Optimal control of third grade fluids with multiplicative noise

TL;DR

This paper develops an optimal control framework for third grade non-Newtonian fluids affected by multiplicative noise, establishing existence, stability, and optimality conditions despite the challenges posed by nonlinearity and stochasticity.

Contribution

It introduces a novel stochastic control approach for complex non-Newtonian fluids, including existence of optimal controls and derivation of necessary optimality conditions.

Findings

Existence of an optimal control pair for the stochastic fluid system.

Derivation of the Gâteaux derivative of the control-to-state map.

Establishment of necessary optimality conditions and duality relations.

Abstract

This work aims to control the dynamics of certain non-Newtonian fluids in a bounded domain of $\mathbb{R}^d$, $d=2,3$ perturbed by a multiplicative Wiener noise, the control acts as a predictable distributed random force, and the goal is to achieve a predefined velocity profile under a minimal cost. Due to the strong nonlinearity of the stochastic state equations, strong solutions are available just locally in time, and the cost functional includes an appropriate stopping time. First, we show the existence of an optimal pair. Then,we show that the solution of the stochastic forward linearized equation coincides with the G\^ateaux derivative of the control-to-state mapping, after establishing some stability results. Next, we analyse the backward stochastic adjoint equation; where the uniqueness of solution holds only when $d=2$. Finally, we establish a duality relation and deduce the necessary optimality conditions.