# Existence of solution for an optimal control problem associated to the Ginzburg-Landau system in superconductivity

## TL;DR

This paper proves the existence of solutions for an optimal control problem involving the Ginzburg-Landau system in superconductivity, considering boundary controls and external magnetic fields, using advanced analysis tools.

## Contribution

It establishes a global existence result for the control problem associated with the Ginzburg-Landau equations in superconductivity, incorporating boundary control and external magnetic fields.

## Key findings

- Existence of solutions for the control problem.
- Application of Friedrichs Curl Inequality and Rellich-Kondrashov Theorem.
- Model includes boundary control and external magnetic field.

## Abstract

This article develops a global existence result for the solution of an optimal control problem associated to the Ginzburg-Landau system. This main result is based on standard tools of analysis and functional analysis, such as the Friedrichs Curl Inequality and the Rellich-Kondrashov Theorem. In the concerning model, we consider the presence of an external magnetic field and the control variable is a complex function acting on the super-conducting sample boundary. Finally the state variables are the Ginzburg-Landau order parameter and the magnetic potential, defined on domains properly specified.