*2.3. 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].

$$d\mathbf{x}(t) = [A(t)\mathbf{x}(t)dt + B(t)u(t) + \int\_0^t M(t, \mathbf{s})\mathbf{x}(s)ds]dt$$

$$+ [\mathbb{C}(t)\mathbf{x}(t) + D(t)u(t)]dw(t), t \ge 0,\tag{14}$$

where *<sup>x</sup>*(·), *<sup>u</sup>*(·) are the state variable, control variable which take values in <sup>R</sup>*n*, <sup>R</sup>*m*, respectively; for any *<sup>t</sup>*,*<sup>s</sup>* <sup>∈</sup> [0, *<sup>τ</sup>*] with *<sup>τ</sup>* <sup>∈</sup> [0, <sup>∞</sup>), *<sup>A</sup>*(*t*), *<sup>M</sup>*(*t*,*s*), *<sup>C</sup>*(*t*) <sup>∈</sup> <sup>R</sup>*n*×*n*, and *<sup>B</sup>*(*t*), *<sup>D</sup>*(*t*) <sup>∈</sup> <sup>R</sup>*n*×*m*; *<sup>w</sup>*(*t*) is 1-dimensional Wiener process. System (14) is denoted by [*A*(·), *<sup>M</sup>*(·, ·), *<sup>C</sup>*(·); *B*(·), *D*(·)].

The following is definition of controllability for system (14).

**Definition 4.** *For any τ*0, *τ*(*τ*<sup>0</sup> ≤ *τ*)*, the following system*

$$d\mathbf{x}(t) = [A(t)\mathbf{x}(t)dt + B(t)u(t) + \int\_{\tau\_0}^{t} M(t, \mathbf{s})\mathbf{x}(\mathbf{s})d\mathbf{s}]dt$$

$$+ [\mathbb{C}(t)\mathbf{x}(t) + D(t)u(t)]dw(t), t \ge 0,\tag{15}$$

*<sup>τ</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>τ*<sup>0</sup> , *<sup>P</sup>*, <sup>R</sup>*n*)*, is called exactly controllable on* [*τ*0, *<sup>τ</sup>*]*, if for any <sup>τ</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>τ*<sup>0</sup> , *<sup>P</sup>*, <sup>R</sup>*n*)*, <sup>τ</sup>* <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>τ*, *<sup>P</sup>*, <sup>R</sup>*n*)*, there exists a control <sup>u</sup>*(·) <sup>∈</sup> *<sup>L</sup>*2([*τ*0, *<sup>τ</sup>*], <sup>Ω</sup>, <sup>R</sup>*m*)*, such that the solution x*(·, *τ*0, *xτ*<sup>0</sup> , *u*(·)) *to system (15) with initial condition x*(*τ*0) = *xτ*<sup>0</sup> *satisfies x*(*τ*, *τ*0, *xτ*<sup>0</sup> , *u*(·)) = *x<sup>τ</sup> a.s.*

Throughout this subsection, we introduce the following basic hypothesis:

$$(A(\cdot), \mathbb{C}(\cdot)) \in L^{\infty}([0, \tau], \Omega, \mathbb{R}^{n \times n}), \\ \mathcal{M}(\cdot, \cdot) \in L^{\infty}([0, \tau]; L^{\infty}([0, \tau], \Omega, \mathbb{R}^{n \times n})),$$

$$B(\cdot)\_{\prime} D(\cdot) \in L^{\infty}([0, \tau], \Omega, \mathbb{R}^{n \times m}).$$

2.3.1. Time Invariant Systems

In this subsection, we discuss system (14) with time invariant matrices: i.e.,

$$[A(\cdot), M(\cdot, \cdot), \mathbb{C}(\cdot); B(\cdot), D(\cdot)] = [A, M, \mathbb{C}; B, D].$$

To consider the exact controllability of system [*A*, *M*, *C*; *B*, *D*], we adopt the partial controllability of controlled system as follows:

$$dx(t) = [A\_0(t)x(t)dt + B\_0(t)u(t)]dt + [A\_1(t)x(t)dt + B\_1(t)u(t)]dw(t), t \ge 0. \tag{16}$$

For fixed *<sup>τ</sup>* <sup>≥</sup> 0 and a matrix *<sup>Q</sup>* <sup>∈</sup> <sup>R</sup>*l*×*n*, define *<sup>X</sup><sup>τ</sup>* <sup>=</sup> {*<sup>ξ</sup>* <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>τ*, *<sup>P</sup>*, <sup>R</sup>*<sup>l</sup>* ) : *ξ*(*ω*) ∈ ran(*Q*)}.

**Definition 5.** *Let a matrix <sup>Q</sup>* <sup>∈</sup> <sup>R</sup>*l*×*<sup>n</sup> be given. System (16) is called <sup>Q</sup>*−*partially controllable on* [0, *<sup>τ</sup>*]*, if for any <sup>x</sup>*<sup>0</sup> <sup>∈</sup> <sup>R</sup>*n*, *<sup>ξ</sup>* <sup>∈</sup> *<sup>X</sup>τ*, *there exists a <sup>u</sup>*(·) <sup>∈</sup> *<sup>L</sup>*2([*τ*0, *<sup>τ</sup>*], <sup>Ω</sup>, <sup>R</sup>*m*)*, such that the solution x*(·, *x*0, *u*(·)) *to system (16) with the initial condition x*(0) = *x*<sup>0</sup> *satisfies Qx*(*τ*, *x*0, *u*(·)) = *ξ a.s.*

Setting

$$\eta(\cdot) = \int\_0^\cdot \mathfrak{x}(s)ds, \mathfrak{z}(\cdot) = \begin{bmatrix} \mathfrak{x}\_1(\cdot) \\ \eta(\cdot) \end{bmatrix}, A\_0 = \begin{bmatrix} A & M \\ I\_\mathbb{II} & 0 \end{bmatrix},$$

$$B\_0 = \begin{bmatrix} B \\ & 0 \end{bmatrix}, \mathfrak{C}\_0 = \begin{bmatrix} \mathfrak{C} & 0 \\ 0 & 0 \end{bmatrix}, D\_0 = \begin{bmatrix} B \\ 0 \end{bmatrix},$$

we can rewrite system [*A*, *M*, *C*; *B*, *D*] as follows:

$$dy(t) = [A\_0(t)y(t)dt + B\_0(t)u(t)]dt + [C\_0y(t)dt + D\_0u(t)]dw(t), t \ge 0. \tag{17}$$

The following results were obtained in [61].

**Theorem 10** ([61])**.** *System* [*A*, *M*, *C*; *B*, *D*] *is exactly controllable on* [0, *τ*] *with x*(0) = *x*<sup>0</sup> *if and only if system (17) is* [*In*, 0]−*partially controllable on* [0, *<sup>τ</sup>*] *with y*(0)=[*x<sup>T</sup>* <sup>0</sup> , 0*T*] *T.*

**Theorem 11** ([61])**.** *If system* [*A*, *M*, *C*; *B*, *D*] *is exactly controllable on* [0, *τ*]*, then* rank*D* = *n*.

In what follows, we tend to present a rank criterion ensuring system [*A*, *M*, *C*; *B*, *D*] s exact controllability. By Theorem 11, from now on, we suppose that rank*D* = *n*. Then, there exists an invertible *<sup>K</sup>* <sup>∈</sup> <sup>R</sup>*m*×*m*, such that *DK* = [*In*, 0]. Set

$$\mu(\cdot) = K \begin{bmatrix} \mu\_1(\cdot) \\ \mu\_2(\cdot) \end{bmatrix} + J\mathcal{Y}(\cdot),\\BK = [B\_1, B\_2]\_{\prime\prime}$$

where *<sup>B</sup>*1<sup>∈</sup> *<sup>R</sup>n*×*n*, *<sup>B</sup>*<sup>2</sup> <sup>∈</sup> *<sup>R</sup>n*×(*m*−*n*), and *<sup>J</sup>* <sup>∈</sup> *<sup>R</sup>m*×2*n*. Then, system (17) turns to

$$dy(t) = \left\{ \begin{bmatrix} A\_0 + \begin{bmatrix} Bf - B\_1([\mathbb{C}, 0] + Df \end{bmatrix} \end{bmatrix} \right\} y(t) \\
$$

$$+ \begin{bmatrix} B\_1 \\ 0 \end{bmatrix} [u\_1(t) + ([\mathbb{C}, 0] + Df)y(t)] + \begin{bmatrix} B\_2 \\ 0 \end{bmatrix} u\_2(t) \right\} dt \\
$$

$$+ \begin{bmatrix} I\_{\mathbb{H}} \\ 0 \end{bmatrix} [u\_1(t) + ([\mathbb{C}, 0] + Df)y(t)] dw(t). \tag{18}$$

Take

$$
\tilde{A}\_0 = A\_0 + \begin{bmatrix} Bf - B\_1([\mathbb{C}, 0] + Df) \\ 0 \end{bmatrix}, \\
v(\cdot) = u(\cdot) + ([\mathbb{C}, 0] + Df)y(\cdot).
$$

Then, system (17) or (18) can be rewritten as

$$d\begin{bmatrix} \mathbf{x}(t) \\ \eta(t) \end{bmatrix} = [\tilde{A}\_0 \begin{bmatrix} \mathbf{x}(t) \\ \eta(t) \end{bmatrix} + \begin{bmatrix} B\_1 \\ 0 \end{bmatrix} v(t) + \begin{bmatrix} B\_2 \\ 0 \end{bmatrix} u\_2(t)]dt$$

$$+ \begin{bmatrix} I\_n \\ 0 \end{bmatrix} v(t)dw(t), t \ge 0. \tag{19}$$

In order to discuss the exact controllability of (19), we need to introduce the following stochastic linear differential equation

$$d\begin{bmatrix} \mathbf{x}(t) \\ \eta(t) \end{bmatrix} = [\tilde{A}\_0 \begin{bmatrix} \mathbf{x}(t) \\ \eta(t) \end{bmatrix} + \begin{bmatrix} B\_1 \\ 0 \end{bmatrix} v(t) + \begin{bmatrix} B\_2 \\ 0 \end{bmatrix} u\_2(t)] dt$$

$$+ \begin{bmatrix} I\_n \\ 0 \end{bmatrix} v(t) dw(t), t \in [0, \tau], \mathbf{x}(\tau) = \mathbf{0}, \eta(0) = \mathbf{0}. \tag{20}$$

Let

$$L = -([0, I\_{\boldsymbol{n}}]e^{-\tilde{A}\_0^T \boldsymbol{\tau}}[0, I\_{\boldsymbol{n}}]^T)^{-1}[0, I\_{\boldsymbol{n}}]e^{-\tilde{A}\_0^T \boldsymbol{\tau}}[I\_{\boldsymbol{n}}, 0]^T,$$

$$L\_0 = [I\_{\boldsymbol{n}}, L^T]\_{\prime}B\_0 = \begin{bmatrix} B\_2 \\ 0 \end{bmatrix}, \tilde{B}\_0 = \begin{bmatrix} B\_1 & 0 \\ 0 & 0 \end{bmatrix}.$$

The determinant of a square matrix *F* will be denoted by det*F*. The following result was obtained in [61].

**Theorem 12** ([61])**.** *Suppose that for any <sup>u</sup>*2(·) <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>τ</sup>*], <sup>Ω</sup>, <sup>R</sup>*m*−*n*) *system (20) admits a unique solution, and*

$$\det(\left[0, I\_{\boldsymbol{\eta}}\right]e^{-\hat{A}\_0^T \boldsymbol{\tau}} \left[0, I\_{\boldsymbol{\eta}}\right]^T) \neq \mathbf{0}, \forall t \in \left[0, \boldsymbol{\tau}\right]$$

*holds. Then system* [*A*, *M*, *C*; *B*, *D*] *is exactly controllable if, and only if, the following rank condition holds:*

$$\text{rank}[L\_0B\_0, L\_0\tilde{A}\_0B\_0, L\_0\tilde{B}\_0B\_0, L\_0\tilde{A}\_0\tilde{B}\_0B\_0, L\_0\tilde{B}\_0\tilde{A}\_0B\_0\cdots] = n\dots$$
