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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Duality and uniqueness of convex solutions to stationary Hamilton-Jacobi equations


Author: Rafal Goebel
Journal: Trans. Amer. Math. Soc. 357 (2005), 2187-2203
MSC (2000): Primary 49N15, 49L99; Secondary 49M29
DOI: https://doi.org/10.1090/S0002-9947-05-03817-1
Published electronically: January 21, 2005
MathSciNet review: 2140437
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Abstract: Value functions for convex optimal control problems on infinite time intervals are studied in the framework of duality. Hamilton-Jacobi characterizations and the conjugacy of primal and dual value functions are of main interest. Close ties between the uniqueness of convex solutions to a Hamilton-Jacobi equation, the uniqueness of such solutions to a dual Hamilton-Jacobi equation, and the conjugacy of primal and dual value functions are displayed. Simultaneous approximation of primal and dual infinite horizon problems with a pair of dual problems on finite horizon, for which the value functions are conjugate, leads to sufficient conditions on the conjugacy of the infinite time horizon value functions. Consequently, uniqueness results for the Hamilton-Jacobi equation are established. Little regularity is assumed on the cost functions in the control problems, correspondingly, the Hamiltonians need not display any strict convexity and may have several saddle points.


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Additional Information

Rafal Goebel
Affiliation: Center for Control Engineering and Computation, University of California, Santa Barbara, California 93106
Address at time of publication: 3518 NE 42 St., Seattle, Washington 98105
Email: rafal@ece.ucsb.edu

DOI: https://doi.org/10.1090/S0002-9947-05-03817-1
Keywords: Stationary Hamilton-Jacobi equation, convex value function, optimal control, infinite horizon, dual problem, concave-convex Hamiltonian
Received by editor(s): April 3, 2003
Published electronically: January 21, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.