Existence of Timewise Optimal Feedback Controls for the Stochastic Navier Stokes Equation in 2D

We show that there exist optimal feedback controls $\phi^v$ and $\phi^h$ for specific types of cost functionals $J^V(\phi) = \mathbb{E} \sup_{ t \in [0,T]}(\varphi^V(\mathcal{L}[t,u_\phi(t), \Phi(t)]))$ and $J^H(\phi) = \mathbb{E} \sup_{ t \in [0,T]}(\varphi^H(\mathcal{L}[t,u_\phi(t), \Phi(t)]))$ respectively of the SNSE in 2D on an open bounded nonperiodic domain $\mathcal{O}$ given that the control set $\Phi$ is compact, where $\phi^V(x) = \log(1 + \log(1 + x))$ and $\phi^H(x) = x^{1-\epsilon}$ with $0 < \epsilon <1$.