Optimal control of two dimensional third grade fluids

TL;DR

This paper investigates the optimal control of flows governed by two-dimensional third grade fluid equations, establishing well-posedness, linearization, stability, and optimality conditions, including uniqueness under certain cost conditions.

Contribution

It provides the first comprehensive analysis of optimal control problems for third grade fluids with boundary conditions, including existence, uniqueness, and optimality conditions.

Findings

Existence and uniqueness of solutions to the control problem.

Derivation of first order optimality conditions.

Uniqueness of the coupled system under high cost intensity.

Abstract

The aim of this work is to study the optimal control problems of flows governed by the incompressible third grade fluid equations with Navier-slip boundary conditions. After recalling a result on the well-posedness of the state equations, we study the existence and the uniqueness of solution to the linearized state and adjoint equations. Furthermore, we present a stability result for the state, and show that the solution of the linearized equation coincides with the G\^ateaux derivative of the control-to-state mapping. Next, we prove the existence of an optimal solution and establish the first order optimality conditions. Finally, an uniqueness result of the coupled system constituted by the state equation, the adjoint equation and the first order optimality condition is established, under sufficiently large intensity of the cost.