Continuity of Cost Functional and Optimal FeedbackControls for the Stochastic Navier Stokes Equation in2D

We show the continuity of a specific cost functional $J(\phi) = \mathbb{E} \sup_{ t \in [0,T]}(\varphi(\mathcal{L}[t,u_\phi(t), \phi(t)]))$ of the SNSE in 2D on an open bounded nonperiodic domain $\mathcal{O}$ with respect to a special set of feedback controls $\{\phi_n\}_{n \geq 0}$, where $\varphi(x) =\log(1 + x)^{1-\epsilon}$ with $0 < \epsilon <1$.