5.3.2. Controllability of System (56)

In this part, the exact controllability and approximate controllability of system (56) are discussed by using GE-evolution operator in the sense of mild solution in Hilbert spaces. In order to discuss the controllability, we introduce the following concepts.

Hilbert space {*v*(*t*) ∈ *U* : *C*(*t*)*v*(*t*) ∈ *A*(*D*0)} is still denoted by *U*.

Controllability operator *C<sup>T</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>*0) and Controllability Gramian *G<sup>b</sup> <sup>c</sup>* : *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*0) <sup>→</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*0) associated with system (56) are defined as

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

$$\mathcal{G}\_c^b z = \int\_0^b \mathcal{U}(b, \tau) \mathcal{C}(\tau) \mathcal{C}^\*(\tau) \mathcal{U}^\*(b, \tau) E(z|F\_\tau) d\tau,$$

respectively. It is obvious that operators *C<sup>b</sup>* <sup>0</sup> and *<sup>G</sup><sup>b</sup> <sup>c</sup>* are bounded linear operators, and the dual

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

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>*∗(*τ*)*U*∗(*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>*0) and

$$G\_c^b = \mathcal{C}\_0^b \mathcal{C}\_0^{b\*} \dots$$

**Definition 28.** *(a) Time varying stochastic singular system (56) is said 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>*0)*, xb* <sup>∈</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*0)*, 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 (56) satisfies x*(*T*, *x*0) = *xb;*

*(b) Time varying stochastic singular system (56) is said to be approximately controllable on* [0, *<sup>b</sup>*]*, if for any state xb* <sup>∈</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*0)*, 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>*0)*, and any* <sup>&</sup>gt; <sup>0</sup>*, there exists a v* <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) *to (56) satisfies*

$$\|\varkappa(b,\varkappa\_0) - \varkappa\_b\|\_{L^2(\Omega, F\_{b\ast}P, \overline{D\_0})} < \epsilon.$$

The following results were obtained in [80].

**Theorem 42** ([80])**.** *The necessary and sufficient conditions for time-varying stochastic singular system (56) to be exactly controllable on* [0, *b*] *are* ran*C<sup>b</sup>* <sup>0</sup> = *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*0).

**Theorem 43** ([80])**.** *Time varying stochastic singular system (56) is exactly controllable on* [0, *b*] *if, and only if, one of the following conditions is true: (a)* < *G<sup>b</sup>* 2

*<sup>c</sup> <sup>z</sup>*, *<sup>z</sup>* >*L*2(Ω,*Fb*,*P*,*D*0)≥ *<sup>γ</sup>z*-*<sup>L</sup>*2(Ω,*Fb*,*P*,*D*0) *for some <sup>γ</sup>* <sup>&</sup>gt; <sup>0</sup> *and all*

$$z \in L^2(\Omega, F\_{\mathsf{b}\prime}P, \overline{D\_0});$$

*(b)* lim*λ*→0<sup>+</sup> -(*λI* + *G<sup>T</sup> <sup>c</sup>* )−<sup>1</sup> <sup>−</sup> (*G<sup>T</sup> <sup>c</sup>* )−1-*<sup>L</sup>*(*L*2(Ω,*Fb*,*P*,*D*0),*L*2(Ω,*Fb*,*P*,*D*0)) = <sup>0</sup>*; (c)* lim*λ*→0<sup>+</sup> *λ*(*λI* + *G<sup>T</sup> <sup>c</sup>* )−1-*<sup>L</sup>*(*L*2(Ω,*Fb*,*P*,*D*0),*L*2(Ω,*Fb*,*P*,*D*0)) = <sup>0</sup>*; (d)* -*Cb*<sup>∗</sup> <sup>0</sup> *z*-*<sup>L</sup>*2([0,*b*],Ω,*U*) ≥ *γz*-*<sup>L</sup>*2(Ω,*Fb*,*P*,*D*0) *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>*0); *(e)* ker(*Cb*<sup>∗</sup> <sup>0</sup> ) = {0} *and* ran(*Cb*<sup>∗</sup> <sup>0</sup> ) *is closed.*

**Theorem 44** ([80])**.** *The necessary and sufficient conditions for time varying stochastic singular system (56) to be approximately controllable on* [0, *T*] *are 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*0)<sup>&</sup>gt; <sup>0</sup> *for all z* <sup>∈</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*0), *<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>T</sup> <sup>c</sup>* )−1*x*, *<sup>z</sup>* <sup>&</sup>gt;*L*2(Ω,*Fb*,*P*,*D*0)<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>*0)*; (c)* lim*λ*→0<sup>+</sup> *λ*(*λI* + *G<sup>T</sup> <sup>c</sup>* )−1*z*-*<sup>L</sup>*2(Ω,*Fb*,*P*,*D*0) <sup>=</sup> <sup>0</sup> *for all z* <sup>∈</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*0)*; (d)* ker(*Cb*<sup>∗</sup> <sup>0</sup> ) = {0}.

The details of applicable example can be found in [80].

*5.4. Stochastic GE-Evolution Operator Method for a Class of Time Invariant Systems*

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

$$A\mathbf{x}(t) = B\mathbf{x}(t)dt + \mathbb{C}\mathbf{v}(t)dt + D\mathbf{x}(t)dw(t), \\ t \ge \mathbf{0}, \\ \mathbf{x}(0) = \mathbf{x}\_{0\star} \tag{58}$$

where *x*(*t*) is the state process valued in *H*, *v*(*t*) is the control process valued in *U*, *w*(*t*) is the one-dimensional standard Wiener process, *<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, *H*, *U* are Banach spaces; *A*, *D* ∈ *B*(*H*), *C* ∈ *B*(*U*, *H*), *B* : dom(*B*) ⊆ *H* → *H* is a linear operator. The organization of this subsection is as follows. Firstly, the concept of stochastic GE-evolution operator is introduced, and the mild solution to system (58) is given by stochastic GE-evolution operator. Secondly, The exact controllability and approximate controllability of (58) are discussed by stochastic GE-evolution operator in the sense of mild solution in Banach spaces, respectively.
