On the uniqueness of the optimal control for 2-dimensional second grade fluids

TL;DR

This paper investigates the optimal control problem for 2D second grade non-Newtonian fluids, establishing second-order optimality conditions and proving the global uniqueness of solutions under specific conditions.

Contribution

It provides the first proof of global uniqueness for the solutions of the first-order optimality system in this context.

Findings

Established second-order sufficient optimality conditions.

Proved global uniqueness of the optimality system solutions.

Analyzed the control problem for 2D second grade fluids with boundary conditions.

Abstract

We study an optimal control problem with a quadratic cost functional for non-Newtonian fluids of differential type. More precisely, we consider the system governing the evolution of a second grade fluid filling a two-dimensional bounded domain, supplemented with a Navier slip boundary condition, and under certain assumptions on the size of the initial data and parameters of the model, we prove the second-order sufficient optimality conditions. Furthermore, we establish a global uniqueness result for the solutions of the first-order optimality system.