Optimal control of Newtonian fluids in a stochastic environment

TL;DR

This paper develops an optimal control framework for stochastic Navier-Stokes equations in 2D, addressing boundary control with uncertainty, and derives conditions for optimality in a stochastic environment.

Contribution

It introduces a novel approach to boundary control of stochastic fluid dynamics, establishing well-posedness, duality, and optimality conditions for the problem.

Findings

Proved stability of the stochastic state solution.

Established well-posedness of the linearized and adjoint equations.

Derived first-order optimality conditions for the control problem.

Abstract

We consider a velocity tracking problem for stochastic Navier-Stokes equations in a 2D-bounded domain. The control acts on the boundary through an injection-suction device with uncertainty, which acts in accordance with the non-homogeneous Navier-slip boundary conditions. After establishing a suitable stability result for the solution of the stochastic state equation, we prove the well-posedness of the stochastic linearized state equation and show that the G\^ateaux derivative of the control-to-state mapping corresponds to the unique solution of the linearized equation. Next, we study the stochastic backward adjoint equation and establish a duality relation between the solutions of the forward linearized equation and the backward adjoint equation. Finally, we derive the first-order optimality conditions.