*2.1. Lp-Exact Controllability*

In 2017, Wang et al. consider the controllability of the following stochastic linear differential equation in [59]:

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

where *<sup>A</sup>*, *Ck*: [0, *<sup>τ</sup>*] <sup>×</sup> <sup>Ω</sup> <sup>→</sup> <sup>R</sup>*n*×*<sup>n</sup>* and *<sup>B</sup>*, *Dk*: [0, *<sup>τ</sup>*] <sup>×</sup> <sup>Ω</sup> <sup>→</sup> <sup>R</sup>*n*×*m*(*<sup>k</sup>* <sup>=</sup> 1, 2, ··· , *<sup>d</sup>*) are suitable matrix-valued processes; *x*(*t*) is the state process valued in R*<sup>n</sup>* and *u*(*t*) is the control process valued in <sup>R</sup>*m*; {*wk*(*t*): (*<sup>k</sup>* <sup>=</sup> 1, 2, ··· , *<sup>d</sup>*)} is a system of independent one-dimensional standard Wiener processes, *w*(*t*)=(*w*1(*t*), ··· , *wd*(*t*)). We will denote system (1) by [*A*(·), *C*(·); *B*(·), *D*(·)], with *C*(·)=(*C*1(·), ··· , *Cd*(·)) and *D*(·)=(*D*1(·), ··· , *Dd*(·)).

For the convenience of narration, the following notations and concepts are introduced. *Lp <sup>F</sup>*(Ω; *<sup>L</sup>q*([0, *<sup>τ</sup>*]; *<sup>H</sup>*)) is the set of all processes *<sup>x</sup>*(·) valued in *<sup>H</sup>*, such that

$$\|\|\mathfrak{x}(\cdot)\|\|\_{L^p\_F(\Omega; \mathbb{L}^q([0,\tau]; H))} = [E(\int\_0^\tau \|\|\mathfrak{x}(t)\|\|\_H^q dt)^{p/q}]^{1/p} < \infty,$$

$$L^p\_F(\Omega; L^p([0,\tau]; H)) = L^p\_F([0,\tau]; H), p \in [1,\infty].$$

*Lp <sup>F</sup>*(Ω; *C*([0, *τ*]; *H*)) is the set of all processes *x*(·) valued in *H*, such that for almost *ω* ∈ Ω, *t* → *x*(*t*, *ω*) is continuous and

$$\|\|\mathfrak{x}(\cdot)\|\|\_{L^p\_F(\Omega; \mathbb{C}([0,\tau]; H))} = [E(\sup\_{t \in [0,\tau]} ||\mathfrak{x}(t)||\_H^p)]^{1/p} < \infty.$$

In the similar manner, one can define

*L*∞ *<sup>F</sup>* (Ω; *<sup>L</sup>*∞([0, *<sup>τ</sup>*]; *<sup>H</sup>*)) and *<sup>L</sup>*<sup>∞</sup> *<sup>F</sup>* (Ω; *C*([0, *τ*]; *H*)).

**Hypothesis 1.** *The* <sup>R</sup>*n*×*n*−*valued processes A*(·), *Ck*(·) *satisfy*

$$(A(\cdot), \mathbb{C}\_k(\cdot) \in L\_F^{\infty}(\Omega; L^{\infty}([0, r]; \mathbb{R}^{n \times n}))(k = 1, \cdot, \cdot, d).$$

**Hypothesis 2.** *For some μ* ∈ (1, ∞] *and σ* ∈ (2, ∞]*, the following hold:*

$$B(\cdot) \in L\_F^{\mu}(\Omega; L^{\frac{2r}{s+2}}([0,\tau]; \mathbb{R}^{n \times m})), \mu \in (1,\infty], \sigma \in (2,\infty),$$

$$B(\cdot) \in L\_F^{\mu}(\Omega; L^2([0,\tau]; \mathbb{R}^{n \times m})), \mu \in (1,\infty], \sigma = \infty,$$

$$D\_1(\cdot), \cdot, \cdot, D\_d(\cdot) \in L\_F^{\mu}(\Omega; L^{\sigma}([0,\tau]; \mathbb{R}^{n \times m})).$$

Now, we introduce the following definition.

**Definition 1.** *(i) A process u*(*t*)(*t* ∈ [0, *τ*]) *is called a feasible control of system (1) if under <sup>u</sup>*(*t*)*, for any <sup>x</sup>*<sup>0</sup> <sup>∈</sup> <sup>R</sup>*n, system (1) admits a unique strong solution <sup>x</sup>*(*t*) <sup>∈</sup> *<sup>L</sup>*<sup>1</sup> *<sup>F</sup>*(Ω; *<sup>C</sup>*([0, *<sup>τ</sup>*]; <sup>R</sup>*n*)) *satisfying x*(0) = *x*0*. The set of feasible controls is denoted by U*[0, *τ*]*;*

*(ii) A control u*(*t*) <sup>∈</sup> *<sup>U</sup>*[0, *<sup>τ</sup>*] *is said to be Lp*−*feasible for system (1) if*

$$p \ge 1, B(\cdot)u(\cdot) \in L\_F^p(\Omega; L^1([0, \tau]; \mathbb{R}^n)), D\_k(\cdot)u(\cdot) \in L\_F^p(\Omega; L^2([0, \tau]; \mathbb{R}^{n \times n}))$$

*holds true. The set of Lp*−*feasible controls is denoted by Up*[0, *<sup>τ</sup>*]*;*

*(iii) System (1) is said to be <sup>L</sup>p*−*exactly controllable by <sup>U</sup>*[0, *<sup>τ</sup>*] *on* [0, *<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>p*(Ω, *<sup>F</sup>τ*, *<sup>P</sup>*, <sup>R</sup>*n*)*, there exists a <sup>u</sup>*(·) <sup>∈</sup> *<sup>U</sup>*[0, *<sup>τ</sup>*] *such that the solution <sup>x</sup>*(·) <sup>∈</sup> *<sup>L</sup>*<sup>1</sup> *<sup>F</sup>*(Ω; *<sup>C</sup>*([0, *<sup>τ</sup>*]; <sup>R</sup>*n*)) *of (1) with x*(0) = *<sup>x</sup>*<sup>0</sup> *satisfies x*(*τ*) = *<sup>ξ</sup>.*

2.1.1. The Case *D*(·) = 0

In this case, we consider system [*A*(·), *C*(·); *B*(·), 0], i.e., the state equation is

$$d\mathbf{x}(t) = [A(t)\mathbf{x}(t) + B(t)u(t)]dt + \sum\_{k=1}^{d} \mathbb{C}\_{k}(t)\mathbf{x}(t)dw\_{k}(t), t \ge 0. \tag{2}$$

Thus, the control *<sup>u</sup>*(·) does not appear in the diffusion. The *<sup>L</sup>p*−exact controllability of system (2) was discussed and the following results were obtained in [59].

**Theorem 1** ([59])**.** *Let Hypothesis 1 hold. Let*

$$B(t)B(t)^T \ge \delta I\_{\mathfrak{n}\_\prime} t \in [0, \mathfrak{r}]\_\prime a.s.\_\prime$$

*for some <sup>δ</sup>* <sup>&</sup>gt; <sup>0</sup>*. Then for any <sup>p</sup>* <sup>&</sup>gt; <sup>1</sup>*, system (2) is <sup>L</sup>p*−*exactly controllable on* [0, *<sup>τ</sup>*] *by <sup>U</sup>p*<sup>−</sup> <sup>=</sup> <sup>∩</sup>*q*∈(0,*p*)*Uq*[0, *<sup>τ</sup>*].

**Theorem 2** ([59])**.** *Let Hypothesis 1 hold. Suppose there exists a continuous differentiable function <sup>f</sup>* : [0, *<sup>τ</sup>*] <sup>→</sup> <sup>R</sup>*n*, *f*(*t*)-<sup>R</sup>*<sup>n</sup>* <sup>=</sup> 1, *for all t* <sup>∈</sup> [0, *<sup>τ</sup>*] *such that f*(*t*)*TB*(*t*) = <sup>0</sup>*. Additionally, let*

$$\mathbb{C}\_{k}(\cdot) \in L\_{F}^{\infty}(\Omega; \mathbb{C}([0, \tau]; \mathbb{R}^{n \times n})), 1 \le k \le d. \tag{3}$$

*Then for any p* <sup>&</sup>gt; 1, *system (2) is not Lp*−*exactly controllable on* [0, *<sup>τ</sup>*] *by Up*[0, *<sup>τ</sup>*]*.*

**Corollary 1** ([59])**.** *Let Hypothesis 1 and (3) hold. Let B* <sup>∈</sup> <sup>R</sup>*n*×*m*.

*(i) If for some <sup>p</sup>* <sup>&</sup>gt; <sup>1</sup>*, system* [*A*(·), *<sup>C</sup>*(·); *<sup>B</sup>*, 0] *is <sup>L</sup>p*−*exactly controllable on* [0, *<sup>τ</sup>*] *by Up*[0, *τ*]*, then*

$$\text{rank}B = n,\tag{4}$$

*where* rank*B denotes the rank of B;*

*(ii) If (4) holds, then for any <sup>p</sup>* <sup>&</sup>gt; <sup>1</sup>*, system* [*A*(·), *<sup>C</sup>*(·); *<sup>B</sup>*, 0] *is <sup>L</sup>p*−*exactly controllable on* [0, *τ*] *by Up*−[0, *τ*]*.*

The above result shows that the gap between condition (4) and the *<sup>L</sup>p*−exact controllability of system [*A*(·), *<sup>C</sup>*(·); *<sup>B</sup>*, 0] (by *<sup>U</sup>p*[0, *<sup>τ</sup>*], or *<sup>U</sup>p*−[0, *<sup>τ</sup>*]) is very small.

2.1.2. The Case rank*D*(·) = *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:

$$D(t)D(t)^T \ge \delta I\_{n\prime} a.s., a.e.t \in [0, \tau]. \tag{5}$$

In this case, [*D*(*t*)*D*(*t*)*T*] <sup>−</sup><sup>1</sup> exists and uniformly bounded. We define

$$\bar{A}(t) = A(t) - B(t)D(t)^T [D(t)D(t)^T]^{-1} \mathbb{C}(t),$$

$$\tilde{B}(t) = B(t)\{I\_{\mathbb{H}} - D(t)^T [D(t)D(t)^T]^{-1} D(t)\}, \\ \tilde{D}(t) = B(t)D(t)^T [D(t)D(t)^T]^{-1}.$$

and introduce the following controlled system:

$$d\mathbf{x}(t) = [\bar{A}(t)\mathbf{x}(t) + \mathcal{B}(t)v(t) + \mathcal{D}(t)z(t)]dt + z(t)dw(t), \\ t \in [0, \pi], \\ \mathbf{x}(0) = \mathbf{x}\_0,\tag{6}$$

with *x*(*t*) being the state and (*v*(·), *z*(·)) being the control. For system (6), we need the following set and definition:

$$
\tilde{\mathcal{U}}^p[0,\tau] = \{ v(\tau) : \tilde{B}(\tau)v(\tau) \in L^p\_F(\Omega; L^1([0,\tau]; \mathbb{R}^n)) \}.
$$

**Definition 2.** *System (6) is said to be exactly null-controllable by*

$$\tilde{\mathcal{U}}^p[0,\tau] \times L\_F^p(\Omega; L^2([0,\tau]; \mathbb{R}^n))$$

*on the* [0, *<sup>τ</sup>*], *if for any x*<sup>0</sup> <sup>∈</sup> <sup>R</sup>*n*, *there exists a pair*

$$(v(\cdot), z(\cdot)) \in \tilde{\mathcal{U}}^p[0, \tau] \times L\_F^p(\Omega; L^2([0, \tau]; \mathbb{R}^n)),$$

*such that the solution x*(·) *to*

$$d\mathbf{x}(t) = [\tilde{A}(t)\mathbf{x}(t) + \tilde{B}(t)\mathbf{v}(t) + \tilde{D}(t)\mathbf{z}(t)]dt + \mathbf{z}(t)dw(t), \mathbf{t} \in [0, \pi],$$

$$\mathbf{x}(0) = \mathbf{x}\_0, \mathbf{x}(\tau) = \tilde{\xi},\tag{7}$$

*under* (*v*(*τ*), *z*(*τ*)) *satisfies x*(*τ*) = 0*.*

The following results were obtained in [59].

**Theorem 3** ([59])**.** *Let Hypothesis 1 and (5) hold. Suppose*

$$\tilde{A}(t) \in L\_F^{\infty}(\Omega; L^{1+\varepsilon}([0, \mathfrak{r}]; \mathbb{R}^{n \times n})),\\\tilde{D}(t) \in L\_F^{\infty}(\Omega; L^2([0, \mathfrak{r}]; \mathbb{R}^{n \times n})),$$

*where* <sup>&</sup>gt; <sup>0</sup> *is a given constant. Then system (1) is <sup>L</sup>p*−*exactly controllable on* [0, *<sup>τ</sup>*] *by <sup>U</sup>p*[0, *<sup>τ</sup>*] *if and only if system (6) is Lp*−*exactly controllable on* [0, *<sup>τ</sup>*] *by <sup>U</sup>*˜ *<sup>p</sup>*[0, *<sup>τ</sup>*] <sup>×</sup> *<sup>L</sup><sup>p</sup> <sup>F</sup>*(Ω; *<sup>L</sup>*2([0, *<sup>τ</sup>*]; <sup>R</sup>*n*))*.*

**Theorem 4** ([59])**.** *Let Hypothesis 1 and (5) hold. Suppose*

$$\tilde{A}(t) \in L\_F^{\infty}(\Omega; L^{1+\varepsilon}([0, \tau]; \mathbb{R}^{n \times n})),\\\tilde{B}(t) \in L\_F^{\max\{2, p\} + \varepsilon}(\Omega; L^{2+\varepsilon}([0, \tau]; \mathbb{R}^{n \times m})),$$

$$\tilde{D}(t) \in L\_F^{\infty}(\Omega; L^{2+\varepsilon}([0, \tau]; \mathbb{R}^{n \times n})),\tag{8}$$

*where* > 0 *is a given constant. Then the following are equivalent:*

*(i) System (6) is Lp*−*exactly controllable on* [0, *<sup>τ</sup>*] *by <sup>U</sup>*˜ *<sup>p</sup>*[0, *<sup>τ</sup>*] <sup>×</sup> *<sup>L</sup><sup>p</sup> <sup>F</sup>*(Ω; *<sup>L</sup>*2([0, *<sup>τ</sup>*]; <sup>R</sup>*n*))*;*

*(ii) System (6) is exactly null-controllable on* [0, *<sup>τ</sup>*] *by <sup>U</sup>*˜ *<sup>p</sup>*[0, *<sup>τ</sup>*] <sup>×</sup> *<sup>L</sup><sup>p</sup> <sup>F</sup>*(Ω; *<sup>L</sup>*2([0, *<sup>τ</sup>*]; <sup>R</sup>*n*))*; (iii) Matrix G defined below is invertible:*

$$G = E \int\_0^\tau \chi(t)\vec{B}(t)\vec{B}(t)^T\boldsymbol{\chi}(t)^T dt,\tag{9}$$

*where Y*(·) *is the adapted solution to the following stochastic linear equation:*

$$d\mathcal{Y}(t) = -\mathcal{Y}(t)\bar{A}(t)dt - \mathcal{Y}(t)\mathcal{D}(t)dw(t), t \ge 0,\\ \mathcal{Y}(0) = I\_n.$$

**Theorem 5** ([59])**.** *Let Hypothesis 1, (5), and (8) hold. Then system (1) is <sup>L</sup>p*−*exactly controllable on* [0, *τ*] *by Up*[0, *τ*] *if and only if G defined by (9) is invertible.*

In the above, we have discussed the two extreme cases: either *D*(·) = 0 or rank*D*(·) = *n*. The case in between remains open. Therefore, we have the following open problem.

**Problem 1.** *If* 0 < rank*D*(·) < *n, what are the conditions under which system (1) can be <sup>L</sup>p*−*exactly controlled?*

## 2.1.3. Duality and Observability Inequality

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

**Theorem 6** ([59])**.** *Let hypotheses 1 and 2 hold. Then system (1) is <sup>L</sup>p*−*exactly controllable on* [0, *τ*] *by Up*,*μ*,*σ*[0, *τ*] *if and only if there exists a δ* > 0 *such that the following, called an observability inequality holds:*

$$\| |B(\cdot)^T y(\cdot) + \sum\_{k}^{d} D\_k(\cdot) z\_k(\cdot) \|\_{\underline{L}I^{p,q,\boldsymbol{\nu}}[0,\mathbb{T}]^\*} \geq \delta \| \eta \|\_{L^q(\Omega, F\_{\boldsymbol{\tau}}, P, \mathbb{R}^n)^\*} \forall \eta \in L^q(\Omega, F\_{\boldsymbol{\tau}}, P, \mathbb{R}^n),$$

*where*

$$\begin{split} \mathcal{U}^{p,\mu,\sigma}[0,\tau] &= L\_{F}^{\frac{pp}{p-p}}(\Omega; L^{\frac{2\mu}{\sigma-2}}([0,\tau]; \mathbb{R}^{m})), p \in [1,\mu), \mu \in (1,\infty], \sigma \in (2,\infty), \\ \mathcal{U}^{p,\mu,\sigma}[0,\tau] &= L\_{F}^{p}(\Omega; L^{\frac{2\mu}{\sigma-2}}([0,\tau]; \mathbb{R}^{m})), p \in [1,\mu), \mu = \infty, \sigma \in (2,\infty), \\ \mathcal{U}^{p,\mu,\sigma}[0,\tau] &= L\_{F}^{\frac{pp}{p-p}}(\Omega; L^{2}([0,\tau]; \mathbb{R}^{m})), p \in [1,\mu), \mu \in [1,\infty], \sigma = \infty, \\ \mathcal{U}^{p,\mu,\sigma}[0,\tau] &= L\_{F}^{p}(\Omega; L^{2}([0,\tau]; \mathbb{R}^{m})), p \in [1,\mu), \mu = \sigma = \infty; \end{split}$$

*Up*,*μ*,*σ*[0, *τ*] <sup>∗</sup> *denotes the adjoint space of Up*,*μ*,*σ*[0, *<sup>τ</sup>*]*;* (*y*(·), *<sup>z</sup>*(·)) *(with z*(·)=(*z*1(·), ··· , *zd*(·))*) is the unique adapted solution to the following system:*

$$dy(t) = -[A(t)^T y(t) + \sum\_{k=1}^d \mathbb{C}\_k(t)^T z\_k(t)]dt + \sum\_{k=1}^d z\_k(t)dw\_k(t), \\ t \in [0, \tau], \\ y(\tau) = \eta. \tag{10}$$

Now, we introduce the following definition which makes the name "observability inequality" aforementioned meaningful.

**Definition 3.** *Let Hypothesis 1 hold and* (*y*(*t*), *z*(*t*)) *be the adapted solution to system (10) with <sup>η</sup>* <sup>∈</sup> *<sup>L</sup>q*(Ω, *<sup>F</sup>τ*, *<sup>P</sup>*, <sup>R</sup>*n*). *(i) For the pair (B(*·*),D(*·*)) with <sup>B</sup>*(·), *Dk*(·) <sup>∈</sup> *<sup>L</sup>*<sup>1</sup> *<sup>F</sup>*([0, *<sup>τ</sup>*]; <sup>R</sup>*n*×*m*)(*<sup>k</sup>* <sup>=</sup> 1, 2, ··· , *d*) *and D*(·)=(*D*1(·), ··· , *Dd*(·)), *the map*

$$\eta \to K^\* \eta = B(\cdot)^T y(\cdot) + \sum\_{k}^{d} D\_k(\cdot)^T z\_k(\cdot)$$

*is called an <sup>Y</sup>*[0, *<sup>τ</sup>*]−*observer of (10) if <sup>K</sup>*∗*<sup>η</sup>* <sup>∈</sup> *<sup>Y</sup>*[0, *<sup>τ</sup>*], <sup>∀</sup>*<sup>η</sup>* <sup>∈</sup> *<sup>L</sup>q*(Ω, *<sup>F</sup>τ*, *<sup>P</sup>*, <sup>R</sup>*m*), *where <sup>Y</sup>*[0, *<sup>τ</sup>*] *is a subspace of L*<sup>1</sup> *<sup>F</sup>*([0, *<sup>τ</sup>*]; <sup>R</sup>*m*). *System (10), together with the observer of (10) is denoted by* [*A*(·)*T*, *<sup>C</sup>*(·)*T*; *<sup>B</sup>*(·)*T*, *<sup>D</sup>*(·)*T*];

*(ii) Subsystem* [*A*(·)*T*, *<sup>C</sup>*(·)*T*; *<sup>B</sup>*(·)*T*, *<sup>D</sup>*(·)*T*] *is said to be <sup>L</sup>q*−*exactly observable by <sup>Y</sup>*[0, *<sup>τ</sup>*] *observations if from the observation <sup>K</sup>*<sup>∗</sup> <sup>∈</sup> *<sup>Y</sup>*[0, *<sup>τ</sup>*], *the terminal value <sup>η</sup>* <sup>∈</sup> *<sup>L</sup>q*(Ω, *<sup>F</sup>τ*, *<sup>P</sup>*, <sup>R</sup>*n*) *of <sup>y</sup>*(·) *at <sup>τ</sup> can be uniquely determined, i.e., the map <sup>K</sup>*<sup>∗</sup> : *<sup>L</sup>q*(Ω, *<sup>F</sup>τ*, *<sup>P</sup>*, <sup>R</sup>*n*) <sup>→</sup> *<sup>Y</sup>*[0, *<sup>τ</sup>*] *admits a bounded inverse.*

With the above definition, the following result was obtained in [59]:

**Theorem 7** ([59])**.** *Let Hypotheses 1 and 2 hold true. Then, system (1) is <sup>L</sup>p*−*exactly controllable on* [0, *<sup>τ</sup>*] *by <sup>U</sup>p*,*μ*,*σ*[0, *<sup>τ</sup>*] *if and only if system* [*A*(·)*T*, *<sup>C</sup>*(·)*T*; *<sup>B</sup>*(·)*T*, *<sup>D</sup>*(·)*T*] *is <sup>L</sup>p*−*exactly observable by Up*,*μ*,*σ*[0, *τ*] ∗ *observations.*
