**3. 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]:

$$dx(t) = [Ax(t) + Bu(t)]dt + \sum\_{k=1}^{\infty} \mathbb{C}\_k x(t) dw\_{1,k}(t) + \sum\_{j=1}^{\infty} D\_j u(t) dw\_{2,j}(t), \\ x(0) = x\_{0,} \tag{22}$$

where *x*(·) is the state process valued in *H*, *u*(·) is the control process valued in *H*, *A* : dom(*A*) ⊆ *H* → *H* is the infinitesimal generator of a *C*0−semigroup in *H* (the Hilbert space with product < ·, · >), *B* ∈ *B*(*H*) (the space of all bounded linear operators on *H*); *Ck*, *Dk* <sup>∈</sup> *<sup>B</sup>*(*H*) for each *<sup>i</sup>* <sup>∈</sup> <sup>N</sup> and

$$\sum\_{k=1}^{\infty} \left\|{\mathbb{C}\_k}\right\|\_{B(H)}^2 < +\infty \,\,\sum\_{k=1}^{\infty} \left\|{D\_k}\right\|\_{B(H)}^2 < +\infty \,\,\forall$$

the countable set {*w*1,*k*, *<sup>w</sup>*2,*j*, *<sup>k</sup>*, *<sup>j</sup>* <sup>∈</sup> <sup>N</sup>} consists of independent standard Wiener processes defined on the stochastic basis (Ω, *F*, {*Ft*}, *P*).

Given any Hilbert space *<sup>H</sup>*, We denote by *<sup>C</sup>*2([0, *<sup>τ</sup>*], <sup>Ω</sup>, *Ft*, *<sup>H</sup>*) the space of all *<sup>ξ</sup>* <sup>∈</sup> *L*2([0, *τ*], Ω, *Ft*, *H*) such that *ξ* has a modification in *C*([0, *τ*]; *L*2(Ω, *F*, *P*, *H*)), where

*<sup>L</sup>*2(Ω, *<sup>F</sup>*, *<sup>P</sup>*, *<sup>H</sup>*) = {*<sup>x</sup>* : *<sup>x</sup>* is *<sup>F</sup>*−adapted process valued in *<sup>H</sup>* with norm

$$(E(\|\|\mathbf{x}\|\_H^2))^{1/2} < +\infty[.$$

As it is well known (see for instance [62]) for any initial data *<sup>x</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, *<sup>H</sup>*) and any control *<sup>u</sup>* <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>τ</sup>*], <sup>Ω</sup>, *Ft*, *<sup>H</sup>*) there exists a unique mild solution *<sup>x</sup>* <sup>∈</sup> *<sup>C</sup>*2([0, *<sup>τ</sup>*], <sup>Ω</sup>, *Ft*, *<sup>H</sup>*) 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).

**Definition 6.** *For <sup>τ</sup>* <sup>&</sup>gt; <sup>0</sup>*, the state system (22) is <sup>τ</sup>*−*null controllable if for each <sup>x</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, *<sup>H</sup>*) *there exists <sup>u</sup>* <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>τ</sup>*], <sup>Ω</sup>, *Ft*, *<sup>H</sup>*) *such that the solution <sup>x</sup>*(*τ*, *<sup>x</sup>*0, *<sup>u</sup>*) = 0, *<sup>P</sup>*−*almost surely. Moreover, the system is null controllable if it is τ*−*null controllable for each τ* > 0*.*

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 *<sup>J</sup>* <sup>⊂</sup> <sup>R</sup><sup>+</sup> is an interval (bounded or unbounded), we denote by *Cs*(*J*; <sup>Σ</sup>+(*H*)) the space of all maps *<sup>Q</sup>* : *<sup>J</sup>* <sup>→</sup> <sup>Σ</sup>+(*H*), such that *<sup>Q</sup>*(·)*<sup>v</sup>* is continuous in *<sup>H</sup>* for every *<sup>v</sup>* <sup>∈</sup> *<sup>H</sup>*.

**Definition 7.** *We say that Y* <sup>∈</sup> *Cs*((0, <sup>∞</sup>); <sup>Σ</sup>+(*H*)) *is a mild solution of the Riccati equation*

$$\frac{dY(t)}{dt} = A^\*Y(t) + Y(t)A - Y(t)B[I + \sum\_{j=1}^{\infty} D\_j^\*Y(t)D\_j]^{-1}B^\*Y(t)$$

$$+\sum\_{j=1}^{\infty} \mathbf{C}\_{j}^{\*} \mathbf{Y}(t) \mathbf{C}\_{j} + \mathbf{S}\_{\prime} \mathbf{Y}(0) = +\infty \tag{23}$$

*if*

*(i) For each δ* ∈ (0, +∞),*Y*(· + *δ*) *is a mild solution of*

$$\begin{aligned} \frac{dY(t)}{dt} &= A^\*Y(t) + Y(t)A - Y(t)B[I + \sum\_{j=1}^{\infty} D\_j^\*Y(t)D\_j]^{-1}B^\*Y(t) \\\\ &+ \sum\_{j=1}^{\infty} \mathcal{C}\_j^\*Y(t)\mathcal{C}\_j + S, Y(0) = Y(\delta) \in \Sigma^+(H); \\ &\dots \end{aligned}$$

*(ii)* lim(*t*,*z*)→(0,*v*) < *<sup>Y</sup>*(*t*)*z*, *<sup>z</sup>* >= +<sup>∞</sup> *for all v* ∈ *<sup>H</sup>*, *<sup>v</sup>* = <sup>0</sup>*.*

The following result was obtained in [62]:

**Theorem 15** ([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 *Ft* <sup>=</sup> *<sup>σ</sup> <sup>w</sup>*1,*k*(*s*), *<sup>w</sup>*2,*k*(*s*),*<sup>s</sup>* <sup>∈</sup> [0, *<sup>t</sup>*], *<sup>k</sup>* <sup>∈</sup> <sup>N</sup> and introduce the following backward stochastic differential equation:

$$dp(t) = -[A^\*p(t) + \sum\_{k=1}^{\infty} \mathbb{C}\_k^\* q\_{1,k}(t)]dt + \sum\_{k=1}^{\infty} q\_{1,k}(t)dw\_{1,k}(t)$$

$$+ \sum\_{j=1}^{\infty} q\_{2,k}(t)dw\_{2,j}(t), p(\tau) = p\_\tau.$$

The following duality approach was obtained in [62]:

**Theorem 16** ([62])**.** *The following statements are equivalent:*

*(i) System (1) is τ*−*null controllable;*

*(ii) There exists a constant <sup>C</sup><sup>τ</sup>* <sup>&</sup>gt; <sup>0</sup>*, such that for all <sup>p</sup><sup>τ</sup>* <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>τ*, *<sup>P</sup>*, *<sup>H</sup>*) *the following observability relation holds:*

$$\|\|p(0)\|\|\_{L^{2}(\Omega,\mathbb{F}\_{0},P,H)}^{2} \leq \mathbb{C}\_{\tau}E\int\_{0}^{\tau} \|\|B^{\*}p(t) + \sum\_{k=1}^{\infty}D\_{k}^{\*}q\_{2,k}(t)\|\_{L^{2}(\Omega,\mathbb{F}\_{0},P,H)}^{2}dt.$$

**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*

$$\|\|p(\mathbf{r})\|\|\_{L^{2}(\Omega,\mathbb{F}\_{\mathbf{r}},\mathbf{P},\mathbf{H})}^{2} \leq C\_{\mathsf{T}}E\int\_{0}^{\mathsf{T}} \|\|\boldsymbol{B}^{\*}\boldsymbol{p}(\mathbf{t}) + \sum\_{k=1}^{\infty}D\_{k}^{\*}q\_{2,k}(\mathbf{t})\|\_{L^{2}(\Omega,\mathbb{F}\_{\mathbf{r}},\mathbf{P},\mathbf{H})}^{2}d\mathbf{t}.$$

See [62] (p. 392) for the applicable example.

**Problem 5.** *How about the controllability of the following system?*

$$\begin{aligned} dx(t) &= [A(t)x(t) + B(t)u(t)]dt + \sum\_{k=1}^{\infty} \mathbb{C}\_k(t)x(t)dw\_{1,k}(t) \\ &+ \sum\_{j=1}^{\infty} D\_j(t)u(t)dw\_{2,j}(t), x(0) = x\_{0'} \end{aligned}$$

*where A*(*t*) : dom(*A*(*t*)) ⊆ *H* → *H is the generator of an evolution operator in the Hilbert space H, B*(*t*) : dom(*B*(*t*)) ⊂ *U* → *H is unbounded, U is a Hilbert space; Ck*(*t*) ∈ *P*([0, *τ*], *B*(*H*)), *Dk*(*t*) <sup>∈</sup> *<sup>P</sup>*([0, *<sup>τ</sup>*], *<sup>B</sup>*(*U*, *<sup>H</sup>*)), *for each <sup>i</sup>* <sup>∈</sup> <sup>N</sup>*, <sup>P</sup>*([0, *<sup>τ</sup>*], *<sup>B</sup>*(*U*, *<sup>H</sup>*)) = {*C*(·) <sup>∈</sup> *<sup>B</sup>*(*U*, *<sup>H</sup>*) : *<sup>C</sup>*(·)*<sup>z</sup> is continuous for every z* ∈ *U and* sup0≤*t*≤*τ*-*C*(*t*)-*<sup>B</sup>*(*U*,*H*) < +∞}; *and*

$$\sum\_{k=1}^{\infty} \sup\_{0 \le t \le \tau} \|\mathbb{C}\_k(t)\|\_{\mathcal{B}(H)}^2 < +\infty, \sum\_{k=1}^{\infty} \sup\_{0 \le t \le \tau} \|D\_k(t)\|\_{\mathcal{B}(\mathcal{U},H)}^2 < +\infty.$$

*B*(*U*, *H*) *denotes the set of all bounded linear operators from U to H; the countable set*

$$\{w\_{1,k}, w\_{2,j}, k, j \in \mathbb{N}\}$$

*consists of independent standard Wiener processes defined on the stochastic basis* (Ω, *F*, {*Ft*}, *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]:

$$dx(t) = [Ax(t) + Bu(t)]dt + \mathbb{C}x(t)dw(t), \\ x(0) = \mathbf{x}\_{0\prime} \tag{24}$$

where *x*(*t*) is the state process valued in *H*, *u*(*t*) is the control process valued in *U*, *x*(0) = *<sup>x</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, *<sup>H</sup>*), *<sup>w</sup>*(*t*) is a standard Wiener process valued in *<sup>W</sup>*, and *<sup>A</sup>* : *<sup>D</sup>*(*A*) <sup>⊆</sup> *H* → *H* is the infinitesimal generator of a *C*0−semigroup on *H*; *B* ∈ *B*(*U*, *H*), *C* ∈ *B*(*H*, *B*(*W*, *H*)); *H*, *U*, *W* are separable Hilbert spaces. System (24) admits a unique mild solution *<sup>x</sup>*(*t*, *<sup>x</sup>*0, *<sup>u</sup>*) <sup>∈</sup> *<sup>L</sup>*<sup>2</sup> *<sup>F</sup>*(Ω; *C*([0, *τ*]; *H*)).

We introduce the following backward stochastic system as our adjoint system to obtain sufficient conditions.

$$dy(t) = -[A^\*y(t) + C^\*z(t)]dt + z(t)dw(t), \\ y(\tau) = \eta,\tag{25}$$

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*.

**Definition 8.** *For <sup>τ</sup>* <sup>&</sup>gt; <sup>0</sup>*, system (24) is null controllable at <sup>τ</sup> if for each <sup>x</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, *<sup>H</sup>*), *there exists u* ∈ *U such that x*(*τ*, *x*0, *u*) = 0, *P* − *a*.*s*.

*System (24) is approximately controllable at <sup>τ</sup> if for each <sup>x</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, *<sup>H</sup>*), *there exists <sup>u</sup>* <sup>∈</sup> *U such that* {*x*(*τ*, *<sup>x</sup>*0, *<sup>u</sup>*), *<sup>u</sup>* <sup>∈</sup> *<sup>U</sup>*} <sup>=</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>τ*, *<sup>P</sup>*, *<sup>H</sup>*), *<sup>P</sup>* <sup>−</sup> *<sup>a</sup>*.*s*.

*System (24) is approximately null controllable at <sup>τ</sup> if for each <sup>x</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, *<sup>H</sup>*), *there exists u* ∈ *U such that x*(*τ*, *x*0, *u*) *can be arbitrarily close to 0, P* − *a*.*s*.

The following results were obtained in [63].

**Theorem 17** ([63])**.** *System (24) is null controllable if, and only if, there exists a positive constant c, such that*

$$\|\|y(0)\|\|\_{L^2(\Omega,\mathbb{F}\_0,P,H)}^2 \le c \int\_0^\tau \|\|B^\*y(s)\|\|\_{L^2(\Omega,\mathbb{F}\_\tau,P,H)}^2 ds.$$

**Theorem 18** ([63])**.** *Let* (*y*(*t*), *z*(*t*)) *denote the solution of (25).*

*(i) System (24) is approximate controllable at τ if and only if for every* (*y*(*t*), *z*(*t*)) *such that B*∗*y*(*t*) = 0 *we have* (*y*(*t*), *z*(*t*)) = 0, *t* ∈ [0, *τ*], *P* − *a*.*s*.;

*(ii) System (24) is approximate null controllable at τ if, and only if, for every y*(*t*) *such that B*∗*y*(*t*) = 0 *we have y*(0) = 0, *t* ∈ [0, *τ*], *P* − *a*.*s*.;

The illustrative example can be found in [63] (p. 601).

**Problem 6.** *If A*, *B*, *C are A*(*t*), *B*(*t*), *C*(*t*)*, respectively, and A*(*t*) : dom(*A*(*t*)) ⊆ *H* → *H is the generator of an evolution operator; B*(*t*), *C*(*t*) *are unbounded in (24), how about the controllability of this system?*

In 2019, Dou and Lu studied the partial approximate controllability for the following system in [64]:

$$A\,dy(t) - A(t)y(t)dt = (A\_1(t)y(t) + Bu(t))dt$$

$$+ A\_2(t)y(t)dw(t), t \in (0, \tau], y(0) = y\_{0\prime} \tag{26}$$

here *A*(*t*) is a linear operator on *H*, which generates strongly continuous evolution operator; *<sup>A</sup>*1(*t*), *<sup>A</sup>*2(*t*) <sup>∈</sup> *<sup>L</sup>*∞([0, *<sup>τ</sup>*]; *<sup>B</sup>*(*H*)), *<sup>B</sup>* <sup>∈</sup> *<sup>B</sup>*(*U*, *<sup>H</sup>*); *<sup>U</sup>*, *<sup>H</sup>* are separable Hilbert spaces; *<sup>u</sup>* <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>τ</sup>*], *Ft*, *<sup>P</sup>*, *<sup>U</sup>*), *<sup>y</sup>*<sup>0</sup> <sup>∈</sup> *<sup>H</sup>*, *<sup>w</sup>*(*t*) is a one-dimensional standard Wiener process. In (26), *<sup>y</sup>* 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.

$$dz(t) - A(t)^{\*}z(t)dt = -(A\_1^{\*}z(t) + A\_2^{\*}Z(t))dt + Z(t)dw(t), \\ t \in (0, \tau], \\ z(\tau) = z\_{\tau}, \tag{27}$$

where the final datum *<sup>z</sup><sup>τ</sup>* <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>τ*, *<sup>P</sup>*, *<sup>H</sup>*).

In what follows, we denoted by (z,Z) the mild solution to (27) (the definition of mild solution is in the ordinary sense).

**Definition 9.** *We say that (27) fulfills the unique continuation property (UCP) with respect to B*<sup>∗</sup> *if z* = *Z* = 0 *in H for a.e.* (*t*, *ω*) ∈ [0, *τ*] × Ω*, provided that B*∗*z* = 0 *in U for a.e.* (*t*, *ω*) ∈ [0, *τ*] × Ω*.*

$$
\bar{z}(t) + A(t)^\* \bar{z}(t) = -A\_1(t)^\* \bar{z}(t), \\
t \in [t\_0, \tau], \bar{z}(\tau) = \bar{z}\_{\tau}, \tag{28}
$$

where the final data*z*˜*<sup>τ</sup>* ∈ *H* and *t*<sup>0</sup> ∈ [0, *τ*].

**Definition 10.** *We say that (28) fulfills UCP if z*˜ = 0 *in H for a.e. t* ∈ [*t*0, *τ*]*, provided that B*∗*z*˜ = 0 *for a.e. t* ∈ [*t*0, *τ*]*.*

**Hypothesis 3.** *Solutions to (28) fulfill the UPC for any t*<sup>0</sup> ∈ [0, *τ*]*.*

Denoted by *hk*(*x*) the *<sup>k</sup>*th Hermite polynomial (see [64]). For *<sup>k</sup>* <sup>∈</sup> <sup>N</sup> ∪ {0}, let

$$H\_k = \text{span}\{h\_k(\int\_0^\tau l(t)dw(t)) : l \in L^2([0,\tau], \mathbb{R}), \|l\|\_{L^2([0,\tau],\mathbb{R})} = 1\}.$$

We have that *<sup>H</sup>*<sup>0</sup> = R, *Hk* and *Hr* are orthogonal subspaces of *<sup>L</sup>*2(Ω, *<sup>F</sup>τ*, *<sup>P</sup>*, R) for *k* = *r* and

$$L^2(\Omega, F\_{\tau\prime}P\_\prime \mathbb{R}) = \oplus\_{k=0}^{\infty} H\_k \cdot \mathbb{R}$$

For *<sup>k</sup>* <sup>∈</sup> <sup>N</sup> ∪ {0}, denote by *Hk*(*H*) the closed subspace of *<sup>L</sup>*2(Ω, *<sup>F</sup>τ*, *<sup>P</sup>*, *<sup>H</sup>*) generated by *H* valued random variable of the form ∑*<sup>r</sup> <sup>j</sup>*=<sup>1</sup> *ljvj*(*<sup>r</sup>* <sup>∈</sup> <sup>N</sup>), *lj* <sup>∈</sup> *Hk*, and *vj* <sup>∈</sup> *<sup>H</sup>*. Let {*ej*}<sup>∞</sup> *j*=1 be an orthonormal basis of *H*. It is easy to see that

$$H\_k(H) = \{ \sum\_{j=1}^{\infty} l\_j e\_j \, : \, \{ l\_j \}\_{j=1}^{\infty} \subset H\_k, E \sum\_{j=1}^{\infty} |l\_j|^2 < +\infty \}.$$

*<sup>H</sup>*0(*H*) = *<sup>H</sup>*, *Hk*(*H*) and *Hr*(*H*) are orthogonal subspaces of *<sup>L</sup>*2(Ω, *<sup>F</sup>τ*, *<sup>P</sup>*, *<sup>H</sup>*) for *<sup>k</sup>* <sup>=</sup> *<sup>r</sup>* and

$$L^2(\Omega, F\_{\tau \prime} P\_\prime H) = \oplus\_{k=0}^{\infty} H\_k(H).$$

Write

$$L\_m^2(\Omega, F\_{\tau}, P, H) = \oplus\_{k=0}^m H\_k(H).$$

Clearly *L*<sup>2</sup> *<sup>m</sup>*(Ω, *Fτ*, *P*, *H*) is a closed subspace of *L*2(Ω, *Fτ*, *P*, *H*). Denote by Γ*<sup>m</sup>* the orthogonal projection from *L*2(Ω, *Fτ*, *P*, *H*) to *L*<sup>2</sup> *<sup>m</sup>*(Ω, *Fτ*, *P*, *H*).

**Definition 11.** *System (26) is said to be m*−*approximately controllable if for any* > 0, *y*<sup>0</sup> ∈ *H and <sup>y</sup>*<sup>1</sup> <sup>∈</sup> *<sup>L</sup>*<sup>2</sup> *<sup>m</sup>*(Ω, *<sup>F</sup>τ*, *<sup>P</sup>*, *<sup>H</sup>*)*, there is a control <sup>u</sup>* <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>τ</sup>*], <sup>Ω</sup>, *<sup>U</sup>*)*, such that the corresponding mild solution fulfills that* -Γ*m*(*y*(*τ*, *y*0, *u*) − *y*1)-*<sup>L</sup>*2(Ω,*Fτ*,*P*,*H*) < .

*The system (26) is said to be partially approximately controllable if it is m*−*approximately controllable for all m* <sup>∈</sup> <sup>N</sup>*.*

To study the above controllability problem, we need the following notion.

**Definition 12.** *Equation (27) is said to fulfill the m-unique continuation property (m-UCP) if <sup>z</sup>* <sup>=</sup> *<sup>Z</sup>* <sup>=</sup> <sup>0</sup> *in <sup>H</sup> for a.e.* (*t*, *<sup>ω</sup>*) <sup>∈</sup> [0, *<sup>τ</sup>*] <sup>×</sup> <sup>Ω</sup>*, provided that <sup>z</sup><sup>τ</sup>* <sup>∈</sup> *<sup>L</sup>*<sup>2</sup> *<sup>m</sup>*(Ω, *Fτ*, *P*, *H*) *and B*∗*z* = 0 *in U for a.e.* (*t*, *ω*) ∈ [0, *τ*] × Ω*.*

*Equation (27) is said to fulfill the partial UCP if it fulfills m-UCP for all m* <sup>∈</sup> <sup>N</sup>*.*

The following results were obtained in [64].

**Theorem 19** ([64])**.** *(i) System (26) is m-approximately controllable if and only if (27) fulfills the m-UCP;*

*(ii) System (26) is partially approximately controllable if and only if (27) fulfills the partial UCP.*

**Theorem 20** ([64])**.** *Suppose that Hypothesis 3 holds. Then system (26) is partially approximate controllable.*

**Problem 7.** *If B is B*(*t*)*, and A*1(*t*), *B*(*t*), *A*2(*t*) *are unbounded in (26), how about the controllability of this system?*

## **4. 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].

$$Ldx(t) = [Mx(t) + Bu(t)]dt + [\mathbb{C}x(t) + Du(t)]dw(t), \ge (0 = \mathbf{x}\_0),\tag{29}$$

where *<sup>L</sup>*, *<sup>M</sup>*, *<sup>C</sup>* <sup>∈</sup> <sup>R</sup>*n*×*n*, det*<sup>L</sup>* <sup>=</sup> 0; *<sup>B</sup>*, *<sup>D</sup>* <sup>∈</sup> <sup>R</sup>*n*×*m*, *<sup>x</sup>*(*t*) is the state process valued in <sup>R</sup>*n*, *<sup>u</sup>*(*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* <sup>−</sup> *<sup>M</sup>*) is not identically zero (*<sup>s</sup>* <sup>∈</sup> <sup>R</sup>). Let us begin by stating the definition of exact controllability.

**Definition 13.** *System (29) is called exactly controllable at time <sup>τ</sup> if for any <sup>x</sup>*<sup>0</sup> <sup>∈</sup> <sup>R</sup>*<sup>n</sup> and <sup>ξ</sup>* <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>τ*, *<sup>P</sup>*, <sup>R</sup>*n*)*, there exists at least one admissible control <sup>u</sup>*(·) <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>τ</sup>*], <sup>Ω</sup>, <sup>R</sup>*m*)*, such* *that the corresponding trajectory x*(·) *satisfies the initial condition x*(0) = *x*<sup>0</sup> *and the terminal condition x*(*τ*) = *ξ, a.s.*

The following result was obtained in [65,66].

**Theorem 21** ([65,66])**.** *(i) A necessary condition for exact controllability of (29) is*

$$\text{rank}\mathcal{K}\_1 = n - \sigma;\tag{30}$$

*(ii) Let the condition (30) hold. A necessary and sufficient condition for exact controllability of (29) is*

$$\text{rank}G\_{\mathbb{T}} = n - \sigma.$$

*Here, Gτ is the Gramian matrix defined as*

$$G\_{\rm tr} = E \int\_0^{\tau} \Phi(t) \bar{K}\_{12} \bar{K}\_{12}^T \Phi(t)^T dt.$$

*where* Φ(*t*) *is the unique solution to the matrix stochastic differential equation*

$$d\Phi(t) = -\Phi(t)[\tilde{\mathsf{N}}dt + \mathcal{K}\_{11}dw(t)], \Phi(0) = I.$$

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].

**Theorem 22** ([67])**.** *(i) A necessary condition for exact controllability of (29) is*

$$\text{rank}\ddot{K}\_1 = n - q - 2p; \tag{31}$$

*(ii) Let the condition (31) hold. A necessary and sufficient condition for exact controllability of (29) is*

$$\text{rank}G\_{\mathbb{T}} = n - q - 2p.$$

*Here, Gτ is the Gramian matrix defined as*

$$G\_{\tau} = E \int\_{0}^{\tau} \Phi(t) \mathcal{K}\_{12} \mathcal{K}\_{12}^{T} \Phi(t)^{T} dt.$$

*where* Φ(*t*) *is the unique solution to the matrix stochastic differential equation*

$$d\Phi(t) = -\Phi(t)[\tilde{N}dt + \tilde{K}\_{11}dw(t)], \Phi(0) = I.$$

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 *<sup>P</sup>*1, *<sup>Q</sup>* <sup>∈</sup> <sup>R</sup>*n*×*<sup>n</sup>* such that the following condition is satisfied:

$$P\_1 L Q = \begin{bmatrix} I\_{n\_1} & 0 \\ 0 & N \end{bmatrix}, P\_1 M Q = \begin{bmatrix} B\_1 & 0 \\ 0 & I\_{n\_2} \end{bmatrix} \prime$$

$$P\_1 B = \begin{bmatrix} \mathcal{C}\_1 \\ \mathcal{C}\_2 \end{bmatrix}, P\_1 \mathcal{C} Q = \begin{bmatrix} D\_1 & 0 \\ 0 & 0 \end{bmatrix}, P\_1 D = \begin{bmatrix} \mathcal{G}\_1 \\ 0 \end{bmatrix} \prime \tag{32}$$

where *<sup>N</sup>* <sup>∈</sup> <sup>R</sup>*n*2×*n*<sup>2</sup> denotes a nilpotent matrix with order *<sup>h</sup>*, i.e., *<sup>h</sup>* <sup>=</sup> min{*<sup>k</sup>* : *<sup>k</sup>* <sup>≥</sup> 1, *<sup>N</sup><sup>k</sup>* <sup>=</sup> <sup>0</sup>}; *<sup>B</sup>*1, *<sup>D</sup>*<sup>1</sup> <sup>∈</sup> <sup>R</sup>*n*1×*n*<sup>1</sup> , *<sup>C</sup>*1, *<sup>G</sup>*<sup>1</sup> <sup>∈</sup> <sup>R</sup>*n*1×*m*, *<sup>C</sup>*<sup>2</sup> <sup>∈</sup> <sup>R</sup>*n*2×*m*, and *<sup>n</sup>*<sup>1</sup> <sup>+</sup> *<sup>n</sup>*<sup>2</sup> <sup>=</sup> *<sup>n</sup>*. Let & *<sup>x</sup>*<sup>1</sup> *x*2 ' = *Q*−1*x*, system (29) is equivalent to

$$d\mathbf{x}\_1(t) = (B\_1\mathbf{x}\_1(t) + C\_1\boldsymbol{\mu}(t))dt + (D\_1\mathbf{x}\_1(t) + G\_1\boldsymbol{\mu}(t))dw(t), \\ \mathbf{x}\_1(0) = \mathbf{x}\_{10}, \tag{33}$$

$$\text{Ndx}\_2(t) = \text{x}\_2(t)dt + \text{C}\_2\text{u}(t)dt,\\ \text{x}\_2(0) = \text{x}\_{20}.\tag{34}$$

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

$$(sN - I\_{\text{tr}\_2})X\_2(s) = Nx\_{20} + \mathcal{C}\_2 \mathcal{U}(s). \tag{35}$$

**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 x*(*t*) = *Q* & *x*1(*t*) *x*2(*t*) ' *is called the impulse solution of Equation (29).*

Let Φ(*t*) be the solution of system

$$d\Phi(t) = (B\_1 dt + D\_1 dw(t))\Phi(t), \Phi(0) = I\_{n\_1 \prime} \tag{36}$$

**Definition 15.** *(Exact Null Controllability) System (33) and (34) is said to be exactly null controllable on* [0, *<sup>τ</sup>*] *if for any* & *<sup>x</sup>*<sup>10</sup> *x*20 ' <sup>∈</sup> *<sup>R</sup>n*, *there exists <sup>u</sup>* <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>τ</sup>*], <sup>Ω</sup>, *<sup>R</sup>m*), *such that (33) and (34) has a unique solution* & *<sup>x</sup>*1(*t*) *x*2(*t*) ' *satisfying the initial condition* & *<sup>x</sup>*1(0) *x*2(0) ' = & *x*<sup>10</sup> *x*20 ' *in addition to the terminal condition* & *<sup>x</sup>*1(*τ*) *x*2(*τ*) ' = 0.

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*<sup>1</sup> = 0*, then the necessary condition for (33) to be exactly null controllable on* [0, *τ*] *is that*

$$E\left(\int\_0^\tau f^2(t)\Phi^{-1}(t)\mathbf{C}\_1(\Phi^{-1}(t)\mathbf{C}\_1)^\mathsf{T}dt\right) \tag{37}$$

*is invertible for any real valued polynomial f*(*t*) *not identical zero.*

Let rank*G*<sup>1</sup> = *n*1; let *u*(*t*) = *M*<sup>1</sup> & 0 *v*(*t*) ' , *z*(*t*) = *D*1*x*1(*t*), where *M*<sup>1</sup> denotes an *m* × *m* matrix, which satisfies *G*1*M*<sup>1</sup> = [*In*<sup>1</sup> 0], and *v*(*t*) denotes an (*m* − *n*1)−dimension vector. For the above *u*(*t*), system (33) and (34) is equivalent to

$$-d\mathbf{x}\_1(t) = (F\_1\mathbf{x}\_1(t) + F\_2\mathbf{z}(t) + F\_3\mathbf{v}(t))dt - z(t)dw(t), \mathbf{x}\_1(0) = \mathbf{x}\_{10} \tag{38}$$

$$\text{tr}\_2(t) = -\text{C}\_2 M\_1 \begin{bmatrix} 0 \\ v(t) \end{bmatrix}, t > 0,\tag{39}$$

where

$$F\_1 = D\_1 - B\_1,\\ F\_2 = -I\_{\mathbb{N}\_1},\\ F\_3 v(t) = -\mathbb{C}\_1 M\_1 \begin{bmatrix} 0 \\ v(t) \end{bmatrix}.$$

Let Ψ(*t*) denote the solution of system

$$d\Psi(t) = \Psi(t)(F\_1 dt + F\_2 dw(t)), \Psi(0) = I\_{n\_1}.$$

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,*

$$E(\int\_0^\tau f^2(t)\Psi^{-1}(t)F\_3(\Psi^{-1}(t)F\_3)^\mathsf{T}dt),$$

*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].

$$A d\mathbf{x}(t) = B\mathbf{x}(t)dt + \mathbb{C}u(t)dt + D\mathbf{x}(t)dw(t), \mathbf{x}(0) = \mathbf{x}\_0,\tag{40}$$

$$y(t) = Gx(t),\tag{41}$$

where *<sup>x</sup>*(*t*) <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>τ</sup>*], <sup>Ω</sup>, <sup>R</sup>*n*) is the state vector, *<sup>u</sup>*(*t*) <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>τ</sup>*], <sup>Ω</sup>, <sup>R</sup>*m*) is the control vector, *<sup>w</sup>*(*t*) is one dimensional standard Wiener process, *<sup>x</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, <sup>R</sup>*n*) is a given random variable, *<sup>y</sup>*(*t*) <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>τ</sup>*], <sup>Ω</sup>, <sup>R</sup>*<sup>l</sup>* ) is the measurement output.

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.

In order to discusses the impulse controllability and impulse observability for stochastic singular system (40), let us introduce the class *Hn* of all processes *<sup>f</sup>*(*t*) <sup>∈</sup> *<sup>L</sup>*2([0, <sup>+</sup>∞), <sup>Ω</sup>, <sup>R</sup>*n*), such that


$$(E\|f(t)\|\|\_{\mathbb{R}^n}^2)^{1/2} \le M\_0 e^{\alpha t}, t \ge 0.$$

In the following, *Ck*(*J*, Ω, R*n*) denotes the set of all *k* times continuously differentiable stochastic processes *<sup>x</sup>*(*t*) <sup>∈</sup> *<sup>L</sup>*2(*J*, <sup>Ω</sup>, <sup>R</sup>*n*), such that *<sup>x</sup>*(*i*)(*t*) <sup>∈</sup> *<sup>L</sup>*2(*J*, <sup>Ω</sup>, <sup>R</sup>*n*)(*<sup>i</sup>* <sup>=</sup> 0, 1, ··· , *k*)(*J* = [0, *τ*]*or*[0, +∞); we assume that there are a pair of non-singular matrices *<sup>P</sup>*1, *<sup>Q</sup>* <sup>∈</sup> <sup>R</sup>*n*×*n*, such that the following condition is satisfied

$$\begin{cases} \begin{array}{c} P\_1 A Q = \begin{bmatrix} I\_{\eta\_1} & 0 \\ 0 & N \end{bmatrix}, P\_1 B Q = \begin{bmatrix} B\_1 & 0 \\ 0 & I\_{\eta\_2} \end{bmatrix} \end{cases} \end{cases} \tag{42}$$

$$\begin{cases} \begin{array}{c} P\_1 C = \begin{bmatrix} C\_1 \\ C\_2 \end{bmatrix}, P\_1 D Q = \begin{bmatrix} D\_1 & 0 \\ 0 & 0 \end{bmatrix}, G Q = \begin{bmatrix} G\_1 & G\_2 \end{bmatrix} \end{cases} \tag{42}$$

where *<sup>N</sup>* <sup>∈</sup> <sup>R</sup>*n*2×*n*<sup>2</sup> is a nilpotent, the index of nilpotency of *<sup>N</sup>* is denoted by *<sup>h</sup>*, i.e., *<sup>h</sup>* <sup>=</sup> min{*<sup>k</sup>* : *<sup>k</sup>* is a positive integer, *<sup>k</sup>* <sup>≥</sup> 1, *<sup>N</sup><sup>k</sup>* <sup>=</sup> <sup>0</sup>}, *<sup>B</sup>*<sup>1</sup> <sup>∈</sup> <sup>R</sup>*n*1×*n*<sup>1</sup> , *<sup>C</sup>*<sup>1</sup> <sup>∈</sup> <sup>R</sup>*n*1×*m*, *<sup>C</sup>*<sup>2</sup> <sup>∈</sup> <sup>R</sup>*n*2×*m*, *<sup>D</sup>*<sup>1</sup> <sup>∈</sup> <sup>R</sup>*n*1×*n*<sup>1</sup> , *<sup>G</sup>*<sup>1</sup> <sup>∈</sup> <sup>R</sup>*l*×*n*<sup>1</sup> , *<sup>G</sup>*<sup>2</sup> <sup>∈</sup> <sup>R</sup>*l*×*n*<sup>2</sup> , *<sup>n</sup>*<sup>1</sup> <sup>+</sup> *<sup>n</sup>*<sup>2</sup> <sup>=</sup> *<sup>n</sup>*. Let & *<sup>x</sup>*1(*t*) *x*2(*t*) ' = *Q*−1*x*(*t*), system (40) and (41) is equivalent to

*dx*1(*t*)=(*B*1*x*1(*t*) + *C*1*u*(*t*))*dt* + *D*1*x*1(*t*)*dw*(*t*), *x*1(0) = *x*10, (43)

$$y\_1(t) = G\_1 x\_1(t),\tag{44}$$

$$\mathbf{N}d\mathbf{x}\_2(t) = \mathbf{x}\_2(t)dt + \mathbf{C}\_2\mathbf{u}\_2(t)dt,\\ \mathbf{x}\_2(0) = \mathbf{x}\_{20} \tag{45}$$

$$y\_2(t) = G\_2 \mathbf{x}\_2(t). \tag{46}$$

Let Φ(*t*) be the solution of system

$$d\Phi(t) = (B\_1 dt + D\_1 dw(t))\Phi(t), \Phi(0) = I\_{n\_1 \nu}$$

the following results were obtained in [69]

**Theorem 25** ([69])**.** *If <sup>u</sup>* <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>τ</sup>*], <sup>Ω</sup>, <sup>R</sup>*m*) *is a bounded Borel measurable function, then subsystem (43) has a unique solution on* [0, *<sup>τ</sup>*] *with any <sup>x</sup>*<sup>10</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, <sup>R</sup>*n*<sup>1</sup> ), *and the solution is given by the stochastic process*

$$\mathbf{x}\_{1}(t) = \Phi(t)\mathbf{x}\_{10} + \Phi(t)\int\_{0}^{t} \Phi^{-1}(s)\mathbf{C}\_{1}u(s)ds.\tag{47}$$

**Theorem 26** ([69])**.** *For any <sup>x</sup>*<sup>20</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, <sup>R</sup>*n*<sup>2</sup> ), *<sup>u</sup>* <sup>∈</sup> *<sup>C</sup>h*−1([0, <sup>+</sup>∞), <sup>Ω</sup>, <sup>R</sup>*m*) *and <sup>u</sup>*(*i*) <sup>∈</sup> *Hm*(*i* = 0, 1, ··· , *h* − 1)*, subsystem (45) has a unique impulse solution, which is given by*

$$\mathbf{x}\_{2}(t) = -\sum\_{i=1}^{h-1} \delta^{(i-1)}(t) [N^{i}\mathbf{x}\_{20} + \sum\_{k=i}^{h-1} N^{k}\mathbb{C}\_{2}\boldsymbol{\mu}^{(k-i)}(0)] - \sum\_{i=0}^{h-1} N^{i}\mathbb{C}\_{2}\boldsymbol{\mu}^{(i)}(t),\tag{48}$$

*where <sup>δ</sup>*(*t*) *is the Dirac function, <sup>δ</sup>*(*i*−<sup>1</sup>)(*t*) *is the* (*<sup>i</sup>* <sup>−</sup> <sup>1</sup>)*th derivative of <sup>δ</sup>*(*t*)*.*

**Theorem 27** ([69])**.** *Assume that (40) and (41) is equivalent to (43)–(46),*

$$\mu \in \mathbb{C}^{h-1}([0, +\infty), \Omega, \mathbb{R}^m)$$

*is a bounded Borel measurable function, and <sup>u</sup>*(*i*) <sup>∈</sup> *Hm*(*<sup>i</sup>* <sup>=</sup> 0, 1, ··· , *<sup>h</sup>* <sup>−</sup> <sup>1</sup>). *Then, for any <sup>x</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, <sup>R</sup>*n*), *system (40) has a unique impulse solution on* [0, *<sup>τ</sup>*], *which is given by*

$$\mathbf{x}(t) = Q \begin{bmatrix} \mathbf{x}\_1(t) \\ \mathbf{x}\_2(t) \end{bmatrix} \tag{49}$$

*where x*1(*t*) *and x*2(*t*) *are given by (47) and (48), respectively.*

**Definition 16.** *System (40) is called impulse controllable, if for any <sup>x</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, <sup>R</sup>*n*)*, there exists a bounded Borel measurable function <sup>u</sup>* <sup>∈</sup> *<sup>C</sup>h*([0, <sup>+</sup>∞), <sup>Ω</sup>, <sup>R</sup>*m*) *and <sup>u</sup>*(*i*) <sup>∈</sup> *Hm*(*<sup>i</sup>* <sup>=</sup> 0, 1, ··· , *<sup>h</sup>* <sup>−</sup> <sup>1</sup>)*, such that the coefficient vectors of <sup>δ</sup>*(*i*)(*t*), *<sup>i</sup>* <sup>=</sup> 0, 1, ··· , *<sup>h</sup>* <sup>−</sup> 2, *in the solution formula (49) are all zero.*

The following results were obtained in [69].

**Theorem 28** ([69])**.** *System (40) is impulse controllable if, and only if, subsystem (45) is impulse controllable.*

**Theorem 29** ([69])**.** *Subsystem (45) is impulse controllable if and only if for any*

$$\mathfrak{x}\_{20} \in L^2(\Omega, F\_{0\prime}P\_{\prime} \mathbb{R}^{n\_2}),$$

*there exists a bounded Borel measurable function u* <sup>∈</sup> *<sup>C</sup>h*−1([0, <sup>+</sup>∞), <sup>Ω</sup>, <sup>R</sup>*m*) *and u*(*i*) <sup>∈</sup> *Hm*(*<sup>i</sup>* <sup>=</sup> 0, 1, ··· , *h* − 1)*, such that*

$$N\mathfrak{x}\_{20} + \sum\_{i=0}^{h-2} N^{i+1} \mathbb{C}\_2 \mathfrak{u}^{(i)}(0) = 0.1$$

**Theorem 30** ([69])**.** *System (40) is impulse controllable if, and only if,*

$$\text{ran}(N) = \text{ran}([N\mathbb{C}\_2 \quad \cdots \quad N^{h-1}\mathbb{C}\_2]),$$

*where* ran(*N*) = {*<sup>y</sup>* : *<sup>y</sup>* <sup>=</sup> *Nz*, *<sup>z</sup>* <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, <sup>R</sup>*n*<sup>2</sup> )}, ran([*NC*<sup>2</sup> ··· *<sup>N</sup>h*−1*C*2]) = {*<sup>y</sup>* : <sup>∃</sup>*α<sup>k</sup>* <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, <sup>R</sup>*m*), *<sup>k</sup>* <sup>=</sup> 1, 2, ··· , *<sup>h</sup>* <sup>−</sup> 1, *<sup>y</sup>* <sup>=</sup> <sup>∑</sup>*h*−<sup>1</sup> *<sup>k</sup>*=<sup>1</sup> *<sup>N</sup>kC*2*αk*}.

Now, we discuss the impulse observability of system (40) and (41). Without loss of generality, let *u*(*t*) ≡ 0.

**Definition 17.** *System (40) and (41) with subsystem (43)–(46) is called impulse observable if, y*2(*t*)|*t*=<sup>0</sup> = 0 *implies x*2(*t*)|*t*=<sup>0</sup> = 0.

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].

**Theorem 31** ([69])**.** *Subsystem (43) and (44) is always impulse observable.*

**Theorem 32** ([69])**.** *System (40) and (41) is impulse observable if, and only if, one of the following conditions holds:*

*(i) Subsystem (45) and (46) is impulse observable;*

*(ii)*

$$\ker(\begin{bmatrix} G\_2N\\ G\_2N^2\\ \vdots\\ G\_2N^{\text{fl}} \end{bmatrix}) = \ker(N).$$

*where* ker(*N*) = {*<sup>x</sup>* : *Nx* <sup>=</sup> 0, *<sup>x</sup>* <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, *<sup>R</sup>n*<sup>2</sup> )},

$$\ker(\begin{bmatrix} G\_2N\\ G\_2N^2\\ \vdots\\ G\_2N^h \end{bmatrix}) = \{ \mathbf{x} : \begin{bmatrix} G\_2N\\ G\_2N^2\\ \vdots\\ G\_2N^h \end{bmatrix} \mathbf{x} = \mathbf{0}, \mathbf{x} \in L^2(\Omega, F\_{\mathbf{0}}, P, R^{v\_2}) \}.$$

For the impulse observability and impulse controllability, the so-called dual principle holds, which reveals the close relation between impulse observability and impulse controllability.

In order to introduce the dual principle for system (40) and (41), let us first introduce the dual system.

**Definition 18.** *The following system*

$$\begin{cases} A^T dz(t) = B^T z(t)dt + G^T v(t)dt + D^T z(t)dw(t), \\ w\_0(t) = \mathbb{C}^T z(t), \end{cases} \tag{50}$$

*is called the dual system of the system (40) and (41).*

The following dual principle was obtained in [69].

**Theorem 33** ([69])**.** *Let (50) be the dual system of system (40) and (41). Then, system (40) and (41) is impulse observable (impulse controllable) if, and only if, its dual system (50) is impulse controllable (impulse observable).*

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 <sup>L</sup>p*−*exact controllability for the following stochastic singular linear system?*

$$L d\mathbf{x}(t) = [A(t)\mathbf{x}(t) + B(t)u(t)]dt + \sum\_{k=1}^{d} [\mathbb{C}\_k(t)\mathbf{x}(t) + D\_k(t)u(t)]dw\_k(t), \\ t \ge 0, \\ \mathbf{x}(0) = \mathbf{x}\_0.$$

*where L as defined in (29); A*(*t*), *B*(*t*), *Ck*(*t*), *Dk*(*t*) *as defined in (1).*
