
x
T
λ  (α)

tr
r

M M t
+  B
(T – s)α– ds
(T – s)α– ES (T – s)f s, xrρ(s,xrs ) ds
λ (α) 

r

M M t
+  B
(T – s)α– ES (T – s)g s, xrρ(s,xrs ) ds
λ (α) 
n

MB M n EIk x tk– ds


λ (α)
k=
T
 


M M
≤  B
Eφ̃(s) ds + M Eφ() + M Mh  + r∗
ExT  +
λ (α)

+
M M n M M T α–
MB M T α Mf  + r∗ +  B
Mg  + r∗ +  B 
MIk r



λ α (α)
λ (α)(α – )
λ (α)
n
+
k=
Zhang et al. Advances in Difference Equations (2015) 2015:91
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and

r < Exr t r n

r  

+ M n
≤ M E φ() + E h x
EIk xr tkr– 
k=
tr

r α– r t –s
+ E
S t – s Bu(s) ds

tr
r α– r r

t –s
+ E
S t – s f s, xρ(s,xrs ) ds
+ E

tr r
t –s
α–


r
S t – s g s, xρ(s,xrs ) dw(s)
r
n

≤ M Eφ() + Mh  + r∗ + M n
MIk (r)
k=
α– λ 
T M
tr – s
Eu (s) ds

α (α)

r

T α M t r α– t –s
Ef s, xrρ(s,xrs ) ds
+
α  (α) 
tr
r α– r

M
+ 
t –s
Eg s, xρ(s,xrs ) ds
(α) 

α
+
MB
tr n


T α M MB ≤ M Eφ() + Mh  + r∗ + M n
MIk (r) +
Euλ (s)
α   (α)
k=
α
+

T M
Mf  + r
α   (α)
∗
T
M
Mg  + r∗ .

(α – ) (α)
α–
+

Dividing both sides by r and taking the limit as r → ∞, we obtain
n
α 
α– 
M
M
T
T
 ≤  M Mh +   Mf +
Mg H + M n
γk
α (α)
(α – )  (α)
k=
M M T α
× +   B  ,
λ (α)α
which is a contradiction to our assumption. Thus, for each r > , there exists some positive
number r such that Br ⊂ Br .
Next, we denote (x) =  (x) +  (x), where
 (x)(t) =
t
(t – s)α– S (t – s)f (s, xρ(s,xs ) ) ds +

<tk <t
(t – s)α– S (t – s)g(s, xρ(s,xs ) ) dw(s),

t
 (x)(t) = T (t) φ() – h(x) +
+
t

(t – s)α– S (t – s)Buλ (s) ds
T (t – tk )Ik x tk– .
Zhang et al. Advances in Difference Equations (2015) 2015:91
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Step :  (x) is contractive. Let x, y ∈ Br , then

E (x)(t) –  (y)(t)
t

α–
≤ E
(t
–
s)
S
(t
–
s)
f
(s,
x
)
–
f
(s,
y
)
ds
ρ(s,x
)
ρ(s,y
)
s
s

t

α–
+ E
(t
–
s)
S
(t
–
s)
g(s,
x
)
–
g(s,
y
)
dw(s)
ρ(s,x
)
ρ(s,y
)
s
s


M T α t
≤ 
(t – s)α– Ef (s, xρ(s,xs ) ) – f (s, yρ(s,ys ) ) ds
(α)α 
t

M
+ 
(t – s)α– Eg(s, xρ(s,xs ) ) – g(s, yρ(s,ys ) ) ds
(α) 


M T α M T α–
Lg xρ(s,xs ) – yρ(s,ys ) Lf xρ(s,xs ) – yρ(s,ys ) + 


(α)α
(α)(α – )
 α
α–

M
T
T

Lg H  sup Ex(s) – y(s)
≤ 
Lf +
(α) α 
α – 
≤s≤T
 α
α–


T
M
T

Lg H  x(s) – y(s) := L x(s) – y(s) ,
≤ 
Lf +

(α) α
α – 
≤
where L < , hence  x is contractive.
Step :  maps bounded sets to bounded sets in Br . We have

E (x)(t)


≤ ET (t) φ() – h(x) + ET (t – tk )Ik x tk– t

α–
λ
+ E
(t
–
s)
S
(t
–
s)Bu
(s)
ds

n


∗

≤ M E φ() + Mh  + r + M n
MIk Ex tk– 
+
T α M MB
α  (α)
k=
t

(t – s)α– Euλ (s) ds

n


T α M MB Euλ .
MIk r +
≤ M Eφ() + Mh  + r∗ + M n


α (α)
k=
Therefore, for each x ∈ Br , E (x)(t) is bounded.
Step :  is equicontinuous. Let  < t < t + h ≤ T, where t, t + h and |h| is sufficiently
small. By means of Hölder’s inequality and the compactness of T and S , we have

E x(t + h) –  x(t)

≤ E T (t + h) – T (t) φ() – h(x) 
n
– + E
T (t + h – tk ) – T (t – tk ) Ik x tk k=
t

α–
α–
λ
(t
+
h
–
s)
+ E
–
(t
–
s)
S
(t
+
h
–
s)Bu
(s)
ds

Zhang et al. Advances in Difference Equations (2015) 2015:91
+ E
t+h
t
Page 9 of 12

(t + h – s)α– S (t + h – s)Buλ (s) ds
t

λ
α–
+ E
(t
–
s)
S
(t
+
h
–
s)
–
S
(t
–
s)
Bu
(s)
ds

n


M M ≤   Eφ() + Mh  + r∗ + n 
MIk r +   B Euλ (s) hα
α (α)
k=
+
    MB T α λ 
M MB Euλ (s) (t + h)α – t α – hα +
Eu (s) .


α (α)
α
Consequently, E x(t + h) –  x(t) →  as h → .
Step : V (t) = { x(t), x ∈ Br } is relatively compact in Br . For any ∈ (, t) and δ > ,
we define
,δ x(t)
∞
=
ξα (θ )S t α θ φ() – h(x) dθ +
δ
<tk <t δ
t–
∞
∞
+α

= S α δ
∞
ξα (θ )S (t – tk )α θ Ik x tk– dθ
θ (t – s)α– ξα (θ )S (t – s)α θ Buλ (s) dθ ds
δ
ξα (θ )S t α θ – α δ φ() – h(x) dθ
δ
+ S α δ
∞
<tk <t δ
+ αS α δ

t–
ξα (θ )S (t – tk )α θ – α δ Ik x tk– dθ
∞
θ (t – s)α– ξα (θ )S (t – s)α θ – α δ Buλ (s) dθ ds.
δ
Because of the compactness of S( α δ), the set V ,δ (t) = {,δ x(t), x ∈ Br } is relatively compact for every ∈ (, t). Furthermore, ,δ x(t) is convergent to  x(t) as →  and δ → ,
then V (t) = { x(t), x ∈ Br } is also relatively compact. By the Arzela-Ascoli theorem, 
is completely continuous. So, by means of the Krasnoselskii fixed point theorem, the operator has a fixed point, which is a mild solution of (.).
Theorem . Assume the linear stochastic system (.) is of approximate controllability,
and the conditions of Theorem . hold. Further, assume that the functions f and g are
bounded uniformly, then the fractional nonlinear stochastic system (.) is approximately
controllable.
Proof Let xλ be a solution of (.), then it is not difficult to see that
T
λ λ
T –
φ̃(s) dw(s) – T (T) φ() – h x (T)
Ex̃T +
x (T) = x̃T – λ λI + 

T
+λ
s

+λ

–
(T – s)α– λI + sT S (T – s)f s, xλρ(s,xλ ) ds
T
–
(T – s)α– λI + sT S (T – s)g s, xλρ(s,xλ ) dw(s)
s
– T (T – tk )Ik x tk– .
+ λ λI + T
<tk <T
Zhang et al. Advances in Difference Equations (2015) 2015:91
Page 10 of 12
In view of the uniformly boundedness of f and g, there are subsequences still denoted by
f (s, xλ (s)) and g(s, xλ (s)), which converge weakly to f (s) and g(s). On the other hand, the
operator λ(λI + sT )– →  strongly from the assumption of Theorem . as λ → + for
all s ∈ [, T], moreover, λ(λI + sT )– < . Thus, the Lebesgue dominated convergence
theorem and the compactness of S and T yield

Exλ (T) – x̃T T –
Ex̃T +
≤ Eλ λI + 

T
+ E
(T – s)

T
+ E

(T – s)

T
+ E

λ 
φ̃(s) dw(s) – T (T) φ() – h x (T) – λ λI + sT · S (T – s) f s, xλρ(s,xλ ) – f (s) ds
α– 
s
–
(T – s)α– λ λI + sT S (T – s)f (s) ds
T
+ E
T

– λ λI + sT · S (T – s) g s, xλρ(s,xλ ) – g(s) dw(s)
α– s
–
(T – s)α– λ λI + sT S (T – s)g(s) dw(s)
–

+ Eλ λI + T T (T – tk )Ik xλ tk– → ,


as λ → .
This gives the approximate controllability of (.), which completes the proof.
4 Example
Example . As a simple application, we consider a control system governed by the fractional stochastic partial differential equation with state-dependent delay that is impulsive
and of the form
⎧
t
∂
c α
Dt x(t, z) = ∂z
⎪
 x(t, z) + μ(t, z) + –∞ a(s – t)x(s – ρ (t)ρ (x(t)), z) ds
⎪
⎪
t
⎪
⎪
,
+ [ –∞ b(s – t)x(s – ρ (t)ρ (x(t)), z) ds] dβ(t)
⎪
dt
⎪
⎪
⎨
t ∈ [, T]\{t , t , . . . , tn },
⎪
x(t, ) = x(t, π) = ,
⎪
⎪
⎪
⎪
⎪
x(τ , z) = φ(τ , z), τ ≤ , z ∈ [, π],
⎪
⎪
tk
⎩
p(tk – s)x(s, z) dz, k = , , . . . , n,
x(tk , z) = –∞
(.)
where  < t < t < · · · < tn < T are prefixed numbers, ρi : [, ∞) → [, ∞), i = , , a, b :
R → R are continuous. Let β(t) stands for a standard one-dimensional Wiener process
in H = L [, π] defined on a stochastic space (, F , P) and φ ∈ B = PC × L (g, H) (g :
(–∞, –r] → R is a positive function) be the phase space used in Hino et al. [].
Let H = U = L [, π] and define the operator A by Aω = ω with the domain D(A) =

{ω(·) ∈ L [, π], ω, ω are absolutely continuous,
ω ∈ L [, π], ω() = ω(π)}. Then Aω =
∞


sin nz is the orthogonal basis of eigenvecn= –n ω, en en , ω ∈ D(A), where en (z) =
π
tors of A. Clearly, A generates a compact analytic semigroup {S(t)}t> in H and it can be
–n t
ω, en en , ω ∈ X.
written by S(t)ω = ∞
n= e
Zhang et al. Advances in Difference Equations (2015) 2015:91
Page 11 of 12
Defining the maps ρ, f , g : J × B → H by ρ(t, ϕ)(z) = t – ρ (t)ρ (ϕ(, z)), f (t, ϕ)(z) =


tk
–∞ a(s)ϕ(s, z) ds, g(t, ϕ)(z) = –∞ b(s)ϕ(s, z) ds, Ik (x)(z) =  p(tk – s)x(s, z) dz. Taking B :
U → H by Bu(t)(z) = μ(t, z),  ≤ z ≤ π , u ∈ U, where μ : [, T] × [, π] → H is continuous.
Now, under the above preparations, we can represent the partial stochastic differential
equation (.) in the abstract form (.). Moreover, f and g are bounded linear operators,
f ≤ Mf , g ≤ Mg , and Ik ≤ MIk . All the conditions of Theorem . are fulfilled, so
we can claim that the system (.) is approximately controllable.
Remark . This paper has investigated the approximate controllability of fractional
stochastic impulsive functional differential system with state-dependent delay in Hilbert
spaces. Firstly we have constructed the control function by using the stochastic analysis
technique, it is very important for us to study the approximate controllability. On basis of
this control function, a set of sufficient conditions for the approximate controllability of
the control system has been obtained with the help of the strong continuous semigroup,
and the Krasnoselskii fixed point theorem.
The differential control systems with delay often arise in applications, the control system
with state-dependent delay considered in this paper can be come bake to the usual control
system with delay as ρ(t, xt ) = t. Therefore, our main results are the generalizations and
continuations of the recent results on the differential control systems with delay. Moreover, one can consider the neutral stochastic functional differential equations according
to the method in this paper.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Each of the authors, XZ, CZ, and CG, contributed to each part of this work equally. All authors read and approved the final
manuscript.
Author details
1
Department of Mathematics, Nanchang University, Nanchang, 330031, P.R. China. 2 Department of Mathematics,
Swansea University, Swansea, SA2 8PP, UK.
Acknowledgements
The authors would like to thank referee for valuable comments and careful reading which lead to a great improvement of
the article. This work has been supported by the National Natural Science Foundation of China (11361042, 11461045,
11071108), the TianYuan Special Funds of the National Natural Science Foundation of China (11426129, 11326099), and
the Provincial Natural Science Foundation of Jiangxi, China (20142BAB211004, 20132BAB201001, 20142BAB211020).
Received: 2 December 2014 Accepted: 10 February 2015
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