&(x,t)\in{\mathbb{T}}^2\times (0,T),\\[4pt]
\partial_{(0,t]}^\alpha m - \Delta m - {{\rm{div}}}(mH_p(x,Du)) = 0&(x,t)\in{\mathbb{T}}^2\times (0,T),\\[4pt]
u(x,T)=u_T(x),\,m(x,0)=m_0(x), \\[4pt]
\int_{{\mathbb{T}}^2}m_0(x)dx=1,
\end{cases}$$ where ${\mathbb{T}}^2$ denotes the unit torus in ${\mathbb{R}}^2$ and $\partial_{(0,t]}^\alpha\cdot$, $\partial_{[t,T)}^\alpha\cdot$ are the forward and backward time-fractional Caputo derivatives defined, for $\alpha\in (0,1)$, by $$\begin{aligned}
&\partial_{(0,t]}^\alpha m(x,t) = \frac{1}{\Gamma(1-\alpha)}\int_{0}^{t}\frac{\partial_s m(x,s)}{(t-s)^{\alpha}}ds, \label{eq:caputo_forward}\\
&\partial_{[t,T)}^\alpha u(x,t) = -\frac{1}{\Gamma(1-\alpha)}\int_{t}^{T}\frac{\partial_su(x,s)}{(s-t)^{\alpha}}ds.\label{eq:caputo_backward}\end{aligned}$$ The system is composed by time-fractional backward Hamilton-Jacobi-Bellman and forward Fokker-Planck equations with the corresponding final and initial conditions. If the coupling cost $f[m]$ admits a primitive $F(x,m)$, can be derived via convex duality from an optimal control problem of $$\inf_{v,m}\int_0^t\int_{{\mathbb{T}}^2} (mL(x,-v)+F(x,m))dx\,ds+\int_{{\mathbb{T}}^2}(I^{1-\alpha}_{(0,t]}m)(T)dx,$$ subject to the constraint given by the time-fractional Fokker-Planck equation $$\label{frac_fp}
\begin{cases}
\partial_{(0,t]}^\alpha m - \Delta m - {{\rm{div}}}(mv) = 0,&(x,t)\in{\mathbb{T}}^2\times (0,T),\\[4pt]
m(x,0)=m_0(x),\\[4pt]
\int_{{\mathbb{T}}^2}m_0(x)dx=1.
\end{cases}$$ Here the forward Riemann-Liouville fractional integral is defined as $$I^{1-\alpha}_{(0,t]}m(t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t\frac{m(s)}{(t-s)^\alpha}ds.$$ System is derived in [@q], where the Fokker-Planck equation is considered in its weak formulation via a general fractional Sobolev space framework. The wellposedness of the coupled system is obtained using compactness and Schauder fixed-point argument. Time fractional advection-diffusion equations have been extensively used for studying evolution of probability of particles governed by subdiffusion behavior. This refers to the trajectory of a single particle is modeled by non Markovian stochastic process where 