*5.2. GE-Semigroup Method for a Class of Time Invariant Systems*

In this subsection, we discuss the controllability of the following time invariant stochastic singular linear system by using GE-semigroup in the sense of mild solution in Banach and Hilbert spaces, respectively,

$$A d\mathbf{x}(t) = B\mathbf{x}(t)dt + \mathbb{C}v(t)dt + Ddw(t), \mathbf{x}(0) = \mathbf{x}\_0, t \ge 0,\tag{54}$$

where *x*(*t*) is the state process valued in *H*, *v*(*t*) is the control process valued in *U*, *w*(*t*) is the standard Wiener process on *<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 Banach or Hilbert spaces; *A* ∈ *B*(*H*), *C* ∈ *B*(*U*, *H*), *D* ∈ *B*(*Z*, *H*), *B* : dom(*B*) ⊆ *H* → *H* is a linear operator. This subsection is organized as follows. Firstly, the GE-semigroup is introduced and the mild solution of (54) is obtained; Secondly, the controllability of (54) is discussed in Banach spaces; Thirdly, the controllability of (54) is discussed in Hilbert spaces.
