摘要：Let U, V and W be three Hilbert spaces and let B H $B^{H}$ be a W-valued fractional Brownian motion with Hurst index H ∈ ( 1 2 , 1 ) $H\in(\frac),,1)$ . In this paper, we consider the approximate controllability of the Sobolev-type fractional stochastic differential equation { D t α c [ L x ( t ) ] = A x ( t ) + f ( t , x t ) + B u ( t ) + σ ( t ) d d t B H ( t ) , t ∈ ( 0 , T ] , x ( t ) = ϕ ( t ) , t ∈ ( − ∞ , 0 ] , $$\textstyle\begin{cases} {}^{\mathrm{c}}D^{\alpha}_{t}[Lx(t)]=Ax(t)+f(t,x_{t})+ Bu(t)+\sigma(t)\frac {d}{dt}B^{H}(t), &t\in(0,T], \\ x(t)=\phi(t), &t\in(-\infty,0], \end{cases} $$ where D α c ${}^{\mathrm{c}}D^{\alpha}$ is the Caputo fractional derivative of order α ∈ ( 1 − H , 1 ) $\alpha\in(1-H,1)$ , the time history x t : ( − ∞ , 0 ] → x t ( θ ) = x ( t + θ ) $x_{t}:(-\infty,0]\rightarrow x_{t}(\theta)=x(t+\theta)$ with t > 0 $t>0$ belonging to the phase space B h $\mathscr{B}_{h}$ , the control function u ( ⋅ ) $u(\cdot)$ is given in L 2 ( [ 0 , T ] , V ) $L^,([0,T],V)$ , B is a bounded linear operator from V into U. Under some suitable conditions, we show that the system is approximately controllable on [ 0 , T ] $[0,T]$ and we give an example to illustrate the theory.
