The large time profile for Hamilton–Jacobi–Bellman equations

dc.contributor.authorGomes, Diogo A.
dc.contributor.authorMitake, Hiroyoshi
dc.contributor.authorTran, Hung V.
dc.contributor.departmentApplied Mathematics and Computational Science Program
dc.contributor.departmentComputer, Electrical and Mathematical Science and Engineering (CEMSE) Division
dc.contributor.institutionGraduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan.
dc.contributor.institutionDepartment of Mathematics, University of Wisconsin Madison, Van Vleck Hall, 480 Lincoln Drive, Madison, WI 53706, USA.
dc.date.accepted2021-11-14
dc.date.accessioned2021-12-01T10:19:17Z
dc.date.available2020-06-22T08:34:47Z
dc.date.available2021-12-01T10:19:17Z
dc.date.issued2021-11-30
dc.date.posted2020-06-08
dc.date.published-online2021-11-30
dc.date.published-print2022-12
dc.date.submitted2021-02-03
dc.description.abstractHere, 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.
dc.description.sponsorshipWe 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.
dc.eprint.versionPost-print
dc.identifier.arxivid2006.04785
dc.identifier.citationGomes, 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
dc.identifier.doi10.1007/s00208-021-02320-5
dc.identifier.issn0025-5831
dc.identifier.issn1432-1807
dc.identifier.journalMathematische Annalen
dc.identifier.urihttp://hdl.handle.net/10754/663760
dc.publisherSpringer Science and Business Media LLC
dc.relation.urlhttps://link.springer.com/10.1007/s00208-021-02320-5
dc.rightsArchived with thanks to Mathematische Annalen
dc.rights.embargodate2022-11-30
dc.titleThe large time profile for Hamilton–Jacobi–Bellman equations
dc.typeArticle
display.details.left<span><h5>Embargo End Date</h5>2022-11-30<br><br><h5>Type</h5>Article<br><br><h5>Authors</h5><a href="https://repository.kaust.edu.sa/search?query=orcid.id:0000-0002-3129-3956&spc.sf=dc.date.issued&spc.sd=DESC">Gomes, Diogo A.</a> <a href="https://orcid.org/0000-0002-3129-3956" target="_blank"><img src="https://repository.kaust.edu.sa/server/api/core/bitstreams/82a625b4-ed4b-40c8-865a-d6a5225a26a4/content" width="16" height="16"/></a><br><a href="https://repository.kaust.edu.sa/search?spc.sf=dc.date.issued&spc.sd=DESC&f.author=Mitake, Hiroyoshi,equals">Mitake, Hiroyoshi</a><br><a href="https://repository.kaust.edu.sa/search?spc.sf=dc.date.issued&spc.sd=DESC&f.author=Tran, Hung V.,equals">Tran, Hung V.</a><br><br><h5>KAUST Department</h5><a href="https://repository.kaust.edu.sa/search?spc.sf=dc.date.issued&spc.sd=DESC&f.department=Applied Mathematics and Computational Science Program,equals">Applied Mathematics and Computational Science Program</a><br><a href="https://repository.kaust.edu.sa/search?spc.sf=dc.date.issued&spc.sd=DESC&f.department=Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division,equals">Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division</a><br><br><h5>KAUST Grant Number</h5>OSR-CRG2017-3452<br><br><h5>Preprint Posting Date</h5>2020-06-08<br><br><h5>Online Publication Date</h5>2021-11-30<br><br><h5>Print Publication Date</h5>2022-12<br><br><h5>Date</h5>2021-11-30<br><br><h5>Submitted Date</h5>2021-02-03</span>
display.details.right<span><h5>Abstract</h5>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.<br><br><h5>Citation</h5>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<br><br><h5>Acknowledgements</h5>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.<br><br><h5>Publisher</h5><a href="https://repository.kaust.edu.sa/search?spc.sf=dc.date.issued&spc.sd=DESC&f.publisher=Springer Science and Business Media LLC,equals">Springer Science and Business Media LLC</a><br><br><h5>Journal</h5><a href="https://repository.kaust.edu.sa/search?spc.sf=dc.date.issued&spc.sd=DESC&f.journal=Mathematische Annalen,equals">Mathematische Annalen</a><br><br><h5>DOI</h5><a href="https://doi.org/10.1007/s00208-021-02320-5">10.1007/s00208-021-02320-5</a><br><br><h5>arXiv</h5><a href="https://arxiv.org/abs/2006.04785">2006.04785</a><br><br><h5>Additional Links</h5>https://link.springer.com/10.1007/s00208-021-02320-5</span>
kaust.acknowledged.supportUnitKAUST baseline fund
kaust.acknowledged.supportUnitOSR
kaust.grant.numberOSR-CRG2017-3452
kaust.personGomes, Diogo A.
orcid.authorGomes, Diogo A.::0000-0002-3129-3956
orcid.authorMitake, Hiroyoshi
orcid.authorTran, Hung V.
orcid.id0000-0002-3129-3956
refterms.dateFOA2020-06-22T08:37:30Z
Files
Original bundle
Now showing 1 - 1 of 1
Loading...
Thumbnail Image
Name:
GMT_revision.pdf
Size:
246.23 KB
Format:
Adobe Portable Document Format
Description:
Accepted manuscript

Version History

Now showing 1 - 2 of 2
VersionDateSummary
2*
2021-12-01 10:17:35
Published in journal
2020-06-22 08:34:47
* Selected version