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Comparison principle for Walsh's spider HJB equations with non linear local time Kirchhoff's boundary transmission

Isaac Ohavi

TL;DR

This work addresses a Walsh spider HJB system posed on a star-shaped network with a non-linear local-time Kirchhoff boundary at the vertex, establishing a comparison principle for viscosity solutions and hence uniqueness. The authors introduce a novel vertex testing mechanism built from parametric ODEs that captures the envelope of the Kirchhoff-speed and employs a local-time derivative with respect to $l$ to absorb transmission errors. Notably, the result does not rely on vanishing viscosity at the vertex and extends to unbounded local time, as well as to strong Neumann-type boundary formulations. The methodological advance enhances the theory of HJB systems on networks with discontinuous coefficients and local-time dynamics, with potential implications for diffusion control, scattering, and quantum trajectory models.

Abstract

The main purpose of this work is to obtain a comparison principle for viscosity solutions of a system of elliptic Walsh's spider Hamilton-Jacobi-Bellman equations, possessing a new boundary condition called non linear local-time Kirchhoff's transmission. The main idea is to build test functions at the neighborhood of the vertex solutions of ODE, with well-designed coefficients. The key point is to impose a 'local-time' derivative at the vertex absorbing the error term induced by - what we decide to call here - the Kirchhoff's speed of the Hamiltonians.

Comparison principle for Walsh's spider HJB equations with non linear local time Kirchhoff's boundary transmission

TL;DR

This work addresses a Walsh spider HJB system posed on a star-shaped network with a non-linear local-time Kirchhoff boundary at the vertex, establishing a comparison principle for viscosity solutions and hence uniqueness. The authors introduce a novel vertex testing mechanism built from parametric ODEs that captures the envelope of the Kirchhoff-speed and employs a local-time derivative with respect to to absorb transmission errors. Notably, the result does not rely on vanishing viscosity at the vertex and extends to unbounded local time, as well as to strong Neumann-type boundary formulations. The methodological advance enhances the theory of HJB systems on networks with discontinuous coefficients and local-time dynamics, with potential implications for diffusion control, scattering, and quantum trajectory models.

Abstract

The main purpose of this work is to obtain a comparison principle for viscosity solutions of a system of elliptic Walsh's spider Hamilton-Jacobi-Bellman equations, possessing a new boundary condition called non linear local-time Kirchhoff's transmission. The main idea is to build test functions at the neighborhood of the vertex solutions of ODE, with well-designed coefficients. The key point is to impose a 'local-time' derivative at the vertex absorbing the error term induced by - what we decide to call here - the Kirchhoff's speed of the Hamiltonians.
Paper Structure (9 sections, 4 theorems, 184 equations)

This paper contains 9 sections, 4 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Origins and motivations:
  3. On the terminology of the system \ref{['eq PDE 0']} and its possible extensions:
  4. Main contributions and novelties:
  5. Review of literature:
  6. Organisation of the paper:
  7. Notations and Definitions
  8. A short example for the construction of test functions in the linear case
  9. Proof of Theorem \ref{['th: compa theorem principale']}

Key Result

Theorem 2.2

(Comparison Theorem.) Assume assumption $(\mathcal{H})$. Let $v\in \mathcal{C}(\mathcal{N}_R\times [0,K])$ a continuous viscosity sub solution and $u\in \mathcal{C}(\mathcal{N}_R\times [0,K])$ a continuous viscosity super solution of the Walsh's spider HJB system - $\mathcal{W}_{alsh}(\mathbb{S})$ - Then we have:

Theorems & Definitions (9)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • Proposition 4.1
  • proof
  • proof
  • proof