2.3.2. 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.

Case I. *M*(*t*,*s*) = *M*1(*t*)*M*2(*s*), 0 ≤ *s* ≤ *t* ≤ *τ*, and

$$M\_1(\cdot), M\_2(\cdot) \in L^{\infty}([0,\tau], \Omega, \mathbb{R}^{n \times n})\,.$$

In this case, we can set

$$\eta(\cdot) = \int\_0^\cdot M\_2(s)\mathbf{x}(s)ds, \mathbf{y}(\cdot) = \begin{bmatrix} \chi\_1(\cdot) \\ \eta(\cdot) \end{bmatrix}, A\_0(\cdot) = \begin{bmatrix} A(\cdot) & M\_1(\cdot) \\ M\_2(\cdot) & 0 \end{bmatrix},$$

$$B\_0(\cdot) = \begin{bmatrix} B(\cdot) \\ 0 \end{bmatrix}, \mathbf{C}\_0(\cdot) = \begin{bmatrix} \mathbf{C}(\cdot) & 0 \\ 0 & 0 \end{bmatrix}, D\_0(\cdot) = \begin{bmatrix} B(\cdot) \\ 0 \end{bmatrix}.$$

Hence, time varying system [*A*(·), *M*(·, ·), *C*(·); *B*(·), *D*(·)] s exact controllability turns to the [*In*, 0]−partial controllability of the following linear system without memory term:

$$dy(t) = \left[A\_0(t)y(t)dt + B\_0(t)u(t)\right]dt + \left[\mathcal{C}\_0y(t) + D\_0u(t)\right]dw(t), t \ge 0. \tag{21}$$

The following result provides an equivalent condition ensuring system (21)'s [*In*, 0]− partial controllability (see [61] (Theorem 3.1)).

**Theorem 13** ([61])**.** *Assume that M*(*t*,*s*) = *M*1(*t*)*M*2(*s*), 0 ≤ *s* ≤ *t* ≤ *τ. Then the following two statements are equivalent:*

*(i) System (21) is* [*In*, 0]−*partially controllable on [0, τ];*

*(ii) There exists a positive c such that the following observability inequality holds*

$$\|\|\tilde{\mathcal{G}}\|\|\_{L^{2}(\Omega,\mathbb{F}\_{\mathbf{t}},P,\mathbb{R}^{n})} \leq c\|\|B\_{0}(\cdot)^{T}Y(\cdot) + D\_{0}(\cdot)^{T}Z(\cdot)\|\_{L^{2}([0,\mathbf{r}],\Omega,\mathbb{R}^{m})^{\prime}}$$

*for all <sup>ξ</sup>* <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>τ</sup>*], <sup>Ω</sup>, <sup>R</sup>*n*), *and* (*Y*(·), *<sup>Z</sup>*(·)) *solve the following equation:*

$$dY(t) = [A\_0(t)^T Y(t)dt + \mathbb{C}\_0(t)^T Z(t)]dt + Z(t)dw(t), \\ t \in [0, \tau], \\ Y(\tau) = [I\_n, 0]^T \tilde{\xi}.$$

**Remark 3.** *Theorem 13 can be used to determine some stochastic system's exact controllability (see [61] (Example 3.2)).*

Case II. *<sup>M</sup>*(*t*,*s*) = *<sup>M</sup>*(*<sup>t</sup>* <sup>−</sup> *<sup>s</sup>*), 0 <sup>≤</sup> *<sup>s</sup>* <sup>≤</sup> *<sup>t</sup>* <sup>≤</sup> *<sup>τ</sup>*, and *<sup>M</sup>*(*τ*) <sup>∈</sup> *<sup>L</sup>*∞([0, *<sup>τ</sup>*], <sup>Ω</sup>, <sup>R</sup>*n*×*n*).

In this case, for the stochastic system [*A*(·), *M*(·, ·), *C*(·); *B*(·), *D*(·)], we can present the following sufficient condition (see [61] (Proposition 3.4)).

**Theorem 14** ([61])**.** *Assume that M*(*t*,*s*) = *M*(*t* − *s*), 0 ≤ *s* ≤ *t* ≤ *τ*, *and*

$$M(\mathfrak{r}) \in L^{\infty}([0,\mathfrak{r}], \Omega, \mathbb{R}^{n \times n}).$$

*If system* [*A*(·), *M*(·, ·), *C*(·); *B*(·), *D*(·)] *is exactly controllable on* [*τ*0, *τ*]*, for some τ*<sup>0</sup> ∈ (0, *τ*)*, then system* [*A*(·), *M*(·, ·), *C*(·); *B*(·), *D*(·)] *is exactly controllable on* [0, *τ*]*.*

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.

**Problem 3.** *Find a <sup>u</sup>*(·) <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>τ</sup>*], <sup>Ω</sup>, <sup>R</sup>*m*) *in general case such that the system (14) is exactly controllable.*

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