Optimal control of third grade fluids with multiplicative noise

¹ Scuola Normale Superiore, Piazza dei Cavalieri 7, Pisa, Italy
² Center for Mathematics and Applications (NovaMath), NOVA SST and Department of Mathematics, NOVA SST, Portugal

^* Corresponding author: tahraouiyacine@yahoo.fr

Received: 19 March 2023
Accepted: 2 January 2025

Abstract

This work aims to control the dynamics of certain non-Newtonian fluids in a bounded domain of 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 Gâteaux 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.