The large time profile for Hamilton–Jacobi–Bellman equations

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

doi:10.1007/s00208-021-02320-5

The large time profile for Hamilton–Jacobi–Bellman equations

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

Research output: Contribution to journal › Article › peer-review

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.

Original language	English (US)
Journal	Mathematische Annalen
DOIs	https://doi.org/10.1007/s00208-021-02320-5
State	Published - Nov 30 2021

Bibliographical note

KAUST Repository Item: Exported on 2022-01-27
Acknowledged KAUST grant number(s): OSR-CRG2017-3452
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.

ASJC Scopus subject areas

General Mathematics

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10.1007/s00208-021-02320-5

https://repository.kaust.edu.sa/bitstream/10754/663760/3/GMT_revision.pdf

Cite this

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title = "The large time profile for Hamilton–Jacobi–Bellman equations",

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.",

author = "Gomes, {Diogo A.} and Hiroyoshi Mitake and Tran, {Hung V.}",

note = "KAUST Repository Item: Exported on 2022-01-27 Acknowledged KAUST grant number(s): OSR-CRG2017-3452 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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AU - Gomes, Diogo A.

AU - Mitake, Hiroyoshi

AU - Tran, Hung V.

N1 - KAUST Repository Item: Exported on 2022-01-27 Acknowledged KAUST grant number(s): OSR-CRG2017-3452 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.

PY - 2021/11/30

Y1 - 2021/11/30

N2 - 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.

AB - 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.

UR - http://hdl.handle.net/10754/663760

UR - https://link.springer.com/10.1007/s00208-021-02320-5

U2 - 10.1007/s00208-021-02320-5

DO - 10.1007/s00208-021-02320-5

M3 - Article

SN - 0025-5831

JO - Mathematische Annalen

JF - Mathematische Annalen

ER -

The large time profile for Hamilton–Jacobi–Bellman equations

Abstract

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ASJC Scopus subject areas

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