# Optimal control for two-dimensional stochastic second grade fluids

## TL;DR

This paper develops an optimal control framework for two-dimensional stochastic second grade fluids, establishing existence, uniqueness, and optimality conditions for controlling non-Newtonian fluid flows under stochastic perturbations.

## Contribution

It introduces a novel stochastic control approach for non-Newtonian fluids, deriving the first order optimality conditions and analyzing the associated stochastic equations.

## Key findings

- Existence and uniqueness of the control-to-state map derivative.
- Well-posedness of the stochastic linearized and adjoint equations.
- Derivation of first order optimality conditions for the control problem.

## Abstract

This article deals with a stochastic control problem for certain fluids of non-Newtonian type. More precisely, the state equation is given by the two-dimensional stochastic second grade fluids perturbed by a multiplicative white noise. The control acts through an external stochastic force and we search for a control that minimizes a cost functional. We show that the G\^ateaux derivative of the control to state map is a stochastic process being the unique solution of the stochastic linearized state equation. The well-posedness of the corresponding stochastic backward adjoint equation is also established, allowing to derive the first order optimality condition.