Controllability of a stochastic functional differential equation driven by a fractional Brownian motion - Jingqi Han; Litan Yan (Advances in Difference Equations, 2018)
Abstract Let U, V and W be three Hilbert spaces and let BH $B^{H}$ be a W-valued fractional Brownian motion with Hurst index H∈(12,1) $H\in(\frac{1}{2},1)$. In this paper, we consider the approximate controllability of the Sobolev-type fractional stochastic differential equation {Dtαc[Lx(t)]=Ax(t)+f(t,xt)+Bu(t)+σ(t)ddtBH(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 Dαc ${}^{\mathrm{c}}D^{\alpha}$ is the Caputo fractional derivative of order α∈(1−H,1) $\alpha\in(1-H,1)$, the time history xt:(−∞,0]→xt(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 Bh $\mathscr{B}_{h}$, the control function u(⋅) $u(\cdot)$ is given in L2([0,T],V) $L^{2}([0,T],V)$, 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.