Controllability of a stochastic functional differential equation driven by a fractional Brownian motion

Let U, V and W be three Hilbert spaces and let BH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B^{H}$\end{document} be a W-valued fractional Brownian motion with Hurst index H∈(12,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H\in(\frac{1}{2},1)$\end{document}. 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],\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textstyle\begin{cases} {}^{\mathrm{c}}D^{\alpha}_{t}[Lx(t)]=Ax(t)+f(t,x_{t})+ Bu(t)+\sigma(t)\frac {d}{dt}B^{H}(t), &t\in(0,T], \\ x(t)=\phi(t), &t\in(-\infty,0], \end{cases} $$\end{document} where Dαc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${}^{\mathrm{c}}D^{\alpha}$\end{document} is the Caputo fractional derivative of order α∈(1−H,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha\in(1-H,1)$\end{document}, the time history xt:(−∞,0]→xt(θ)=x(t+θ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x_{t}:(-\infty,0]\rightarrow x_{t}(\theta)=x(t+\theta)$\end{document} with t>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t>0$\end{document} belonging to the phase space Bh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathscr{B}_{h}$\end{document}, the control function u(⋅)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u(\cdot)$\end{document} is given in L2([0,T],V)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{2}([0,T],V)$\end{document}, 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]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[0,T]$\end{document} and we give an example to illustrate the theory.