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

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 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 Gâteaux derivative of the control-to-state mapping coincides with the solution of the linearized state equation. Finally, we prove the existence of an optimal solution and establish the first-order necessary optimality conditions.