Control-Translated Finsler-type structure and Anisotropic Ginzburg-Landau models

TL;DR

This paper extends the Finsler--Ginzburg--Landau framework by incorporating a control translation in the tangent bundle, enabling manipulation of vortex configurations in anisotropic superconductors through a rigorous geometric and variational approach.

Contribution

It introduces a control-translated Finsler structure that couples external control fields with anisotropic Ginzburg-Landau models, providing a new geometric framework for vortex manipulation.

Findings

Established convexity, coercivity, and regularity of the energy functional.

Proved existence of controlled minimizers via variational methods.

Derived a $ ext{Gamma}$-convergence result for vortex interaction energy under control translation.

Abstract

This paper develops a geometric and analytical extension of the Finsler--Ginzburg--Landau framework by introducing a distributed control field acting as a translation in the tangent bundle. Within this formulation, the classical Tonelli Lagrangian is deformed into a control--translated Finsler structure, whose Legendre dual induces a uniformly elliptic operator and a convex energy functional preserving the essential variational features of the anisotropic model. This approach provides a rigorous analytical setting for coupling external control fields with the intrinsic Finsler geometry of anisotropic superconductors. The study establishes the convexity, coercivity, and regularity properties of the induced energy functional and proves the existence of controlled minimizers through variational arguments on admissible configurations. In the asymptotic regime as the Ginzburg--Landau parameter tends to zero, a detailed $\Gamma$--convergence analysis yields a renormalized energy $W_u$ governing vortex interactions under control translation, quantifying the modification of the Green kernel and the self-energy due to the field $u(x)$. The results demonstrate that the control translation preserves the underlying Finsler structure while introducing a new geometric degree of freedom for manipulating and stabilizing vortex configurations.