A Bellman approach for two-domains optimal control problems in ℝN

G. Barles; A. Briani; E. Chasseigne

ESAIM: Control, Optimisation and Calculus of Variations (2013)

  • Volume: 19, Issue: 3, page 710-739
  • ISSN: 1292-8119

Abstract

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This article is the starting point of a series of works whose aim is the study of deterministic control problems where the dynamic and the running cost can be completely different in two (or more) complementary domains of the space ℝN. As a consequence, the dynamic and running cost present discontinuities at the boundary of these domains and this is the main difficulty of this type of problems. We address these questions by using a Bellman approach: our aim is to investigate how to define properly the value function(s), to deduce what is (are) the right Bellman Equation(s) associated to this problem (in particular what are the conditions on the set where the dynamic and running cost are discontinuous) and to study the uniqueness properties for this Bellman equation. In this work, we provide rather complete answers to these questions in the case of a simple geometry, namely when we only consider two different domains which are half spaces: we properly define the control problem, identify the different conditions on the hyperplane where the dynamic and the running cost are discontinuous and discuss the uniqueness properties of the Bellman problem by either providing explicitly the minimal and maximal solution or by giving explicit conditions to have uniqueness.

How to cite

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Barles, G., Briani, A., and Chasseigne, E.. "A Bellman approach for two-domains optimal control problems in ℝN." ESAIM: Control, Optimisation and Calculus of Variations 19.3 (2013): 710-739. <http://eudml.org/doc/272800>.

@article{Barles2013,
abstract = {This article is the starting point of a series of works whose aim is the study of deterministic control problems where the dynamic and the running cost can be completely different in two (or more) complementary domains of the space ℝN. As a consequence, the dynamic and running cost present discontinuities at the boundary of these domains and this is the main difficulty of this type of problems. We address these questions by using a Bellman approach: our aim is to investigate how to define properly the value function(s), to deduce what is (are) the right Bellman Equation(s) associated to this problem (in particular what are the conditions on the set where the dynamic and running cost are discontinuous) and to study the uniqueness properties for this Bellman equation. In this work, we provide rather complete answers to these questions in the case of a simple geometry, namely when we only consider two different domains which are half spaces: we properly define the control problem, identify the different conditions on the hyperplane where the dynamic and the running cost are discontinuous and discuss the uniqueness properties of the Bellman problem by either providing explicitly the minimal and maximal solution or by giving explicit conditions to have uniqueness.},
author = {Barles, G., Briani, A., Chasseigne, E.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {optimal control; discontinuous dynamic; Bellman equation; viscosity solutions; discontinuous dynamics},
language = {eng},
number = {3},
pages = {710-739},
publisher = {EDP-Sciences},
title = {A Bellman approach for two-domains optimal control problems in ℝN},
url = {http://eudml.org/doc/272800},
volume = {19},
year = {2013},
}

TY - JOUR
AU - Barles, G.
AU - Briani, A.
AU - Chasseigne, E.
TI - A Bellman approach for two-domains optimal control problems in ℝN
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2013
PB - EDP-Sciences
VL - 19
IS - 3
SP - 710
EP - 739
AB - This article is the starting point of a series of works whose aim is the study of deterministic control problems where the dynamic and the running cost can be completely different in two (or more) complementary domains of the space ℝN. As a consequence, the dynamic and running cost present discontinuities at the boundary of these domains and this is the main difficulty of this type of problems. We address these questions by using a Bellman approach: our aim is to investigate how to define properly the value function(s), to deduce what is (are) the right Bellman Equation(s) associated to this problem (in particular what are the conditions on the set where the dynamic and running cost are discontinuous) and to study the uniqueness properties for this Bellman equation. In this work, we provide rather complete answers to these questions in the case of a simple geometry, namely when we only consider two different domains which are half spaces: we properly define the control problem, identify the different conditions on the hyperplane where the dynamic and the running cost are discontinuous and discuss the uniqueness properties of the Bellman problem by either providing explicitly the minimal and maximal solution or by giving explicit conditions to have uniqueness.
LA - eng
KW - optimal control; discontinuous dynamic; Bellman equation; viscosity solutions; discontinuous dynamics
UR - http://eudml.org/doc/272800
ER -

References

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