Hamilton-Jacobi-Bellman equations on graphs

TL;DR

This paper investigates Hamilton-Jacobi-Bellman equations on graphs, establishing conditions for solutions, unifying various assumptions, and highlighting their common structure as monotone functions of differences, with applications to Bellman-Isaacs representations.

Contribution

It provides a comprehensive framework for existence and uniqueness of solutions to HJB equations on graphs, unifies previous assumptions, and connects these equations to nonlocal elliptic integro-differential equations.

Findings

Conditions for existence and uniqueness of solutions.

Most operators have a Bellman-Isaacs representation.

Operators share a common structure as monotone functions.

Abstract

Here, we study Hamilton-Jacobi-Bellman equations on graphs. These are meant to be the analog of any of the following types of equations in the continuum setting of partial differential and nonlocal integro-differential equations: Hamilton-Jacobi (typically first order and local), Hamilton-Jacobi-Bellmann-Isaacs (first, second, or fractional order), and elliptic integro-differential equations (nonlocal equations). We give conditions for the existence and uniqueness of solutions of these equations, and work through a long list of examples in which these assumptions are satisfied. This work is meant to accomplish three goals: complement and unite earlier assumptions and arguments focused more on the Hamilton-Jacobi type structure; import ideas from nonlocal elliptic integro-differential equations; and argue that nearly all of the operators in this family enjoy a common structure of being a monotone function of the differences of the unknown, plus ``lower order'' terms. This last goal is tied to the fact that most of the examples in this family can be proven to have a Bellman-Isaacs representation as a min-max of linear operators with a graph Laplacian structure.