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Journal of Convex Analysis 09 (2002), No. 2, 463--473
Copyright Heldermann Verlag 2002

Generalized Conjugacy in Hamilton-Jacobi Theory for Fully Convex Langrangians

Rafal Goebel
Center for Control Engineering and Computation, Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106-9650, U.S.A.

Ralph Tyrell Rockafellar
Dept. of Mathematics, University of Washington, Seattle, WA 98195-4350, U.S.A.

Control problems with fully convex Lagrangians and convex initial costs are considered. Generalized conjugacy and envelope representation in terms of a dualizing kernel are employed to recover the initial cost from the value function at some fixed future time, leading to a generalization of the cancellation rule for inf-convolution. Such recovery is possible subject to persistence of trajectories of a generalized Hamiltonian system, associated with the Lagrangian. Global analysis of Hamiltonian trajectories is carried out, leading to conditions on the Hamiltonian, and the corresponding Lagrangian, guaranteeing persistence of the trajectories.

Keywords: Convex value functions, inf-convolution, Hopf-Lax formula, nonsmooth Hamiltonian dynamics, generalized conjugacy, fully convex Langrangians, optimal control.

MSC: 49L20, 93C10; 49N15

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