Review of the Latest Progress in Controllability of Stochastic Linear Systems and Stochastic GE-Evolution Operator

: According to the spatial dimension, equation type, and time sequence, the latest progress in controllability of stochastic linear systems and some unsolved problems are introduced. Firstly, the exact controllability of stochastic linear systems in ﬁnite dimensional spaces is discussed. Secondly, the exact, exact null, approximate, approximate null, and partial approximate controllability of stochastic linear systems in inﬁnite dimensional spaces are considered. Thirdly, the exact, exact null and impulse controllability of stochastic singular linear systems in ﬁnite dimensional spaces are investigated. Fourthly, the exact and approximate controllability of stochastic singular linear systems in inﬁnite dimensional spaces are studied. At last, the controllability and observability for a type of time-varying stochastic singular linear systems are studied by using stochastic GE-evolution operator in the sense of mild solution in Banach spaces, some necessary and sufﬁcient conditions are obtained, the dual principle is proved to be true, an example is given to illustrate the validity of the theoretical results obtained in this part, and a problem to be solved is introduced. The main purpose of this paper is to facilitate readers to fully understand the latest research results concerning the controllability of stochastic linear systems and the problems that need to be further studied, and attract more scholars to engage in this research.


Introduction
Since Kalman published the seminal paper [1], the controllability of stochastic systems has become a central problem in the study of mathematical control theory, a large number of academic papers have been published. For representative papers, see references . However, even for the controllability of stochastic linear systems, there are still many important problems to be solved. In this paper, we discuss the latest development of controllability of stochastic linear systems and raise some unsolved issues. According to the spatial dimension, equation type and time sequence, the rest of the paper is organized as follows. In Section 2, the following contents are introduced concerning the controllability of stochastic linear systems in finite dimensional spaces: (i) The L p −exact controllability and exact observability are discussed; (ii) The exact controllability by feedback controller is considered; (iii) The exact controllability of the stochastic linear systems with memory is investigated; (iv) Some theoretical results for these concepts are given and four important problems to be solved are put forward. In Section 3, the controllability of stochastic linear systems in infinite dimensional spaces is considered: (i) The null controllability is investigated by using C 0 −semigroup in the sense of mild solution in Hilbert spaces; (ii) The approximate controllability and approximate null controllability are discussed by using C 0 −semigroup in the sense of mild solution in Hilbert spaces; (iii) The partial approximate controllability is studied by using evolution operator in the sense of mild solution in Hilbert spaces; (iv) According to these theories, three problems that need to be studied are raised. In Section 4, the controllability of stochastic singular linear systems in finite dimensional spaces is dealt with: (i) The exact controllability is considered by using Gramian matrix; (ii) The exact null controllability is studied by using Gramian matrix; (iii) The impulse controllability and impulse observability are investigated in the sense of impulse solution; (iv) A problem that needs to be discussed is put forward. In Section 5, the controllability of stochastic singular linear systems in infinite dimensional spaces is studied: (i) The exact controllability for a type of time invariant systems is considered by using C 0 −semigroup in the sense of strong solution in Hilbert spaces; (ii) The exact controllability and approximate controllability for a type of time invariant systems are investigated by using GE-semigroup in the sense of mild solution in Banach and Hilbert spaces, respectively; (iii) The exact controllability and approximate controllability for a type of time-varying systems are dealt with by using GE-evolution operator in the sense of mild solution in Hilbert spaces; (iv) The exact controllability and approximate controllability for a type of time invariant systems are considered by using stochastic GE-evolution operator in the sense of mild solution in Banach spaces; (v) The exact controllability, approximate controllability, exact observability, and approximate observability for a type of time-varying systems are studied by using stochastic GE-evolution operator in the sense of mild solution in Banach spaces. Some necessary and sufficient conditions concerning these concepts are obtained, the dual principle is proved to be true, an example is given to illustrate the validity of the theoretical results obtained in this part, and a problem to be solved is raised.
The main idea of this paper is to introduce the latest progress for the controllability of stochastic linear systems and the mathematical methods applied in this field, including GE-semigroup, GE-evolution operator, stochastic GE-evolution operator and so on. The main purpose of this paper is to facilitate readers to fully understand the latest research results concerning the controllability of stochastic linear systems and the problems that need to be further studied, and attract more scholars to engage in this research. Notations.
(Ω, F, {F t }, P) is a complete probability space with filtration {F t } satisfying the usual condition (i.e., the filtration contains all P−null sets and is right continuous); all processes are {F t }−adapted; w(t) is a standard Wiener process defined on (Ω, F, {F t }, P); E denotes the mathematical expectation; R n is the n−dimensional real Euclidean space with the standard norm · R n , R n×m is the space of all (n × m) real matrices; I n ∈ R n×n denotes the identical matrix; T denotes the transpose of a vector or a matrix; H = R n , R n×m, etc, and p ∈ , Ω, H) where each element x(·) is essentially bounded; Let A be a linear operator. dom(A), ker(A) and ran(A) denote its domain, kernel and range, respectively; I denotes the identical operator. Other mathematical symbols involved in this paper will be properly explained in the discussion.

Exact Controllability of Finite Dimensional Stochastic Linear Systems
In this section, we discuss the latest development of exact controllability of finite dimensional stochastic linear systems.

The Case rankD(·) = n
In this case, we let d = 1, i.e., the Wiener process is one-dimensional. The case d > 1 can be discussed similarly. For system [A(·), C(·); B(·), D(·)], we assume the following: In this case, [D(t)D(t) T ] −1 exists and uniformly bounded. We definẽ and introduce the following controlled system: with x(t) being the state and (v(·), z(·)) being the control. For system (6), we need the following set and definition: (6) is said to be exactly null-controllable bỹ
The following results were obtained in [59].

Theorem 4 ([59]). Let Hypothesis 1 and (5) hold. Supposẽ
where > 0 is a given constant. Then the following are equivalent: where Y(·) is the adapted solution to the following stochastic linear equation:  (9) is invertible.
In the above, we have discussed the two extreme cases: either D(·) = 0 or rankD(·) = n. The case in between remains open. Therefore, we have the following open problem. Problem 1. If 0 < rankD(·) < n, what are the conditions under which system (1) can be L p −exactly controlled?

Duality and Observability Inequality
In this subsection, we introduce the dual principle for system (1). The following result was obtained in [59].

Exact Controllability by Feedback Controller
In 2018, Barbu and Tubaro consider the exact controllability by feedback controller of the following stochastic linear system in [60]: with the final target x(τ) = ξ, where A(·), B(·) ∈ C([0, ∞); R n×m ); for some γ > 0, The problem we address here is the following.
Let F ∈ C([0, τ]; R n×n ) be the solution to equation By the substitution x(t) = F(t)z(t) one transforms via Ito's formula equation (see [60] for details) (11) into stochastic differential equation In (12), we take as u the feedback controller whereα ∈ L 2 (Ω, F T , P, R), z τ ∈ L 2 (Ω, F T , P, R n ) are given and z τ = F(τ) −1 ξ; sign : R n → R n is the multivalued mapping signy = y y R n if y = 0, signy = {β ∈ R n : β R n ≤ 1} if y = 0. Arguing as in the proof of Proposition 3.1 in [60], it follows that (12) has unique absolutely continuous solution z(t). We note that if z(t) is an F t −adapted solution to (12) and (13) then x(t) = F(t)z(t) is the solution to closed loop system (11) with feedback control The following results were obtained in [60]. Theorem 8 ([60]). Let τ > 0, x 0 ∈ R n and ξ ∈ L 2 (Ω, F τ , P, R n ) be arbitrary but fixed. Then there isα ∈ L 2 (Ω, F T , P, R), such that the controller (13) steers x 0 in z τ , in time τ, with probability one.

Remark 1.
It should be noted that, under the assumption of the Theorem 8, the solution z(t) to (12) is not adapted. Therefore, the solution x(t) = F(t)z(t) to system (11) is not F t −adapted. Hence, further research is needed on Problem 2. Theorem 9 ([59]). Consider system (11) where A ∈ R n×n , B ∈ R n×m , 1 ≤ m ≤ n is time independent and satisfy the Kalman rank condition rank[B, AB, · · · , A n−1 B] = n. Assume also that d = 1, C 1 = C and C 2 = aC, C(R n ) ⊂ B(R m ) for some a ∈ R. Let τ > 0 and x 0 ∈ R n be arbitrary but fixed. Then there is an F t −adapted controller u ∈ L 2 ([0, τ], Ω, R m ) which steers x 0 in origin, in time τ, with probability one.

Remark 2.
One might suspect that the controller u steering x 0 in origin can be found in feedback form but the problem is open.

Exact Controllability of Stochastic Differential Equation with Memory
In 2020, Wang and Zhou consider the exact controllability of the following controlled stochastic linear differential equation with a memory in [61].
where x(·), u(·) are the state variable, control variable which take values in R n , R m , respectively; for any t, The following is definition of controllability for system (14).

Time Invariant Systems
In this subsection, we discuss system (14) with time invariant matrices: i.e., To consider the exact controllability of system [A, M, C; B, D], we adopt the partial controllability of controlled system as follows: For fixed τ ≥ 0 and a matrix Q ∈ R l×n , define X τ = {ξ ∈ L 2 (Ω, F τ , P, R l ) : ξ(ω) ∈ ran(Q)}.
Then, system (17) or (18) can be rewritten as In order to discuss the exact controllability of (19), we need to introduce the following stochastic linear differential equation Let The determinant of a square matrix F will be denoted by detF.
The following result was obtained in [61].

Time Varying System
In this part, we discuss time varying stochastic linear system with memory terms, and tend to to provide some criteria. In Section 2.3.1, we can present some criteria ensuring system [A, M, C; B, D] s exact controllability. However, for time variant systems even for systems without memory terms, it is difficult to list those criteria. However, for some special systems, we still can make a try.
The applicable example of this part can be found in [61] (p. 9). According to the above discussion, further research is needed on the following problems.
, Ω, R m ) in general case such that the system (14) is exactly controllable.

Problem 4.
How to discuss the L p −exact controllability for system (14)?

Controllability of Infinite Dimensional Stochastic Linear Systems
In this section, we discuss the latest development of controllability of infinite dimensional stochastic linear systems.
In 2001, Sirbu and Tessitore discussed the null controllability of the following general infinite dimensional linear stochastic differential equation in [62]: where x(·) is the state process valued in H, u(·) is the control process valued in H, A : the countable set {w 1,k , w 2,j , k, j ∈ N} consists of independent standard Wiener processes defined on the stochastic basis (Ω, F, {F t }, P). Given any Hilbert space H, We denote by As it is well known (see for instance [62]) for any initial data x 0 ∈ L 2 (Ω, F 0 , P, H) and any control u ∈ L 2 ([0, τ], Ω, F t , H) there exists a unique mild solution x ∈ C 2 ([0, τ], Ω, F t , H) of (22). When needed, we will denote the mild solution of (22) by x(·, x 0 , u) (the definition of mild solution is in the ordinary sense).
We recall a classical result on linear quadratic games for Equation (22). By Σ + (H) we denote the space of all self-adjoint, non-negative, bounded linear operators on H. Moreover, if J ⊂ R + is an interval (bounded or unbounded), we denote by C s (J; Σ + (H)) the space of all maps Q : Definition 7. We say that Y ∈ C s ((0, ∞); Σ + (H)) is a mild solution of the Riccati equation The following result was obtained in [62]: ). The following conditions are equivalent: (i) The Riccati Equation (23) has a mild solution; (ii) The state system (22) is null controllable.
We assume that F t = σ w 1,k (s), w 2,k (s), s ∈ [0, t], k ∈ N and introduce the following backward stochastic differential equation: The following duality approach was obtained in [62]: ). The following statements are equivalent: (ii) There exists a constant C τ > 0, such that for all p τ ∈ L 2 (Ω, F τ , P, H) the following observability relation holds:

Remark 4.
We can give the similar characterization for the exact controllability on the interval [0, τ]. This is equivalent to the stronger observability inequality See [62] (p. 392) for the applicable example.

Problem 5.
How about the controllability of the following system?
where A(t) : dom(A(t)) ⊆ H → H is the generator of an evolution operator in the Hilbert space consists of independent standard Wiener processes defined on the stochastic basis (Ω, F, {F t }, P).
In 2015, Shen et al. studied the exact null controllability, approximate controllability and approximate null controllability of the following linear stochastic system in [63]: where . We introduce the following backward stochastic system as our adjoint system to obtain sufficient conditions.
where A * , C * denote the adjoint operators of A, C, respectively. For any η ∈ H, system (25) admits a unique mild solution (y(t), z(t)). In (25) y(t) can be interpreted as an evolution process of the fair price, whereas z(t) as the related consumption and portfolio process.

Remark 5.
When C is unbounded, the situation will be more complex.
The closure of a set S will be denoted by S.
The following results were obtained in [63].
In 2019, Dou and Lu studied the partial approximate controllability for the following system in [64]: here A(t) is a linear operator on H, which generates strongly continuous evolution operator; is a one-dimensional standard Wiener process. In (26), y is the state process valued in H and u is the control process valued in U. In what follows, y(·, y 0 , u) denotes the mild solution to (26).
In order to discuss the partial approximate controllability of (26), we introduce the following equations and concepts.
In what follows, we denoted by (z,Z) the mild solution to (27) (the definition of mild solution is in the ordinary sense).

Hypothesis 3. Solutions to (28) fulfill the UPC for any t
Denoted by h k (x) the kth Hermite polynomial (see [64]). For k ∈ N ∪ {0}, let We have that H 0 = R, H k and H r are orthogonal subspaces of L 2 (Ω, F τ , P, R) for k = r and be an orthonormal basis of H. It is easy to see that H 0 (H) = H, H k (H) and H r (H) are orthogonal subspaces of L 2 (Ω, F τ , P, H) for k = r and (26) is said to be m−approximately controllable if for any > 0, y 0 ∈ H and
The system (26) is said to be partially approximately controllable if it is m−approximately controllable for all m ∈ N.
To study the above controllability problem, we need the following notion. (27) is said to fulfill the m-unique continuation property (m-UCP) if

Definition 12. Equation
Equation (27) is said to fulfill the partial UCP if it fulfills m-UCP for all m ∈ N.
The following results were obtained in [64]. (26) (26), how about the controllability of this system?

Controllability of Finite Dimensional Stochastic Singular Linear Systems
Stochastic singular linear systems are also called stochastic implicit systems, stochastic differential algebraic systems, stochastic descriptor systems, stochastic degenerate systems, and stochastic generalized systems, etc. Controllability is the important concept for stochastic singular linear systems. So far, however, few results have been obtained. In this section, we discuss the latest development of controllability of finite dimensional stochastic singular linear systems.
In 2013, Gashi and Pantelous studied the exact controllability of the following stochastic singular linear system in [65,66].
where L, M, C ∈ R n×n , detL = 0; B, D ∈ R n×m , x(t) is the state process valued in R n , u(t) is the state process valued in R m , w(t) is a one-dimensional standard Wiener process, (L, M) is regular, i.e., matrix pencil det(sL − M) is not identically zero (s ∈ R). Let us begin by stating the definition of exact controllability. (29) is called exactly controllable at time τ if for any x 0 ∈ R n and ξ ∈ L 2 (Ω, F τ , P, R n ), there exists at least one admissible control u(·) ∈ L 2 ([0, τ], Ω, R m ), such that the corresponding trajectory x(·) satisfies the initial condition x(0) = x 0 and the terminal condition x(τ) = ξ, a.s.
Theorem 21 ([65,66]). (i) A necessary condition for exact controllability of (29) is (ii) Let the condition (30) hold. A necessary and sufficient condition for exact controllability of (29) is Here, G τ is the Gramian matrix defined as where Φ(t) is the unique solution to the matrix stochastic differential equation For the detail see [65] (Theorem 4) and [65] (Theorem 2). In 2015, Gashi and Pantelous studied the exact controllability of the stochastic singular linear system (29) on the basis of [65,66] in [67], in which L is skew-symmetric and M is symmetric. The following result was obtained in [67]. (29) is

Theorem 22 ([67]). (i) A necessary condition for exact controllability of
(ii) Let the condition (31) hold. A necessary and sufficient condition for exact controllability of (29) is Here, G τ is the Gramian matrix defined as where Φ(t) is the unique solution to the matrix stochastic differential equation For the detail see [67] (Theorem 5). See [67] (p. 9) for practical example. In 2021, Ge and Ge considered the exact null controllability of stochastic singular linear system (29).
Here, we assume that there are a pair of nonsingular deterministic and constant matrices P 1 , Q ∈ R n×n such that the following condition is satisfied: where N ∈ R n 2 ×n 2 denotes a nilpotent matrix with order h, i.e., h = min{k : k ≥ 1, N k = 0}; Now, we consider the initial value problem (34). In the following, assume that the solution to (33) is the strong solution in the ordinary sense and (34) admits the stochastic Laplace transform (see [68]). Applying the stochastic Laplace transform to (34), we have Definition 14. (Impulse Solution) Suppose that x 2 (t) is the inverse stochastic Laplace transform of X 2 (s) obtained from (35). Then, x 2 (t) is the impulse solution to (34) in the sense of the stochastic Laplace transform, or simply, the impulse solution to (34). In this case, if x 1 (t) denotes the solution to (33), then is called the impulse solution of Equation (29).
Let Φ(t) be the solution of system (33) and (34) is said to be exactly null control-

Definition 15. (Exact Null Controllability) System
, Ω, R m ), such that (33) and (34) has a unique solution x 1 (t) x 2 (t) satisfying the initial condition x 20 in addition to the terminal condition It is obvious that if (33) and (34) is exactly null controllable, so is (33) and (34). In general, if N = 0, then (33) and (34) is not necessarily exactly null controllable. Consequently, we assume that N = 0 in the following.
The following result was obtained in [68].

Theorem 23 ([68]
). If G 1 = 0, then the necessary condition for (33) to be exactly null controllable is invertible for any real valued polynomial f (t) not identical zero.
The following result was obtained in [68].
Theorem 24 ([68]). System (38) and (39) is exactly null controllable on [0, T] if, and only if, is invertible for any real valued polynomial f (t) not identical to zero.
The practical example can be found in [68] (supplementary file).
In 2021, Ge considered the impulse controllability and impulse observability of the following stochastic singular linear system in [69]. where For a stochastic singular system, impulse terms may exist in the solution. In a practical system, the impulse terms are generally undesirable because strong impulse behavior may impede the working of the system or even damage the system. Therefore, the impulse terms must be eliminated by imposing appropriate controls. In view of this fact, in this part, the concepts of impulse controllability and impulse observability for stochastic singular system (40) is considered.

Definition 16. System
The following results were obtained in [69].

Definition 17. System
Impulse observability guarantees the ability to uniquely determine the impulse behavior in solution from information of the impulse behavior in output, and focuses on the impulse terms that take infinite values in the solution.
The following results were obtained in [69].
In order to introduce the dual principle for system (40) and (41), let us first introduce the dual system.

Definition 18. The following system
is called the dual system of the system (40) and (41).
The following dual principle was obtained in [69].
An illustrative example is given in [69] (p. 908). Furthermore, in 2021, Ge discussed the exact observability for a kind of stochastic singular linear systems in the sense of impulse solution. Some necessary and sufficient conditions were obtained. See [70] (Theorems 3.1 and 3.3) for details. Problem 8. How to discuss the L p −exact controllability for the following stochastic singular linear system?

Controllability of Infinite Dimensional Stochastic Singular Linear Systems
In this section the latest development of the controllability of infinite dimensional stochastic singular linear systems is discussed by using the methods of C 0 −semigroup, GE-semigroup, GE-evolution operator, and stochastic GE-evolution operator, respectively. Some necessary and sufficient conditions concerning the controllability are introduced.

C 0 −Semigroup Method for a Class of Time Invariant Systems in Hilbert Spaces
In 2015, Liaskos et al. studied the exact controllability of the following stochastic singular linear system by using the C 0 −semigroup method in the sense of strong solution in Hilbert spaces in [71].
In order to introduce the exact controllability, make the following assumptions and preparations.
Let H, U, K be separable and infinite dimensional Hilbert spaces, x(t) be the state process valued in H, u(t) be the control process valued in U, and w(t) be a U−valued standard Wiener process in (51). The closure of an operator S will be denoted by S. We use the notation S ⊥ for the orthogonal complement of a set S and for the restriction of the operator A to a linear subset S the symbol A| S . For the coefficients L, M, C, f , B, ξ involved in (51), the following assumptions and definitions should be considered.  From the above, the controlled stochastic singular linear system (51) has a unique strong solution x u (t), t ∈ [0, τ], which admits the form: Definition 20. Stochastic singular linear system (51) is called exactly controllable at time τ > 0, if for any ξ which is D−valued random variable P − a.s., with ξ ∈ L 2 (Ω, F 0 , PH) and for any ξ τ which is also a D−valued random variable P − a.s., with ξ ∈ L 2 (Ω, F τ , P, H), there exists at least one control u ∈ L 2 ([0, τ], Ω, K), such that the corresponding strong solution x u (t), which admits the form of (52), satisfies the initial condition x u (0) = ξ and the terminal condition x u (τ) = ξ τ .
The following result was obtain in [71].
Then there exists at least one u ∈ L 2 ([0, τ], Ω, K), such that the corresponding strong solution x u (t), which admits the form of (52), satisfies the initial condition x u (0) = ξ and the terminal condition x u (τ) = ξ τ and hence stochastic singular linear system (51) is exactly controllable.
See [71] for the details of practical example. In 2018, Liaskos et al. studied the exact controllability of the stochastic singular linear system (51) by using the C 0 −semigroup method in the sense of strong solution in Hilbert spaces in [72].
Suppose that (A 1 )-(A 6 ) hold true, and Then, the controlled stochastic singular linear system (51) has a unique strong solution x u (t), t ∈ [0, τ], which admits the form: where S 2 (t) is the C 0 −semigroup generated by the operator M 0 (L ⊥ ) −1 .

GE-Semigroup Method for a Class of Time Invariant Systems
In this subsection, we discuss the controllability of the following time invariant stochastic singular linear system by using GE-semigroup in the sense of mild solution in Banach and Hilbert spaces, respectively, where x(t) is the state process valued in H, v(t) is the control process valued in U, w(t) is the standard Wiener process on Z, x 0 ∈ L 2 (Ω, This subsection is organized as follows. Firstly, the GE-semigroup is introduced and the mild solution of (54) is obtained; Secondly, the controllability of (54) is discussed in Banach spaces; Thirdly, the controllability of (54) is discussed in Hilbert spaces. (54) In this part, the existence and uniqueness of the mild solution to system (54) are considered by GE-semigroup theory.

Definition 21 ([73-77]). Suppose {U(t) : t ≥ 0} is one parameter family of bounded linear operators in Banach space H, and A is a bounded linear operator. If
If the GE-semigroup U(t) satisfies for arbitrary x ∈ H, then it is called strongly continuous on H.

then B is called a generator of GE-semigroup U(t) induced by A.
Now, we consider the initial value problem (54).
In the following, we suppose that Proposition 1 holds true.

Controllability of System (54) in Banach Spaces
In this following, we discuss the exact (approximate) controllability of system (54) in Banach spaces. Some necessary and sufficient conditions are given.

Definition 24. (a) Stochastic singular system (54) is said to be exactly controllable on
(b) Stochastic singular system (54) is said to be approximately controllable on [0, b], if for any state x b ∈ L 2 (Ω, F b , P, D 1 ), any initial state x 0 ∈ L 2 (Ω, F 0 , P, D 1 ), and any > 0, there exists a v ∈ L 2 ([0, b], Ω, U), such that the mild solution x(t, x 0 ) satisfies In order to discuss the controllability, we introduce the following concepts. Banach space {v(t) ∈ U : Cv(t) ∈ A(D 1 )} is still denoted by U. (54) is defined as

Controllability operator
It is obvious that operator C b 0 is a bounded linear operator, and its dual The following results were obtained in [76].

Controllability of System (54) in Hilbert Spaces
In this following, we discuss the exact (approximate) controllability of system (54) in Hilbert spaces. Some necessary and sufficient conditions are given. In order to discuss the controllability, we introduce the following operator.
Theorem 40 ([77]). The necessary and sufficient condition for the stochastic singular linear system (54) to be exactly controllable on [0, b] is that one of the following conditions is true: Theorem 41 ([77]). The necessary and sufficient condition for the stochastic singular linear system (54) to be approximately controllable on [0, b] is that one of the following conditions is true:

GE-Evolution Operator Method for a Class of Time-Varying Systems
In this subsection, we discuss the controllability of the following time varying stochastic singular linear system by using GE-evolution operator in Hilbert spaces,  H)); x(t) is the state process valued in H, v(t) is the control process in U, w(t) is the stand Wiener process valued in Z, x 0 ∈ L 2 (Ω, F 0 , P, H) is a given random variable, H, U, Z are Hilbert spaces. This subsection is organized as follows. Firstly, the GE-evolution operator is introduced and the mild solution of (56) is obtained; Secondly, the controllability of (56) is discussed by GE-evolution operator in the sense of mild solution in Hilbert spaces. (56) In the following, we discuss mild solution of time varying stochastic singular system (56) according to GE-evolution operator. First of all, we recall the GE-evolution operator, and then the mild solution of (56) is given.

GE-Evolution Operator and Mild Solution of System
In the following, we suppose that Proposition 2 holds true. (56) In this part, the exact controllability and approximate controllability of system (56) are discussed by using GE-evolution operator in the sense of mild solution in Hilbert spaces. In order to discuss the controllability, we introduce the following concepts.

Controllability of System
Hilbert space {v(t) ∈ U : (56) are defined as respectively. It is obvious that operators C b 0 and G b c are bounded linear operators, and the dual

Stochastic GE-Evolution Operator Method for a Class of Time Invariant Systems
In this subsection, we discuss the controllability of the following time varying stochastic singular linear system by using stochastic GE-evolution operator in Banach spaces, where x(t) is the state process valued in H, v(t) is the control process valued in U, w(t) is the one-dimensional standard Wiener process, x 0 ∈ L 2 (Ω, F 0 , P, H) is a given random variable, H, U are Banach spaces; A, D ∈ B(H), C ∈ B(U, H), B : dom(B) ⊆ H → H is a linear operator. The organization of this subsection is as follows. Firstly, the concept of stochastic GE-evolution operator is introduced, and the mild solution to system (58) is given by stochastic GE-evolution operator. Secondly, The exact controllability and approximate controllability of (58) are discussed by stochastic GE-evolution operator in the sense of mild solution in Banach spaces, respectively. (58) In the following, the stochastic GE-evolution operator is introduced, and the mild solution of system (58) is give by stochastic GE-evolution operator.

Stochastic GE-Evolution Operator and Mild Solution of System
if it has the following properties: (iii) S(s, s) = S 0 , 0 ≤ s ≤ b, and S(t, r)AS(r, s) = S(t, s) for any 0 ≤ s ≤ r ≤ t ≤ b, where S 0 ∈ B(H) is a steady operator independent of s; (iv) For any ξ ∈ H, (t, s) → S(t, s)ξ is mean square continuous from ∆ T into H.
In the following, we always suppose that B is a generator of GE-semigroup U(t) induced by A. Now, we consider the mild solution of stochastic singular linear system (58).

Definition 31.
We say that stochastic GE-evolution operator S(t, s) induced by A is related to the linear homogeneous equation if x(t) = S(t, s)Ax 0 is the mild solution to (60) with x(s) = S(s, s)Ax 0 = x 0 for arbitrary x 0 ∈ L 2 (Ω, F 0 , P, D 1 ).
In the following, we suppose that there exists a stochastic GE-evolution operator S(t, s) induced by A related to (60) and Lemma 2 holds true. Furthermore, we suppose that the following estimates hold for any 0 ≤ s ≤ t ≤ b and ξ ∈ L 2 (Ω, F s , P, D 1 ) : (Ω,F s ,P,D 1 ) . We can obtain the following theorem.
Theorem 45 ([81]). The mild solution x(t, x 0 ) to (58) can be written in the form (58) In the following, we discuss the exact and approximate controllability of stochastic singular linear system (58) by using stochastic GE-evolution operator theory, some criteria are obtained. In order to discuss the controllability, we introduce the following concepts.
The practical example can be found in [81] if there is a need.

Stochastic GE-Evolution Operator Method for a Class of Time-Varying Systems
In this subsection, we study the controllability and observability of the following time varying stochastic singular linear system by using stochastic GE-evolution operator in Banach spaces, where v(t) is the state process valued in Y 1 , u(t) is the control process valued in Y 2 , w(t) is the one-dimensional standard Wiener process, v 0 ∈ L 2 (Ω, F 0 , P, Y 1 ) is a given random variable, x(t) is the output process valued in Y 3 , Y 1 , Y 2 , Y 3 are Banach spaces; , O 5 (t) are deterministic and constant operators; This subsection is organized as follows. Firstly, the mild solution of (62) is obtained by stochastic GE-evolution operator; Secondly, the exact controllability of (62) is discussed by using stochastic GE-evolution operator in the sense of mild solution in Banach spaces; Thirdly, the approximate controllability of (62) is discussed by using stochastic GE-evolution operator in the sense of mild solution in Banach spaces; Fourthly, the observability of (62) is studied, and the dual principle is given; At last, we give an example to illustrate the validity of the theoretical results obtained in this subsection. (62) In this part, we always suppose that O 2 (t) is a generator of GE-evolution operator V(t, s) induced by O 1 and

Mild Solution of System
Now, we consider the mild solution of time varying stochastic singular linear Equation (62). (62) has a unique mild solution, which is given by (63)

Lemma 3. Time varying stochastic singular Equation
, Ω, D)), and (V 0 O 2 (t))| D satisfies following assumptions: (P 1 ) For t ∈ [0, b], (λI + (V 0 O 2 (t))| D ) −1 exists for all λ with Reλ ≤ 0 and there is a constant M, such that (P 2 ) There exist constants L and 0 < α ≤ 1, such that Proof. First of all, according to Theorem 6.1 of [82] (see P.150 of [82]), we have that For any ξ(t) ∈ Y 11 define and Therefore, if b is sufficient small, P 1 is a contraction and it is easy to see that its unique fixed point can be identified as the mild solution to time varying stochastic singular Equation (62). The case of general b can be handled in a standard way.
Theorem 50. Suppose that stochastic GE-evolution operator G(t, s) induced by O 1 is related to the linear homogeneous time varying stochastic singular equation Lemma 3 holds true, and the following estimates hold for any 0 ≤ s ≤ t ≤ b and ξ ∈ L 2 (Ω, F s , P, D) : Then, the mild solution v(t, v 0 ) to time varying stochastic singular Equation (62) can be written in the form Proof. Since G(t, 0)O 1 v 0 and G(t, s)O 3 (s)u(s) are mild solutions of time varying stochastic singular Equation (64) with v(0) = v 0 and v(s) = G(s, s)O 3 (s)u(s), respectively, we have that We have to prove that the process v(t, v 0 ) in (65) is a solution to the integral Equation (63). By the representation of v(τ, v 0 ), we have where the stochastic Fubini theorem is given in Theorem 4.33 of [83]. Therefore, which proves (63).
In the following, we always assume that time varying stochastic singular Equation (62) has a unique mild solution in the form of (65).
In order to obtain the criteria of controllability, the following concepts are introduced.
associated with time varying stochastic singular Equation (62) is defined as It is obvious that operator Q b C is a bounded linear operator, and its dual . where y * ∈ L 2 (Ω, F b , P, (D) * ). (62) In this part, we discuss the exact controllability of time varying stochastic singular Equation (62) by stochastic GE-evolution operator theory, some criteria are obtained. (62) is called to be exactly controllable on

Definition 34. Time varying stochastic singular Equation
such that the mild solution v(t, v 0 ) to time varying stochastic singular Equation (62) satisfies From the Definition 34, we can obtain the following theorem immediately.
Proof. (a) ⇒ (b) Notice that (a) implies that Q b * C is injective. To prove that Q b * C has closed range, assume that Q b * C y * n is a Cauchy sequence in L 2 ([0, b], Ω, Y * 2 ), then (a) implies that y * n is a Cauchy sequence in L 2 (Ω, F b , P, (D) * ). Since Q b * C is a bounded linear operator, if lim n→+∞ y * n = y * , then lim n→+∞ Q b * C y * n = Q b * C y * and so Q b * C has closed range. (b)⇒(a). (b) shows that Q b * C has an algebraic inverse with domain equal to ran(Q b * C ). Since ran(Q b * C ) is closed, it is a Banach space under the norm of L 2 ([0, b], Ω, Y * 2 ), i.e., By Corollary A.3.50 of [84], we have that (Q b * C ) −1 is bounded on this range, i.e., there exists a γ > 0, such that for every u * ∈ ran(Q b * C ). Substituting u * = C T * 0 y * proves (a). It remains to show that (a) is equivalent to exact controllability of time varying stochastic singular Equation (62).
Necessity. Assume that time varying stochastic singular Equation (62) is exactly controllable. By Theorem 51, we have ran(Q b C ) = L 2 (Ω, F b , P, D). If Q b C is a one to one operator, then (Q b C ) −1 exists on L 2 (Ω, F b , P, D). According to the continuity of operator Q b C we have that (Q b C ) −1 is a closed operator. From the closed graph theorem, we obtain that (Q b C ) −1 is a bounded linear operator on L 2 (Ω, F b , P, D), i.e., This implies that there exists γ b > 0, such that Assume y * ∈ L 2 (Ω, F b , P, (D) * ), then Therefore, for all y 0 ∈ L 2 (Ω, F b , P, D), we find that (66), we obtain that . This implies that (a) holds.
If Q b C is not a one to one operator, then A factor space is defined as follows ,Ω,Y 2 ) . If we define operator 1 is a bijective operator. It can be seen from the above proof that According to the definition of Y 21 and Q b 1 , we obtain This implies that (a) holds. Sufficiency. Assume (a). It is need to prove that if y ∈ L 2 (Ω, F b , P, D), then y ∈ ranQ b C .
, Ω, Y * 2 )). For y ∈ L 2 (Ω, F b , P, D), we can define a functional f on ranQ b * C satisfying This implies that f is linear for Q b * C g * . According to (a), if and lim Therefore, f is continuous linear functional on By Hahn-Banach theorem, we have that f can be extended as a continuous linear functional on According to (67) and (68), we obtain that for every g * ∈ L 2 (Ω, F b , P, (D) * ), Hence y = Q b C u, i.e., ran(Q b C ) = L 2 (Ω, F b , P, D). From Theorem 51, time varying stochastic singular Equation (62) is exactly controllable. (62) In this section, we discuss the approximate controllability of time varying stochastic singular Equation (62). Some necessary and sufficient conditions are obtained. (62) is called to be approximately controllable on [0, b], if for any state v b ∈ L 2 (Ω, F b , P, D), any initial state v 0 ∈ L 2 (Ω, F 0 , P, D), and any > 0, existence u ∈ L 2 ([0, b], Ω, Y 2 ) makes that the mild solution v(t, v 0 ) to time varying stochastic singular Equation (62) satisfies

Definition 35. Time varying stochastic singular Equation
It is obvious that the necessary and sufficient conditions for the time varying stochastic singular Equation (62) to be approximately controllable on [0, b] are Theorem 53. Time varying stochastic singular Equation (62) is approximate controllable on [0, b] if, and only if, one of the following conditions holds: Proof. It is obvious that (a) is equivalent to (b). We only need to prove that (b) is equivalent to approximate controllability of time varying stochastic singular linear Equation (62).

Observability
Consider the following time varying stochastic singular equation and its dual time varying stochastic singular equation For the time varying stochastic singular Equation (70) is defined by In the case of Definition 36, the state v 0 can be uniquely and continuously constructed from the knowledge of the output x(t) in L 2 ([0, b], Ω, Y 3 ). In the case of Definition 37, the state v 0 can be uniquely constructed from the knowledge of the output x(t) in L 2 ([0, b], Ω, Y 3 ).
We can obtain the following dual principle. Proof. Here, we only prove the case of exact observability. Since

An Illustrative Example
In this part, we give an example to illustrate the effectiveness of the obtained results. According to [72], in input-output economics, many models were established to describe the real economics. The economics Leontief dynamic input-output model can be extended as an ordinary differential equation of the form: in Banach space Y 1 , where O 1 ∈ B(Y 1 ) and O 2 (t) : dom(O 2 (t)) ⊆ Y 1 → Y 1 is a linear and possibly unbounded operator, O 3 (t), O 5 (t) ∈ P([0, b], B(Y 1 )), while v(t) and u(t) are state process and control process valued in Y 1 , respectively, for t ≥ 0. However, in reality, there are many unpredicted parameters and different types of uncertainty that have not been implemented in the mathematical modelling process of this equation. Nonetheless, according to [85,86], we can consider a stochastic version of the singular Equation (72) with the one-dimensional standard Wiener process w(t) used to model the uncertainties of the form: where O 4 (t) ∈ P([0, b], B(Y 1 )). This stochastic version of the input-output model is a time varying stochastic singular equation in Banach space Y 1 of the form (62). We consider the following unforced time varying stochastic singular equation, i.e., u(t) = 0 in time varying stochastic singular Equation (73): Time varying stochastic singular Equation (74) is the form of time varying stochastic singular linear Equation (70). In what follows, we will verify the effectiveness of Corollary 3. If for some concrete engineering practice, the following data are taken in time varying stochastic singular Equation (74): where U 1 , U 2 are identical operators in Banach spaces Y 11 , Y 12 , respectively. Time varying stochastic singular Equation (74) can be written as x(t) = 7(t + 1) 2 U 1 0 0 0 where v 1 (t) v 2 (t) ∈ Y 11 ⊕ Y 12 = Y 1 . We can find that D = Y 11 . According to [87], we can obtain G(t, s) = exp[− 3 2 t 2 − t + 3 2 s 2 + s + t s (2r) 1/2 w(r)ds]U 1 0 0 0 .
In this section, we have discussed the controllability of some types of stochastic singular linear systems. However, the following problems still need to be studied.

Problem 9.
How about the controllability of the following system? the countable set {w 1,k , w 2,j , k, j ∈ N} consists of independent standard Wiener processes defined on the stochastic basis (Ω, F, {F t }, P).

Conclusions
We have introduced the latest progress in controllability of stochastic linear systems and put forward some problems that need to be further studied, which includes stochastic linear systems in finite dimensional spaces, stochastic linear systems in infinite dimensional spaces, stochastic singular linear systems in finite dimensional spaces, and stochastic singular linear systems in infinite dimensional spaces. The controllability and observability for a type of time-varying stochastic singular linear systems have been studied by using stochastic GE-evolution operator in the sense of mild solution in Banach spaces, some necessary and sufficient conditions have been obtained, the dual principle has been proved to be true, an example has been given to illustrate the validity of the theoretical results obtained in this part. Readers can easily and comprehensively understand the latest progress concerning the controllability of stochastic linear systems and further problems to be solved. The next research direction is how to solve these problems.