Hamilton-Jacobi equations for optimal control on multidimensional junctions with entry costs

TL;DR

This paper studies an optimal control problem on multidimensional junctions with entry costs, deriving Hamilton-Jacobi equations and proving the uniqueness of the viscosity solution under various controllability conditions.

Contribution

It introduces a new framework for Hamilton-Jacobi equations on junctions with entry costs and establishes uniqueness results under both strong and moderate controllability assumptions.

Findings

Derived Hamilton-Jacobi system for the control problem.

Proved comparison principle for the HJ system.

Established uniqueness of the viscosity solution.

Abstract

We consider an infinite horizon control problem for dynamics constrained to remain on a multidimensional junction with entry costs. We derive the associated system of Hamilton-Jacobi equations (HJ), prove the comparison principle and that the value function of the optimal control problem is the unique viscosity solution of the HJ system. This is done under the usual strong controllability assumption and also under a weaker condition, coined 'moderate controllability assumption'.