Existence of solution for the Ginzburg-Landau system, a related optimal control problem and its computation by the generalized method of lines

The first objective of this work is the development of a global existence result for the Ginzburg-Landau system of superconductivity in the presence of a magnetic field. The results are obtained through standard tools of analysis and the mild hypothesis of boundedness of the infinity norm of the magnetic potential. In a second step we study the existence of solution and computation for a closely related optimal control problem. The solution of an example concerning such a problem is obtained through the generalized method of lines.