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

TL;DR

This paper proves the continuity of a specific cost functional related to the 2D stochastic Navier-Stokes equation under certain feedback controls, contributing to the understanding of control stability in stochastic fluid dynamics.

Contribution

It establishes the continuity of a particular cost functional for the 2D stochastic Navier-Stokes equation with respect to a class of feedback controls, advancing control theory in stochastic PDEs.

Findings

Cost functional is continuous under feedback controls.

Results apply to 2D stochastic Navier-Stokes equations in bounded domains.

Provides a foundation for stable feedback control design in stochastic fluid models.

Abstract

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 $0 < \epsilon <1$.