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Controllability of linear impulsive stochastic systems in Hilbert spaces
Automatica (
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Brief paper
Controllability of linear impulsive stochastic systems in Hilbert spaces✩ Lijuan Shen a,b , Jitao Sun a,1 , Qidi Wu b a
Department of Mathematics, Tongji University, Shanghai 200092, China
b
Department of Control Science and Engineering, Tongji University, Shanghai 201804, China
article
info
Article history: Received 16 December 2011 Received in revised form 14 August 2012 Accepted 2 November 2012 Available online xxxx Keywords: Impulsive stochastic systems Null controllability Quasi-backward stochastic systems Adjoint systems
abstract This paper is concerned with the controllability of linear impulsive stochastic systems (LISSs) in Hilbert spaces. For this class of systems, the concepts of null controllability and approximate null controllability are introduced. To overcome the difficulties related, we construct the adjoint systems and the quasibackward stochastic systems of LISSs. Necessary and sufficient conditions for the null controllability and the approximate null controllability are developed by utilizing our introduced quasi-backward stochastic systems. Furthermore, an equivalence is established between the null controllability of LISSs and some initial state of quasi-backward stochastic systems. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction Control theory is an area of application-oriented mathematics which deals with basic principles underlying the analysis and design of control systems. Conceived by Kalman, the controllability concept has been studied extensively in the fields of finite-dimensional systems, infinite-dimensional systems, hybrid systems, and behavioral systems. One may refer, for instance, to Ho and Niu (2007), Klamka (1991), Sontag (1998) and the references therein. Motivated by the fact that impulsive systems provide a natural framework for mathematical modeling of biology, economics, electronics and telecommunications, their study has received considerable 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 with the control only acting on the discontinuous points. George et al.
✩ This work is supported by the NNSF of China under Grants 61174039, 61034004 and 11201215, and China Postdoctoral Science Foundation (No. 2012M520928). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Akira Kojima under the direction of Editor Ian R. Petersen. E-mail addresses: [email protected] (L. Shen), [email protected] (J. Sun), [email protected] (Q. Wu). 1 Tel.: +86 21 65983241x1307; fax: +86 21 65981985.
0005-1098/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2013.01.036
(2000) modified some results in Leela et al. (1993) and investigated the complete controllability of linear impulsive systems and its perturbed systems. Recently, Guan et al. (2002) and Xie and Wang (2005) obtained some good results for the controllability of linear impulsive systems. In Guan et al. (2002), sufficient conditions are termed as the rank of matrices. And main results in Xie and Wang (2005) are established based on the fact that the reachable set can be expressed as the combination of the minimal invariant subspaces. In the stochastic framework, many efficient tools dealing with controllability have already been developed; see, for example, Picard type iteration (Balachandran, Karthikeyan, & Kim, 2007), contraction mapping principle (Sakthivel, Mahmudov, & Lee, 2009), and Lyapunov approach (Zhao, 2008), for nonlinear stochastic systems. The tools for stochastic linear systems, however, are relatively a few. For example, Klamka (2007, 2008a,b) described the controllability using the algebraic condition similar to those of deterministic systems. The study of backward stochastic differential equations (BSDEs), in the linear case, can be traced back to Bensoussan (1983) and Bismut (1978). But the first well-posedness result for nonlinear BSDEs was proved by Pardoux and Peng (1990) and ever since this paper, BSDEs have been one of the useful tools in the control theory. Peng (1994) firstly defined the exact controllability 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 dimensional linear differential equations. Buckdahn, Quincampoix, and Tessitore (2006) studied the approximate controllability in finite dimensional spaces, which was improved by Goreac (2007) when the control also acted on the noise. Recently, the results in
2
L. Shen et al. / Automatica (
Buckdahn et al. (2006) and Goreac (2007) were generalized by Goreac (2009) from finite dimensional spaces to infinite dimensional cases. There exist some results on the controllability about nonlinear impulsive stochastic systems by using various fixed point theorems. 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 speaking, the controllability of corresponding linear systems is a basic and important assumption in investigating the controllability of nonlinear systems; see, e.g., Sakthivel et al. (2009), Shen and Sun (2011) and Shen et al. (2010). However, to the best of our knowledge, there exist very few results about the controllability of LISSs. Therefore, the study of controllability of LISSs in Hilbert spaces is meaningful and challenging. In this paper, we study the null controllability and the approximate null controllability problem for LISSs. The LISSs studied in this paper are more complicated than the existing systems and have not only impulse effects but also noise terms. In the system analysis, these two factors should be considered at the same time. Meanwhile, we emphasize that in this paper, the control is allowed to act both on the continuous terms and discontinuous points. It should be pointed out that, due to the noise term, the existing methods of linear deterministic impulsive systems, such as those in Guan et al. (2002) and Xie and Wang (2005), are not available for LISSs. On the other hand, BSDEs coupled with the dual method commonly used for stochastic linear systems are not suitable for LISSs either as a result of the impulse effect. From the analysis given above, in the case that the conventional methods cannot apply, a new system should be introduced to study the controllability of LISSs. And its relationship with the state of LISSs should be established to get better results on the controllability of LISSs. In this paper, we will construct the quasi-backward stochastic systems of LISSs. Its relationship with the state of LISSs is then described by a useful generalized Itô lemma. Then with the aid of quasi-backward stochastic systems, sufficient conditions and properties of the controllability of LISSs are provided in Hilbert spaces. In particular, an equivalence will be established between the null controllability of LISSs and the existence of some initial value of the quasi-backward stochastic systems. The organization of the paper is as follows: In Section 2, we construct the adjoint systems and quasi-backward systems of LISSs and 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. In Section 3, some necessary and sufficient conditions for the null controllability and the approximate null controllability will be proposed by investigating the quasi-backward stochastic systems. In Section 4 we provide our conclusions. 2. Preliminaries Throughout this paper, unless otherwise specified, we will employ the following notations. Let {Ω , F , P} be a complete probability space with a filtration {Ft }t ≥0 of {wt } satisfying the usual conditions (i.e. right continuous and F0 containing all P-null sets) and E(·) be the expectation operator with respect to the probability 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 continuous everywhere except finite points τk , at which, y(τk+ ) and
F y(τk− ) exist with y(τk− ) = y(τk )}. Let L2 t Ω , PC (J , E ) denote the space of all Ft -adapted square integrable processes x from Ω into PC (J , E ). Denote L(U , E ) as the space of all bounded linear operators from U to E. We also denote by L2 (Ω , Ft , E ) the space of
)
–
all Ft -measurable square integrable random variables with values in E. If A is an operator, its adjoint operator is denoted by A∗ . In this paper, we focus on the following linear impulsive stochastic systems: dx(t ) = (Ax(t ) + Bu(t ))dt + Cx(t )dw(t ),
(1a)
1x(τk ) = Ik x(τk ) + Dk v(τk ), x(0) = x0 ,
(1b)
k ∈ Jm = {1, 2, . . . , m},
(1c)
where x0 ∈ L2 (Ω , F0 , E ), u, v ∈ U, w is a standard Wiener process valued in W , and A : D(A) → E is the infinitesimal generator 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 bounded linear operators. It is clear by Da Prato and Zabczyk (1992) and the induction method that system (1) admitsa unique mild solution Ft
x(t , x0 , u, {vk }k∈Jm ) ∈ L2
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, {vk }k∈Jm ) = Φ (t )x0 +
t
Φ (t − s)Bu(s)ds 0
+
Φ (t − τk )(Ik x(τk ) + Dk v(τk ))
0<τk
Φ (t − s)Cx(s)dw(s).
+ 0
To study system (1), we first consider its adjoint system as follows: dx˜ (t ) = (Ax˜ (t ) + Bu(t ))dt + C x˜ (t )dw(t ),
(2a)
1x˜ (τk ) = Ik x˜ (τk ),
(2b)
k ∈ Jm ,
x˜ (0) = x˜ 0 ,
(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 the impulsive functions. And with this choice of Ik , the generalized Itô’s lemma (Lemma 2, introduced later) depends only on the control u(t ) and v(τk ), k ∈ Jm . Based on (2), the following system is introduced and will play an important role in obtaining the main results: dy˜ (t ) = (−A∗ y˜ (t ) − C ∗ z (t ))dt + z (t )dw(t ),
1y˜ (τm−(k−1) ) = y˜ (T ) = x˜ 0 .
−Im−(k−1) y˜ (τm+−(k−1) ), ∗
k ∈ Jm ,
(3a) (3b) (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 stochastic system of (1). y(t ) in (3) may be interpreted as an evolution process of the fair price, whereas the stochastic process z (t ) may be interpreted as the related consumption and portfolio process. For convenience denote by ϖ the control in (1) ϖ = (u(t ), {vk }k∈Jm ) ∈ K , and the control operator B = (B, {Dk }k∈Jm ) ∈ L(K ) with Bϖ = (Bu(t ), {Dk vk }k∈Jm ). The space K of ϖ is a Hilbert space with respect to the inner product ⟨·, ·⟩K defined T m as ⟨ϖ1 , ϖ2 ⟩K = 0 ⟨u1 (t ), u2 (t )⟩E dt + k=1 ⟨v1k , v2k ⟩E , for all ϖ1 , ϖ2 ∈ K . Lemma 1. System (3) admits a unique mild solution (˜y(t ), z (t )) for any x˜ 0 ∈ 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(τm+−1 ), z (τm+−1 )) was transferred by impulse into (˜y(τm−1 ), z (τm−1 )) = ((I + ∗ Im y(τm+−1 ), z (τm+−1 )). From the induction method, it follows that −1 )˜ system (3) admits a unique mild solution (˜y(t ), z (t )) on [0, T ].
L. Shen et al. / Automatica (
Remark 2. When the operator C is unbounded, the situation will be more complex. We do not pursue this issue any further. And the reader may refer to Goreac (2009) for some general results in the case of Ik = Dk = 0 with unbounded operator C . Motivated by the definitions in Goreac (2007) and Sirbu and Tessitore (2001), we introduce the following definitions. Definition 1. For T > 0, system (1) is null controllable at T if for each x0 ∈ L2 (Ω , F0 , E ), there exists some ϖ ∈ K such that x(T , x0 , ϖ ) = 0, P-a.s. System (1) is said to be approximately controllable at T if for each x0 ∈ L2 (Ω , F0 , E ), there exists some ϖ ∈ 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 if for each x0 ∈ L2 (Ω , F0 , E ), there exists some ϖ ∈ K such that x(T , x0 , ϖ ) can be arbitrarily close to 0, P-a.s. We now introduce the following lemma and this lemma concerning the generalized Itô lemma characterizes the relationship between 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)⟩ T
⟨Bu(s), y˜ (s)⟩ds
=E
m +E ⟨Dk v(τk ), (I − Fk∗ )˜y(τk )⟩,
(4)
Proof. Consider the dynamics of ⟨x(t ), y˜ (t )⟩, t ̸= τk , then from stochastic calculus we have d⟨x(t ), y˜ (t )⟩ = ⟨Ax + Bu, y˜ (t )⟩dt + ⟨Cx(t ), y˜ (t )⟩dw(t )
∥˜y(0)∥ ≤ c 2
T
∥B y˜ (s)∥ ds + ∗
2
0
m
∥Dk (I − Fk )˜y(τk )∥ ∗
∗
2
.
(6)
k=1
Proof. It is easy to see that the operators LT and MT satisfy LT x0 = Φ (t )x0 +
Φ (t − s)Cx(s, x0 , 0)dw(s) 0
+
t
Φ (t − τk )Ik x(τk , x0 , 0),
0<τk
MT ϖ =
t
Φ (t − s)Bu(s)ds t Φ (t − s)Cx(s, 0, ϖ )dw(s) + 0 + Φ (t − τk )(Ik x(τk , 0, ϖ ) + Dk v(τk )). 0
0<τk
Since A, B, C , Dk are linear operators, we can conclude x(T , x0 , ϖ ) = LT x0 + MT ϖ . Therefore, that (1) is null controllable is equivalent to Da Prato and Zabczyk (1992) Im(LT ) ⊂ Im(MT ), i.e., there exists a constant c > 0 such that (7)
To verify the conclusion, let us show the form of L∗T and MT∗ . Let-
∆⟨x(t ), y˜ (t )⟩|t =τk = ⟨x(τk ), y˜ (τk )⟩ − ⟨x(τk ), y˜ (τk )⟩ +
= ⟨Dk v(τk ), (I − Fk∗ )˜y(τk )⟩.
(5)
By the information of differential equation and the property of Itô formula, we can deduce
E⟨x(T ), y˜ (T )⟩ − E⟨x(0), y˜ (0)⟩ T
Ed⟨x(s), y˜ (s)⟩ + E
0
m
∆⟨x(t ), y˜ (t )⟩|t =τk
k=1 T
⟨Bu(s), y˜ (s)⟩ds + E 0
And that completes the proof.
(8)
Similarly, taking ϖ = 0 in (4), we will obtain (9)
Substituting (9) and (8) into (7), it follows ∥˜y(0)∥2 ≤ c ( ∗ ∗ (s)∥2 ds + m y(τk )∥2 ). k=1 ∥Dk (I − Fk )˜
and for the impulse time τk , k ∈ Jm , it is not hard to check from (3b) that y˜ (τk+ ) = (I + Ik∗ )−1 y˜ (τk ), and 1y˜ (τk ) = ((I + Ik∗ )−1 − I )˜y(τk ) = −(I + Ik∗ )−1 Ik∗ y˜ (τk ), based on which, we get +
MT∗ x˜ 0 = (B∗ y˜ (·), {D∗k (I − Fk∗ )˜y(·)}k∈Jm ) ∈ K , P-a.s. L∗T x˜ 0 = y˜ (0).
+ ⟨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 ),
=E
Theorem 1. System (1) is null controllable if and only if there exists a positive constant c such that
T
with Fk = Ik (I + Ik )−1 .
3
ting x0 = 0 in (4) yields E⟨ϖ , MT∗ x˜ 0 ⟩K = E 0 ⟨u(s), B∗ y˜ (s)⟩E ds + E m ∗ ∗ y(τk )⟩E , which implies, k=1 ⟨v(τk ), Dk (I − Fk )˜
k=1
=
–
∥L∗T x˜ 0 ∥2E ≤ c ∥MT∗ x˜ 0 ∥2K .
0
)
m ⟨Dk v(τk ), (I − Fk∗ )˜y(τk )⟩. k =1
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 MT ϖ = x(T , 0, ϖ ). Motivated by Sirbu and Tessitore (2001), the null controllability can be characterized in terms of the state of quasi-backward stochastic systems (3).
T 0
∥B∗ y˜
Example 1. Consider an example in the form of (1). Define Fk as (Fk z )(t ) = z (t ) − z (t + τk ), then I − Fk , k = 1, 2, . . . , m is the shift operator and has the adjoint I − Fk∗ given by
((I − Fk∗ )z )(t ) =
z (t − (τk − τk−1 )), 0, 0 ≤ t < τk .
t ≥ τk ,
(10)
Let B = Dk be the orthogonal projection
(Dk z )(t ) = (Bz )(t ) =
0, 0 ≤ t < t0 , z (t ), t > t0 ,
(11)
t0 is a fixed, positive constant. So if t < t0 , we have where t ∗˜ ∥ B y (s)∥2 ds + 0<τk
4
L. Shen et al. / Automatica (
(ii) System (1) is approximately null controllable if and only if Im(LT ) ⊂ Im(MT ), or, equivalently, ker MT∗ ⊂ ker L∗T . That is, every solution (˜y(t ), z (t )) of (3) such that (B∗ y˜ (t ), ∗ {Dk (I − Fk∗ )˜y(τk )}k∈Jm ) = 0 will satisfy y˜ (0) = 0. And that completes the proof. Example 2. Consider the same system introduced in Example 1. By Theorem 2, this system is approximately controllable at T if and only if for every (˜y(t ), z (t )) such that (B∗ y˜ (t ), {D∗k (I − Fk∗ )˜y(τk )}k∈Jm ) = 0 we have (˜y(t ), z (t )) = 0, t ∈ [0, T ], P-a.s. So if t < t0 , we have (B∗ y˜ (t ), {D∗k (I − Fk∗ )˜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 − Fk∗ )˜y(τk )}k∈Jm ) = (˜y(t ), {˜y(τk−1 )}k∈Jm ) = 0. The following results provide a necessary condition of the null controllability 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 at T , then there exists some positive constant c2 such that
∥ − E⟨x(0), y˜ (0)⟩∥
∥B∗ y˜ (s)∥2 ds +
0
m
21 ∥D∗k (I − Fk∗ )˜y(τk )∥2
.
(12)
k=1
Proof. If system (1) is null controllable at T , by Definition 1 there exists a ϖ ∈ K such that x(T , x0 , ϖ ) = 0, P-a.s. Then From Lemma 2 we have
−E⟨x(0), y˜ (0)⟩ = E
T
⟨Bu(s), y˜ (s)⟩ds 0
m +E ⟨Dk v(τk ), (I − Fk∗ )˜y(τk )⟩, k=1
and by the Cauchy–Schwarz inequality, we obtain
∥ − E⟨x(0), y˜ (0)⟩∥ T ≤E ∥B∗ y˜ (s)∥2 ds
∥u(s)∥2 ds 0
m
+E
∥v(τk )∥2
k=1
∥B∗ y˜ (s)∥2 ds + 0
×
m
dx(t ) = (Ax(t ) + By˜ (t ))dt + Cx(t )dw(t ),
1x(τk ) = Ik x(τk ) + Dk y˜ (τk ), x(0) = x0 ,
k ∈ Jm ,
(13)
will satisfy x(T ) = 0. Now we define a bounded linear operator A : E → E such that Ay˜ T = −x0 . From (4), A will satisfy
E⟨Ay˜ T , y˜ T ⟩ = E⟨−x(0), y˜ (0)⟩
T
m ∗ ∗ ⟨v(τk ), Dk (I − Fk )˜y(τk )⟩ ⟨u(s), B y˜ (s)⟩ds + ∗
=E 0
k=1 T
⟨˜y(s), B y˜ (s)⟩ds + ∗
=E 0
≥ ∥B∥E
T
∥˜y(s)∥ ds + 2
0
τ1
⟨˜y(τk ), Dk (I − Fk )˜y(τk )⟩ ∗
∗
k=1
≥ αE
m
m
∥˜y(τk )∥
2
k=1
∥˜y(s)∥2 ds ≥ ατ1 E∥˜yT ∥2 .
∥D∗k (I − Fk∗ )˜y(τk )∥2 21 ∥D∗k (I − Fk∗ )˜y(τk )∥2
k=1
∥v(τk )∥ + 2
∥u(s)∥ ds 2
−E⟨x0 , y˜ (0)⟩ T m =E ⟨β ϵ y˜ ϵ (s), y˜ ϵ (s)⟩ds + E ⟨∆ϵk y˜ ϵ (τk ), y˜ ϵ (τk )⟩ 0
k=1 T
≤ c2 E
12 m ⟨B2 y˜ ϵ (s), y˜ ϵ (s)⟩ds + ⟨∆2k y˜ ϵ (τk ), y˜ ϵ (τk )⟩
0
1 2 Letting c2 = ∥ϖ ∥K completes the proof.
Theorem 4. Assume A∗ = −A, B∗ = B, D∗ = D, Ik = Ik∗ , (B, Dk ) ≥ 0 and x0 be an initial value of (1). System (1) is null controllable at T if and only if there exists some positive constant c2 such that (12) holds.
.
0
k=1
Under some conditions, we will find that the necessary condition in Lemma 3 can also be sufficient.
21
T
Remark 3. Theorem 3 has characterized the null controllability of (1) in terms of the existence of some initial value of (3). And from this point of view, the null controllability of (1) is equivalent to ‘‘a certain individual observability’’ of quasi-backward stochastic systems (3).
21
m
Thus A is coercive on E , P-a.s., which implies AE = E , P-a.s. Therefore, given any x0 , there exists some y˜ T satisfying Ay˜ T = −x0 . And this y˜ T is what we search for. That completes the proof.
with β ϵ , {∆ϵk }k∈Jm in (13) and denote its solution by x˜ ϵ (t ). By Theorem 3 there exists y˜ ϵ ∈ E such that xϵ (T ) = 0. Moreover, by definition of β ϵ , ∆ϵk and (12), we get
k=1
T
≤E
that the mild solution x(t ) of the following system,
Proof. The necessity can be seen in Lemma 3. Now let us prove the sufficiency. To verify this assertion, we define for each ϵ > 0, β ϵ = B2 + ϵ I and ∆ϵk = D2k (I − Fk )2 + ϵ I. Replace B, {Dk }k∈Jm
21
T
0 m
–
0
T
≤ c2 E
)
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 = Ik∗ and (B, Dk ) ≥ α > 0. There exists some y˜ T such that (1) is null controllable with control ϖ = (˜y(t ), {˜y(τk )}k∈Jm ). Here (˜y(t ), z (t )) is the solution to (3) with initial value y˜ (0) = y˜ T corresponding to x˜ (T ) = y˜ T of system (2). Proof. Theorem 3 states that there exists y˜ T ∈ E such that the mild solution y˜ (t ) of (3) satisfying y˜ (0) = y˜ T (˜x(T ) = y˜ T ) can serve as the control to steer (1) from x0 to 0. In brief, it can be described as
k=1 T
≤ c2 E
12 m ϵ ϵ ϵ ⟨β y˜ (s), y˜ (s)⟩ds + ⟨∆k y˜ (τk ), y˜ (τk )⟩ , ϵ
ϵ ϵ
0
k=1
which implies E 0 ⟨β y (s), y (s)⟩ds + E k=1 ⟨∆ϵk y˜ ϵ (τk ), y˜ ϵ (τk )⟩ ≤ c22 . √ It is easy to see that ϵ(˜yϵ (t ), {˜yϵ (τk )}k∈Jm ) and (By˜ ϵ (t ), {Dk (I − Fk )˜yϵ (τk )}k∈Jm ) are bounded in K . As a result, assume
T
ϵ ˜ϵ
˜ϵ
m
(By˜ ϵ (t ), {Dk y˜ ϵ (τk )}k∈Jm ) → ϖ ϵ ˜ϵ
(14) ϵ ϵ ˜
in K , then obviously, (β y (t ), {∆k y (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 thus completed.
L. Shen et al. / Automatica (
Remark 4. Similar to Theorem 3, Theorem 4 has also characterized the relation between the null controllability of (1) and the state of quasi-backward stochastic systems (3) in the form of inequality. 4. Conclusion We have investigated the null controllability and approximate null controllability of LISSs. By investigating quasi-backward stochastic systems of LISSs, we obtained some sufficient and necessary conditions of controllability of LISSs. In particular, an equivalence has been established between the null controllability of (1) and the existence of some initial value of the quasi-backward stochastic systems. Acknowledgments The authors are grateful to AE and the anonymous reviewers for their detailed comments and suggestions, which have helped to improve the quality of this paper.
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Pardoux, E., & Peng, S. G. (1990). Adapted solution of a backward stochastic differential equation. Systems & Control Letters, 14, 55–61. Peng, S. G. (1994). Backward stochastic differential equation and exact controllability of stochastic control systems. Progress in Natural Science, 4(3), 274–284. Sakthivel, R., Mahmudov, N. I., & Lee, S. G. (2009). Controllability of non-linear impulsive stochastic systems. International Journal of Control, 82(5), 801–807. Shen, L. J., Shi, J. P., & Sun, J. T. (2010). Complete controllability of impulsive stochastic integro-differential systems. Automatica, 46(6), 1068–1073. Shen, L. J., & Sun, J. T. (2011). Approximate controllability of stochastic impulsive systems with control-dependent coefficients. IET Control Theory & Applications, 5(16), 1889–1894. Shen, L. J., & Sun, J. T. (2012). Approximate controllability of abstract stochastic impulsive systems with multiple time-varying delays. International Journal of Robust and Nonlinear Control, http://dx.doi.org/10.1002/rnc.2789. Sirbu, M., & Tessitore, G. (2001). Null controllability of an infinite dimensional SDE with state- and control-dependent noise. Systems & Control Letters, 44(5), 385–394. Sontag, E. D. (1998). Mathematical control theory: deterministic finite dimensional systems (2nd ed.). New York: Springer-Verlag. Tessitore, G. (1996). Existence, uniqueness and space regularity of the adapted solutions of a backward SPDE. Stochastic Analysis and Applications, 14(4), 461–486. Xie, G. M., & Wang, L. (2005). Controllability and observability of a class of linear impulsive systems. Journal of Mathematical Analysis and Applications, 304, 336–355. Zhao, P. (2008). Practical stability, controllability and optimal control of stochastic Markovian jump systems with time-delays. Automatica, 44, 3120–3125.
References Balachandran, K., Karthikeyan, S., & Kim, J. H. (2007). Controllability of semilinear stochastic integrodifferential systems. Kybernetika, 43(1), 31–44. Bensoussan, A. (1983). Stochastic maximum principle for distributed parameter system. Journal of the Franklin Institute, 315, 387–406. Benzaid, Z., & Sznaier, M. (1994). Constrained controllability of linear impulse differential systems. IEEE Transactions on Automatic Control, 39, 1064–1066. Bismut, J. M. (1978). An introductory approach to duality in optimal stochastic control. SIAM Review, 20, 62–78. Buckdahn, R., Quincampoix, M., & Tessitore, G. (2006). A characterization of approximately controllable linear stochastic differential equations. In G. Da Prato, & L. Tubaro (Eds.), Lecture notes in pure and a. math.: vol. 245. Stoch. PDEs and applications (pp. 253–260). London: Chapman & Hall. Chen, L. J., & Sun, J. T. (2006). Nonlinear boundary value problem of first order impulsive functional differential equations. Journal of Mathematical Analysis and Applications, 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 on controllability of impulsive systems. Journal of Mathematical Analysis and Applications, 241(2), 276–283. Goreac, D. (2007). Approximate controllability for linear stochastic differential equations with control acting on the noise. In Applied analysis and differential equations (pp. 153–164). Singapore: World Scientific. Goreac, D. (2009). Approximate controllability for linear stochastic differential equations in infinite dimensions. Applied Mathematics and Optimization, 60, 105–132. Guan, Z. H., Qian, T. H., & Yu, X. H. (2002). Controllability and observability of linear time-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 uncertain stochastic systems via sliding-mode control. IEEE Transactions on Fuzzy Systems, 15(3), 350–358. Klamka, J. (1991). Controllability of dynamical systems. Netherlands: Kluwer Academic 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 of systems with multiple delays in control. Applied Mathematics and Computation, 206(2), 704–715. Klamka, J. (2008b). Stochastic controllability of systems with variable delay in control. Bulletin of the Polish Academy of Sciences, Technical Sciences, 56(3), 279–284. Lakshmikantham, V., Bainov, D. D., & Simeonov, P. S. (1989). Theory of impulsive differential equations. Singapore: World Scientific. Leela, S., McRae, F. A., & Sivasundaram, S. (1993). Controllability of impulsive differential 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 stochastic differential delay equations with nonlinear impulsive effects. Journal of the Franklin Institute, 347(7), 1186–1198. Liu, J., Liu, X. Z., & Xie, W. C. (2011). Input-to-state stability of impulsive and switching hybrid systems with time-delay. Automatica, 47(5), 899–908.
Lijuan Shen received the B.S. degree from Henan Normal University, Xinxiang, in 2003, the M.S. degree from Dalian University of Technology in Dalian in 2006, and the Ph.D. degree from Tongji University, Shanghai, in 2011. From 2006 to 2012, she was employed at Luoyang Normal University, and since 2012 she has been a postdoctoral researcher at Tongji University in Shanghai. Her research interests include existence and control analysis of stochastic differential systems.
Jitao Sun was born in Jiangsu, China, in 1963. He received the B.Sc. degree in Mathematics from the Nanjing University, China, in 1983, and the Ph.D. degree in Control Theory and Control Engineering from the South China University of Technology, China, in 2002, respectively. He was with Anhui University of Technology from July 1983 to September 1997. From September 1997 to April 2000, he was with Shanghai Tiedao University. In April 2000, he joined the Department of Mathematics, Tongji University, Shanghai, China. From March 2004 to June 2004, he was a Senior Research Assistant in the Centre for Chaos Control and Synchronization, City University of Hong Kong, China. From February 2005 to May 2005, he was a Research Fellow in the Department of Applied Mathematics, City University of Hong Kong, China. From July 2005 to September 2005, 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 Research Fellow in the Department of Electrical & Computer Engineering, National University of Singapore, Singapore, respectively. From November 2009 to May 2010, he was a Visiting Scholar in the Department of Mathematics, College of William & Mary, USA. He is currently a Professor at the Tongji University. Prior to this, he was a Professor at Anhui University of Technology and Shanghai Tiedao University from 1995 to 2000, respectively. He is the author or coauthor of more than 140 journals papers. His recent research interests include impulsive control, time delay systems, hybrid systems, and systems biology. Prof. Sun is the Member of Technical Committee on Nonlinear Circuits and Systems, Part of the IEEE Circuits and Systems Society, and reviewer of Mathematical Reviews on AMS. Qidi Wu received her B.S. and M.S. from Tsinghua University in 1970 and 1981 respectively, and the Ph.D.degree from the department of electrical engineering, Federal Institute of Technology Zurich in 1986. Since 1986 she joined Tongji 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 Committee of Engineering Education, Director of Informationization Expert Committee of Shanghai Government. Her research interests include control theory and application, planning and scheduling of complex manufacturing systems, system engineering and engineering management.
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Contents lists available at SciVerse ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Brief paper
Controllability of linear impulsive stochastic systems in Hilbert spaces✩ Lijuan Shen a,b , Jitao Sun a,1 , Qidi Wu b a
Department of Mathematics, Tongji University, Shanghai 200092, China
b
Department of Control Science and Engineering, Tongji University, Shanghai 201804, China
article
info
Article history: Received 16 December 2011 Received in revised form 14 August 2012 Accepted 2 November 2012 Available online xxxx Keywords: Impulsive stochastic systems Null controllability Quasi-backward stochastic systems Adjoint systems
abstract This paper is concerned with the controllability of linear impulsive stochastic systems (LISSs) in Hilbert spaces. For this class of systems, the concepts of null controllability and approximate null controllability are introduced. To overcome the difficulties related, we construct the adjoint systems and the quasibackward stochastic systems of LISSs. Necessary and sufficient conditions for the null controllability and the approximate null controllability are developed by utilizing our introduced quasi-backward stochastic systems. Furthermore, an equivalence is established between the null controllability of LISSs and some initial state of quasi-backward stochastic systems. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction Control theory is an area of application-oriented mathematics which deals with basic principles underlying the analysis and design of control systems. Conceived by Kalman, the controllability concept has been studied extensively in the fields of finite-dimensional systems, infinite-dimensional systems, hybrid systems, and behavioral systems. One may refer, for instance, to Ho and Niu (2007), Klamka (1991), Sontag (1998) and the references therein. Motivated by the fact that impulsive systems provide a natural framework for mathematical modeling of biology, economics, electronics and telecommunications, their study has received considerable 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 with the control only acting on the discontinuous points. George et al.
✩ This work is supported by the NNSF of China under Grants 61174039, 61034004 and 11201215, and China Postdoctoral Science Foundation (No. 2012M520928). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Akira Kojima under the direction of Editor Ian R. Petersen. E-mail addresses: [email protected] (L. Shen), [email protected] (J. Sun), [email protected] (Q. Wu). 1 Tel.: +86 21 65983241x1307; fax: +86 21 65981985.
0005-1098/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2013.01.036
(2000) modified some results in Leela et al. (1993) and investigated the complete controllability of linear impulsive systems and its perturbed systems. Recently, Guan et al. (2002) and Xie and Wang (2005) obtained some good results for the controllability of linear impulsive systems. In Guan et al. (2002), sufficient conditions are termed as the rank of matrices. And main results in Xie and Wang (2005) are established based on the fact that the reachable set can be expressed as the combination of the minimal invariant subspaces. In the stochastic framework, many efficient tools dealing with controllability have already been developed; see, for example, Picard type iteration (Balachandran, Karthikeyan, & Kim, 2007), contraction mapping principle (Sakthivel, Mahmudov, & Lee, 2009), and Lyapunov approach (Zhao, 2008), for nonlinear stochastic systems. The tools for stochastic linear systems, however, are relatively a few. For example, Klamka (2007, 2008a,b) described the controllability using the algebraic condition similar to those of deterministic systems. The study of backward stochastic differential equations (BSDEs), in the linear case, can be traced back to Bensoussan (1983) and Bismut (1978). But the first well-posedness result for nonlinear BSDEs was proved by Pardoux and Peng (1990) and ever since this paper, BSDEs have been one of the useful tools in the control theory. Peng (1994) firstly defined the exact controllability 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 dimensional linear differential equations. Buckdahn, Quincampoix, and Tessitore (2006) studied the approximate controllability in finite dimensional spaces, which was improved by Goreac (2007) when the control also acted on the noise. Recently, the results in
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Buckdahn et al. (2006) and Goreac (2007) were generalized by Goreac (2009) from finite dimensional spaces to infinite dimensional cases. There exist some results on the controllability about nonlinear impulsive stochastic systems by using various fixed point theorems. 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 speaking, the controllability of corresponding linear systems is a basic and important assumption in investigating the controllability of nonlinear systems; see, e.g., Sakthivel et al. (2009), Shen and Sun (2011) and Shen et al. (2010). However, to the best of our knowledge, there exist very few results about the controllability of LISSs. Therefore, the study of controllability of LISSs in Hilbert spaces is meaningful and challenging. In this paper, we study the null controllability and the approximate null controllability problem for LISSs. The LISSs studied in this paper are more complicated than the existing systems and have not only impulse effects but also noise terms. In the system analysis, these two factors should be considered at the same time. Meanwhile, we emphasize that in this paper, the control is allowed to act both on the continuous terms and discontinuous points. It should be pointed out that, due to the noise term, the existing methods of linear deterministic impulsive systems, such as those in Guan et al. (2002) and Xie and Wang (2005), are not available for LISSs. On the other hand, BSDEs coupled with the dual method commonly used for stochastic linear systems are not suitable for LISSs either as a result of the impulse effect. From the analysis given above, in the case that the conventional methods cannot apply, a new system should be introduced to study the controllability of LISSs. And its relationship with the state of LISSs should be established to get better results on the controllability of LISSs. In this paper, we will construct the quasi-backward stochastic systems of LISSs. Its relationship with the state of LISSs is then described by a useful generalized Itô lemma. Then with the aid of quasi-backward stochastic systems, sufficient conditions and properties of the controllability of LISSs are provided in Hilbert spaces. In particular, an equivalence will be established between the null controllability of LISSs and the existence of some initial value of the quasi-backward stochastic systems. The organization of the paper is as follows: In Section 2, we construct the adjoint systems and quasi-backward systems of LISSs and 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. In Section 3, some necessary and sufficient conditions for the null controllability and the approximate null controllability will be proposed by investigating the quasi-backward stochastic systems. In Section 4 we provide our conclusions. 2. Preliminaries Throughout this paper, unless otherwise specified, we will employ the following notations. Let {Ω , F , P} be a complete probability space with a filtration {Ft }t ≥0 of {wt } satisfying the usual conditions (i.e. right continuous and F0 containing all P-null sets) and E(·) be the expectation operator with respect to the probability 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 continuous everywhere except finite points τk , at which, y(τk+ ) and
F y(τk− ) exist with y(τk− ) = y(τk )}. Let L2 t Ω , PC (J , E ) denote the space of all Ft -adapted square integrable processes x from Ω into PC (J , E ). Denote L(U , E ) as the space of all bounded linear operators from U to E. We also denote by L2 (Ω , Ft , E ) the space of
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all Ft -measurable square integrable random variables with values in E. If A is an operator, its adjoint operator is denoted by A∗ . In this paper, we focus on the following linear impulsive stochastic systems: dx(t ) = (Ax(t ) + Bu(t ))dt + Cx(t )dw(t ),
(1a)
1x(τk ) = Ik x(τk ) + Dk v(τk ), x(0) = x0 ,
(1b)
k ∈ Jm = {1, 2, . . . , m},
(1c)
where x0 ∈ L2 (Ω , F0 , E ), u, v ∈ U, w is a standard Wiener process valued in W , and A : D(A) → E is the infinitesimal generator 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 bounded linear operators. It is clear by Da Prato and Zabczyk (1992) and the induction method that system (1) admitsa unique mild solution Ft
x(t , x0 , u, {vk }k∈Jm ) ∈ L2
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, {vk }k∈Jm ) = Φ (t )x0 +
t
Φ (t − s)Bu(s)ds 0
+
Φ (t − τk )(Ik x(τk ) + Dk v(τk ))
0<τk
Φ (t − s)Cx(s)dw(s).
+ 0
To study system (1), we first consider its adjoint system as follows: dx˜ (t ) = (Ax˜ (t ) + Bu(t ))dt + C x˜ (t )dw(t ),
(2a)
1x˜ (τk ) = Ik x˜ (τk ),
(2b)
k ∈ Jm ,
x˜ (0) = x˜ 0 ,
(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 the impulsive functions. And with this choice of Ik , the generalized Itô’s lemma (Lemma 2, introduced later) depends only on the control u(t ) and v(τk ), k ∈ Jm . Based on (2), the following system is introduced and will play an important role in obtaining the main results: dy˜ (t ) = (−A∗ y˜ (t ) − C ∗ z (t ))dt + z (t )dw(t ),
1y˜ (τm−(k−1) ) = y˜ (T ) = x˜ 0 .
−Im−(k−1) y˜ (τm+−(k−1) ), ∗
k ∈ Jm ,
(3a) (3b) (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 stochastic system of (1). y(t ) in (3) may be interpreted as an evolution process of the fair price, whereas the stochastic process z (t ) may be interpreted as the related consumption and portfolio process. For convenience denote by ϖ the control in (1) ϖ = (u(t ), {vk }k∈Jm ) ∈ K , and the control operator B = (B, {Dk }k∈Jm ) ∈ L(K ) with Bϖ = (Bu(t ), {Dk vk }k∈Jm ). The space K of ϖ is a Hilbert space with respect to the inner product ⟨·, ·⟩K defined T m as ⟨ϖ1 , ϖ2 ⟩K = 0 ⟨u1 (t ), u2 (t )⟩E dt + k=1 ⟨v1k , v2k ⟩E , for all ϖ1 , ϖ2 ∈ K . Lemma 1. System (3) admits a unique mild solution (˜y(t ), z (t )) for any x˜ 0 ∈ 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(τm+−1 ), z (τm+−1 )) was transferred by impulse into (˜y(τm−1 ), z (τm−1 )) = ((I + ∗ Im y(τm+−1 ), z (τm+−1 )). From the induction method, it follows that −1 )˜ system (3) admits a unique mild solution (˜y(t ), z (t )) on [0, T ].
L. Shen et al. / Automatica (
Remark 2. When the operator C is unbounded, the situation will be more complex. We do not pursue this issue any further. And the reader may refer to Goreac (2009) for some general results in the case of Ik = Dk = 0 with unbounded operator C . Motivated by the definitions in Goreac (2007) and Sirbu and Tessitore (2001), we introduce the following definitions. Definition 1. For T > 0, system (1) is null controllable at T if for each x0 ∈ L2 (Ω , F0 , E ), there exists some ϖ ∈ K such that x(T , x0 , ϖ ) = 0, P-a.s. System (1) is said to be approximately controllable at T if for each x0 ∈ L2 (Ω , F0 , E ), there exists some ϖ ∈ 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 if for each x0 ∈ L2 (Ω , F0 , E ), there exists some ϖ ∈ K such that x(T , x0 , ϖ ) can be arbitrarily close to 0, P-a.s. We now introduce the following lemma and this lemma concerning the generalized Itô lemma characterizes the relationship between 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)⟩ T
⟨Bu(s), y˜ (s)⟩ds
=E
m +E ⟨Dk v(τk ), (I − Fk∗ )˜y(τk )⟩,
(4)
Proof. Consider the dynamics of ⟨x(t ), y˜ (t )⟩, t ̸= τk , then from stochastic calculus we have d⟨x(t ), y˜ (t )⟩ = ⟨Ax + Bu, y˜ (t )⟩dt + ⟨Cx(t ), y˜ (t )⟩dw(t )
∥˜y(0)∥ ≤ c 2
T
∥B y˜ (s)∥ ds + ∗
2
0
m
∥Dk (I − Fk )˜y(τk )∥ ∗
∗
2
.
(6)
k=1
Proof. It is easy to see that the operators LT and MT satisfy LT x0 = Φ (t )x0 +
Φ (t − s)Cx(s, x0 , 0)dw(s) 0
+
t
Φ (t − τk )Ik x(τk , x0 , 0),
0<τk
MT ϖ =
t
Φ (t − s)Bu(s)ds t Φ (t − s)Cx(s, 0, ϖ )dw(s) + 0 + Φ (t − τk )(Ik x(τk , 0, ϖ ) + Dk v(τk )). 0
0<τk
Since A, B, C , Dk are linear operators, we can conclude x(T , x0 , ϖ ) = LT x0 + MT ϖ . Therefore, that (1) is null controllable is equivalent to Da Prato and Zabczyk (1992) Im(LT ) ⊂ Im(MT ), i.e., there exists a constant c > 0 such that (7)
To verify the conclusion, let us show the form of L∗T and MT∗ . Let-
∆⟨x(t ), y˜ (t )⟩|t =τk = ⟨x(τk ), y˜ (τk )⟩ − ⟨x(τk ), y˜ (τk )⟩ +
= ⟨Dk v(τk ), (I − Fk∗ )˜y(τk )⟩.
(5)
By the information of differential equation and the property of Itô formula, we can deduce
E⟨x(T ), y˜ (T )⟩ − E⟨x(0), y˜ (0)⟩ T
Ed⟨x(s), y˜ (s)⟩ + E
0
m
∆⟨x(t ), y˜ (t )⟩|t =τk
k=1 T
⟨Bu(s), y˜ (s)⟩ds + E 0
And that completes the proof.
(8)
Similarly, taking ϖ = 0 in (4), we will obtain (9)
Substituting (9) and (8) into (7), it follows ∥˜y(0)∥2 ≤ c ( ∗ ∗ (s)∥2 ds + m y(τk )∥2 ). k=1 ∥Dk (I − Fk )˜
and for the impulse time τk , k ∈ Jm , it is not hard to check from (3b) that y˜ (τk+ ) = (I + Ik∗ )−1 y˜ (τk ), and 1y˜ (τk ) = ((I + Ik∗ )−1 − I )˜y(τk ) = −(I + Ik∗ )−1 Ik∗ y˜ (τk ), based on which, we get +
MT∗ x˜ 0 = (B∗ y˜ (·), {D∗k (I − Fk∗ )˜y(·)}k∈Jm ) ∈ K , P-a.s. L∗T x˜ 0 = y˜ (0).
+ ⟨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 ),
=E
Theorem 1. System (1) is null controllable if and only if there exists a positive constant c such that
T
with Fk = Ik (I + Ik )−1 .
3
ting x0 = 0 in (4) yields E⟨ϖ , MT∗ x˜ 0 ⟩K = E 0 ⟨u(s), B∗ y˜ (s)⟩E ds + E m ∗ ∗ y(τk )⟩E , which implies, k=1 ⟨v(τk ), Dk (I − Fk )˜
k=1
=
–
∥L∗T x˜ 0 ∥2E ≤ c ∥MT∗ x˜ 0 ∥2K .
0
)
m ⟨Dk v(τk ), (I − Fk∗ )˜y(τk )⟩. k =1
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 MT ϖ = x(T , 0, ϖ ). Motivated by Sirbu and Tessitore (2001), the null controllability can be characterized in terms of the state of quasi-backward stochastic systems (3).
T 0
∥B∗ y˜
Example 1. Consider an example in the form of (1). Define Fk as (Fk z )(t ) = z (t ) − z (t + τk ), then I − Fk , k = 1, 2, . . . , m is the shift operator and has the adjoint I − Fk∗ given by
((I − Fk∗ )z )(t ) =
z (t − (τk − τk−1 )), 0, 0 ≤ t < τk .
t ≥ τk ,
(10)
Let B = Dk be the orthogonal projection
(Dk z )(t ) = (Bz )(t ) =
0, 0 ≤ t < t0 , z (t ), t > t0 ,
(11)
t0 is a fixed, positive constant. So if t < t0 , we have where t ∗˜ ∥ B y (s)∥2 ds + 0<τk
4
L. Shen et al. / Automatica (
(ii) System (1) is approximately null controllable if and only if Im(LT ) ⊂ Im(MT ), or, equivalently, ker MT∗ ⊂ ker L∗T . That is, every solution (˜y(t ), z (t )) of (3) such that (B∗ y˜ (t ), ∗ {Dk (I − Fk∗ )˜y(τk )}k∈Jm ) = 0 will satisfy y˜ (0) = 0. And that completes the proof. Example 2. Consider the same system introduced in Example 1. By Theorem 2, this system is approximately controllable at T if and only if for every (˜y(t ), z (t )) such that (B∗ y˜ (t ), {D∗k (I − Fk∗ )˜y(τk )}k∈Jm ) = 0 we have (˜y(t ), z (t )) = 0, t ∈ [0, T ], P-a.s. So if t < t0 , we have (B∗ y˜ (t ), {D∗k (I − Fk∗ )˜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 − Fk∗ )˜y(τk )}k∈Jm ) = (˜y(t ), {˜y(τk−1 )}k∈Jm ) = 0. The following results provide a necessary condition of the null controllability 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 at T , then there exists some positive constant c2 such that
∥ − E⟨x(0), y˜ (0)⟩∥
∥B∗ y˜ (s)∥2 ds +
0
m
21 ∥D∗k (I − Fk∗ )˜y(τk )∥2
.
(12)
k=1
Proof. If system (1) is null controllable at T , by Definition 1 there exists a ϖ ∈ K such that x(T , x0 , ϖ ) = 0, P-a.s. Then From Lemma 2 we have
−E⟨x(0), y˜ (0)⟩ = E
T
⟨Bu(s), y˜ (s)⟩ds 0
m +E ⟨Dk v(τk ), (I − Fk∗ )˜y(τk )⟩, k=1
and by the Cauchy–Schwarz inequality, we obtain
∥ − E⟨x(0), y˜ (0)⟩∥ T ≤E ∥B∗ y˜ (s)∥2 ds
∥u(s)∥2 ds 0
m
+E
∥v(τk )∥2
k=1
∥B∗ y˜ (s)∥2 ds + 0
×
m
dx(t ) = (Ax(t ) + By˜ (t ))dt + Cx(t )dw(t ),
1x(τk ) = Ik x(τk ) + Dk y˜ (τk ), x(0) = x0 ,
k ∈ Jm ,
(13)
will satisfy x(T ) = 0. Now we define a bounded linear operator A : E → E such that Ay˜ T = −x0 . From (4), A will satisfy
E⟨Ay˜ T , y˜ T ⟩ = E⟨−x(0), y˜ (0)⟩
T
m ∗ ∗ ⟨v(τk ), Dk (I − Fk )˜y(τk )⟩ ⟨u(s), B y˜ (s)⟩ds + ∗
=E 0
k=1 T
⟨˜y(s), B y˜ (s)⟩ds + ∗
=E 0
≥ ∥B∥E
T
∥˜y(s)∥ ds + 2
0
τ1
⟨˜y(τk ), Dk (I − Fk )˜y(τk )⟩ ∗
∗
k=1
≥ αE
m
m
∥˜y(τk )∥
2
k=1
∥˜y(s)∥2 ds ≥ ατ1 E∥˜yT ∥2 .
∥D∗k (I − Fk∗ )˜y(τk )∥2 21 ∥D∗k (I − Fk∗ )˜y(τk )∥2
k=1
∥v(τk )∥ + 2
∥u(s)∥ ds 2
−E⟨x0 , y˜ (0)⟩ T m =E ⟨β ϵ y˜ ϵ (s), y˜ ϵ (s)⟩ds + E ⟨∆ϵk y˜ ϵ (τk ), y˜ ϵ (τk )⟩ 0
k=1 T
≤ c2 E
12 m ⟨B2 y˜ ϵ (s), y˜ ϵ (s)⟩ds + ⟨∆2k y˜ ϵ (τk ), y˜ ϵ (τk )⟩
0
1 2 Letting c2 = ∥ϖ ∥K completes the proof.
Theorem 4. Assume A∗ = −A, B∗ = B, D∗ = D, Ik = Ik∗ , (B, Dk ) ≥ 0 and x0 be an initial value of (1). System (1) is null controllable at T if and only if there exists some positive constant c2 such that (12) holds.
.
0
k=1
Under some conditions, we will find that the necessary condition in Lemma 3 can also be sufficient.
21
T
Remark 3. Theorem 3 has characterized the null controllability of (1) in terms of the existence of some initial value of (3). And from this point of view, the null controllability of (1) is equivalent to ‘‘a certain individual observability’’ of quasi-backward stochastic systems (3).
21
m
Thus A is coercive on E , P-a.s., which implies AE = E , P-a.s. Therefore, given any x0 , there exists some y˜ T satisfying Ay˜ T = −x0 . And this y˜ T is what we search for. That completes the proof.
with β ϵ , {∆ϵk }k∈Jm in (13) and denote its solution by x˜ ϵ (t ). By Theorem 3 there exists y˜ ϵ ∈ E such that xϵ (T ) = 0. Moreover, by definition of β ϵ , ∆ϵk and (12), we get
k=1
T
≤E
that the mild solution x(t ) of the following system,
Proof. The necessity can be seen in Lemma 3. Now let us prove the sufficiency. To verify this assertion, we define for each ϵ > 0, β ϵ = B2 + ϵ I and ∆ϵk = D2k (I − Fk )2 + ϵ I. Replace B, {Dk }k∈Jm
21
T
0 m
–
0
T
≤ c2 E
)
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 = Ik∗ and (B, Dk ) ≥ α > 0. There exists some y˜ T such that (1) is null controllable with control ϖ = (˜y(t ), {˜y(τk )}k∈Jm ). Here (˜y(t ), z (t )) is the solution to (3) with initial value y˜ (0) = y˜ T corresponding to x˜ (T ) = y˜ T of system (2). Proof. Theorem 3 states that there exists y˜ T ∈ E such that the mild solution y˜ (t ) of (3) satisfying y˜ (0) = y˜ T (˜x(T ) = y˜ T ) can serve as the control to steer (1) from x0 to 0. In brief, it can be described as
k=1 T
≤ c2 E
12 m ϵ ϵ ϵ ⟨β y˜ (s), y˜ (s)⟩ds + ⟨∆k y˜ (τk ), y˜ (τk )⟩ , ϵ
ϵ ϵ
0
k=1
which implies E 0 ⟨β y (s), y (s)⟩ds + E k=1 ⟨∆ϵk y˜ ϵ (τk ), y˜ ϵ (τk )⟩ ≤ c22 . √ It is easy to see that ϵ(˜yϵ (t ), {˜yϵ (τk )}k∈Jm ) and (By˜ ϵ (t ), {Dk (I − Fk )˜yϵ (τk )}k∈Jm ) are bounded in K . As a result, assume
T
ϵ ˜ϵ
˜ϵ
m
(By˜ ϵ (t ), {Dk y˜ ϵ (τk )}k∈Jm ) → ϖ ϵ ˜ϵ
(14) ϵ ϵ ˜
in K , then obviously, (β y (t ), {∆k y (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 thus completed.
L. Shen et al. / Automatica (
Remark 4. Similar to Theorem 3, Theorem 4 has also characterized the relation between the null controllability of (1) and the state of quasi-backward stochastic systems (3) in the form of inequality. 4. Conclusion We have investigated the null controllability and approximate null controllability of LISSs. By investigating quasi-backward stochastic systems of LISSs, we obtained some sufficient and necessary conditions of controllability of LISSs. In particular, an equivalence has been established between the null controllability of (1) and the existence of some initial value of the quasi-backward stochastic systems. Acknowledgments The authors are grateful to AE and the anonymous reviewers for their detailed comments and suggestions, which have helped to improve the quality of this paper.
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Lijuan Shen received the B.S. degree from Henan Normal University, Xinxiang, in 2003, the M.S. degree from Dalian University of Technology in Dalian in 2006, and the Ph.D. degree from Tongji University, Shanghai, in 2011. From 2006 to 2012, she was employed at Luoyang Normal University, and since 2012 she has been a postdoctoral researcher at Tongji University in Shanghai. Her research interests include existence and control analysis of stochastic differential systems.
Jitao Sun was born in Jiangsu, China, in 1963. He received the B.Sc. degree in Mathematics from the Nanjing University, China, in 1983, and the Ph.D. degree in Control Theory and Control Engineering from the South China University of Technology, China, in 2002, respectively. He was with Anhui University of Technology from July 1983 to September 1997. From September 1997 to April 2000, he was with Shanghai Tiedao University. In April 2000, he joined the Department of Mathematics, Tongji University, Shanghai, China. From March 2004 to June 2004, he was a Senior Research Assistant in the Centre for Chaos Control and Synchronization, City University of Hong Kong, China. From February 2005 to May 2005, he was a Research Fellow in the Department of Applied Mathematics, City University of Hong Kong, China. From July 2005 to September 2005, 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 Research Fellow in the Department of Electrical & Computer Engineering, National University of Singapore, Singapore, respectively. From November 2009 to May 2010, he was a Visiting Scholar in the Department of Mathematics, College of William & Mary, USA. He is currently a Professor at the Tongji University. Prior to this, he was a Professor at Anhui University of Technology and Shanghai Tiedao University from 1995 to 2000, respectively. He is the author or coauthor of more than 140 journals papers. His recent research interests include impulsive control, time delay systems, hybrid systems, and systems biology. Prof. Sun is the Member of Technical Committee on Nonlinear Circuits and Systems, Part of the IEEE Circuits and Systems Society, and reviewer of Mathematical Reviews on AMS. Qidi Wu received her B.S. and M.S. from Tsinghua University in 1970 and 1981 respectively, and the Ph.D.degree from the department of electrical engineering, Federal Institute of Technology Zurich in 1986. Since 1986 she joined Tongji 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 Committee of Engineering Education, Director of Informationization Expert Committee of Shanghai Government. Her research interests include control theory and application, planning and scheduling of complex manufacturing systems, system engineering and engineering management.