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

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

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

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topMalisoff, 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 -

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