Hamilton-Jacobi-Bellman equations on graphs
Nicolò Forcillo, Jun Kitagawa, Russell W. Schwab
TL;DR
This work develops a rigorous framework for Hamilton-Jacobi-Bellman equations on graphs, proving well-posedness for a broad class of operators that satisfy a global comparison property (GCP). The authors show that many graph-based operators admit a min–max (Bellman) representation and establish comparison and existence results via a Perron-type construction, under assumptions that guarantee monotonicity and boundary control. They connect the discrete theory to continuum nonlocal elliptic equations and Cald_Ette’s graph-HJB approach, illustrating the framework with concrete examples including graph Laplacians, Markov-control Bellman problems, and graph eikonal-type operators. The paper also provides convex-analytic and min–max decompositions, clarifying when discrete operators align with Isaacs-type structures and how distance functions on graphs underpin key constructions. Overall, the results offer a unified, robust methodology for analyzing well-posedness of HJB-type equations on graphs with potential applications in control, distance computations, and nonlocal graph-based models.
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.