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

Y. Alipour Fakhri

TL;DR

This work develops a rigorous geometric framework for anisotropic Ginzburg--Landau models by embedding a distributed control field as a translation in the tangent bundle, yielding a control-translated Tonelli Lagrangian and a uniformly elliptic operator $\Delta_u$ that preserves the convex variational structure. A detailed $\Gamma$-convergence analysis shows that, as $\varepsilon\to 0$, vortex interactions are governed by a renormalized energy $W_u$ that depends on the control through the Green kernel $G_u$ and a self-interaction potential $\Phi_u$. The paper proves well-posedness and regularity of the controlled GL system, derives the Euler--Lagrange and gradient-flow dynamics, and establishes optimal control results with adjoint formalisms and reduced limits. Consequently, the control translation provides a geometric degree of freedom to manipulate and stabilize vortex configurations in anisotropic superconductors, while preserving the underlying Finsler structure. The framework opens avenues for time-dependent control, feedback strategies, and computational approaches in engineered anisotropic GL-type systems.

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 $Γ$--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.

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

TL;DR

This work develops a rigorous geometric framework for anisotropic Ginzburg--Landau models by embedding a distributed control field as a translation in the tangent bundle, yielding a control-translated Tonelli Lagrangian and a uniformly elliptic operator that preserves the convex variational structure. A detailed -convergence analysis shows that, as , vortex interactions are governed by a renormalized energy that depends on the control through the Green kernel and a self-interaction potential . The paper proves well-posedness and regularity of the controlled GL system, derives the Euler--Lagrange and gradient-flow dynamics, and establishes optimal control results with adjoint formalisms and reduced limits. Consequently, the control translation provides a geometric degree of freedom to manipulate and stabilize vortex configurations in anisotropic superconductors, while preserving the underlying Finsler structure. The framework opens avenues for time-dependent control, feedback strategies, and computational approaches in engineered anisotropic GL-type systems.

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 --convergence analysis yields a renormalized energy governing vortex interactions under control translation, quantifying the modification of the Green kernel and the self-energy due to the field . 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.

Paper Structure

This paper contains 5 sections, 18 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Control-Translated Tonelli Structure and Deformed Energy Functional
  3. Stability and Renormalized Energy in Control-Translated Tonelli Structures
  4. Variational Control and Vortex Dynamics in Tonelli-Finsler Geometry
  5. Optimal Control of the Tonelli-Finsler Energy

Key Result

Theorem 2.1

For every fixed $x\in M$ and $\xi\in T_x^*M$, the convex conjugate of $\phi_u$ satisfies and where $L_x^{-1}$ is the inverse of the classical Legendre map $\xi=\partial_y(\tfrac{1}{2}F(x,y)^2)$. The map $L_{x,u}(y):=\partial_y\phi_u(x,y)$ is a smooth diffeomorphism between $T_xM$ and $T_x^*M$ for all bounded $u$.

Theorems & Definitions (36)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 26 more