Automatica ( ) –

Contents lists available at SciVerse ScienceDirect

Automatica

journal homepage: www.elsevier.com/locate/automatica

Brief paper

Controllability of linear impulsive stochastic systems inHilbert spaces

Lijuan Shen a,b, Jitao Sun a,1, Qidi Wu b

a Department of Mathematics, Tongji University, Shanghai 200092, Chinab Department of Control Science and Engineering, Tongji University, Shanghai 201804, China

a r t i c l e i n f o

Article history:Received 16 December 2011Received in revised form14 August 2012Accepted 2 November 2012Available online xxxx

Keywords:Impulsive stochastic systemsNull controllabilityQuasi-backward stochastic systemsAdjoint systems

a b s t r a c t

This paper is concerned with the controllability of linear impulsive stochastic systems (LISSs) in Hilbertspaces. For this class of systems, the concepts of null controllability and approximate null controllabilityare introduced. To overcome the difficulties related, we construct the adjoint systems and the quasi-backward stochastic systems of LISSs. Necessary and sufficient conditions for the null controllability andthe approximate null controllability are developed by utilizing our introduced quasi-backward stochasticsystems. Furthermore, an equivalence is established between the null controllability of LISSs and someinitial state of quasi-backward stochastic systems.

© 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Control theory is an area of application-oriented mathemat-ics which deals with basic principles underlying the analysisand design of control systems. Conceived by Kalman, the con-trollability concept has been studied extensively in the fields offinite-dimensional systems, infinite-dimensional systems, hybridsystems, and behavioral systems. Onemay refer, for instance, to Hoand Niu (2007), Klamka (1991), Sontag (1998) and the referencestherein.

Motivated by the fact that impulsive systems provide a naturalframework for mathematical modeling of biology, economics,electronics and telecommunications, their study has receivedconsiderable attention (Benzaid& Sznaier, 1994; Chen&Sun, 2006;George, Nandakumaran, & Arapostathis, 2000; Guan, Qian, & Yu,2002; Lakshmikantham, Bainov, & Simeonov, 1989; Leela, McRae,& Sivasundaram, 1993; Li, Sun, & Sun, 2010; Liu, Liu, & Xie, 2011;Shen & Sun, 2012; Xie & Wang, 2005). Benzaid and Sznaier (1994)studied the null controllability of linear impulsive systems withthe control only acting on the discontinuous points. George et al.

Thiswork is supported by the NNSF of China under Grants 61174039, 61034004and 11201215, and China Postdoctoral Science Foundation (No. 2012M520928).The material in this paper was not presented at any conference. This paper wasrecommended for publication in revised form by Associate Editor Akira Kojimaunder the direction of Editor Ian R. Petersen.

E-mail addresses: lijuan2425@163.com (L. Shen), sunjt@sh163.net (J. Sun),wuqidi@moe.edu.cn (Q. Wu).1 Tel.: +86 21 65983241x1307; fax: +86 21 65981985.

(2000)modified some results in Leela et al. (1993) and investigatedthe complete controllability of linear impulsive systems and itsperturbed systems. Recently, Guan et al. (2002) and Xie and Wang(2005) obtained some good results for the controllability of linearimpulsive systems. In Guan et al. (2002), sufficient conditions aretermed as the rank of matrices. And main results in Xie and Wang(2005) are established based on the fact that the reachable setcan be expressed as the combination of the minimal invariantsubspaces.

In the stochastic framework, many efficient tools dealing withcontrollability have already been developed; see, for example, Pi-card type iteration (Balachandran, Karthikeyan, & Kim, 2007), con-traction mapping principle (Sakthivel, Mahmudov, & Lee, 2009),and Lyapunov approach (Zhao, 2008), for nonlinear stochastic sys-tems. The tools for stochastic linear systems, however, are rela-tively a few. For example, Klamka (2007, 2008a,b) described thecontrollability using the algebraic condition similar to those of de-terministic systems. The study of backward stochastic differentialequations (BSDEs), in the linear case, can be traced back to Ben-soussan (1983) and Bismut (1978). But the first well-posedness re-sult for nonlinear BSDEs was proved by Pardoux and Peng (1990)and ever since this paper, BSDEs have been one of the useful toolsin the control theory. Peng (1994) firstly defined the exact control-lability of stochastic control systems from the viewpoint of BSDEs.By using BSDEs and Riccati equations, Sirbu and Tessitore (2001)was concerned with the exact null controllability of infinite di-mensional linear differential equations. Buckdahn, Quincampoix,and Tessitore (2006) studied the approximate controllability in fi-nite dimensional spaces, which was improved by Goreac (2007)when the control also acted on the noise. Recently, the results in

0005-1098/$ – see front matter© 2013 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2013.01.036

2 L. Shen et al. / Automatica ( ) –

Buckdahn et al. (2006) and Goreac (2007) were generalized byGoreac (2009) from finite dimensional spaces to infinite dimen-sional cases.

There exist some results on the controllability about nonlinearimpulsive stochastic systems by using various fixed point theo-rems. More details can be seen in Sakthivel et al. (2009) and Shen,Shi, and Sun (2010) for complete controllability and Shen and Sun(2011) for weaker approximate controllability. Generally speak-ing, the controllability of corresponding linear systems is a basicand important assumption in investigating the controllability ofnonlinear systems; see, e.g., Sakthivel et al. (2009), Shen and Sun(2011) and Shen et al. (2010). However, to the best of our knowl-edge, there exist very few results about the controllability of LISSs.Therefore, the study of controllability of LISSs in Hilbert spaces ismeaningful and challenging.

In this paper, we study the null controllability and theapproximate null controllability problem for LISSs. The LISSsstudied in this paper are more complicated than the existingsystems and have not only impulse effects but also noise terms.In the system analysis, these two factors should be consideredat the same time. Meanwhile, we emphasize that in this paper,the control is allowed to act both on the continuous terms anddiscontinuous points.

It should be pointed out that, due to the noise term, the existingmethods of linear deterministic impulsive systems, such as thosein Guan et al. (2002) and Xie and Wang (2005), are not availablefor LISSs. On the other hand, BSDEs coupled with the dual methodcommonly used for stochastic linear systems are not suitable forLISSs either as a result of the impulse effect. From the analysis givenabove, in the case that the conventional methods cannot apply,a new system should be introduced to study the controllabilityof LISSs. And its relationship with the state of LISSs should beestablished to get better results on the controllability of LISSs.

In this paper, we will construct the quasi-backward stochasticsystems of LISSs. Its relationship with the state of LISSs is thendescribed by a useful generalized Itô lemma. Then with the aidof quasi-backward stochastic systems, sufficient conditions andproperties of the controllability of LISSs are provided in Hilbertspaces. In particular, an equivalence will be established betweenthe null controllability of LISSs and the existence of some initialvalue of the quasi-backward stochastic systems.

The organization of the paper is as follows: In Section 2, weconstruct the adjoint systems and quasi-backward systems of LISSsand prove the existence of quasi-backward stochastic systems.In addition, a useful generalized Itô lemma will be obtained,which is a key tool for the subsequent stochastic dual analysis. InSection 3, some necessary and sufficient conditions for the nullcontrollability and the approximate null controllability will beproposed by investigating the quasi-backward stochastic systems.In Section 4 we provide our conclusions.

2. Preliminaries

Throughout this paper, unless otherwise specified, we will em-ploy the following notations. Let Ω, F , P be a complete prob-ability space with a filtration Ftt≥0 of wt satisfying the usualconditions (i.e. right continuous and F0 containing all P-null sets)and E(·) be the expectation operator with respect to the probabil-ity measure P. The state space E (with norm ∥ ·∥ and product ⟨·, ·⟩)as well as the control state space U are separable Hilbert spaces.PC(J, E) = y(t)| the function y(t) from J = [0, T ] into E is con-tinuous everywhere except finite points τk, at which, y(τ+

k ) and

y(τ−

k ) exist with y(τ−

k ) = y(τk). Let LFt2

Ω, PC(J, E)

denote

the space of all Ft-adapted square integrable processes x from Ω

into PC(J, E). DenoteL(U, E) as the space of all bounded linear op-erators from U to E. We also denote by L2(Ω, Ft , E) the space of

all Ft-measurable square integrable random variables with valuesin E. If A is an operator, its adjoint operator is denoted by A∗.

In this paper, we focus on the following linear impulsivestochastic systems:

dx(t) = (Ax(t) + Bu(t))dt + Cx(t)dw(t), (1a)1x(τk) = Ikx(τk) + Dkv(τk), k ∈ Jm = 1, 2, . . . ,m, (1b)x(0) = x0, (1c)

where x0 ∈ L2(Ω, F0, E), u, v ∈ U , w is a standard Wienerprocess valued in W , and A : D(A) → E is the infinitesimalgenerator of a C0-semigroup Φ(t), t ≥ 0 in E. Furthermore, Ik ∈

L(E, E), B,Dk ∈ L(U, E), C ∈ L(E, L(W , E)) are both boundedlinear operators. It is clear by Da Prato and Zabczyk (1992) and theinduction method that system (1) admits a unique mild solutionx(t, x0, u, vkk∈Jm) ∈ L

Ft2

Ω, PC(J, E)

, t ∈ J , for any x0 ∈

L2(Ω, F0, E), u, vk ∈ U , which can be described as

x(t, x0, u, vkk∈Jm) = Φ(t)x0 +

t

0Φ(t − s)Bu(s)ds

+

0<τk<t

Φ(t − τk)(Ikx(τk) + Dkv(τk))

+

t

0Φ(t − s)Cx(s)dw(s).

To study system (1), we first consider its adjoint system asfollows:

dx(t) = (Ax(t) + Bu(t))dt + Cx(t)dw(t), (2a)

1x(τk) = Ikx(τk), k ∈ Jm, (2b)

x(0) = x0, (2c)

where I is the identity operator, I+Ik is invertible, and so is (I+Ik)∗.

Remark 1. The main difference between (1) and (2) lies in theimpulsive functions. Andwith this choice of Ik, the generalized Itô’slemma (Lemma 2, introduced later) depends only on the controlu(t) and v(τk), k ∈ Jm.

Based on (2), the following system is introduced and will play animportant role in obtaining the main results:

dy(t) = (−A∗y(t) − C∗z(t))dt + z(t)dw(t), (3a)

1y(τm−(k−1)) = −I∗m−(k−1)y(τ+

m−(k−1)), k ∈ Jm, (3b)

y(T ) = x0. (3c)

It is obvious that (3a) is, in fact, the backward stochastic system of(1a) and (2a). And here we call (3) the quasi-backward stochasticsystem of (1). y(t) in (3) may be interpreted as an evolutionprocess of the fair price, whereas the stochastic process z(t) maybe interpreted as the related consumption and portfolio process.

For convenience denote by ϖ the control in (1) ϖ = (u(t),vkk∈Jm) ∈ K , and the control operator B = (B, Dkk∈Jm) ∈

L(K) with Bϖ = (Bu(t), Dkvkk∈Jm). The space K of ϖ is aHilbert space with respect to the inner product ⟨·, ·⟩K definedas ⟨ϖ1, ϖ2⟩K =

T0 ⟨u1(t), u2(t)⟩Edt +

mk=1⟨v1k, v2k⟩E , for all

ϖ1, ϖ2 ∈ K .

Lemma 1. System (3) admits a unique mild solution (y(t), z(t)) forany x0 ∈ E.

Proof. Proceeding exactly as in Tessitore (1996), we get that dy =

(−A∗y(t) − C∗z(t))dt + z(t)dw(t) has a unique mild solution(y(t), z(t)) for t ∈ (τm−1, T ]. At t = τm−1, (y(τ+

m−1), z(τ+

m−1))was transferred by impulse into (y(τm−1), z(τm−1)) = ((I +

I∗m−1)y(τ+

m−1), z(τ+

m−1)). From the inductionmethod, it follows thatsystem (3) admits a unique mild solution (y(t), z(t)) on [0, T ].

L. Shen et al. / Automatica ( ) – 3

Remark 2. When the operator C is unbounded, the situation willbemore complex.We do not pursue this issue any further. And thereader may refer to Goreac (2009) for some general results in thecase of Ik = Dk = 0 with unbounded operator C .

Motivated by the definitions in Goreac (2007) and Sirbu andTessitore (2001), we introduce the following definitions.

Definition 1. For T > 0, system (1) is null controllable at T if foreach x0 ∈ L2(Ω, F0, E), there exists some ϖ ∈ K such thatx(T , x0, ϖ) = 0, P-a.s. System (1) is said to be approximatelycontrollable at T if for each x0 ∈ L2(Ω, F0, E), there exists someϖ ∈ K such that x(T , x0, ϖ),ϖ ∈ K = L2(Ω, FT , E), P-a.s.

Similarly, system (1) is approximately null controllable at T iffor each x0 ∈ L2(Ω, F0, E), there exists some ϖ ∈ K such thatx(T , x0, ϖ) can be arbitrarily close to 0, P-a.s.

We now introduce the following lemma and this lemma concern-ing the generalized Itô lemma characterizes the relationship be-tween solutions of (1), (3) and the control operator.

Lemma 2. Let x(t), (y(t), z(t)) be the solutions of (1) and (3),respectively, then

E⟨x(T ), y(T )⟩ − E⟨x(0), y(0)⟩

= E T

0⟨Bu(s), y(s)⟩ds

+ Em

k=1

⟨Dkv(τk), (I − F∗

k )y(τk)⟩, (4)

with Fk = Ik(I + Ik)−1.

Proof. Consider the dynamics of ⟨x(t), y(t)⟩, t = τk, then fromstochastic calculus we have

d⟨x(t), y(t)⟩ = ⟨Ax + Bu, y(t)⟩dt + ⟨Cx(t), y(t)⟩dw(t)+ ⟨Cx(t), z(t)⟩dt + ⟨x(t), z(t)⟩dw(t)+ ⟨x(t), −A∗y(t) − C∗z(t)⟩dt

= ⟨Bu(t), y(t)⟩dt + ⟨x(t), z(t)⟩dw(t)+ ⟨Cx(t), y(t)⟩dw(t),

and for the impulse time τk, k ∈ Jm, it is not hard to check from (3b)that y(τ+

k ) = (I+ I∗k )−1y(τk), and1y(τk) = ((I+ I∗k )

−1− I)y(τk) =

−(I + I∗k )−1I∗k y(τk), based on which, we get

∆⟨x(t), y(t)⟩|t=τk = ⟨x(τ+

k ), y(τ+

k )⟩ − ⟨x(τk), y(τk)⟩

= ⟨Dkv(τk), (I − F∗

k )y(τk)⟩. (5)

By the information of differential equation and the property ofItô formula, we can deduce

E⟨x(T ), y(T )⟩ − E⟨x(0), y(0)⟩

=

T

0Ed⟨x(s), y(s)⟩ + E

mk=1

∆⟨x(t), y(t)⟩|t=τk

= E T

0⟨Bu(s), y(s)⟩ds + E

mk=1

⟨Dkv(τk), (I − F∗

k )y(τk)⟩.

And that completes the proof.

3. Controllability

For fixed T > 0 define two bounded linear operators LT : E →

L2(Ω, FT , E) and MT : K → L2(Ω, FT , E) by LT x0 = x(T , x0, 0)and MTϖ = x(T , 0, ϖ).

Motivated by Sirbu and Tessitore (2001), the null controllabilitycan be characterized in terms of the state of quasi-backwardstochastic systems (3).

Theorem 1. System (1) is null controllable if and only if there existsa positive constant c such that

∥y(0)∥2≤ c

T

0∥B∗y(s)∥2ds +

mk=1

∥D∗

k(I − F∗

k )y(τk)∥2

. (6)

Proof. It is easy to see that the operators LT andMT satisfy

LT x0 = Φ(t)x0 +

t

0Φ(t − s)Cx(s, x0, 0)dw(s)

+

0<τk<t

Φ(t − τk)Ikx(τk, x0, 0),

MTϖ =

t

0Φ(t − s)Bu(s)ds

+

t

0Φ(t − s)Cx(s, 0, ϖ)dw(s)

+

0<τk<t

Φ(t − τk)(Ikx(τk, 0, ϖ) + Dkv(τk)).

Since A, B, C,Dk are linear operators, we can conclude x(T , x0, ϖ)= LT x0+MTϖ . Therefore, that (1) is null controllable is equivalentto Da Prato and Zabczyk (1992) Im(LT ) ⊂ Im(MT ), i.e., there existsa constant c > 0 such that

∥L∗

T x0∥2E ≤ c∥M∗

T x0∥2K . (7)

To verify the conclusion, let us show the form of L∗

T andM∗

T . Let-ting x0 = 0 in (4) yieldsE⟨ϖ,M∗

T x0⟩K = E T0 ⟨u(s), B∗y(s)⟩Eds+Em

k=1⟨v(τk),D∗

k(I − F∗

k )y(τk)⟩E , which implies,

M∗

T x0 = (B∗y(·), D∗

k(I − F∗

k )y(·)k∈Jm) ∈ K, P-a.s. (8)

Similarly, taking ϖ = 0 in (4), we will obtain

L∗

T x0 = y(0). (9)

Substituting (9) and (8) into (7), it follows ∥y(0)∥2≤ c(

T0 ∥B∗y

(s)∥2ds +m

k=1 ∥D∗

k(I − F∗

k )y(τk)∥2).

Example 1. Consider an example in the form of (1). Define Fk as(Fkz)(t) = z(t)− z(t + τk), then I − Fk, k = 1, 2, . . . ,m is the shiftoperator and has the adjoint I − F∗

k given by

((I − F∗

k )z)(t) =

z(t − (τk − τk−1)), t ≥ τk,0, 0 ≤ t < τk.

(10)

Let B = Dk be the orthogonal projection

(Dkz)(t) = (Bz)(t) =

0, 0 ≤ t < t0,z(t), t > t0,

(11)

where t0 is a fixed, positive constant. So if t < t0, we have t0 ∥B∗y(s)∥2ds+

0<τk<t ∥D

∗

k(I − F∗

k )y(τk)∥2= 0 and this system

is not null controllable on [0, t] by Theorem 1.

Theorem 2. Let (y(t), z(t)) denote the solution of (3).(i) System (1) is approximately controllable at T if and only if for

every (y(t), z(t)) such that (B∗y(t), D∗

k(I − F∗

k )y(τk)k∈Jm) = 0we have (y(t), z(t)) = 0, t ∈ [0, T ], P-a.s.

(ii) System (1) is approximately null controllable at T if and only iffor every y(t) such that (B∗y(t), D∗

k(I − F∗

k )y(τk)k∈Jm) = 0 wehave y(0) = 0, t ∈ [0, T ], P-a.s.

Proof. (i) That system (1) is approximately controllable is equiva-lent to ImMT is dense in L2(Ω, FT , E). That means, the kernel ofM∗

T , denoted by KerM∗

T , is trivial in L2(Ω, FT , E). Or equivalently,M∗

T x0 = 0 can deduce x0 = 0. By Lemma 1, x0 = 0 if and only if(y(t), z(t)) = 0, t ∈ [0, T ], P-a.s.

4 L. Shen et al. / Automatica ( ) –

(ii) System (1) is approximately null controllable if and only ifIm(LT ) ⊂ Im(MT ), or, equivalently, kerM∗

T ⊂ ker L∗

T .That is, every solution (y(t), z(t)) of (3) such that (B∗y(t),

D∗

k(I − F∗

k )y(τk)k∈Jm) = 0 will satisfy y(0) = 0. And that com-pletes the proof.

Example 2. Consider the same system introduced in Example 1.By Theorem 2, this system is approximately controllable at Tif and only if for every (y(t), z(t)) such that (B∗y(t), D∗

k(I −

F∗

k )y(τk)k∈Jm) = 0 we have (y(t), z(t)) = 0, t ∈ [0, T ], P-a.s. So ift < t0, we have (B∗y(t), D∗

k(I − F∗

k )y(τk)k∈Jm) = 0 for arbitrary(y(t), z(t)) and this system is not approximately controllable on[0, t]P-a.s. However, this system is approximately controllable on[0, t] for any t > τm, since (B∗y(t), D∗

k(I − F∗

k )y(τk)k∈Jm) =

(y(t), y(τk−1)k∈Jm) = 0.

The following results provide a necessary condition of the nullcontrollability of (1).

Lemma 3. Let x(t) be a solution of (1) with initial value x0 and(y(t), z(t)) be a solution of (3). If system (1) is null controllable atT , then there exists some positive constant c2 such that

∥ − E⟨x(0), y(0)⟩∥

≤ c2E

T

0∥B∗y(s)∥2ds +

mk=1

∥D∗

k(I − F∗

k )y(τk)∥2

12

. (12)

Proof. If system (1) is null controllable at T , by Definition 1 thereexists a ϖ ∈ K such that x(T , x0, ϖ) = 0, P-a.s. Then FromLemma 2 we have

−E⟨x(0), y(0)⟩ = E T

0⟨Bu(s), y(s)⟩ds

+ Em

k=1

⟨Dkv(τk), (I − F∗

k )y(τk)⟩,

and by the Cauchy–Schwarz inequality, we obtain

∥ − E⟨x(0), y(0)⟩∥

≤ E

T

0∥B∗y(s)∥2ds

T

0∥u(s)∥2ds

12

+ E

m

k=1

∥v(τk)∥2

mk=1

∥D∗

k(I − F∗

k )y(τk)∥2

12

≤ E

T

0∥B∗y(s)∥2ds +

mk=1

∥D∗

k(I − F∗

k )y(τk)∥2

12

×

m

k=1

∥v(τk)∥2+

T

0∥u(s)∥2ds

12

.

Letting c2 = ∥ϖ∥

12K completes the proof.

We continue with results that will deal with the case when(B,Dk) ≥ α > 0.

Theorem 3. Assume A∗= −A, B∗

= B,D∗= D, Ik = I∗k and

(B,Dk) ≥ α > 0. There exists some yT such that (1) is null control-lablewith controlϖ = (y(t), y(τk)k∈Jm). Here (y(t), z(t)) is the so-lution to (3) with initial value y(0) = yT corresponding to x(T ) = yTof system (2).

Proof. Theorem3 states that there exists yT ∈ E such that themildsolution y(t) of (3) satisfying y(0) = yT (x(T ) = yT ) can serve asthe control to steer (1) from x0 to 0. In brief, it can be described as

that the mild solution x(t) of the following system,

dx(t) = (Ax(t) + By(t))dt + Cx(t)dw(t),

1x(τk) = Ikx(τk) + Dky(τk), k ∈ Jm, (13)x(0) = x0,

will satisfy x(T ) = 0.Now we define a bounded linear operator A : E → E such that

AyT = −x0. From (4), A will satisfy

E⟨AyT , yT ⟩ = E⟨−x(0), y(0)⟩

= E

T

0⟨u(s), B∗y(s)⟩ds +

mk=1

⟨v(τk),D∗

k(I − F∗

k )y(τk)⟩

= E

T

0⟨y(s), B∗y(s)⟩ds +

mk=1

⟨y(τk),D∗

k(I − F∗

k )y(τk)⟩

≥ ∥B∥E

T

0∥y(s)∥2ds +

mk=1

∥y(τk)∥2

≥ αE τ1

0∥y(s)∥2ds ≥ ατ1E∥yT∥2.

Thus A is coercive on E, P-a.s., which implies AE = E, P-a.s.Therefore, given any x0, there exists some yT satisfyingAyT = −x0.And this yT is what we search for. That completes the proof.

Remark 3. Theorem 3 has characterized the null controllability of(1) in terms of the existence of some initial value of (3). And fromthis point of view, the null controllability of (1) is equivalent to‘‘a certain individual observability’’ of quasi-backward stochasticsystems (3).

Under some conditions, we will find that the necessary conditionin Lemma 3 can also be sufficient.

Theorem 4. Assume A∗= −A, B∗

= B,D∗= D, Ik = I∗k , (B,Dk) ≥

0 and x0 be an initial value of (1). System (1) is null controllable at T ifand only if there exists some positive constant c2 such that (12) holds.Proof. The necessity can be seen in Lemma 3. Now let us provethe sufficiency. To verify this assertion, we define for each ϵ > 0,βϵ

= B2+ ϵI and ∆ϵ

k = D2k(I − Fk)2 + ϵI . Replace

B, Dkk∈Jm

with

βϵ, ∆ϵ

kk∈Jm

in (13) and denote its solution by xϵ(t). By

Theorem 3 there exists yϵ∈ E such that xϵ(T ) = 0. Moreover,

by definition of βϵ, ∆ϵk and (12), we get

−E⟨x0, y(0)⟩

= E T

0⟨βϵ yϵ(s), yϵ(s)⟩ds + E

mk=1

⟨∆ϵk y

ϵ(τk), yϵ(τk)⟩

≤ c2E

T

0⟨B2yϵ(s), yϵ(s)⟩ds +

mk=1

⟨∆2k y

ϵ(τk), yϵ(τk)⟩

12

≤ c2E

T

0⟨βϵ yϵ(s), yϵ(s)⟩ds +

mk=1

⟨∆ϵk y

ϵ(τk), yϵ(τk)⟩

12

,

which implies E T0 ⟨βϵ yϵ(s), yϵ(s)⟩ds + E

mk=1⟨∆

ϵk y

ϵ(τk),

yϵ(τk)⟩ ≤ c22 .It is easy to see that

√ϵ(yϵ(t), yϵ(τk)k∈Jm) and (Byϵ(t), Dk(I−

Fk)yϵ(τk)k∈Jm) are bounded in K . As a result, assume

(Byϵ(t), Dkyϵ(τk)k∈Jm) → ϖ (14)

in K , then obviously, (βϵ yϵ(t), ∆ϵk y

ϵ(τk)k∈Jm) → (B, Dk(I −

Fk)k∈Jm)ϖ , weakly in K . By passing to the limit, system (1)is null controllable with ϖ defined in (14). The proof is thuscompleted.

L. Shen et al. / Automatica ( ) – 5

Remark 4. Similar to Theorem3, Theorem4has also characterizedthe relation between the null controllability of (1) and the state ofquasi-backward stochastic systems (3) in the form of inequality.

4. Conclusion

We have investigated the null controllability and approximatenull controllability of LISSs. By investigating quasi-backwardstochastic systems of LISSs, we obtained some sufficient andnecessary conditions of controllability of LISSs. In particular, anequivalence has been established between the null controllabilityof (1) and the existence of some initial value of the quasi-backwardstochastic systems.

Acknowledgments

The authors are grateful to AE and the anonymous reviewersfor their detailed comments and suggestions, which have helpedto improve the quality of this paper.

References

Balachandran, K., Karthikeyan, S., & Kim, J. H. (2007). Controllability of semilinearstochastic integrodifferential systems. Kybernetika, 43(1), 31–44.

Bensoussan, A. (1983). Stochastic maximum principle for distributed parametersystem. Journal of the Franklin Institute, 315, 387–406.

Benzaid, Z., & Sznaier, M. (1994). Constrained controllability of linear impulsedifferential systems. IEEE Transactions on Automatic Control, 39, 1064–1066.

Bismut, J. M. (1978). An introductory approach to duality in optimal stochasticcontrol. SIAM Review, 20, 62–78.

Buckdahn, R., Quincampoix, M., & Tessitore, G. (2006). A characterization ofapproximately controllable linear stochastic differential equations. In G. DaPrato, & L. Tubaro (Eds.), Lecture notes in pure and a. math.: vol. 245. Stoch. PDEsand applications (pp. 253–260). London: Chapman & Hall.

Chen, L. J., & Sun, J. T. (2006). Nonlinear boundary value problem of first orderimpulsive functional differential equations. Journal of Mathematical Analysis andApplications, 318(2), 726–741.

Da Prato, G., & Zabczyk, J. (1992). Stochastic equations in infinite dimensions.Cambridge: Cambridge University Press.

George, R. K., Nandakumaran, A. K., & Arapostathis, A. (2000). A note oncontrollability of impulsive systems. Journal of Mathematical Analysis andApplications, 241(2), 276–283.

Goreac, D. (2007). Approximate controllability for linear stochastic differentialequations with control acting on the noise. In Applied analysis and differentialequations (pp. 153–164). Singapore: World Scientific.

Goreac, D. (2009). Approximate controllability for linear stochastic differentialequations in infinite dimensions. Applied Mathematics and Optimization, 60,105–132.

Guan, Z. H., Qian, T. H., & Yu, X. H. (2002). Controllability and observability of lineartime-varying impulsive systems. IEEE Transactions on Automatic Control, 49(8),1198–1208.

Ho, D. W. C., & Niu, Y. G. (2007). Robust fuzzy design for nonlinear uncertainstochastic systems via sliding-mode control. IEEE Transactions on Fuzzy Systems,15(3), 350–358.

Klamka, J. (1991). Controllability of dynamical systems. Netherlands: KluwerAcademic Publishers.

Klamka, J. (2007). Stochastic controllability of linear systems with state delays.International Journal of Applied Mathematics and Computer Science, 17(1), 5–13.

Klamka, J. (2008a). Stochastic controllability and minimum energy control ofsystems with multiple delays in control. Applied Mathematics and Computation,206(2), 704–715.

Klamka, J. (2008b). Stochastic controllability of systems with variable delay incontrol. Bulletin of the Polish Academy of Sciences, Technical Sciences, 56(3),279–284.

Lakshmikantham, V., Bainov, D. D., & Simeonov, P. S. (1989). Theory of impulsivedifferential equations. Singapore: World Scientific.

Leela, S., McRae, F. A., & Sivasundaram, S. (1993). Controllability of impulsivedifferential equations. Journal of Mathematical Analysis and Applications, 177,24–30.

Li, C. X., Sun, J. T., & Sun, R. Y. (2010). Stability analysis of a class of stochasticdifferential delay equations with nonlinear impulsive effects. Journal of theFranklin Institute, 347(7), 1186–1198.

Liu, J., Liu, X. Z., & Xie, W. C. (2011). Input-to-state stability of impulsive andswitching hybrid systems with time-delay. Automatica, 47(5), 899–908.

Pardoux, E., & Peng, S. G. (1990). Adapted solution of a backward stochasticdifferential equation. Systems & Control Letters, 14, 55–61.

Peng, S. G. (1994). Backward stochastic differential equation and exact controllabil-ity of stochastic control systems. Progress in Natural Science, 4(3), 274–284.

Sakthivel, R., Mahmudov, N. I., & Lee, S. G. (2009). Controllability of non-linearimpulsive stochastic systems. International Journal of Control, 82(5), 801–807.

Shen, L. J., Shi, J. P., & Sun, J. T. (2010). Complete controllability of impulsivestochastic integro-differential systems. Automatica, 46(6), 1068–1073.

Shen, L. J., & Sun, J. T. (2011). Approximate controllability of stochastic impulsivesystems with control-dependent coefficients. IET Control Theory & Applications,5(16), 1889–1894.

Shen, L. J., & Sun, J. T. (2012). Approximate controllability of abstract stochasticimpulsive systems with multiple time-varying delays. International Journal ofRobust and Nonlinear Control, http://dx.doi.org/10.1002/rnc.2789.

Sirbu, M., & Tessitore, G. (2001). Null controllability of an infinite dimensionalSDE with state- and control-dependent noise. Systems & Control Letters, 44(5),385–394.

Sontag, E. D. (1998). Mathematical control theory: deterministic finite dimensionalsystems (2nd ed.). New York: Springer-Verlag.

Tessitore, G. (1996). Existence, uniqueness and space regularity of the adaptedsolutions of a backward SPDE. Stochastic Analysis and Applications, 14(4),461–486.

Xie, G. M., & Wang, L. (2005). Controllability and observability of a class oflinear impulsive systems. Journal of Mathematical Analysis and Applications, 304,336–355.

Zhao, P. (2008). Practical stability, controllability and optimal control of stochasticMarkovian jump systems with time-delays. Automatica, 44, 3120–3125.

Lijuan Shen received the B.S. degree from Henan NormalUniversity, Xinxiang, in 2003, the M.S. degree fromDalian University of Technology in Dalian in 2006, andthe Ph.D. degree from Tongji University, Shanghai, in2011. From 2006 to 2012, she was employed at LuoyangNormal University, and since 2012 she has been apostdoctoral researcher at Tongji University in Shanghai.Her research interests include existence and controlanalysis of stochastic differential systems.

Jitao Sun was born in Jiangsu, China, in 1963. Hereceived the B.Sc. degree inMathematics from the NanjingUniversity, China, in 1983, and the Ph.D. degree in ControlTheory and Control Engineering from the South ChinaUniversity of Technology, China, in 2002, respectively.

He was with Anhui University of Technology from July1983 to September 1997. From September 1997 to April2000, he was with Shanghai Tiedao University. In April2000, he joined the Department of Mathematics, TongjiUniversity, Shanghai, China. From March 2004 to June2004, he was a Senior Research Assistant in the Centre

for Chaos Control and Synchronization, City University of Hong Kong, China. FromFebruary 2005 toMay 2005, hewas a Research Fellow in the Department of AppliedMathematics, City University of Hong Kong, China. From July 2005 to September2005, he was a Visiting Professor in the Faculty of Informatics and Communication,Central Queensland University, Australia. From February 2006 to October 2006,August 2007 to October 2007, and April 2008 to June 2008, he was a ResearchFellow in the Department of Electrical & Computer Engineering, National Universityof Singapore, Singapore, respectively. From November 2009 to May 2010, he was aVisiting Scholar in the Department ofMathematics, College ofWilliam&Mary, USA.He is currently a Professor at the Tongji University. Prior to this, he was a Professorat Anhui University of Technology and Shanghai Tiedao University from 1995 to2000, respectively. He is the author or coauthor of more than 140 journals papers.His recent research interests include impulsive control, time delay systems, hybridsystems, and systems biology.

Prof. Sun is the Member of Technical Committee on Nonlinear Circuitsand Systems, Part of the IEEE Circuits and Systems Society, and reviewer ofMathematical Reviews on AMS.

Qidi Wu received her B.S. and M.S. from Tsinghua Univer-sity in 1970 and 1981 respectively, and the Ph.D.degreefrom the department of electrical engineering, Federal In-stitute of Technology Zurich in 1986. Since 1986 she joinedTongji University as professor of electrical engineering.She was president of Tongji University from 1995 to 2003.At present she is Chair of the National Accreditation Com-mittee of Engineering Education, Director of Information-ization Expert Committee of Shanghai Government. Herresearch interests include control theory and application,planning and scheduling of complex manufacturing sys-

tems, system engineering and engineering management.