The large time profile for Hamilton–Jacobi–Bellman equations

Gomes, Diogo A.; Mitake, Hiroyoshi; Tran, Hung V.

The large time profile for Hamilton–Jacobi–Bellman equations

Files

GMT_revision.pdf (246.23 KB)

Embargo End Date

2022-11-30

Type

Article

Authors

Gomes, Diogo A.

Mitake, Hiroyoshi
Tran, Hung V.

KAUST Department

Applied Mathematics and Computational Science Program
Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

KAUST Grant Number

OSR-CRG2017-3452

Preprint Posting Date

2020-06-08

Online Publication Date

2021-11-30

Print Publication Date

2022-12

Date

2021-11-30

Submitted Date

2021-02-03

Abstract

Here, we study the large-time limit of viscosity solutions of the Cauchy problem for second-order Hamilton–Jacobi–Bellman equations with convex Hamiltonians in the torus. This large-time limit solves the corresponding stationary problem, sometimes called the ergodic problem. This problem, however, has multiple viscosity solutions and, thus, a key question is which of these solutions is selected by the limit. Here, we provide a representation for the viscosity solution to the Cauchy problem in terms of generalized holonomic measures. Then, we use this representation to characterize the large-time limit in terms of the initial data and generalized Mather measures. In addition, we establish various results on generalized Mather measures and duality theorems that are of independent interest.

Citation

Gomes, D. A., Mitake, H., & Tran, H. V. (2021). The large time profile for Hamilton–Jacobi–Bellman equations. Mathematische Annalen. doi:10.1007/s00208-021-02320-5

Acknowledgements

We would like to thank Hitoshi Ishii for his suggestions on the approximations of viscosity solutions and subsolutions in Appendix B. We are grateful to Toshio Mikami for the discussions on Theorem 1.1 and for giving us relevant references on the duality result in Theorem 1.4.

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