A Stochastic Maximum Principle for Control Problems Constrained by the Stochastic Navier-Stokes Equations

TL;DR

This paper develops a stochastic maximum principle for controlling stochastic Navier-Stokes equations, providing necessary and sufficient conditions for optimal controls in multidimensional domains, including 2D and 3D cases.

Contribution

It introduces a novel stochastic maximum principle for Navier-Stokes control problems with Q-Wiener noise, enabling unique solutions for both 2D and 3D domains.

Findings

Derivation of a necessary optimality condition using a backward SPDE.

Establishment of a sufficient optimality condition for the control.

Unique solvability of control problems constrained by stochastic Navier-Stokes equations.

Abstract

We consider the control problem of the stochastic Navier-Stokes equations in multidimensional domains introduced in \cite{ocpc} restricted to noise terms defined by Q-Wiener processes. Using a stochastic maximum principle, we derive a necessary optimality condition to design the optimal control based on an adjoint equation, which is given by a backward SPDE. Moreover, we show that the optimal control satisfies a sufficient optimality condition. As a consequence, we can solve uniquely control problems constrained by the stochastic Navier-Stokes equations especially for two-dimensional as well as for three-dimensional domains.