# Viscosity solutions of the Bellman equation for exit time optimal control problems with non-Lipschitz dynamics

ESAIM: Control, Optimisation and Calculus of Variations (2001)

- 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 (2001): 415-441. <http://eudml.org/doc/90601>.

@article{Malisoff2001,

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; Bellman equation; exit time optimal control; dynamic programming; eikonal equations},

language = {eng},

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/90601},

volume = {6},

year = {2001},

}

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

PY - 2001

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; Bellman equation; exit time optimal control; dynamic programming; eikonal equations

UR - http://eudml.org/doc/90601

ER -

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