Approximate controllability of nonlinear stochastic partial differential systems with infinite delay

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R E S E A R C H

Open Access

Approximate controllability of nonlinear

stochastic partial differential systems with

inﬁnite delay

Hanwen Ning

1*

_{and Guangyan Qing}

2

*_{Correspondence:}

[email protected]

1_{School of Statistics and}

Mathematics, Zhongnan University of Economics and Law, Wuhan, 430073, P.R. China

Full list of author information is available at the end of the article

Abstract

This paper aims to investigate approximate controllability of stochastic nonlinear partial differential systems with infinite delay. In the systems under study, nonlinearity and control variable exist both in drift and diffusion terms, and controllability

problems are considered in the framework with a novel Banach space, which not only leads to some diﬃculties in deriving the properties of interest but also bring some opportunities to study the system in a more general framework. With the help of a new fundamental lemma established in this paper and some useful inequality techniques, some improved results for approximate controllability of stochastic partial diﬀerential systems are obtained by using the Banach contraction theorem without introducing any additional restraints on the terms of the system. An example of stochastic heat equation is also provided to illustrate our results.

Keywords: stochastic; partial diﬀerential systems; approximate controllability; Banach contraction theorem; inﬁnite delay

1 Introduction

Stochastic partial differential systems are usually used to describe physical and engi-neering phenomena such as heat process, population dynamics, chemical reactors, fluid dynamics,etc. and have been widely investigated (for instance, [–] and references therein). As an important concept in control theory, controllability of dynamical systems has been also investigated by many researchers. In [–], Mahmudov developed several concepts of controllability for linear stochastic differential systems and extended the clas-sical theory for deterministic dynamical systems to stochastic cases. The authors in [, ] considered weak, complete, and exact controllability of semilinear stochastic systems and stochastic functional differential in Hilbert spaces, and the obtained results therein can be applied to the stochastic systems with kinds of delays. Moreover, with the help of semigroup theory, sufficient conditions for the controllability of stochastic integrodiffer-ential systems were derived in [, ].

It should be also noted that the controllability problems can be transformed into fixed point problems. Fixed point principles such as Banach contraction theorem, Nussbaum theorem and Schauder fixed point theorem are frequently used and considered as an im-portant technique in solving the controllability problems. By employing an axiomatic defi-nition of the phase space introduced in [, ], Balasubramaniam and Ntouyas []

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tigated controllability of neutral stochastic differential inclusions with infinite delay with the aid of Leray-Schauder nonlinear alternative. Sufficient conditions for controllability of neutral functional integrodifferential infinite delay systems in Banach spaces were stud-ied in [] with the Nussbaum fixed point theorem. Bao and Jiang [] considered the approximate controllability of stochastic partial differential equations with infinite delay. Recently, in [, ], by using the Schauder fixed point theorem and the fixed point theo-rem for condensing maps due to Martelli, the authors derived some sufficient conditions for controllability of functional differential systems with infinite delay in a deterministic case.

However, most of the results for the stochastic systems mentioned above focus on either finite delays or without delays. Since many systems arising from realistic models greatly depend on histories, it is highly relevant to discuss stochastic differential systems with infinite delays, and few works are available for the controllability properties on this case. Therefore, in this paper, we study the approximate controllability of a class of nonlinear stochastic partial differential systems with infinite delays. This kind of system can be found in many engineering practices, especially those relating to continuum physics, vibration control, and thermodynamics [–]. Nonlinearity and control input exist both in the drift and diffusion terms of the underlying equation, and the control problem of interest is considered in the framework with a novel Banach space. This may not only enable the system under study to be a more general case compared to [], but also bring some diffi-culties in moment estimations and in employing the fixed point theorem. To this objective, an important lemma is established, which greatly facilitates the development of the main result of this paper. Together with the aid of some useful inequality formula adopted in the proof, improved approximate controllability results of the systems under study can be obtained without imposing any additional restraints on the system. An example is given to illustrate our results.

The rest of the paper is organized as follows. In Section , we introduce the basic nota-tions and deﬁninota-tions. In Section ., we introduce Banach spacesBhandBαh and prove an

important inequality. In Section ., we give the controllability results. In the last section, an example of stochastic heat equation is given to illustrate our results.

2 Preliminaries

In this section, we will brieﬂy give some basic assumptions and deﬁnitions.

LetKandHbe two separable Hilbert spaces. We denote by| · |and| · |Kthe norms in H andK, respectively, by·,·the scalar product inH.L(K,H) denotes the space of all bounded linear operators fromKintoH. · is also used to denote the norm in an ordi-nary Banach space. Let (,,t,P) be a complete probability space with a ﬁltration{t}

satisfying the usual condition (i.e., the ﬁltration contains all P-null sets and is right con-tinuous). Letwn(t) (n= , , , . . .) be a sequence of real-valued one-dimensional standard

Brownian motions mutually independent on (,,t,P). Set

W(t) =

∞

n=

√

σnwn(t)en, t≥,

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we denote byGits Hilbert-Schmidt norm,i.e.,

G_=trGQG∗.

We ﬁrst give an abstract phase spaceBh.

Assume that h: (–∞, ]→(,∞) is a continuous function withl=_–_∞h(s)ds<∞. Deﬁne

Bh=

φ: (–∞, ]→Hφis continuous on [–a, ] for anya> , and

 –∞

h(s) sup

s≤θ≤

φ(θ)ds<∞

.

Bhis a Banach space endowed with the norm

φBh=

 –∞h(s)

sup

s≤θ≤

φ(θ)ds for∀φ∈Bh.

The above conclusion will be proved in the next section. In the present paper, we are in-terested in the controllability problem of the following system:

⎧ ⎨ ⎩

dx(t) = [–Ax(t) +Bu(t) +f(t,xt,u(t))]dt+g(t,xt,u(t))dw(t), t∈[,T),

x(t) =ξ(t), t∈(–∞, ], ()

where –A is a closed, densely deﬁned linear operator generating an analytic semi-group S(t), t≥ on H. Let  <α < , and deﬁne the Banach space D(Aα_{) with the}

norm xα =|Aαx| for x∈ D(Aα), where D(Aα) denotes the domain of a fractional

power operator Aα _: _H _→ _H _{[]. Denote} _Hα ₌_D₍_Aα_{), deﬁne} _Bα

h ={φ : (–∞, ] → Hα|φis continuous on [–a, ] for anya> , and_–_∞h(s)sup_s≤θ≤_φ(θ)αds<∞}. LetU

be another separable Hilbert space,u(t) is aUvalued process andBis a bounded linear op-erator fromUtoH. We also assume thatf : (R+×Bα

h×U)→Handg: (R+×Bαh×U)→ L_(K,H) are two measurable mappings, whereL_(K,H) denotes the space of bounded linear operators from K to H with the Hilbert-Schmidt norm. For initial datum ξ, let ξ ∈Lp₍_,_,_P_;_Bα

h)≡Lp(,B α

h), and we always assumep> . Moreover, denote histories xt: (–∞, ]→Hbyxt(θ) =x(t+θ) for –∞<θ≤.

Deﬁnition . A stochastic processxis said to be a mild solution of () if the following conditions are satisﬁed.

() x(t,ω)is measurable as a function from[,T]×toH, andx(t)istadapted.

() E|x(t)|p_<_∞_{for each}_t_∈_(–_∞_{, ]}_.

() For eachu∈L

t(;L

p_(,_T_;_U₎₎₍_u₍_t₎_is

tadapted, andE T

 u(t)pdt<∞) the

processxsatisﬁes the following equation:

x(t) =S(t)ξ() +

t



S(t–s)Bu(s)ds+

t



S(t–s)fs,xs,u(s)ds

+

t



S(t–s)gs,xs,u(s)dw(s), t> ,

x=ξ∈Lp

,Bα h

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Remark . The dynamic system is controlled byu(t). Sometimes, control termu(t) af-fects the system not only in a linear manner, but also in a nonlinear way [, ]. Especially for stochastic diﬀerential systems,u(t) exists in both the diﬀusion and drift terms. These cases are all taken into consideration in the present study.

Deﬁnition . System () is said to be approximately controllable on [,T] if

R(T) =Lp(,,P;H),

whereR(T) ={x(T) =x(T,u) :u∈Lp_(,_T_;_U₎_}_.

Lemma .[] For any h∗∈Lp₍_,_,_P_;_H_),_{there exists}_ϕ_∈_Lp₍_;_L_(,_T_;_L

(K,H)))such

that

h∗=Eh∗+

T



ϕ(t)dw(t).

Lemma .[] Ifϕ∈L(,T;L(K,H)),Aαϕ∈L(,T;L(K,H))andϕ(t)k∈Hαfor t≥

and arbitrary k∈K,then

Aα t



ϕ(s)dw(s) =

t



Aαϕ(s)dw(s).

Lemma . Let p> ,∈Lp₍_;_L_(,_T_;_L

(K,H))),we have

E

sup

≤s≤t

_s(r)dw(r)

p

≤cp sup

≤s≤tE

_s(r)dw(r)

p

≤CpE t



(r)_dr p



, t∈[,T],

where cp= (_pp_–)pand Cp= (p_(p– ))p₍ p p–)

p  _.

Lemma .[] Let–A be the inﬁnitesimal generator of an analytic semigroup S(t).If∈

ρ(A),then

() There exist a constantM≥and a real numbera> such that

S(t)h≤Me–at|h| for allt≥andh∈H.

() There exists a constantMαsuch that the fractional power operatorAα_{satisﬁes that} AαS(t)h≤Mαe–att–α|h| for allt≥andh∈H.

() Let <α≤andh∈D(Aα₎_,_{there exists a constant}_Nα_{such that} S(t)h–h≤Nαtα_Aα_h _{for all}_t_≥__and_h_∈_H_.

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(A) ∈ρ(A).

(A) For anyη,η∈Bαh,υ,υ∈Uand≤t≤T, there exists a constantKsuch that

f(t,η,υ) –f(t,η,υ)p+g(t,η,υ) –g(t,η,υ)p_

≤K

η–ηpBα

h+υ–υ p_,

f(t,η,υ)p+g(t,η,υ)p_≤K

 +ηp_Bα h+υ

p_.

(A∗_) For anyη,η∈Bαh,υ,υ∈Uand≤t≤T, there exists a constantKsuch that

_f₍_t_,η,υ) –f(t,η,υ)

p

+g(t,η,υ) –g(t,η,υ)

p



≤K

η–ηp_Bα

h+υ–υ p_,

f(t,η,υ)

p

+g(t,η,υ)

p

≤K.

(A) For each≤s<T, the operatorλ(λI+sT)–→in the strong operator topology as

λ→+_{, where}T s =

T

s S(T–r)BB∗S∗(T–r)dris the controllability Grammian.

3 Main results

3.1 Banach spacesBhandBαh

In this section we prove thatBhandBα

h are two Banach spaces and establish an important

lemma, which will be used in the next section.

It is obvious that · Bh is a norm. Following a similar discussion in [] and [], the

following lemma can be obtained.

Lemma . For anyε> and k> ,there existsδ=δ(ε,k) > such that for anyη,η∈Bh,

ifη–ηBh≤δ,thensup–k≤τ≤|η(τ) –η(τ)| ≤ε.

Lemma . Let Xa={x∈Xais an H valued continuous function deﬁned on[–a, ],where a is a positive constant andsup_–_a≤t≤_|x(t)|<∞},then Xais a Banach space endowed with the normx=sup_–_a≤t≤_|x(t)|.

Lemma . Bhis a Banach space.

LetBT ={x∈BT|xis anHvalued continuoust adapted process deﬁned on (–∞,T], Esup__≤t≤T|x(t)|p<∞andx=Aαξ ∈Lp(,Bh)}. Heret :=for allt≤. We take a

seminorm · Tdeﬁned by

xT=

ExpBh 

p₊_E _sup

≤t≤T x(t)p

 p

.

In the following, an important lemma is established, which will play a crucial role in the development of the main results in the next section.

Lemma . Assume x∈BT,then for t∈[,T],xt∈Lp₍_,_Bh_),_and

lpEx(t)p≤ExtpBh≤

p–_lp_E _sup

≤s≤t

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Proof For anyt∈[,T], we have

In the following, we give the controllability results.

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LetB_T={x|x∈BT,x= ∈Lp(,Bh)}, and for anyx∈BT,

xT=

ExpBh 

p₊_E _sup

≤t≤T

x(t)p

 p

=E sup

≤t≤T

x(t)p

 p

.

It is obvious thatB_Tis a Banach space with the norm·T= (Esup≤t≤T|x(t)|p) 

p_{. For}_ξ₍_t_),

we deﬁne

ξ(t) =

⎧ ⎨ ⎩

Aα_ξ₍_t_), _t_∈_(–_∞_{, ],} S(t)Aα_ξ_(), _t_∈_J_,

andξ(t)∈Lp₍_,_B h). Let

Yλ(t) =Zλ(t) –ξ(t) and Y(t) =Z(t) –ξ(t),

then it can be easily concluded thatY(t)∈B

T. Deﬁne an operatorλonBT×C(J,LP(,

,P;U)) byλ(Y,u)(t) = (Yλ(t),uλ(t)) for (Y,u)∈B_T×C(J,LP(,,P;U)), whereB_T× C(J,LP₍_,_,_P_;_U_{)) is a Banach space with the norm}_·₌_Y

T+u. Then from the

above deﬁnition of (Zλ_,_uλ_{), it holds that} ⎧

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Yλ₍_t_{) =}t

A

α_S₍_t_–_s₎_Buλ₍_s₎_ds₊t

A

α_S₍_t_–_s₎_f₍_s_,_A–α₍_Ys₊_ξ

s),u(s))ds

+_tAα_S₍_t_–_s₎_g₍_s_,_A–α₍_Ys₊_ξ

s),u(s))dw(s), t∈J, Yλ₍_t_{) = ,} _t_∈_(–_∞_{, ],}

uλ₍_t_{) =}_B∗_S∗₍_T_–_t₎₍_λ_I₊T

)–(Eh–S(T)ξ())

–B∗S∗(T–t)_t(λI+T_s)–_S₍_T_–_s₎_f₍_s_,_A–α₍_Y

s+ξs),u(s))ds

–B∗S∗(T–t)_t(λI+T s)–

×[S(T–s)g(s,A–α₍_Y

s+ξs),u(s)) –ϕ(s)]dw(s), t∈J.

()

By Lemma ., we also remark that for anyt∈J,

EA–α(Yt+ξ_t)p_Bα

h =EYt+ξt p Bh

≤EYtBh+ξtBh p

≤p–EYtp_B_h+ p–Eξ_tp_B

h

≤p–EYpBh+  p–_E_ξ



p Bh+ 

p–_lp_E _sup

≤s≤t Y(s)p

+ p–lpE sup

≤s≤t ξ(s)p

≤p–

Eξp_Bα

h+l p_E _sup

≤t≤T

Y(t)p+lpE sup

≤t≤T

S(t)Aαξ()p

≤p–Eξp_Bα h+l

p_Yp

T+lpMpEA α_ξ

()p.

It can be seen that the operatorλhas a ﬁxed point onBT×C(J,Lp(,,P;U)) if and only

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Lemma . Let <α<p_–_p,and(A), (A)hold,then for anyλ> ,the operatorλmaps B_T×C(J,LP₍_,_,_P_;_U₎₎_{into B}

T×C(J,LP(,,P;U)).

Proof We just need to prove thatYλ₍_t_{) and}_uλ₍_t_{) are bounded on}_J_{and continuous in}_Lp

sense onJ. By (A), (A), the Hölder inequality and the Burkholder-Davis-Gundy

inequal-ity, we have

Euλ₍_t₎p ₌_E_B∗_S∗₍_T_–_t₎_λ_I₊T



–

Eh–S(T)ξ()

–B∗S∗(T–t)

t



λI+_sT–S(T–s)fs,A–α(Ys+ξ_s),u(s)ds

–B∗S∗(T–t)

t



λI+_sT–

×S(T–s)gs,A–α(Ys+ξ_s),u(s)–ϕ(s)dw(s)

p

≤p–B∗S∗(T–t)λI+_T–Ehp

+ p–EB∗S∗(T–t)λI+_T–S(T)ξ()p + p–EB∗S∗(T–t)

t



λI+_sT–S(T–s)fs,A–α(Ys+ξ_s),u(s)ds p

+ p–EB∗S∗(T–t)

t



λI+_sT–S(T–s) ×gs,A–α(Ys+ξ_s),u(s)dw(s)

p

+ p–EB∗S∗(T–t)

t



λI+_sT–S(T–s)ϕ(s)dw(s)

p

=I+I+I+I+I.

For each one

I= p–B∗S∗(T–t)

λI+_T–Ehp≤p–BpMp 

λp|Eh| p_,

I= p–EB∗S∗(T–t)

λI+T_–S(T)ξ()p≤p–BpMp 

λpEξ() p

,

I= p–E

B∗S∗(T–t)

t



λI+T_s–S(T–s)fs,A–α(Ys+ξ_s),u(s)ds p

≤p–_Bp_Mp 

λpT p–_E t



fs,A–α₍_Y

s+ξs),u(s) p

ds

≤p–BpMp 

λpT p– t



K

 +EA–α(Ys+ξ_s)p_Bα h+E

u(s)pds

≤p–_Bp_Mp 

λpT p_K



 + p–_E_ξp Bα_h+l

p_Yp

T+lpMpEA

α_ξ_()p₊_up_,

I= p–E

B∗S∗(T–t)

t



λI+T_s–S(T–s)gs,A–α₍_Y

s+ξs),u(s)

dw(s)

p

≤p–BpMpCpE t



λI+_sT–S(T–s)gs,A–α₍_Y

s+ξs),u(s)

 ds

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≤p–_Cp_Bp_Mp 

λp(t–t) p –_E

T



ϕ(s)_ds p



+ p–_C

pBpMp

 λpT

p –_E

T



ϕ(s)_ds p



S∗(t–t) –I

p

.

From the above equations, we can obtain that E|uλ₍_t

) –uλ(t)|p→ ast→t. Then

uλ₍_t_{) is continuous in}_Lp _{sense on}_J_and_uλ₍_t₎_∈_C₍_J_,_Lp₍_,_,_P_;_U_{)). Moreover, for}_Yλ₍_t₎

andt≥, we have

E sup

≤s≤t

Yλ₍_s₎p ₌_E _sup

≤s≤t

_sAα_S₍_s_–_r₎_Buλ₍_r₎_dr

+

s



AαS(s–r)fr,A–α(Yr+ξ_r),u(r)dr

+

s



AαS(s–r)gr,A–α(Yr+ξ_r),u(r)dw(r)

p

≤p–E sup

≤s≤t

_sAαS(s–r)Buλ(r)dr p

+ p–E sup

≤s≤t

_sAαS(s–r)fr,A–α(Yr+ξ_r),u(r)dr p

+ p–E sup

≤s≤t s



AαS(s–r)gr,A–α(Yr+ξ_r),u(r)dw(r)

p

≤I+I+I.

Let _p+_q= . From Lemma ., it can be obtained that

I= p–E sup ≤s≤t

s



AαS(s–r)Buλ(r)dr p

≤p–E sup

≤s≤t s



AαS(s–r)Buλ(r)dr p

≤p–_E _sup ≤s≤t

s



Mαe–a(s–r)₍_s_–_r₎–α_Buλ₍_r₎_dr p

≤p–BpMpαE sup

≤s≤t s



(s–r)–αqds p

q s



uλ(r)pdr

= p–BpMp_αE sup

≤s≤t

s–αq

 –αq p

q s



uλ₍_r₎p_dr

= p–BpMpαE

t–αq

 –αq p

q t



uλ(s)pds

≤p–BpMpα

T–αq

 –αq p

q

Tuλp,

I= p–E sup ≤s≤t

_sAαS(s–r)fr,A–α(Yr+ξ_r),u(r)dr p

≤p–E sup

≤s≤t s



(12)

(13)

(14)

(15)

+ p–CpMpαN p

α(t–t)pα

p– p

(T_)–pα–p

 –_pαp_–

p– p

×TK

 + p–Eξp_Bα h+l

p_Yp

T+lpMpEA α_ξ

()p+up.

Then it is clear thatE|Yλ₍_t

) –Yλ(t)|p→ ast→t, thereforeYλ(t) is continuous inLp

sense.

From all inequalities above, we conclude that the operatorλmapsB_T×C(J,LP(,

,P;U)) intoB_T×C(J,LP(,,P;U)). The proof is completed.

Lemma . Let <α<p_–_p,and(A), (A)and Lp with p lower case hold,then for any

λ> ,the operatorλhas a unique ﬁxed point in B_T×C(J,LP(,,P;U)).

Proof The proof is based on the Banach ﬁxed point theorem for contractions. In the proof, we takeλ(Y,u)(t) = (Yλ(t),uλ(t)) andλ(Y,u)(t) = (Yλ(t),uλ(t)). Then we have

Euλ_(t) –uλ_(t)p =EB∗S∗(T–t)

t



λI+_sT–S(T–s)fs,A–α(Ys+ξs),u(s)

–fs,A–α(Ys+ξs),u(s)

ds

+B∗S∗(T–t)

t



λI+_sT–S(T–s)gs,A–α(Ys+ξs),u(s)

–gs,A–α₍_Y

s+ξs),u(s)

dw(s)

p

≤p–EB∗S∗(T–t)

t



λI+T_s–S(T–s)fs,A–α(Ys+ξs),u(s)

ds

p

+ p–_E_B∗_S∗₍_T_–_t₎ t



λI+T_s–S(T–s)gs,A–α₍_Y

s+ξs),u(s)

–gs,A–α(Ys+ξs),u(s)

dw(s)

p

≤p–BpMp 

λpT p–_E t



fs,A–α(Ys+ξs),u(s)

p ds

+ p–_Bp_Mp 

λpT p –_E

t



_g

s,A–α(Ys+ξs),u(s)

–gs,A–α(Ys+ξs),u(s)p_ds

≤p–BpMp 

λpT p–_K



t



EA–α(Ys+ξs) –A–α(Ys+ξs) p Bα_h

+Eu(s) –u(s)

p

ds+ p–BpMp 

λpT p –

×K

t



EA–α(Ys+ξs) –A–α(Ys+ξs) p

Bα_h+Eu(s) –u(s) p

(16)

≤p–BpMp 

which yields that there exists a constantKsuch that

(17)

(18)

(19)

(20)

≤p–

E sup

≤t≤T

Y_nλ(t) –Y_nλ(t)p+ sup

≤t≤T

Eunλ_ (t) –unλ_ (t)p

≤LY–YpT+u–up

≤LY–YT+u–u

p

,

which implies

n_λ(Y,u) –nλ(Y,u)≤L

 p_Y

–Y+u–u

<Y–Y+u–u

=(Y,u) – (Y,u).

This shows that for large enoughn,n_λis a contraction. Thus,λhas a unique ﬁxed point

inB_T×C(J,Lp₍_,_,_P_,_U_{)). We complete the proof.} Remark . The general Banach spaceB

T×C(J,Lp(,,P;U)) and the operatorλare

constructed to transform the controllability problem into a fixed point problem. It shall be emphasized that the fundamental Lemma . is purposely established, which plays an important role in obtaining the boundedness of the operator and in solving the problems brought by the infinite delays. Although the control input and nonlinearity exist both in the drift and diffusion terms of the system, which makes it complicated to prove the con-traction property of the operator, some inequality techniques are deliberately adopted as demonstrated in the proof of Lemma ., which can effectively overcome the difficulty. With these techniques, the main conclusion will be established in a more general space with less restrictive conditions.

For any λ> , let (Yλ_,_uλ_{) be the ﬁxed point of}

λ, Zλ(t) =Yλ(t) +ξ(t) andxλ(t) = A–α_Zλ₍_t_{). Then (}_Zλ₍_t_),_uλ₍_t_{)) is the unique ﬁxed point of the operator}

λ, andxλ(t) is a

solution of system (). We also have

xλ(t) =S(t)ξ() +_tS∗(T–t)λI+T_–Eh∗–S(T)ξ() +

t



I–_stS∗(T–t)λI+_sT–S(T–t)S(T–s)fs,xλ_s,uλ(s)ds

+

t



I–_stS∗(T–t)λI+_sT–S(T–t)S(T–s)gs,xλ_s,uλ(s)dw(s)

+

t



_stS∗(T–t)λI+_sT–ϕ(s)dw(s), t∈J,

xλ(t) =ξ(t), t≤.

Lettingt=Tin the equation above, it can be obtained that

xλ(T) =h∗–λλI+T_–Eh–S(T)ξ() –λ

T



λI+_sT–S(T–s)fs,xλ_s,uλ(s)ds

–λ

T



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Theorem . Under hypotheses(A), (A∗),and(A),system()is approximately

control-lable on[,T].

Proof By (A∗_),|f(t,η,u)|p+g(t,η,u)p ≤K in [,T]×for anyη∈Bα_h andu∈

U. Then there exists a subsequence, still denoted by{f(s,xλ

s,uλ(s)),g(s,xλs,uλ(s))}, weakly

converging to some{f(s),g(s)}inH×L

(K,H). By (A), we also haveλ(λI+Ts)– ≤

for ≤s≤T[]. From the compactness ofS(t), we obtain that, in [,T]×,

S(T–s)fs,xλ_s,uλ(s)→S(T–s)f(s),

S(T–s)gs,xλ_s,uλ(s)→S(T–s)g(s).

Meanwhile

Exλ(T) –h∗p

≤p–λλI+_T–Eh–S(T)ξ()p

+ p–E T



λλI+_T–S(T–s)fs,xλ_s,uλ(s)–f(s)ds p

+ p–E T



λλI+_T–S(T–s)f(s)ds p

+ p–E T



λλI+_T–S(T–s)gs,xλ s,uλ(s)

–g(s)_ds p



+ p–E T



λλI+_T–S(T–s)g(s)_ds p



+ p–E T



λλI+_T–ϕ(s)_ds p



.

By the dominated convergence theorem and (A) (i.e.,λ(λI+Ts)–→ in strong

oper-ator topology asλ→+_),_E_|_xλ₍_T_{) –}_h∗_|p_→_{ as}_λ_→_+_{, which implies the approximate}

controllability of system (). The proof is completed.

Remark . It shall be noted that a similar but rather preliminary result has been devel-oped in [], where the initial datum of the systemξ∈Lp₍_,_Ch

α) and the delays are

consid-ered in the spaceLp₍_,_Ch

α), whereCαh={x∈C(R–,Hα)| 

–∞h(s)sups≤τ≤x(τ)

p

αds<∞}.

Since

 –∞h(s)

sup

s≤τ≤

x(τ)_αds=

 –∞h(s)



q_h₍_s₎p _sup s≤τ≤

x(τ)_αds

≤ 

–∞h(s)ds 

q 

–∞h(s)

sup

s≤τ≤

x(τ)p_αds 

p

=lq

 –∞h(s)

sup

s≤τ≤

x(τ)p_αds 

p

,

it is clear thatBα

h⊃Chα. That is, a more general space is studied in this paper. Moreover,

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 <α< p_–_p. Clearly, the result in [] could not be well applied to the cases which are studied in this paper.

4 An example

Heat equation describes the ﬂow of heat by conduction through a stationary homoge-neous, isotropic material. As an application of our results, we consider a heat equation system described by the following stochastic partial diﬀerential equation:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

dv(t,x) = (–_∂x∂v(t,x) +b(x)u(t) + t

–∞e(s–t)v(s,x)ds)dt

+ (_–t_∞e(s–t)_v₍_s_,_x₎_ds₊_log_{( +}_|_u₍_t₎_|₎₎_dw₍_t_),

t∈J= [,T], ≤x≤π,

v(t, ) =v(t,π) = , t≥,

v(s,x) =ξ(s,x)∈Lp₍_,_Bα

h), s∈(–∞, ], ≤x≤π.

()

Herev(t,x) denotes the temperature at timet.u(t) is the control term to enable the system temperature achieve the target value approximately for a given timeT. LetH=L_(,_π₎

be endowed with the usual norm · L,h(s) =es, 

∞h(s)ds=. Note that there exists a

complete orthonormal set{en},n≥, of eigenvectors ofAwithen(x) =



π. The analytic

semigroupS(t),t≥, is generated byAsuch that

Aρ=

∞

n=

nρ,enen, ρ∈D(A),

S(t)ρ=

∞

n=

e–ntρ,enen, ρ∈H.

We deﬁneAα_(actually_|_A_|α_{) for the self-adjoint operator}_A_{by the classical spectral}

theo-rem, and it can be obtained that

|A|α_e–At_ρ₌ ∞

n=

nαe–ntρ,enen,

which yields

_Aα

e–Atρ=

∞

n=

nαe–ntρ,en

=

∞

n=

ntαe–(n–a)tρ,en.

On the other hand, for anyζ,ζandζ∈Hα, we have

ζ_L = π



ζ(x)dx

=

∞

n=

(23)

≤

∞

n=

nαζ,en

=Aα_ζ₌_ζ

α,

and

ζ–ζL = π



ζ(x) –ζ(x) 

dx

=

∞

n=

ζ–ζ,en

≤

∞

n=

nα_ζ

–ζ,en

=Aαζ=ζ–ζα.

For anyφ∈Bh,φ(s)(x) =φ(s,x), (s,x)∈(–∞, ]×[,π], it is clear thatf(t,φ) =g(t,φ) = 

–∞esφ(s)(x)ds, and forφ,ψ∈Bh, f(t,φ) –f(t,ψ)_L =

π

  –∞

es_φ₍_s₎₍_x_{) –}_ψ₍_s₎₍_x₎_ds 

dx 



≤ 

–∞e

s_ds 

 π

  –∞e

s_φ₍_s₎₍_x_{) –}_ψ₍_s₎₍_x₎_{ds dx} 



=  

 –∞

es π



φ(s)(x) –ψ(s)(x)dx ds 



=  

 –∞

esφ(s) –ψ(s)_L 



≤ 



 –∞e

s_sup s≤τ≤

φ(s) –ψ(s)_L 

≤ 

–∞

es sup

s≤τ≤

φ(s) –ψ(s)_Lds

= 

φ–ψBαh.

Then the functionsf(t,φ) andg(t,φ) are globally Lipschitz continuous inφ∈Bα_hand uni-formly bounded. On the other hand, it is known that the deterministic linear system cor-responding to () is approximately controllable on every [,t],t> , provided that for

n= , , . . . ,

π



b(x)en(x)dx= .

By Theorem ., we can ensure that system () is approximately controllable on [,T].

Remark . In system (), considering the initial datumξ, if letξ(s)α=e– 

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5 Conclusion

In this paper, improved approximate controllability results of a class of stochastic partial differential systems with infinite delays are obtained in a general case by using a fixed point theorem, stochastic analysis techniques and an important lemma established, which fills a gap of the research area of control theory for stochastic functional partial differential sys-tems. Moreover, intuitively, under more hypothesis, the controllability results established in this study could be also extended to the case of neutral type, which are frequently used to characterize some Burgers equations, vibration equations, Navier-Stokes equations,etc.

[]. Thus, it is an important and interesting topic to further study controllability prob-lems of the neutral stochastic partial diﬀerential systems, and the probprob-lems will be focused on in our future studies.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally in writing this article. All authors read and approved the ﬁnal manuscript.

Author details

1_{School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan, 430073, P.R. China.}2_Key

Laboratory of Advanced Technology for Materials Synthesis and Processing, Wuhan University of Technology, Wuhan, P.R. China.

Acknowledgements

The authors sincerely thank the reviewers for their valuable comments and the editor for kind help on this paper. This research was supported by the National Natural Science Foundation of China (Grant No. 11301544).

Received: 3 November 2014 Accepted: 3 March 2015 References

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