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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

Andrzej Święch
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332

Atanas Stefanov
Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045

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
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