Issue ESAIM: COCV Volume 6, 2001 Page(s) 415 - 441 DOI 10.1051/cocv:2001116

DOI: 10.1051/cocv:2001116


ESAIM: COCV, June 2001, Vol. 6, pp. 415-441

Viscosity Solutions of the Bellman Equation for Exit Time Optimal Control Problems with Non-Lipschitz Dynamics

Michael Malisoff

Department of Systems Science and Mathematics, Washington University in Saint Louis, One Brookings Drive, Campus Box 1040, Saint Louis, MO 63130-4899, U.S.A.; (malisoff@zach.wustl.edu)

(Received April 19, 2000. Revised December 22, 2000.)

Abstract
We study the Bellman equation for undiscounted exit time optimal control problems with fully nonlinear Lagrangians and fully nonlinear dynamics using the dynamic programming approach. We allow problems whose non-Lipschitz dynamics admit more than one solution trajectory for some choices of open loop controls and initial positions. We prove a uniqueness theorem which characterizes the value functions of these problems as the unique viscosity solutions of the corresponding Bellman equations that satisfy appropriate boundary conditions. We deduce that the value function for Sussmann's Reflected Brachystochrone Problem for an arbitrary singleton target is the unique viscosity solution of the corresponding Bellman equation in the class of functions which are continuous in the plane, null at the target, and bounded below. Our results also apply to degenerate eikonal equations, and to problems whose targets can be unbounded and whose Lagrangians vanish for some points in the state space which are outside the target, including Fuller's Example.

AMS Subject: 49L25, 93Cxx.

Key words: Viscosity solutions, dynamical systems, Reflected Brachystochrone Problem.


© EDP Sciences, SMAI 2001

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