Optimal control of first-order Hamilton-Jacobi equations with linearly bounded Hamiltonian

TL;DR

This paper studies the optimal control of first-order Hamilton-Jacobi equations with convex Hamiltonians of linear growth, establishing existence and characterization of minimizers and solutions within a mean field game framework.

Contribution

It introduces a relaxed optimization approach for controlling front propagation and characterizes minimizers as weak solutions to a coupled PDE system, advancing the understanding of mean field game models.

Findings

Existence of minimizers in a relaxed setting

Characterization of minimizers as weak solutions

Existence and partial uniqueness of solutions to the PDE system

Abstract

We consider the optimal control of solutions of first order Hamilton-Jacobi equations, where the Hamiltonian is convex with linear growth. This models the problem of steering the propagation of a front by constructing an obstacle. We prove existence of minimizers to this optimization problem as in a relaxed setting and characterize the minimizers as weak solutions to a mean field game type system of coupled partial differential equations. Furthermore, we prove existence and partial uniqueness of weak solutions to the PDE system. An interpretation in terms of mean field games is also discussed. Keywords: Hamilton-Jacobi equations, optimal control, nonlinear PDE, viscosity solutions, front propagation, mean field games