Hamilton-Jacobi-Bellman equations on graphs
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| Format: | Preprint |
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2025
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| _version_ | 1866911258976976896 |
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| author | Forcillo, Nicolò Kitagawa, Jun Schwab, Russell W. |
| author_facet | Forcillo, Nicolò Kitagawa, Jun Schwab, Russell W. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_07653 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hamilton-Jacobi-Bellman equations on graphs Forcillo, Nicolò Kitagawa, Jun Schwab, Russell W. Analysis of PDEs 35B51, 35R09, 45K05, 47G20, 49L25, 60J75 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. |
| title | Hamilton-Jacobi-Bellman equations on graphs |
| topic | Analysis of PDEs 35B51, 35R09, 45K05, 47G20, 49L25, 60J75 |
| url | https://arxiv.org/abs/2511.07653 |