5.4.1. Stochastic GE-Evolution Operator and Mild Solution of System (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.

**Definition 29** ([81])**.** *Let* Δ*<sup>b</sup>* = {(*t*,*s*) : 0 ≤ *s* ≤ *t* ≤ *b*}*. A family of stochastic operators* {*S*(*t*,*s*) : (*t*,*s*) ∈ Δ*b*} *on H is said to be a stochastic GE-evolution operator induced by A on* [0, *b*] *if it has the following properties: (i) S* : Δ*<sup>b</sup>* × Ω → *B*(*H*) *is strongly measurable; (ii) S*(*t*,*s*) *is strongly Ft*−*measurable for t* ≥ *s; (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*<sup>0</sup> ∈ *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* Δ*<sup>T</sup> 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 30.** *If <sup>v</sup>*(*t*) <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>b</sup>*], <sup>Ω</sup>, *<sup>U</sup>*), *<sup>x</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, *<sup>D</sup>*1), *then the mild solution <sup>x</sup>*(*t*, *<sup>x</sup>*0) <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>b</sup>*], {*Ft*}, *<sup>D</sup>*1) *to (58) is defined by*

$$\mathbf{x}(t, \mathbf{x}\_0) = \mathcal{U}(t) A \mathbf{x}\_0 + \int\_0^t \mathcal{U}(t - \tau) \mathbb{C} \boldsymbol{v}(\tau) d\tau + \int\_0^t \mathcal{U}(t - \tau) D \mathbf{x}(\tau, \mathbf{x}\_0) d\mathbf{w}(\tau), \tag{59}$$

*where L*2([0, *<sup>b</sup>*], {*Ft*}, *<sup>D</sup>*1) *denotes the Banach space of all <sup>D</sup>*1−*valued processes x with norm*

$$||\mathfrak{x}||\_{L^{2}([0,b],\{F\_{l}\},\widetilde{D\_{1}})} = \sup\_{t \in [0,b]} (E||\mathfrak{x}(t)||\_{\widetilde{D\_{1}}}^{2})^{1/2} < +\infty.$$

**Lemma 2** ([81])**.** *If v*(*t*) <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>b</sup>*], <sup>Ω</sup>, *<sup>U</sup>*), *<sup>x</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, *<sup>D</sup>*1);

$$\mathcal{C}v(t) \in A(L^2([0, b], \Omega, \overline{D\_1}))\_{\prime \prime}$$

*then system (58) has a unique mild solution <sup>x</sup>*(*t*, *<sup>x</sup>*0) <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>b</sup>*], {*Ft*}, *<sup>D</sup>*1)*, which is given by (59).*

**Definition 31.** *We say that stochastic GE-evolution operator S*(*t*,*s*) *induced by A is related to the linear homogeneous equation*

$$A d\mathbf{x}(t) = B\mathbf{x}(t)dt + D\mathbf{x}(t)dw(t), \mathbf{x}(s) = \mathbf{x}\_0, 0 \le s \le t \le b,\tag{60}$$

*if x*(*t*) = *S*(*t*,*s*)*Ax*<sup>0</sup> *is the mild solution to (60) with x*(*s*) = *S*(*s*,*s*)*Ax*<sup>0</sup> = *x*<sup>0</sup> *for arbitrary <sup>x</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, *<sup>D</sup>*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 <sup>≤</sup> *<sup>s</sup>* <sup>≤</sup> *<sup>t</sup>* <sup>≤</sup> *<sup>b</sup>* and *<sup>ξ</sup>* <sup>∈</sup> *<sup>L</sup>*2(Ω, *Fs*, *<sup>P</sup>*, *<sup>D</sup>*1) :

$$E\int\_{s}^{t}||S(r,s)\xi||\_{\overline{D\_{1}}}^{2}dr \leq c||\xi||\_{L^{2}(\Omega,\mathbb{F}\_{\*},\overline{D\_{1}})}^{2}.$$

$$\sup\_{r\in[s,t]}E||S(r,s)\xi||\_{\overline{D\_{1}}}^{2} \leq c||\xi||\_{L^{2}(\Omega,\mathbb{F}\_{\*},\overline{D\_{1}})}^{2}.$$

We can obtain the following theorem.

**Theorem 45** ([81])**.** *The mild solution x*(*t*, *x*0) *to (58) can be written in the form*

$$\mathbf{x}(t, \mathbf{x}\_0) = \mathbf{S}(t, 0)A\mathbf{x}\_0 + \int\_0^t \mathbf{S}(t, s)\mathbf{C}\mathbf{v}(s)ds. \tag{61}$$

5.4.2. Controllability of System (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.

Banach space {*v*(*t*) ∈ *U* : *Cv*(*t*) ∈ *A*(*D*1)} is still denoted by *U*.

Controllability operator *C<sup>b</sup>* <sup>0</sup> : *<sup>L</sup>*2([0, *<sup>b</sup>*], <sup>Ω</sup>, *<sup>U</sup>*) <sup>→</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*1) associated with system (58) is defined as

$$\mathbb{C}\_0^b v = \int\_0^b \mathcal{S}(T, \tau) \mathcal{C}v(\tau) d\tau.$$

It is obvious that operator *C<sup>b</sup>* <sup>0</sup> is a bounded linear operator, and the dual

$$C\_0^{b\*}: L^2(\Omega, \mathcal{F}\_{b\prime}P, \overline{D\_0}) \to L^2([0, b], \Omega, U)$$

of *C<sup>b</sup>* <sup>0</sup> is defined by *<sup>C</sup>b*<sup>∗</sup> <sup>0</sup> *<sup>z</sup>* <sup>=</sup> *<sup>C</sup>*∗*S*∗(*b*, *<sup>τ</sup>*)*E*(*z*|*Fτ*), where *<sup>z</sup>* <sup>∈</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*1).

**Definition 32.** *(a) Stochastic singular linear system (58) is called to be exactly controllable on* [0, *<sup>b</sup>*]*, if for all <sup>x</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, *<sup>D</sup>*1)*, xb* <sup>∈</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*1)*, there exists <sup>v</sup>*(*t*) <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>b</sup>*], <sup>Ω</sup>, *<sup>U</sup>*)*, such that the mild solution x*(*t*, *x*0) *to stochastic singular linear system (58) which is given by (61) satisfies x*(*T*, *x*0) = *xb;*

*(b) Stochastic singular linear system (58) is called to be approximately controllable on* [0, *b*]*, if for any state xb* <sup>∈</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*1)*, any initial state <sup>x</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, *<sup>D</sup>*1)*, and any* <sup>&</sup>gt; <sup>0</sup>*, existence <sup>v</sup>* <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>b</sup>*], <sup>Ω</sup>, *<sup>U</sup>*) *makes that the mild solution <sup>x</sup>*(*t*, *<sup>x</sup>*0) *which is given by (61) satisfies*

$$\left\|\left|\mathfrak{x}(b,\mathfrak{x}\_{0})-\mathfrak{x}\_{b}\right\|\right\|\_{L^{2}(\Omega\_{\nu}\mathbb{P}\_{b\sim}\overline{D\_{1}})} < \epsilon.$$

The following results were obtained in [81].

**Theorem 46** ([81])**.** *Stochastic singular system (58) is exactly controllable on* [0, *b*] *if, and only if,* ran(*C<sup>b</sup>* <sup>0</sup>) = *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*1)*.*

**Theorem 47** ([81])**.** *Assume that H and U are reflexive Banach spaces. Stochastic singular system (58) is exactly controllable on* [0, *b*] *if and only if one of the following conditions holds: (a)* -*Cb*<sup>∗</sup> <sup>0</sup> *z*∗-*<sup>L</sup>*2([0,*b*],Ω,*U*∗) ≥ *γz*∗-*<sup>L</sup>*2(Ω,*Fb*,*P*,(*D*1)∗) *for some <sup>γ</sup>* > <sup>0</sup> *and all*

$$z^\* \in L^2(\Omega, F\_{\mathbb{B}^\prime} P\_\prime(\overline{D\_1})^\*);$$

$$(b)\ \ker(\mathbb{C}\_0^{b\*}) = \{0\} \ and \ \text{ran}(\mathbb{C}\_0^{b\*}) \text{ is closed.}$$

**Theorem 48** ([81])**.** *The necessary and sufficient condition for the stochastic singular linear system (58) to be approximately controllable on* [0, *b*] *is* ran(*C<sup>b</sup>* <sup>0</sup>) = *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*1).

**Theorem 49** ([81])**.** *Stochastic singular systems (58) is approximate controllable on* [0, *b*] *if, and only if, one of the following conditions holds:*

*(a)* -*Cb*<sup>∗</sup> <sup>0</sup> *z*∗-*<sup>L</sup>*2([0,*b*],Ω,*U*∗) <sup>&</sup>gt; <sup>0</sup> *for all z*<sup>∗</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*,(*D*1)∗), *<sup>z</sup>*<sup>∗</sup> <sup>=</sup> 0; *(b)* ker(*Cb*<sup>∗</sup> <sup>0</sup> ) = {0}.

The practical example can be found in [81] if there is a need.

*5.5. 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,

$$O\_1 dv(t) = O\_2(t)v(t)dt + O\_3(t)u(t)dt + O\_4(t)v(t)dw(t), \\ t \ge 0, \\ v(0) = v\_0.$$

$$x(t) = \mathcal{O}\_{\mathbb{S}}(t)v(t),\tag{62}$$

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, *<sup>v</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*,*Y*1) is a given random variable, *x*(*t*) is the output process valued in *Y*3, *Y*1,*Y*2,*Y*<sup>3</sup> are Banach spaces;

$$O\_1 \in B(Y\_1), O\_3(t) \in P([0, T], B(Y\_2, Y\_1)), O\_4(t) \in P([0, b], B(Y\_1)),$$

*O*5(*t*) ∈ *P*([0, *b*], *B*(*Y*1,*Y*3)), *O*2(*t*) is a linear operator from dom(*O*2(*t*)) ⊆ *Y*<sup>1</sup> to *Y*1; *O*1, *O*2(*t*), *O*3(*t*),*O*4(*t*),*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.

### 5.5.1. Mild Solution of System (62)

In this part, we always suppose that *O*2(*t*) is a generator of GE-evolution operator *V*(*t*,*s*) induced by *O*<sup>1</sup> and

$$D = \{ v \in \text{dom}\, \mathcal{O}\_2(t) \subseteq \mathcal{Y}\_1, V\_0 \mathcal{O}\_1 v = v\_\prime \}$$

$$\exists \lim\_{h \to 0^+} \frac{O\_1 V(t+h, t)O\_1 - O\_1 V(t, t)O\_1}{h} v, 0 \le t \le b \}$$

is independent of *t*, 0 ≤ *t* ≤ *b*.

Now, we consider the mild solution of time varying stochastic singular linear Equation (62).

**Definition 33.** *If <sup>u</sup>*(*t*) <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>b</sup>*], <sup>Ω</sup>,*Y*2), *<sup>v</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, *<sup>D</sup>*), *then the mild solution <sup>v</sup>*(*t*, *<sup>v</sup>*0) <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>b</sup>*], {*Ft*}, *<sup>D</sup>*) *to time varying stochastic singular Equation (62) is defined by*

$$v(t, v\_0) = V(t, 0)O\_1 v\_0 + \int\_0^t V(t, \tau)O\_3(\tau)u(\tau)d\tau + \int\_0^t V(t, \tau)O\_4(\tau)v(\tau, v\_0)dw(\tau). \tag{63}$$

**Lemma 3.** *Time varying stochastic singular Equation (62) has a unique mild solution, which is given by (63), if u*(*t*) <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>b</sup>*], <sup>Ω</sup>,*Y*2), *<sup>v</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, *<sup>D</sup>*);

$$O\_3(t)u(t) \in O\_1(L^2([0,b],\Omega,\overline{D})),$$

*and* (*V*0*O*2(*t*))|*<sup>D</sup> satisfies following assumptions:*

*(P*1*) For <sup>t</sup>* <sup>∈</sup> [0, *<sup>b</sup>*],(*λ<sup>I</sup>* + (*V*0*O*2(*t*))|*D*)−<sup>1</sup> *exists for all <sup>λ</sup> with Re<sup>λ</sup>* <sup>≤</sup> <sup>0</sup> *and there is a constant M, such that*

$$\| (\lambda I + (V\_0 O\_2(t))|\_D)^{-1} \|\_{B(Y\_1)} \le \frac{M}{|\lambda| + 1} \lambda$$

*for all Reλ* ≤ 0, *t* ∈ [0, *b*], *where I denotes the identical operator on D,* (*V*0*O*2(*t*))|*<sup>D</sup> denotes the restriction of V*0*O*2(*t*) *on D.*

*(P*2*) There exist constants L and* 0 < *α* ≤ 1*, such that*

$$\|\| ((V\_0 O\_2(t))|\_D - (V\_0 O\_2(s))|\_D) ((V\_0 O\_2(\tau))|\_D)^{-1} \|\_{B(Y\_1)} \le L|t - s|^a.$$

*for s*, *t*, *τ* ∈ [0, *b*].

**Proof.** First of all, according to Theorem 6.1 of [82] (see P.150 of [82]), we have that *V*(*t*,*s*)| *<sup>O</sup>*1(*D*) is a unique evolution operator induced by *<sup>O</sup>*<sup>1</sup> with generator *<sup>O</sup>*2(*t*) on *<sup>O</sup>*1(*D*). Let *Y*<sup>11</sup> denote the space of all *D* valued processes *ξ*, such that

$$|\mathfrak{f}|\_{Y\_{11}} = \sup\_{t \in [0,b]} (E ||\mathfrak{f}(t)||\_{\overline{D}}^2)^{1/2} < +\infty.$$

For any *ξ*(*t*) ∈ *Y*<sup>11</sup> define

$$P\_1(\tilde{\xi})(t) = V(t,0)O\_1v\_0 + \int\_0^t V(t,s)O\_3(s)u(s)ds$$

$$+ \int\_0^t V(t,s)O\_4(s)\tilde{\xi}(s)dw(s), t \in [0,b]\_\epsilon$$

and

$$P\_2(\vec{\xi})(t) = \int\_0^t V(t, \mathbf{s}) \mathcal{O}\_4(\mathbf{s}) \vec{\xi}(\mathbf{s}) dw(\mathbf{s}), t \in [0, b].$$

Assume, see (d) of Definition 25, that -*V*(*t*,*s*)-*<sup>B</sup>*(*Y*1) ≤ *M*1, 0 ≤ *s* ≤ *t* ≤ *b*, we have

$$\begin{aligned} \|P\_{\mathbf{2}}(\boldsymbol{\xi})\|\_{Y\_{11}} &\leq \sup\_{\boldsymbol{t}\in[0,b]} (E \int\_0^t \|\boldsymbol{V}(\boldsymbol{t},\boldsymbol{s})O\_4(\boldsymbol{s})\boldsymbol{\xi}(\boldsymbol{s})\|\_{\overline{D}}^2 ds)^{1/2} \\ &\leq M\_1 \|O\_4(\boldsymbol{s})\|\_{P([0,b],B(Y\_1))} b^{1/2} |\boldsymbol{\xi}|\_{Y\_{11}}, t \in [0,b]. \end{aligned}$$

Therefore, if *b* is sufficient small, *P*<sup>1</sup> 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*<sup>1</sup> *is related to the linear homogeneous time varying stochastic singular equation*

$$O\_1 dv(t) = O\_2(t)v(t)dt + O\_4(t)v(t)dw(t), \\ v(s) = v\_0, 0 \le s \le t \le b,\tag{64}$$

*Lemma 3 holds true, and the following estimates hold for any* 0 ≤ *s* ≤ *t* ≤ *b and ξ* ∈ *L*2(Ω, *Fs*, *P*, *D*) :

$$E\int\_{s}^{t}||G(r,s)\xi||\_{\overline{D}}^{2}dr \leq c||\xi||\_{L^{2}(\Omega,\mathbb{F}\_{\boldsymbol{s}},\mathcal{P},\overline{\mathcal{D}})}^{2}.$$

$$\sup\_{r\in[s,t]}E\,||G(r,s)\xi||\_{\overline{D\_{1}}}^{2}\leq c||\xi||\_{L^{2}(\Omega,\mathbb{F}\_{\boldsymbol{s}},\mathcal{P},\overline{\mathcal{D}})}^{2}.$$

*Then, the mild solution v*(*t*, *v*0) *to time varying stochastic singular Equation (62) can be written in the form*

$$v(t, v\_0) = G(t, 0)O\_1 v\_0 + \int\_0^t G(t, s)O\_3(s)u(s)ds.\tag{65}$$

**Proof.** Since *G*(*t*, 0)*O*1*v*<sup>0</sup> and *G*(*t*,*s*)*O*3(*s*)*u*(*s*) are mild solutions of time varying stochastic singular Equation (64) with *v*(0) = *v*<sup>0</sup> and *v*(*s*) = *G*(*s*,*s*)*O*3(*s*)*u*(*s*), respectively, we have that

$$G(t,0)O\_1v\_0 = V(t,0)O\_1v\_0 + \int\_0^t V(t,\tau)O\_4(\tau)G(\tau,0)O\_1v\_0dw(\tau),$$

$$G(t,s)O\_3(s)u(s) = V(t,s)O\_1G(s,s)O\_3(s)u(s) + \int\_s^t V(t,\tau)O\_4(\tau)G(\tau,s)O\_3(s)u(s)dw(\tau),$$

$$= V(t,s)O\_3(s)u(s) + \int\_s^t V(t,\tau)O\_4(\tau)G(\tau,s)O\_3(s)u(s)dw(\tau).$$

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

$$\int\_{0}^{t} V(t,\tau)O\_{4}(\tau)v(\tau,v\_{0})dw(\tau) = \int\_{0}^{t} V(t,\tau)O\_{4}(\tau)G(\tau,0)O\_{1}v\_{0}dw(\tau)$$

$$+ \int\_{0}^{t} V(t,\tau)O\_{4}D(\tau) \left(\int\_{0}^{\tau} G(\tau,s)O\_{3}(s)u(s)ds\right)dw(\tau)$$

$$= G(t,0)O\_{1}v\_{0} - V(t,0)O\_{1}v\_{0} + \int\_{0}^{t} ds \int\_{s}^{t} V(t,\tau)O\_{4}(\tau)G(\tau,s)O\_{3}(s)u(s)dw(\tau)$$

$$= G(t,0)O\_{1}v\_{0} - V(t,0)O\_{1}v\_{0} + \int\_{0}^{t} [G(t,s)O\_{3}(s)u(s) - V(t,s)O\_{1}G(s,s)O\_{3}(s)u(s)]ds$$

$$= G(t,0)O\_{1}v\_{0} - V(t,0)O\_{1}v\_{0} + \int\_{0}^{t} G(t,s)O\_{3}(s)u(s)ds - \int\_{0}^{t} V(t,s)O\_{3}(s)u(s)ds$$

$$\text{in the stochastic Euclidean form is given in Theorem 4.23 of L81. Therefore,}$$

where the stochastic Fubini theorem is given in Theorem 4.33 of [83]. Therefore,

$$v(t, v\_0) = G(t, 0)O\_1 v\_0 + \int\_0^t G(t, s)O\_3(s)u(s)ds$$

$$= V(t, 0)O\_1 v\_0 + \int\_0^t V(t, \tau)O\_3(\tau)u(\tau)d\tau + \int\_0^t V(t, \tau)O\_4(\tau)v(\tau, v\_0)d v(\tau),$$

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. Banach space {*u*(*t*) ∈ *Y*<sup>2</sup> : *O*3(*t*)*u*(*t*) ∈ *O*1(*D*)} is still denoted by *Y*2. Controllability operator

$$\mathcal{Q}\_{\mathbb{C}}^{b}: L^{2}([0,b],\Omega,\mathcal{Y}\_{2}) \to L^{2}(\Omega,F\_{b\prime}P,\overline{D})$$

associated with time varying stochastic singular Equation (62) is defined as

$$\mathcal{Q}\_{\mathbb{C}}^{b}u = \int\_{0}^{T} G(T,\tau)\mathcal{O}\_{3}(\tau)u(\tau)d\tau.$$

It is obvious that operator *Q<sup>b</sup> <sup>C</sup>* is a bounded linear operator, and its dual

$$\mathcal{Q}\_{\mathbb{C}}^{b\*}: L^2(\Omega, \mathbb{F}\_{\mathbb{B}}, P\_\prime(\overline{D})^\*) \to L^2([0, b], \Omega, \mathcal{Y}\_2^\*) $$

is defined by

$$Q\_{\mathbb{C}}^{b\*}y^\* = O\_3^\*(\mathfrak{r})G^\*(b,\mathfrak{r})E(y^\*|F\_{\mathfrak{r}})\dots$$

where *<sup>y</sup>*<sup>∗</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*,(*D*)∗).

5.5.2. Exact Controllability of System (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.

**Definition 34.** *Time varying stochastic singular Equation (62) is called to be exactly controllable on* [0, *<sup>b</sup>*]*, if for all <sup>v</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, *<sup>D</sup>*)*, vb* <sup>∈</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*)*, there exists <sup>u</sup>*(*t*) <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>b</sup>*], <sup>Ω</sup>,*Y*2)*, such that the mild solution v*(*t*, *v*0) *to time varying stochastic singular Equation (62) satisfies v*(*b*, *v*0) = *vb.*

From the Definition 34, we can obtain the following theorem immediately.

**Theorem 51.** *Time varying stochastic singular Equation (62) is exactly controllable on* [0, *b*] *if, and only if,* ran(*Q<sup>b</sup> <sup>C</sup>*) = *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*)*.*

**Theorem 52.** *Assume that Y*<sup>1</sup> *and Y*<sup>2</sup> *are reflexive Banach spaces. Time varying stochastic singular Equation (62) is exactly controllable on* [0, *b*] *if, and only if, one of the following conditions holds: (a)* -*Qb*<sup>∗</sup> *<sup>C</sup> y*∗-*L*2([0,*b*],Ω,*Y*∗ <sup>2</sup> ) ≥ *<sup>γ</sup>y*∗-*<sup>L</sup>*2(Ω,*Fb*,*P*,(*D*)∗) *for some <sup>γ</sup>* > <sup>0</sup> *and all*

$$y^\* \in L^2(\Omega, F\_{\mathsf{b}\prime}P, (\vec{D})^\*);$$

*(b)* ker(*Qb*<sup>∗</sup> *<sup>C</sup>* ) = {0} *and* ran(*Qb*<sup>∗</sup> *<sup>C</sup>* ) *is closed.*

**Proof.** (a) <sup>⇒</sup> (b) Notice that (a) implies that *<sup>Q</sup>b*<sup>∗</sup> *<sup>C</sup>* is injective. To prove that *<sup>Q</sup>b*<sup>∗</sup> *<sup>C</sup>* has closed range, assume that *Qb*<sup>∗</sup> *<sup>C</sup> y*<sup>∗</sup> *<sup>n</sup>* is a Cauchy sequence in *L*2([0, *b*], Ω,*Y*<sup>∗</sup> <sup>2</sup> ), then (a) implies that *y*∗ *<sup>n</sup>* is a Cauchy sequence in *L*2(Ω, *Fb*, *P*,(*D*)∗). Since *Qb*<sup>∗</sup> *<sup>C</sup>* is a bounded linear operator, if lim*n*→+<sup>∞</sup> *y*<sup>∗</sup> *<sup>n</sup>* <sup>=</sup> *<sup>y</sup>*∗, then lim*n*→+<sup>∞</sup> *<sup>Q</sup>b*<sup>∗</sup> *<sup>C</sup> y*<sup>∗</sup> *<sup>n</sup>* = *Qb*<sup>∗</sup> *<sup>C</sup> <sup>y</sup>*<sup>∗</sup> and so *<sup>Q</sup>b*<sup>∗</sup> *<sup>C</sup>* has closed range.

(b)⇒(a). (b) shows that *<sup>Q</sup>b*<sup>∗</sup> *<sup>C</sup>* has an algebraic inverse with domain equal to ran(*Qb*<sup>∗</sup> *<sup>C</sup>* ). Since ran(*Qb*<sup>∗</sup> *<sup>C</sup>* ) is closed, it is a Banach space under the norm of *<sup>L</sup>*2([0, *<sup>b</sup>*], <sup>Ω</sup>,*Y*<sup>∗</sup> <sup>2</sup> ), i.e.,

$$\|\mathfrak{u}^\*\|\_{\text{ran}(Q\_{\mathcal{C}}^{b\*})} = \|\mathfrak{u}^\*\|\_{L^2([0,b],\Omega,Y\_2^\*)}, \mathfrak{u}^\* \in \text{ran}(Q\_{\mathcal{C}}^{b\*}).$$

By Corollary A.3.50 of [84], we have that (*Qb*<sup>∗</sup> *<sup>C</sup>* )−<sup>1</sup> is bounded on this range, i.e., there exists a *γ* > 0, such that

$$\| |(Q\_{\mathbb{C}}^{b\*})^{-1}u^\*| \|\_{L^2(\Omega, F\_b, P\_\*(\overline{D})^\*)} \le \frac{1}{\gamma} \| u^\* \|\_{L^2([0,b], \Omega, Y\_2^\*)} \gamma$$

for every *<sup>u</sup>*<sup>∗</sup> <sup>∈</sup> ran(*Qb*<sup>∗</sup> *<sup>C</sup>* ). Substituting *<sup>u</sup>*<sup>∗</sup> = *<sup>C</sup>T*<sup>∗</sup> <sup>0</sup> *y*<sup>∗</sup> 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<sup>b</sup> <sup>C</sup>*) = *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*).

If *Q<sup>b</sup> <sup>C</sup>* is a one to one operator, then (*Q<sup>b</sup> <sup>C</sup>*)−<sup>1</sup> exists on *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*). According to the continuity of operator *Q<sup>b</sup> <sup>C</sup>* we have that (*Q<sup>b</sup> <sup>C</sup>*)−<sup>1</sup> is a closed operator. From the closed graph theorem, we obtain that (*Q<sup>b</sup> <sup>C</sup>*)−<sup>1</sup> is a bounded linear operator on *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*), i.e.,

$$((Q\_C^b)^{-1} \in B(L^2(\Omega, F\_b, P, \overline{D}), L^2([0, b], \Omega, Y\_2))\,.$$

Therefore

$$((Q\_{\mathbb{C}}^{b})^{-1})^{\*} \in B(L^{2}([0,b],\Omega,\mathcal{Y}\_{2}^{\*}), L^{2}(\Omega,F\_{b},P,(\overline{D})^{\*})).$$

This implies that there exists *γ<sup>b</sup>* > 0, such that

$$\|(({\cal Q\_C^b})^{-1})^\* \upsilon^\*\|\_{L^2(\Omega, {}\_{\cal D}, {}\_{\cal D})^\*)} \le \gamma\_b \|\upsilon^\*\|\_{L^2([0, b] \amalg {}\_{\cal D})}.\tag{66}$$

Assume *<sup>y</sup>*<sup>∗</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*,(*D*)∗), then

$$
\mu^\* = Q\_\mathbb{C}^{b\*} \mathfrak{z}^\* \in L^2([0, b], \Omega, \mathcal{Y}\_2^\*).
$$

Therefore, for all *<sup>y</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*), we find that

$$<\mathcal{Y}\_{0\prime}((Q\_{\mathbb{C}}^{b})^{-1})^{\*}\mu^{\*}> = <\mathcal{Y}\_{0\prime}((Q\_{\mathbb{C}}^{b})^{-1})^{\*}Q\_{\mathbb{C}}^{b\*}\mathcal{Y}^{\*}>$$

$$= <(Q\_{\mathbb{C}}^{T})^{-1}\mathcal{Y}\_{0\prime}Q\_{\mathbb{C}}^{T\*}\mathcal{Y}^{\*}> = <\mathcal{Y}\_{0\prime}\mathcal{y}^{\*}>,$$

where < *y*0, *y*<sup>∗</sup> >= *y*∗(*y*0). From (66), we obtain that

$$\|y^\*\|\|\_{L^2(\Omega,\mathbb{F}\_{\mathbb{B}},\mathbb{P}\_{\mathbb{A}}(\mathbb{D})^\*)} = \sup\|y\_0\|\_{L^2(\Omega,\mathbb{F}\_{\mathbb{B}},\mathbb{D})} = 1 < \mathcal{y}\_{0^\*}y^\* > |$$

$$\leq \|((Q\_{\mathbb{C}}^b)^{-1})^\*u^\*\|\_{L^2(\Omega,\mathbb{F}\_{\mathbb{B}},\mathbb{P}\_{\mathbb{C}}(\mathbb{D})^\*)}$$

$$\leq \gamma\_b \|u^\*\|\_{L^2([0,b],\Omega,Y\_2^\*)} = \gamma\_b \|Q\_{\mathbb{C}}^{b\*}y^\*\|\_{L^2([0,b],\Omega,Y\_2^\*)'}$$

i.e.,

$$\begin{aligned} \|Q\_{\mathcal{C}}^{b\*}y^\*\|\_{L^2([0,b],\Omega,Y\_{\mathcal{D}})} &\geq \frac{1}{\gamma\_b} \|y^\*\|\_{L^2(\Omega,F\_b,P,(\overline{D})^\*)}^2\\ &= \gamma \|y^\*\|\_{L^2(\Omega,F\_b,P,(\overline{D})^\*)'} \end{aligned}$$

where *γ* = <sup>1</sup> *γb* . This implies that (a) holds.

If *Q<sup>b</sup> <sup>C</sup>* is not a one to one operator, then

$$\ker(Q\_{\mathbb{C}}^b) = \{ \mu : \mu \in L^2([0, b], \Omega, \mathbb{Y}\_2), Q\_{\mathbb{C}}^b \mu = 0 \} \neq \{0\} \dots$$

A factor space is defined as follows

$$Y\_{21} = L^2([0, b], \Omega, \mathbb{Y}\_2) / \ker(\mathbb{Q}\_{\mathbb{C}}^b) = \{ u\_1 : u\_1 = \{ u + u\_2 : u\_2 \in \ker(\mathbb{Q}\_{\mathbb{C}}^b) \} \}.$$

For *u*<sup>1</sup> ∈ *Y*21,

$$\|\mu\_1\|\_{\mathcal{Y}\_{21}} = \inf\_{\mu\_2 \in \ker(Q^b\_{\mathcal{C}})} \|\mu + \mu\_2\|\_{L^2([0,b],\Omega,\mathcal{Y}\_2)}.$$

If we define operator

$$Q\_1^b \colon Y\_{21} \to L^2(\Omega, F\_{b\prime}P, \overline{D}), Q\_1^b \mu\_1 = Q\_\mathbb{C}^b \mu\_\prime$$

then

$$Q\_1^b \in B(\mathcal{Y}\_{21\prime}L^2(\Omega, F\_{\mathsf{b}\prime}P, \overline{D}))\_{\prime\prime}$$

and *Q<sup>b</sup>* <sup>1</sup> is a bijective operator. It can be seen from the above proof that

$$\|\|Q\_1^{b\*}y^\*\|\|\_{Y\_{21}^\*} \ge \gamma \|\|y^\*\|\|\_{L^2(\Omega, F\_{b\*}P\_r(\overline{D})^\*)}^2$$

.

According to the definition of *Y*<sup>21</sup> and *Q<sup>b</sup>* <sup>1</sup>, we obtain

$$\|\|Q\_1^{b\*}y^\*\|\|\_{Y\_{21}^\*} = \|\|Q\_C^{b\*}y^\*\|\|\_{L^2([0,b];\Omega,Y\_2^\*)}\cdot \|$$

This implies that (a) holds.

Sufficiency. Assume (a). It is need to prove that if *<sup>y</sup>* <sup>∈</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*), then *<sup>y</sup>* <sup>∈</sup> ran*Q<sup>b</sup> C*. From

$$Q^b\_\mathbb{C} \in B(L^2([0, b], \Omega, \mathcal{Y}\_2), L^2(\Omega, F\_{b\prime}, P, \overline{D}))\_{\prime\prime}$$

we find that

$$\mathcal{Q}\_{\mathbb{C}}^{b\*} \in \mathcal{B}(L^2(\Omega, F\_{b\prime}P\_{\prime}(\overline{D})^\*), L^2([0, b], \Omega, \mathcal{Y}\_2^\*))\,.$$

For *<sup>y</sup>* <sup>∈</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*), we can define a functional *<sup>f</sup>* on ran*Qb*<sup>∗</sup> *<sup>C</sup>* satisfying

$$f(Q\_{\mathbb{C}}^{b\*} \mathbb{g}^\*) = <\mathcal{y}, \mathcal{g}^\* >, \mathcal{g}^\* \in L^2(\Omega, F\_b, P, (\overline{D})^\*). \tag{67}$$

This implies that *f* is linear for *Qb*<sup>∗</sup> *<sup>C</sup> g*∗. According to (a), if

$$\lim\_{n \to \infty} Q\_C^{b\*} g\_n^\* = 0\_r$$

then

$$\lim\_{n \to \infty} \mathcal{g}\_n^\* = 0\_\prime$$

and

$$\lim\_{n \to \infty} f(Q\_C^{b\*} \mathcal{g}\_n^\*) = \lim\_{n \to \infty} < y\_\prime \mathcal{g}\_n^\* > = 0.$$

Therefore, *f* is continuous linear functional on

$$\text{ran}(Q\_{\mathbb{C}}^{b\*}) \subset L^2([0,b], \Omega, \mathcal{Y}\_2^\*).$$

By Hahn–Banach theorem, we have that *f* can be extended as a continuous linear functional on *L*2([0, *b*], Ω,*Y*<sup>∗</sup> <sup>2</sup> ). According to *Y*∗∗ <sup>2</sup> = *Y*2, the existence of

$$\mu \in L^2([0, b], \Omega, \mathcal{Y}\_2) = L^2([0, b], \Omega, \mathcal{Y}\_2^{\*\*})$$

makes

$$f(\mathbb{Q}\_{\mathbb{C}}^{b\*} \mathbb{g}^\*) = <\iota\iota\!\!/ \mathbb{Q}\_{\mathbb{C}}^{b\*} \mathbb{g}^\* > , \mathbb{g}^\* \in L^2(\Omega, \mathbb{F}\_{\mathbb{b}}, P, (\overline{D})^\*). \tag{68}$$

According to (67) and (68), we obtain that for every *<sup>g</sup>*<sup>∗</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*,(*D*)∗),

$$<\mathcal{Y}, \mathcal{g}^\* > = <\mathcal{Q}\_C^b \mathcal{u}\_\prime \mathcal{g}^\* > \dots$$

Hence *y* = *Q<sup>b</sup> <sup>C</sup>u*, i.e.,

$$\text{ran}(Q\_{\mathbb{C}}^b) = L^2(\Omega, F\_{b\prime}P, \overline{D})\dots$$

From Theorem 51, time varying stochastic singular Equation (62) is exactly controllable.

#### 5.5.3. Approximate Controllability of System (62)

In this section, we discuss the approximate controllability of time varying stochastic singular Equation (62). Some necessary and sufficient conditions are obtained.

**Definition 35.** *Time varying stochastic singular Equation (62) is called to be approximately controllable on* [0, *<sup>b</sup>*]*, if for any state vb* <sup>∈</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*)*, any initial state <sup>v</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, *<sup>D</sup>*)*, and any* <sup>&</sup>gt; <sup>0</sup>*, existence <sup>u</sup>* <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>b</sup>*], <sup>Ω</sup>,*Y*2) *makes that the mild solution <sup>v</sup>*(*t*, *<sup>v</sup>*0) *to time varying stochastic singular Equation (62) satisfies*

$$\left\|\left|\boldsymbol{\upsilon}(\boldsymbol{b},\boldsymbol{v}\_{0})-\boldsymbol{v}\_{b}\right\|\_{L^{2}(\Omega,F\_{\boldsymbol{b}},P\_{\boldsymbol{\tau}}\overline{\mathcal{D}})} < \epsilon.$$

$$\text{The first-order coupling between the two-dimensional } \mathcal{N} \text{-matrices is the only possible } \mathcal{N} \text{-matrices with } \mathcal{N} = \{0, 1, 2, \dots, N\} \text{ and } \mathcal{N} = \{0, 1, 2, \dots, N\}.$$

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

$$\overline{\text{ran}(Q\_{\mathbb{C}}^{\overline{b}})} = L^2(\Omega, F\_{b\prime}P, \overline{D}). \tag{69}$$

**Theorem 53.** *Time varying stochastic singular Equation (62) is approximate controllable on* [0, *b*] *if, and only if, one of the following conditions holds:*

$$\begin{aligned} &(a) \; ||Q^{b\*}\_{\mathbb{C}}y^\*||\_{L^2([0,b],\Omega,Y^\*\_{\mathbb{Z}})} > 0 \; for \; all \; y^\* \in L^2(\Omega, F\_{b\*}P, (\overline{D})^\*), y^\* \neq 0;\\ &(b) \; \ker(Q^{b\*}\_{\mathbb{C}}) = \{0\}. \end{aligned}$$

**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). If

$$\overline{\text{ran}(Q\_{\mathbb{C}}^{b})} = L^{2}(\Omega, F\_{b\prime}P, \overline{D}), y^{\*} \in \ker(Q\_{\mathbb{C}}^{b\*})\_{\prime}$$

i.e., *Qb*<sup>∗</sup> *<sup>C</sup> y*<sup>∗</sup> = 0, then

$$<\mu\_\prime Q\_C^{b\*} y^\* > =  , \mu \in L^2([0, b], \Omega, \mathbb{Y}\_2).$$

Since ran(*Q<sup>b</sup> <sup>C</sup>*) = *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*), we have

$$ = 0, y \in L^2(\Omega, P\_{b\prime}, P, \overrightarrow{D}).$$

Therefore, *y*<sup>∗</sup> = 0, i.e., ker(*Qb*<sup>∗</sup> *<sup>C</sup>* ) = {0}. Conversely, if ker(*Qb*<sup>∗</sup> *<sup>C</sup>* ) = {0} but

$$
\overline{\text{ran}(Q\_C^b)} \neq L^2(\Omega, F\_{b\prime}, P, \overline{D}),
$$

then ran(*Q<sup>b</sup> <sup>C</sup>*) is the proper subspace of *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*). According to Hahn–Banach theorem, there exists

$$y^\* \in L^2(\Omega, F\_{b\prime}P\_\prime(\vec{D})^\*), y^\* \neq 0,$$

such that

$$<\mathcal{Q}\_{\mathbb{C}}^{b}\mu, y^\*> := 0, \mu \in L^2([0, b], \Omega, \mathcal{Y}\_2).$$

Thus < *u*, *Qb*<sup>∗</sup> *<sup>C</sup> <sup>y</sup>*<sup>∗</sup> >= 0, i.e., *<sup>Q</sup>b*<sup>∗</sup> *<sup>C</sup> <sup>y</sup>*<sup>∗</sup> = 0. By ker(*Qb*<sup>∗</sup> *<sup>C</sup>* ) = {0}, we find that *y*<sup>∗</sup> = 0. This contradicts *y*<sup>∗</sup> = 0. Therefore,

$$
\overline{\text{ran}(Q\_{\mathcal{C}}^b)} = L^2(\Omega\_\prime F\_{b\prime} P\_\prime \overline{D}).
$$

Hence (69) is true if, and only if, (b) holds, i.e., time varying stochastic singular Equation (62) is approximately controllable on [0, *b*] if, and only if, (b) holds.
