5.2.2. Controllability of System (54) in Banach Spaces

In this following, we discuss the exact (approximate) controllability of system (54) in Banach spaces. Some necessary and sufficient conditions are given.

**Definition 24.** *(a) Stochastic singular system (54) is said to be exactly controllable on* [0, *b*]*, 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 (54) satisfies x*(*T*, *x*0) = *xb;*

*(b) Stochastic singular system (54) is said 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>*, there exists a <sup>v</sup>* <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>b</sup>*], <sup>Ω</sup>, *<sup>U</sup>*)*, such that the mild solution x*(*t*, *<sup>x</sup>*0) *satisfies*

$$\|\mathfrak{x}(b,\mathfrak{x}\_0) - \mathfrak{x}\_b\|\_{L^2(\Omega; F\_{T\_1}, P, \overline{D\_1})} < \epsilon.$$

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

$$C\_0^b: L^2([0, b], \Omega, \mathcal{U}) \to L^2(\Omega, F\_{b\prime}, P, \overline{D\_1})$$

associated with system (54) is defined as

$$\mathbb{C}\_0^b v = \int\_0^b \mathcal{U}(b-\tau)\mathcal{C}v(\tau)d\tau.$$

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

$$C\_0^{b\*}: L^2(\Omega, F\_{b\*}P, (\overline{D\_1})^\*) \to L^2([0, b], \Omega, U^\*)$$

is defined by

$$
\mathbb{C}\_0^{b\*} z^\* = \mathbb{C}^\* \mathcal{U}^\*(b - \tau) E(z^\* | F\_\tau).
$$

where *<sup>z</sup>*<sup>∗</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*,(*D*1)∗).

The following results were obtained in [76].

**Theorem 36** ([76])**.** *Stochastic singular system (54) 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 37** ([76])**.** *Assume that H and U are reflexive Banach spaces. Stochastic singular system (54) is exactly controllable on* [0, *b*] *if, and only if, one of the following conditions hold: (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*

$$z^\* \in L^2(\Omega, F\_{\mathcal{T}'} P\_\prime(\overline{D\_1})^\*);$$

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

**Theorem 38** ([76])**.** *Stochastic singular system (54) is approximately 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 39** ([76])**.** *Stochastic singular system (54) is approximate controllable on* [0, *b*] *if, and only if,*

$$\ker(\mathcal{C}\_0^{b\*}) = \{0\}.$$

See [76] (p. 908) for the illustrative example.

5.2.3. Controllability of System (54) in Hilbert Spaces

In this following, we discuss the exact (approximate) controllability of system (54) in Hilbert spaces. Some necessary and sufficient conditions are given. In order to discuss the controllability, we introduce the following operator.

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

Controllability Gramian operator *G<sup>b</sup> <sup>c</sup>* : *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*1) <sup>→</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*1) in connection with stochastic descriptor linear system (54) is defined as

$$G\_c^b z = \int\_0^b S(b-t) \mathcal{C} \mathcal{C}^\* \mathcal{S}^\*(b-t) E(z|F\_t) dt.$$

The following results were obtained in [77].

**Theorem 40** ([77])**.** *The necessary and sufficient condition for the stochastic singular linear system (54) to be exactly controllable on* [0, *b*] *is that one of the following conditions is true: (a)* < *G<sup>b</sup> <sup>c</sup> <sup>z</sup>*, *<sup>z</sup>* >*L*2(Ω,*Fb*,*P*,*D*1)≥ *<sup>γ</sup>z*-2 *<sup>L</sup>*2(Ω,*Fb*,*P*,*D*1) *for some <sup>γ</sup>* <sup>&</sup>gt; <sup>0</sup> *and all z* <sup>∈</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*1)*; (b)* lim*λ*→0<sup>+</sup> -(*λI* + *G<sup>b</sup> <sup>c</sup>* )−<sup>1</sup> <sup>−</sup> (*G<sup>b</sup> <sup>c</sup>* )−1-*<sup>B</sup>*(*L*2(Ω,*Fb*,*P*,*D*1)),*L*2(Ω,*Fb*,*P*,*D*1)) = <sup>0</sup>*; (c)* lim*λ*→0<sup>+</sup> *λ*(*λI* + *G<sup>b</sup> <sup>c</sup>* )−1-*<sup>B</sup>*(*L*2(Ω,*Fb*,*P*,*D*1),*L*2(Ω,*Fb*,*P*,*D*1)) = <sup>0</sup>*; (d)* ker(*Cb*<sup>∗</sup> <sup>0</sup> ) = {0} *and* ran(*Cb*<sup>∗</sup> <sup>0</sup> ) *is closed.*

**Theorem 41** ([77])**.** *The necessary and sufficient condition for the stochastic singular linear system (54) to be approximately controllable on* [0, *b*] *is that one of the following conditions is true: (a)* < *G<sup>b</sup> <sup>c</sup> <sup>z</sup>*, *<sup>z</sup>* <sup>&</sup>gt;*L*2(Ω,*Fb*,*P*,*D*1)<sup>&</sup>gt; <sup>0</sup> *for all z* <sup>∈</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*1), *<sup>z</sup>* <sup>=</sup> <sup>0</sup>*; (b)* lim*λ*→0<sup>+</sup> <sup>&</sup>lt; *<sup>λ</sup>*(*λ<sup>I</sup>* <sup>+</sup> *<sup>G</sup><sup>b</sup> <sup>c</sup>* )−1*x*, *<sup>z</sup>* <sup>&</sup>gt;*L*2(Ω,*Fb*,*P*,*D*1)<sup>=</sup> <sup>0</sup> *for all x*, *<sup>z</sup>* <sup>∈</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*1)*; (c)* lim*λ*→0<sup>+</sup> *λ*(*λI* + *G<sup>b</sup> <sup>c</sup>* )−1*z*-*<sup>L</sup>*2(Ω,*Fb*,*P*,*D*1) <sup>=</sup> <sup>0</sup> *for all z* <sup>∈</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*1)*.*

*5.3. GE-Evolution Operator Method for a Class of Time-Varying Systems*

In this subsection, we discuss the controllability of the following time varying stochastic singular linear system by using GE-evolution operator in Hilbert spaces,

$$A dx(t) = B(t)x(t)dt + C(t)v(t)dt + D(t)dw(t), \\ x(0) = x\_0, t \ge 0,\tag{56}$$

where *A* ∈ *B*(*H*) is a deterministic and constant operator, *B*(*t*) : dom(*B*(*t*)) ⊆ *H* → *H* is a linear operator (possibly unbounded), *B*(*t*), *C*(*t*), *D*(*t*) are deterministic and time varying operators; *C*(*t*) ∈ *P*([0, *b*], *B*(*U*, *H*)), *D*(*t*) ∈ *P*([0, *b*], *B*(*Z*, *H*)); *x*(*t*) is the state process valued in *H*, *v*(*t*) is the control process in *U*, *w*(*t*) is the stand Wiener process valued in *<sup>Z</sup>*, *<sup>x</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, *<sup>H</sup>*) is a given random variable, *<sup>H</sup>*, *<sup>U</sup>*, *<sup>Z</sup>* are Hilbert spaces. This subsection is organized as follows. Firstly, the GE-evolution operator is introduced and the mild solution of (56) is obtained; Secondly, the controllability of (56) is discussed by GE-evolution operator in the sense of mild solution in Hilbert spaces.

5.3.1. GE-Evolution Operator and Mild Solution of System (56)

In the following, we discuss mild solution of time varying stochastic singular system (56) according to GE-evolution operator. First of all, we recall the GE-evolution operator, and then the mild solution of (56) is given.

**Definition 25** ([78–80])**.** *Let* Δ(*b*) = {(*t*,*s*) : 0 ≤ *s* ≤ *t* ≤ *b*}*. U*(*t*,*s*) : Δ(*b*) → *B*(*H*) *is said to be a GE-evolution operator induced by A on* [0, *b*] *if it has the following properties:*

*(a) U*(*t*,*s*) = *U*(*t*,*r*)*AU*(*r*,*s*), 0 ≤ *s* ≤ *r* ≤ *t* ≤ *b;*

*(b) U*(*s*,*s*) = *U*0, 0 ≤ *s* ≤ *b, where U*<sup>0</sup> *is a definite operator independent of s;*

*GE-evolution operator U*(*t*,*s*) *is said to be strongly continuous on* [0, *b*] *if it has the following property:*

*(c) U*(·,*s*) *is strongly continuous on* [*s*, *b*] *and U*(*t*, ·) *is strongly continuous on* [0, *t*]*;*

*GE-evolution operator U*(*t*,*s*) *is said to be exponential bounded on* [0, *b*] *if it has the following property:*

*(d) There exist M* ≥ 1 *and ω* > 0*, such that*

$$||\mathcal{U}(t,s)||\_{\mathcal{B}(H)} \le Me^{\omega(t-s)}, 0 \le s \le t \le b.$$

**Definition 26** ([78–80])**.** *Assume that U*(*t*,*s*) *is a strongly continuous and exponential bounded GE-evolution operator induced by A. If*

$$B(t)\ge = \lim\_{h \to 0^+} \frac{A\iota I(t+h,t)A - A\iota I(t,t)A}{h}\ge, t \in [0,b],$$

*for every x* ∈ *D*0(*t*)*, where*

$$D\_0(t) = \{ \mathbf{x} : \mathbf{x} \in \text{dom}(B(t)) \subseteq H, \mathcal{U}\_0 A \mathbf{x} = \mathbf{x},$$

$$\exists \lim\_{h \to 0^+} \frac{A \mathcal{U}(t + h, t)A - A \mathcal{U}(t, t)A}{h} \mathbf{x}, t \in [0, b] \},$$

*then B*(*t*) *is called a generator of GE-evolution operator U*(*t*,*s*)*.*

In the following, we always assume that *B*(*t*) is the generator of GE-evolution operator *U*(*t*,*s*) induced by *A* and *D*0(*t*) = *D*<sup>0</sup> is independent of *t*.

Now, we consider the initial value problem (56).

**Definition 27.** *If <sup>x</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, *<sup>D</sup>*0)*, <sup>v</sup>*(*t*) <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>T</sup>*], <sup>Ω</sup>, *<sup>U</sup>*)*; <sup>C</sup>*(*t*)*v*(*t*), *<sup>D</sup>*(*t*)*dw*(*t*) <sup>∈</sup> *A*(*L*2([0, *b*], Ω, *D*0))*, the mild solution x*(*t*, *x*0) *to (56) is defined by*

$$\ln \mathbf{x}(t, \mathbf{x}\_0) = \mathcal{U}(t, 0) A \mathbf{x}\_0 + \int\_0^t \mathcal{U}(t, \tau) \mathbb{C}(\tau) v(\tau) d\tau + \int\_0^t \mathcal{U}(t, \tau) D(\tau) dw(\tau). \tag{57}$$

**Proposition 2** ([80])**.** *There exists unique mild solution x*(*t*, *x*0) *to (56), which is given by (57), 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>*0)*; <sup>C</sup>*(*t*)*v*(*t*), *<sup>D</sup>*(*t*)*dw*(*t*) <sup>∈</sup> *<sup>A</sup>*(*L*2([0, *<sup>b</sup>*], <sup>Ω</sup>, *<sup>D</sup>*0))*, and* (*U*0*B*(*t*))|*D*<sup>0</sup> *satisfies the following assumptions:*

*(P*1*) For <sup>t</sup>* <sup>∈</sup> [0, *<sup>b</sup>*],(*λ<sup>I</sup>* + (*U*0*B*(*t*))|*D*<sup>0</sup> )−<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* > 0*, such that*

$$\|\left(\lambda I + (\mathcal{U}\_0 B(t))|\_{D\_0}\right)^{-1}\|\_{B(H)} \le \frac{M}{|\lambda| + 1},$$

*for all Reλ* ≤ 0, *t* ∈ [0, *b*]. *(P*2*) There exist constants L* > 0 *and* 0 < *α* ≤ 1*, such that*

$$\|\left(\left(\left(Ll\_0B(t)\right)\right)|\_{D\_0} - \left(\left(Ll\_0B(s)\right)|\_{D\_0}\right)\left(\left(Ll\_0B(\tau)\right)|\_{D\_0}\right)^{-1}\|\_{B(H)} \leq L|t-s|^a.$$

*for t*,*s*, *τ* ∈ [0, *b*].

In the following, we suppose that Proposition 2 holds true.
