Complete controllability of impulsive stochastic ... - Semantic Scholar
Automatica 46 (2010) 1068–1073
Contents lists available at ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Brief paper
Complete controllability of impulsive stochastic integro-differential systemsI Lijuan Shen a,b , Junping Shi c , Jitao Sun a,∗ a
Department of Mathematics, Tongji University, Shanghai 200092, China
b
Department of Mathematics, Luoyang Normal University, Luoyang, Henan 471022, China
c
Department of Mathematics, College of William and Mary, Williamsburg, VA, 23187-8795, USA
article
info
Article history: Received 10 May 2009 Received in revised form 1 February 2010 Accepted 28 February 2010 Available online 1 April 2010
abstract This paper is concerned with the controllability of impulsive stochastic integro-differential systems. Sufficient conditions of complete controllability for impulsive stochastic integro-differential systems are obtained by using Schaefer’s fixed point theorem. A numerical example is provided to show the effectiveness of the proposed results. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Impulsive stochastic systems Integro-differential systems Schaefer’s fixed point theorem Complete controllability
1. Introduction It is well known that the concept of controllability plays an important role in control theory and engineering. Controllability has been studied extensively in the fields of finite-dimensional nonlinear systems, infinite-dimensional systems (see e.g., Bemporad, Ferrari-Trecate, & Morari, 2000; Li & Rao, 2003; Mahmudov, 2003). Impulsive systems arise naturally in various fields, such as mechanical systems and biological systems, economics, etc. (see Lakshmikantham, Bainov, & Simeonov, 1989, and the references therein). Impulsive dynamical systems exhibit the continuous evolutions of the states typically described by ordinary differential equations coupled with instantaneous state jumps or impulses. And the presence of impulses implies that the trajectories of the system do not necessarily preserve the basic properties of the non-impulsive dynamical systems. To this end the theory of impulsive differential systems has emerged as an important area of investigation in applied sciences. In the last few years many papers have been published about the controllability of impulsive differential systems. Guan, Qian, and Yu (2002) considered the controllability and observability for a class of time-varying impulsive control systems; Li, Wang, and Zhang (2006) investigated the
I This work is supported by the NNSF of China under grant 60874027, and NSF of Education Department of Henan Province (2008B110010). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor George Yin under the direction of Editor Ian R. Petersen. ∗ Corresponding author. Tel.: +86 21 65983240 1307; fax: +86 21 65981985, +86 21 65982341. E-mail address: [email protected] (J. Sun).
0005-1098/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2010.03.002
controllability of the first-order impulsive functional differential systems in Banach space; Xie and Wang (2004) studied the controllability of switched impulsive control systems; Liu and Marquez (2008) discussed the controllability and observability problem for a class of controlled switching impulsive systems; in Sakthivel, Mahmudov, and Kim (2009), sufficient conditions were formulated for the exact controllability of second-order nonlinear impulsive control differential systems. On the other hand noise is ubiquitous. Systems, both natural and artificial ones, often possess various structures subject to stochastically abrupt changes, which may result from abrupt phenomena such as stochastic failures and repairs of the components, changes in the interconnections of subsystems, sudden environment changes, etc. More details on stochastic differential equations can be found in the books of Mao (1997) and Oksendal (2003). For linear stochastic system the controllability problem of the form dx(t ) = [Ax(t ) + Bu(t )]dt + g (t )dw(t ), x(0) = x0 ,
t ∈ [0, T ],
(1)
has been studied by several authors (e.g., Mahmudov, 2001a,b). Here A, B are both n × n matrices, and g (·) : [0, T ] → Rn×l for n, l ∈ N. For nonlinear stochastic systems there are also many results on the control theory, including Mahmudov (2003), Mahmudov and Zorlu (2003), Wang, Ho, Liu, and Liu (2009) and Niu, Ho, and Wang (2007) dealt with the problem of sliding mode control for a class of nonlinear uncertain stochastic systems with Markovian switching. Dong and Sun (2008) gave a detailed discussion on hybrid control for a class of nonlinear stochastic Markovian switching systems. For the controllability problem there are different methods for various types of nonlinear systems. And the most common
L. Shen et al. / Automatica 46 (2010) 1068–1073
methods for controllability of stochastic systems as we know are: Picard type iteration (e.g. Balachandran, Karthikeyan, & Kim, 2007), contraction mapping principle (e.g., Mahmudov & Zorlu, 2003), and Lyapunov approach (e.g. Zhao, 2008). However, the complete controllability problem of impulsive stochastic integro-differential systems has not been investigated yet, to the best of our knowledge, although Karthikeyan and Balachandran (2009) and Sakthivel, Mahmudov, and Lee (2009) respectively investigated the controllability of impulsive stochastic control systems by using contraction mapping principle, and Subalakshmi and Balachandran (2009) studied the approximate controllability of nonlinear stochastic impulsive systems in Hilbert spaces by using Nussbaum’s fixed point theorem. Based on Schaefer’s fixed point theorem, the proposed work in this paper on the complete controllability of integro-differential systems with both noise perturbations and impulsive effects is new in the literature. This problem is important and challenging in both theory and practice, which has motivated us for this study. In this paper our main aim is to show the complete controllability of the impulsive stochastic integro-differential systems of the form dx(t ) =
Ax(t ) + F
t , x(t ),
2. Preliminaries In this section, we introduce notations, definitions and preliminary facts which are used throughout the paper. Let {Ω , F , P } be a complete probability space with a filtration {Ft }t ≥0 satisfying the usual conditions (i.e. right continuous and F0 containing all P-null sets). E(·) is the expectation with respect F to the measure P. L2 t ([0, T ] × Ω , Rn ) is the Banach space of all square integrable and Ft -adapted processes x(t ) mapping [0, T ] × Rn into Rn equipped with the norm
kxk2L = sup Ekx(t )k2 . t ∈[0,T ]
For the simplicity of considerations we generally assume that the set of admissible control sets is Ft
([0, T ] × Ω , Rn ). Let Φ (t ) = exp(At ). Define the controllability matrix Z T T Φ (T − t )BB∗ Φ ∗ (T − t )dt , 0 ≤ s < t . Ψs = Uad = L2
s
The controllability operator Π0T is the linear transformation from Ft
t
Z
f1 (t , s, x(s))ds,
L2 (2)
([0, T ] × Ω , Rn ) to L2Ft ([0, T ] × Ω , Rn ), associated with
0 t
f2 (t , s, x(s))dw(s) dt + Bu(t )dt + G (t , x(t ), 0 Z t Z t (2) g1 (t , s, x(s))ds, g2 (t , s, x(s))dw(s) dw(t ), Z
0
Π0T {·} =
T
Z
Φ (T − t )BB∗ Φ (T − t )E{·|Ft }dt . 0
For arbitrary x1 process
0
t ∈ [0, T ], t 6= τk , k = 1, 2, . . . , m,
1x(t ) = Ik (x(t )), x(0) = x0 −
Q = x1 − Φ (T )x0 −
t = τ k , k = 1 , 2 , . . . , m,
under some basic assumptions via Schaefer’s fixed point theorem. Here A and B are both n × n matrices; and the functions in the equation are: F : [0, T ] × Rn × Rn × Rn → Rn , G : [0, T ] × Rn × Rn × Rn → Rn×l , f1 , g1 : [0, T ] × [0, T ] × Rn → Rn , f2 , g2 : [0, T ] × [0, T ] × Rn → Rn×l ;
1x(t ) denotes the jump of x at t, i.e. 1x(t ) = x(t + ) − x(t − ) = x(t ) − x(t − ); Ik ∈ C (Rn , Rn ); The initial value x0 is a F0 -measurable random variable with Ekx0 k2 < ∞; u(t ) is a feedback control and w is l-dimensional Wiener process; and Ft is the filtration generated by B(s), 0 ≤ s ≤ t. The system (2) is in a very general form and it covers many possible models with various definitions of f1 , f2 , g1 , g2 . We would like to mention that Balachandran and Karthikeyan (2008) obtained the controllability results of system (2) when Ik = 0. The controllability problem with f2 = g1 = g2 = Ik = 0 was studied by Sakthivel, Kim, and Mahmudov (2006) and Sakthivel et al. (2009) studied the case f1 = f2 = g1 = g2 = 0. The system (2) with f1 = f2 = g1 = g2 = Ik = 0 was investigated by Mahmudov and Zorlu (2003). However, all the papers listed above obtained the controllability results by using the contraction mapping principle which seems to be restrictive in some cases. In Section 4 an example will illustrate it. The paper is organized as follows. In Section 2, some basic notations and preliminary facts are recalled. Some lemmas and the results are given in Section 3. And an example in Section 4 is discussed to illustrate the efficiency of the results. Finally, conclusions are given in Section 5.
1069
∈ L2Ft ([0, T ] × Ω , Rn ) define by Q the T
Z
Φ (T − s)(F˜ x)(s)ds 0
T
Z
˜ )(s)dw(s) − Φ (T − s)(Gx
− 0
m X
Φ (T − τk )Ik (x(τk )),
k=1
where
Z t Z t (F˜ x)(t ) = F t , x(t ), f1 (t , s, x(s))ds, f2 (t , s, x(s))dw(s) , 0 0 Z t Z t ˜ )(t ) = G t , x(t ), (Gx g1 (t , s, x(s))ds, g2 (t , s, x(s))dw(s) . 0
0
Our main results are based on the following fixed point theorem of Schaefer which was discussed and proved in Smart (1980). Lemma 1 (Schaefer’s Theorem). Let (D, k · k) be a normed space, and let the operator A : D → D be completely continuous. Define S (A ) = {x ∈ D : x = λA x, λ ∈ (0, 1)}. Then either (1) the set S (A ) is unbounded, or (2) the operator A has a fixed point in D. Definition 1. The system (2) is completely controllable on [0, T ] if F all the points in L2 t ([0, T ] × Ω , Rn ) can be reached from x0 at time T . Definition 2. A set M ⊂ Rn is said to be quasi-equicontinuous in [0, T ] if for any ε > 0 there exists δ > 0 such that if y ∈ M , t1 , t2 ∈ (τk−1 , τk ], k ∈ N and |t1 − t2 | < δ , then ky(t1 ) − y(t2 )k < ε. This together with the standard result (Conway, 1990, pp. 175–176) yields the following necessary and sufficient conditions F for relative compactness in L2 t ([0, T ] × Ω , Rn ). Lemma 2. The set Λ ⊂ Rn is relatively compact if and only if (a) Λ is uniformly bounded, that is, kxkΛ ≤ B for each x ∈ Λ and some B > 0. (b) Λ is quasi-equicontinuous in [0, T ].
1070
L. Shen et al. / Automatica 46 (2010) 1068–1073
3. Main results
Proof. Substituting (3) into (4) we can obtain that
In this section we discuss the controllability of the stochastic impulsive integro-differential systems (2). For the study of this problem we hence introduce the following hypotheses. (H1 ) The linear control system (1) is completely controllable in [0, T ]. By (H1 ) there exists a positive constant l1 , such that for t ∈ [0, T ] (Mahmudov, 2001a)
x( t ) = Φ ( t ) x0 +
kΦ (t )k2 ≤ l1 . (H2 ) (F˜ , G˜ ) are Ft -adapted with respect to t. And for every positive constant k1 , there exists function q(·) ∈ L1 (R+ , R+ ) such that for every t ∈ [0, T ], ˜ )(t )k2 ) ≤ q(t ). sup (k(F˜ x)(t )k2 + k(Gx kxk≤k1
E
˜ )(s)k )ds exists, (k(F˜ x)(s)k + k(Gx 2
0
and
˜ )(t )k2 ) exists uniformly for t ∈ [0, T ]. E(k(F˜ x)(t )k2 + k(Gx n
kx1 k2 + kx0 k2 ≤ l2 . (H5 ) There exists a continuous nondecreasing function φ : ˜ )(t )k2 ≤ [0, ∞) → [0, ∞) such that Ek(F˜ x)(t )k2 + Ek(Gx Ft 2 a(t )φ(Ekxk ) for all t ∈ [0, T ], x ∈ L2 ([0, T ] × Ω , Rn ), where a(t ) : [0, ∞) → [0, ∞) is an integrable function with Z T Z ∞ ds M1 a(s)ds < , φ( s) 0 c RT where c = 5l1 l2 + 25Ml1 l3 (l2 + l1 l2 + l1 T 0 Ek(F˜ x)(s)k2 ds + RT P Pm 2 2 ˜ )(s)k2 ds+l1 ( m l1 0 Ek(Gx k=1 dk ) )+5l1 ( k=1 dk ) , M1 = 5l1 (T + 1), and l3 is a positive constant satisfying the following Lemma 3. Lemma 3 (Mahmudov & Zorlu, 2003). Assume (H1 ) holds. Then for F every z ∈ L2 t ([0, T ] × Ω , Rn ) there exists some constant l3 such that
Ek(Π0T )−1 k2 ≤ l3
and EkΠ0t z k2 ≤ M Ekz k2 . Ft
([0, T ] × Ω , Rn ) define the
u(t ) = B∗ Φ ∗ (T − t )E{(Π0T )−1 Q |Ft }.
(3)
For arbitrary process x(·) ∈ L2 control
˜ ), For example, In addition we need further hypotheses on (F˜ , G ˜ ˜ (F , G) are Borel-measurable functions satisfying the local Lipschitz condition and the linear growth condition. Under these conditions Liu (2008) showed that there is a unique global solution x(t , t0 , x0 , u) of (2) for any x0 ∈ Rn , t ∈ [0, T ], which can be represented in the following form x(t ) = Φ (t )x0 +
Φ (t − s)[Bu(s) + (F˜ x)(s)]ds 0
t
Z
˜ )(s)dw(s) + Φ (t − s)(Gx
+ 0
X
Φ (t − τk )Ik (x(τk )). (4)
0
Contents lists available at ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Brief paper
Complete controllability of impulsive stochastic integro-differential systemsI Lijuan Shen a,b , Junping Shi c , Jitao Sun a,∗ a
Department of Mathematics, Tongji University, Shanghai 200092, China
b
Department of Mathematics, Luoyang Normal University, Luoyang, Henan 471022, China
c
Department of Mathematics, College of William and Mary, Williamsburg, VA, 23187-8795, USA
article
info
Article history: Received 10 May 2009 Received in revised form 1 February 2010 Accepted 28 February 2010 Available online 1 April 2010
abstract This paper is concerned with the controllability of impulsive stochastic integro-differential systems. Sufficient conditions of complete controllability for impulsive stochastic integro-differential systems are obtained by using Schaefer’s fixed point theorem. A numerical example is provided to show the effectiveness of the proposed results. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Impulsive stochastic systems Integro-differential systems Schaefer’s fixed point theorem Complete controllability
1. Introduction It is well known that the concept of controllability plays an important role in control theory and engineering. Controllability has been studied extensively in the fields of finite-dimensional nonlinear systems, infinite-dimensional systems (see e.g., Bemporad, Ferrari-Trecate, & Morari, 2000; Li & Rao, 2003; Mahmudov, 2003). Impulsive systems arise naturally in various fields, such as mechanical systems and biological systems, economics, etc. (see Lakshmikantham, Bainov, & Simeonov, 1989, and the references therein). Impulsive dynamical systems exhibit the continuous evolutions of the states typically described by ordinary differential equations coupled with instantaneous state jumps or impulses. And the presence of impulses implies that the trajectories of the system do not necessarily preserve the basic properties of the non-impulsive dynamical systems. To this end the theory of impulsive differential systems has emerged as an important area of investigation in applied sciences. In the last few years many papers have been published about the controllability of impulsive differential systems. Guan, Qian, and Yu (2002) considered the controllability and observability for a class of time-varying impulsive control systems; Li, Wang, and Zhang (2006) investigated the
I This work is supported by the NNSF of China under grant 60874027, and NSF of Education Department of Henan Province (2008B110010). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor George Yin under the direction of Editor Ian R. Petersen. ∗ Corresponding author. Tel.: +86 21 65983240 1307; fax: +86 21 65981985, +86 21 65982341. E-mail address: [email protected] (J. Sun).
0005-1098/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2010.03.002
controllability of the first-order impulsive functional differential systems in Banach space; Xie and Wang (2004) studied the controllability of switched impulsive control systems; Liu and Marquez (2008) discussed the controllability and observability problem for a class of controlled switching impulsive systems; in Sakthivel, Mahmudov, and Kim (2009), sufficient conditions were formulated for the exact controllability of second-order nonlinear impulsive control differential systems. On the other hand noise is ubiquitous. Systems, both natural and artificial ones, often possess various structures subject to stochastically abrupt changes, which may result from abrupt phenomena such as stochastic failures and repairs of the components, changes in the interconnections of subsystems, sudden environment changes, etc. More details on stochastic differential equations can be found in the books of Mao (1997) and Oksendal (2003). For linear stochastic system the controllability problem of the form dx(t ) = [Ax(t ) + Bu(t )]dt + g (t )dw(t ), x(0) = x0 ,
t ∈ [0, T ],
(1)
has been studied by several authors (e.g., Mahmudov, 2001a,b). Here A, B are both n × n matrices, and g (·) : [0, T ] → Rn×l for n, l ∈ N. For nonlinear stochastic systems there are also many results on the control theory, including Mahmudov (2003), Mahmudov and Zorlu (2003), Wang, Ho, Liu, and Liu (2009) and Niu, Ho, and Wang (2007) dealt with the problem of sliding mode control for a class of nonlinear uncertain stochastic systems with Markovian switching. Dong and Sun (2008) gave a detailed discussion on hybrid control for a class of nonlinear stochastic Markovian switching systems. For the controllability problem there are different methods for various types of nonlinear systems. And the most common
L. Shen et al. / Automatica 46 (2010) 1068–1073
methods for controllability of stochastic systems as we know are: Picard type iteration (e.g. Balachandran, Karthikeyan, & Kim, 2007), contraction mapping principle (e.g., Mahmudov & Zorlu, 2003), and Lyapunov approach (e.g. Zhao, 2008). However, the complete controllability problem of impulsive stochastic integro-differential systems has not been investigated yet, to the best of our knowledge, although Karthikeyan and Balachandran (2009) and Sakthivel, Mahmudov, and Lee (2009) respectively investigated the controllability of impulsive stochastic control systems by using contraction mapping principle, and Subalakshmi and Balachandran (2009) studied the approximate controllability of nonlinear stochastic impulsive systems in Hilbert spaces by using Nussbaum’s fixed point theorem. Based on Schaefer’s fixed point theorem, the proposed work in this paper on the complete controllability of integro-differential systems with both noise perturbations and impulsive effects is new in the literature. This problem is important and challenging in both theory and practice, which has motivated us for this study. In this paper our main aim is to show the complete controllability of the impulsive stochastic integro-differential systems of the form dx(t ) =
Ax(t ) + F
t , x(t ),
2. Preliminaries In this section, we introduce notations, definitions and preliminary facts which are used throughout the paper. Let {Ω , F , P } be a complete probability space with a filtration {Ft }t ≥0 satisfying the usual conditions (i.e. right continuous and F0 containing all P-null sets). E(·) is the expectation with respect F to the measure P. L2 t ([0, T ] × Ω , Rn ) is the Banach space of all square integrable and Ft -adapted processes x(t ) mapping [0, T ] × Rn into Rn equipped with the norm
kxk2L = sup Ekx(t )k2 . t ∈[0,T ]
For the simplicity of considerations we generally assume that the set of admissible control sets is Ft
([0, T ] × Ω , Rn ). Let Φ (t ) = exp(At ). Define the controllability matrix Z T T Φ (T − t )BB∗ Φ ∗ (T − t )dt , 0 ≤ s < t . Ψs = Uad = L2
s
The controllability operator Π0T is the linear transformation from Ft
t
Z
f1 (t , s, x(s))ds,
L2 (2)
([0, T ] × Ω , Rn ) to L2Ft ([0, T ] × Ω , Rn ), associated with
0 t
f2 (t , s, x(s))dw(s) dt + Bu(t )dt + G (t , x(t ), 0 Z t Z t (2) g1 (t , s, x(s))ds, g2 (t , s, x(s))dw(s) dw(t ), Z
0
Π0T {·} =
T
Z
Φ (T − t )BB∗ Φ (T − t )E{·|Ft }dt . 0
For arbitrary x1 process
0
t ∈ [0, T ], t 6= τk , k = 1, 2, . . . , m,
1x(t ) = Ik (x(t )), x(0) = x0 −
Q = x1 − Φ (T )x0 −
t = τ k , k = 1 , 2 , . . . , m,
under some basic assumptions via Schaefer’s fixed point theorem. Here A and B are both n × n matrices; and the functions in the equation are: F : [0, T ] × Rn × Rn × Rn → Rn , G : [0, T ] × Rn × Rn × Rn → Rn×l , f1 , g1 : [0, T ] × [0, T ] × Rn → Rn , f2 , g2 : [0, T ] × [0, T ] × Rn → Rn×l ;
1x(t ) denotes the jump of x at t, i.e. 1x(t ) = x(t + ) − x(t − ) = x(t ) − x(t − ); Ik ∈ C (Rn , Rn ); The initial value x0 is a F0 -measurable random variable with Ekx0 k2 < ∞; u(t ) is a feedback control and w is l-dimensional Wiener process; and Ft is the filtration generated by B(s), 0 ≤ s ≤ t. The system (2) is in a very general form and it covers many possible models with various definitions of f1 , f2 , g1 , g2 . We would like to mention that Balachandran and Karthikeyan (2008) obtained the controllability results of system (2) when Ik = 0. The controllability problem with f2 = g1 = g2 = Ik = 0 was studied by Sakthivel, Kim, and Mahmudov (2006) and Sakthivel et al. (2009) studied the case f1 = f2 = g1 = g2 = 0. The system (2) with f1 = f2 = g1 = g2 = Ik = 0 was investigated by Mahmudov and Zorlu (2003). However, all the papers listed above obtained the controllability results by using the contraction mapping principle which seems to be restrictive in some cases. In Section 4 an example will illustrate it. The paper is organized as follows. In Section 2, some basic notations and preliminary facts are recalled. Some lemmas and the results are given in Section 3. And an example in Section 4 is discussed to illustrate the efficiency of the results. Finally, conclusions are given in Section 5.
1069
∈ L2Ft ([0, T ] × Ω , Rn ) define by Q the T
Z
Φ (T − s)(F˜ x)(s)ds 0
T
Z
˜ )(s)dw(s) − Φ (T − s)(Gx
− 0
m X
Φ (T − τk )Ik (x(τk )),
k=1
where
Z t Z t (F˜ x)(t ) = F t , x(t ), f1 (t , s, x(s))ds, f2 (t , s, x(s))dw(s) , 0 0 Z t Z t ˜ )(t ) = G t , x(t ), (Gx g1 (t , s, x(s))ds, g2 (t , s, x(s))dw(s) . 0
0
Our main results are based on the following fixed point theorem of Schaefer which was discussed and proved in Smart (1980). Lemma 1 (Schaefer’s Theorem). Let (D, k · k) be a normed space, and let the operator A : D → D be completely continuous. Define S (A ) = {x ∈ D : x = λA x, λ ∈ (0, 1)}. Then either (1) the set S (A ) is unbounded, or (2) the operator A has a fixed point in D. Definition 1. The system (2) is completely controllable on [0, T ] if F all the points in L2 t ([0, T ] × Ω , Rn ) can be reached from x0 at time T . Definition 2. A set M ⊂ Rn is said to be quasi-equicontinuous in [0, T ] if for any ε > 0 there exists δ > 0 such that if y ∈ M , t1 , t2 ∈ (τk−1 , τk ], k ∈ N and |t1 − t2 | < δ , then ky(t1 ) − y(t2 )k < ε. This together with the standard result (Conway, 1990, pp. 175–176) yields the following necessary and sufficient conditions F for relative compactness in L2 t ([0, T ] × Ω , Rn ). Lemma 2. The set Λ ⊂ Rn is relatively compact if and only if (a) Λ is uniformly bounded, that is, kxkΛ ≤ B for each x ∈ Λ and some B > 0. (b) Λ is quasi-equicontinuous in [0, T ].
1070
L. Shen et al. / Automatica 46 (2010) 1068–1073
3. Main results
Proof. Substituting (3) into (4) we can obtain that
In this section we discuss the controllability of the stochastic impulsive integro-differential systems (2). For the study of this problem we hence introduce the following hypotheses. (H1 ) The linear control system (1) is completely controllable in [0, T ]. By (H1 ) there exists a positive constant l1 , such that for t ∈ [0, T ] (Mahmudov, 2001a)
x( t ) = Φ ( t ) x0 +
kΦ (t )k2 ≤ l1 . (H2 ) (F˜ , G˜ ) are Ft -adapted with respect to t. And for every positive constant k1 , there exists function q(·) ∈ L1 (R+ , R+ ) such that for every t ∈ [0, T ], ˜ )(t )k2 ) ≤ q(t ). sup (k(F˜ x)(t )k2 + k(Gx kxk≤k1
E
˜ )(s)k )ds exists, (k(F˜ x)(s)k + k(Gx 2
0
and
˜ )(t )k2 ) exists uniformly for t ∈ [0, T ]. E(k(F˜ x)(t )k2 + k(Gx n
kx1 k2 + kx0 k2 ≤ l2 . (H5 ) There exists a continuous nondecreasing function φ : ˜ )(t )k2 ≤ [0, ∞) → [0, ∞) such that Ek(F˜ x)(t )k2 + Ek(Gx Ft 2 a(t )φ(Ekxk ) for all t ∈ [0, T ], x ∈ L2 ([0, T ] × Ω , Rn ), where a(t ) : [0, ∞) → [0, ∞) is an integrable function with Z T Z ∞ ds M1 a(s)ds < , φ( s) 0 c RT where c = 5l1 l2 + 25Ml1 l3 (l2 + l1 l2 + l1 T 0 Ek(F˜ x)(s)k2 ds + RT P Pm 2 2 ˜ )(s)k2 ds+l1 ( m l1 0 Ek(Gx k=1 dk ) )+5l1 ( k=1 dk ) , M1 = 5l1 (T + 1), and l3 is a positive constant satisfying the following Lemma 3. Lemma 3 (Mahmudov & Zorlu, 2003). Assume (H1 ) holds. Then for F every z ∈ L2 t ([0, T ] × Ω , Rn ) there exists some constant l3 such that
Ek(Π0T )−1 k2 ≤ l3
and EkΠ0t z k2 ≤ M Ekz k2 . Ft
([0, T ] × Ω , Rn ) define the
u(t ) = B∗ Φ ∗ (T − t )E{(Π0T )−1 Q |Ft }.
(3)
For arbitrary process x(·) ∈ L2 control
˜ ), For example, In addition we need further hypotheses on (F˜ , G ˜ ˜ (F , G) are Borel-measurable functions satisfying the local Lipschitz condition and the linear growth condition. Under these conditions Liu (2008) showed that there is a unique global solution x(t , t0 , x0 , u) of (2) for any x0 ∈ Rn , t ∈ [0, T ], which can be represented in the following form x(t ) = Φ (t )x0 +
Φ (t − s)[Bu(s) + (F˜ x)(s)]ds 0
t
Z
˜ )(s)dw(s) + Φ (t − s)(Gx
+ 0
X
Φ (t − τk )Ik (x(τk )). (4)
0
