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

Michael Malisoff

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 6, page 415-441
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Malisoff, Michael. "Viscosity Solutions of the Bellman Equation for Exit Time Optimal Control Problems with Non-Lipschitz Dynamics." ESAIM: Control, Optimisation and Calculus of Variations 6 (2010): 415-441. <http://eudml.org/doc/197298>.

@article{Malisoff2010,
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. },
author = {Malisoff, Michael},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Viscosity solutions; dynamical systems; Reflected Brachystochrone Problem.; viscosity solutions; reflected brachystochrone problem; Bellman equation; exit time optimal control; dynamic programming; eikonal equations},
language = {eng},
month = {3},
pages = {415-441},
publisher = {EDP Sciences},
title = {Viscosity Solutions of the Bellman Equation for Exit Time Optimal Control Problems with Non-Lipschitz Dynamics},
url = {http://eudml.org/doc/197298},
volume = {6},
year = {2010},
}

TY - JOUR
AU - Malisoff, Michael
TI - Viscosity Solutions of the Bellman Equation for Exit Time Optimal Control Problems with Non-Lipschitz Dynamics
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 6
SP - 415
EP - 441
AB - 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.
LA - eng
KW - Viscosity solutions; dynamical systems; Reflected Brachystochrone Problem.; viscosity solutions; reflected brachystochrone problem; Bellman equation; exit time optimal control; dynamic programming; eikonal equations
UR - http://eudml.org/doc/197298
ER -

References

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