# Hamilton-Jacobi equations for optimal control on networks with entry or exit costs

## TL;DR

This paper studies optimal control problems on networks with entry or exit costs, resulting in discontinuous value functions, and characterizes the value function as a unique viscosity solution of a new Hamilton-Jacobi system.

## Contribution

It introduces a novel Hamilton-Jacobi system for networks with entry/exit costs and proves the uniqueness of the viscosity solution using two different comparison principles.

## Key findings

- Established a new Hamilton-Jacobi system for networks with entry/exit costs.
- Proved the uniqueness of the viscosity solution via two distinct comparison principles.
- Extended the theory of optimal control on networks to include discontinuous value functions.

## Abstract

We consider an optimal control on networks in the spirit of the works of Achdou et al. (2013) and Imbert et al. (2013). The main new feature is that there are entry (or exit) costs at the edges of the network leading to a possible discontinuous value function. We characterize the value function as the unique viscosity solution of a new Hamilton-Jacobi system. The uniqueness is a consequence of a comparison principle for which we give two different proofs, one with arguments from the theory of optimal control inspired by Achdou et al. (2014) and one based on partial differential equations techniques inspired by a recent work of Lions and Souganidis (2016).