  This paper investigates the near optimal control for a kind of linear
stochastic control systems governed by the forward backward stochastic
differential equations, where both the drift and diffusion terms are allowed to
depend on controls and the control domain is not assumed to be convex. In the
previous work (Theorem 3.1) of the second and third authors [\textit{%
Automatica} \textbf{46} (2010) 397-404], some problem of near optimal control
with the control dependent diffusion is addressed and our current paper can be
viewed as some direct response to it. The necessary condition of the
near-optimality is established within the framework of optimality variational
principle developed by Yong [\textit{SIAM J. Control Optim.} \textbf{48} (2010)
4119--4156] and obtained by the convergence technique to treat the optimal
control of FBSDEs in unbounded control domains by Wu [% \textit{Automatica}
\textbf{49} (2013) 1473--1480]. Some new estimates are given here to handle the
near optimality. In addition, an illustrating example is discussed as well.
