Global-in-time optimal control of stochastic third-grade fluids with additive noise

TL;DR

This paper develops a framework for globally controlling stochastic third-grade non-Newtonian fluids with additive noise, establishing well-posedness, linearization, and optimality conditions for velocity tracking control.

Contribution

It introduces a novel approach converting the stochastic system into a pathwise deterministic one, proving existence, uniqueness, and optimality conditions for the control problem.

Findings

Proved global well-posedness of the stochastic system.

Established existence and uniqueness of solutions to linearized and adjoint equations.

Derived first-order optimality conditions for the control problem.

Abstract

In this article, we address the velocity tracking control problem for a class of stochastic non-Newtonian fluids. More precisely, we consider the stochastic third-grade fluid equation perturbed by infinite-dimensional additive white noise and defined on the two-dimensional torus $\mathbb{T}^2$. The control acts as a distributed random external force. Taking an \emph{infinite-dimensional Ornstein-Uhlenbeck process}, the stochastic system is converted into an equivalent pathwise deterministic one, which allows to show the well-posedness of the original stochastic system globally in time. The state being a stochastic process with sample paths in $\mathrm{L}^\infty(0,T;\mathbb{H}^3(\mathbb{T}^2))$ and finite moments can be controlled in an optimal way. Namely, we establish the existence and uniqueness of solutions to the corresponding linearized state and adjoint equations. Furthermore, we derive an appropriate stability result for the state equation and verify that the G\^ateaux derivative of the control-to-state mapping coincides with the solution of the linearized state equation. Finally, we establish the first-order optimality conditions and prove the existence of an optimal solution.