  In this paper, we study optimal stochastic control problems for stochastic
systems driven by non-Markov sub-diffusion $B_{L_t}$, which have the mixed
features of deterministic and stochastic controls. Here $B_t$ is the standard
Brownian motion on $R$, and
  $L_t:= \inf\{r>0: S_r>t\}, \quad t\geq 0,$ is the inverse of a subordinator
$S_t$ with drift $\kappa >0$ that is independent of $B_t$.
  We obtain stochastic maximum principles (SMP) for these systems using both
convex and spiking variational methods, depending on whether the convex domain
is convex or not. To derive SMP, we first establish a martingale representation
theorem for sub-diffusions $B_{L_t}$, and then use it to derive the existence
and uniqueness result for the solutions of backward stochastic differential
equations (BSDEs) driven by sub-diffusions, which may be of independent
interest.
  We also derive sufficient SMPs. Application to a linear quadratic system is
given to illustrate the main results of this paper.
