Weak solutions to the time-fractional g-Navier-Stokes equations and optimal control

In this paper we introduce the $g-$Navier-Stokes equations with time-fractional derivative of order $\alpha \in (0,1)$ in domains of $\mathbb R^2$. We then study the existence and uniqueness of weak solutions by means of Galerkin approximation. Finally, an optimal control problem is considered and solved.