[Paper Review] A Lagrangian Approach to Weakly Coupled Hamilton-Jacobi Systems

Hiroyoshi Mitake, Antonio Siconolfi+2 more|arXiv (Cornell University)|Mar 2, 2015

Quantum chaos and dynamical systems3 citations

TL;DR

This paper introduces a Lagrangian framework for weakly coupled Hamilton-Jacobi systems on the torus by constructing action functionals via duality from the Hamiltonians. It characterizes subsolutions through a variational estimate and explicitly constructs maximality-optimized subsolutions, extending weak KAM theory to systems using a random switching framework with cadlag paths and measurable dynamics.

ABSTRACT

We study a class of weakly coupled Hamilton-Jacobi systems with a specific aim to perform a qualitative analysis in the spirit of weak KAM theory. Our main achievement is the definition of a family of related action functionals containing the Lagrangians obtained by duality from the Hamiltonians of the system. We use them to characterize, by means of a suitable estimate, all the subsolutions of the system, and to explicitly represent some subsolutions enjoying an additional maximality property. A crucial step for our analysis is to put the problem in a suitable random frame. Only some basic knowledge of measure theory is required, and the presentation is accessible to readers without background in probability.

Motivation & Objective

• To extend weak KAM theory to weakly coupled Hamilton-Jacobi systems by developing a variational, Lagrangian formulation.
• To characterize all subsolutions of the system through a variational estimate involving dual action functionals.
• To explicitly represent subsolutions with a maximality property using the constructed action functionals.
• To establish a geometric and dynamical counterpart to PDE results by embedding the system in a random switching framework with cadlag paths.
• To provide a probabilistic yet accessible framework using measure-theoretic tools without requiring advanced probability knowledge.

Proposed method

• Define a family of action functionals derived via Legendre-Fenchel duality from the Hamiltonians $ H_i $, forming the core of the Lagrangian approach.
• Model the system's dynamics using $ \mathcal{D}(0,\infty; \mathbb{R}^N) $, the space of $ \{1,\dots,M\} $-valued cadlag paths with the Skorohod metric.
• Introduce a random switching mechanism governed by the coupling matrix $ \Lambda $, which acts as a generator of stochastic matrices.
• Construct a continuous map $ \mathcal{I} $ from cadlag paths to continuous paths in the torus $ \mathbb{T}^N $ via time integration and projection.
• Use measurable shift flows $ \phi_h $ on $ \mathcal{D} $ to analyze time-translation invariance and establish measurability of dynamical operations.
• Apply the dominated convergence theorem and Arzelà-Ascoli theorem to prove continuity of the action functional and the map $ \mathcal{I} $, ensuring variational stability.

Experimental results

Research questions

• RQ1 How can a Lagrangian formulation be constructed for weakly coupled Hamilton-Jacobi systems to extend weak KAM theory?
• RQ2 What variational characterization can be given for all subsolutions of the system using dual action functionals?
• RQ3 Can subsolutions with a maximality property be explicitly represented within this framework?
• RQ4 How does the random switching mechanism, modeled via cadlag paths and the coupling matrix $ \Lambda $, support the variational analysis?
• RQ5 What is the role of the shift flow $ \phi_h $ and measurable structure in ensuring the consistency and continuity of the action functional?

Key findings

• A family of action functionals is defined via duality from the Hamiltonians, enabling a variational characterization of all subsolutions.
• All subsolutions of the system are characterized by a specific variational estimate involving the constructed action functionals.
• A distinguished subsolution with a maximality property—defined by attaining a given value at any point in the Aubry set—is explicitly represented via the action functional.
• The continuity of the action functional is established through the dominated convergence theorem and equicontinuity arguments on the path space.
• The map $ \mathcal{I} $, which lifts cadlag paths to continuous paths in $ \mathbb{T}^N $, is shown to be continuous using Arzelà-Ascoli and dominated convergence.
• The shift flow $ \phi_h $ is proven measurable, supporting the time-homogeneity and dynamical consistency of the framework.