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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Optimal control for a mixed flow of Hamiltonian and gradient type in space of probability measures


Authors: Jin Feng and Andrzej Święch; with Appendix B by Atanas Stefanov
Journal: Trans. Amer. Math. Soc. 365 (2013), 3987-4039
MSC (2010): Primary 35R15, 49L25, 49J20; Secondary 35Q30, 42B20, 49L20, 60F17, 60H10
Published electronically: March 20, 2013
MathSciNet review: 3055687
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we investigate an optimal control problem in the space of measures on $ \mathbb{R}^2$. The problem is motivated by a stochastic interacting particle model which gives the 2-D Navier-Stokes equations in their vorticity formulation as a mean-field equation. We prove that the associated Hamilton-Jacobi-Bellman equation, in the space of probability measures, is well posed in an appropriately defined viscosity solution sense.


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Additional Information

Jin Feng
Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045
Email: jfeng@math.ku.edu

Andrzej Święch
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Email: swiech@math.gatech.edu

Atanas Stefanov
Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045
Email: stefanov@math.ku.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05634-6
PII: S 0002-9947(2013)05634-6
Keywords: Viscosity solutions, mass transport theory, controlled gradient and Hamiltonian flows, Navier-Stokes in 2D
Received by editor(s): October 10, 2010
Received by editor(s) in revised form: May 20, 2011
Published electronically: March 20, 2013
Additional Notes: The research of the first author was partially supported by US ARO grant W911NF-08-1-0064 and by NSF grant DMS 0806434. The research of the second author was partially supported by NSF grant DMS 0856485; the research of the third author was partially supported by NSF grant DMS 0701802. The first and second authors have also been supported by the American Institute of Mathematics through a SQuaRE program. The authors thank the referee for useful remarks and references.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.