Strong well-posedness, stability and optimal control theory for a mathematical model for magneto-viscoelastic fluids

TL;DR

This paper establishes the well-posedness, stability, and optimal control of a 2D magneto-viscoelastic fluid model, including existence of solutions, stability estimates, and optimal control conditions with practical magnetic coil configurations.

Contribution

It provides the first rigorous analysis of strong solutions, stability, and optimal control for a coupled magneto-viscoelastic fluid model in two dimensions.

Findings

Proved global strong solutions exist for the model.

Derived stability estimates with respect to external magnetic fields.

Established existence and optimality conditions for control using magnetic coils.

Abstract

In this article, we study the strong well-posedness, stability and optimal control of an incompressible magneto-viscoelastic fluid model in two dimensions. The model consists of an incompressible Navier--Stokes equation for the velocity field, an evolution equation for the deformation tensor, and a gradient flow equation for the magnetization vector. First, we prove that the model under consideration posseses a global strong solution in a suitable functional framework. Second, we derive stability estimates with respect to an external magnetic field. Based on the stability estimates we use the external magnetic field as the control to minimize a cost functional of tracking-type. We prove existence of an optimal control and derive first-order necessary optimality conditions. Finally, we consider a second optimal control problem, where the external magnetic field, which represents the control, is generated by a finite number of fixed magnetic field coils.