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A Bellman approach for two-domains optimal control problems inℝN∗

Published online by Cambridge University Press:  03 June 2013

G. Barles ,
A. Briani  and
E. Chasseigne
G. Barles
Affiliation:
Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 7350), Fédération Denis Poisson (FR CNRS 2964), Université François Rabelais, Parc de Grandmont, 37200 Tours, France. Guy.Barles@lmpt.univ-tours.fr; Ariela.Briani@lmpt.univ-tours.fr; Emmanuel.Chasseigne@lmpt.univ-tours.fr
A. Briani
Affiliation:
Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 7350), Fédération Denis Poisson (FR CNRS 2964), Université François Rabelais, Parc de Grandmont, 37200 Tours, France. Guy.Barles@lmpt.univ-tours.fr; Ariela.Briani@lmpt.univ-tours.fr; Emmanuel.Chasseigne@lmpt.univ-tours.fr
E. Chasseigne
Affiliation:
Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 7350), Fédération Denis Poisson (FR CNRS 2964), Université François Rabelais, Parc de Grandmont, 37200 Tours, France. Guy.Barles@lmpt.univ-tours.fr; Ariela.Briani@lmpt.univ-tours.fr; Emmanuel.Chasseigne@lmpt.univ-tours.fr
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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.

Keywords

Optimal controldiscontinuous dynamicBellman Equationviscosity solutions
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 3 , July 2013 , pp. 710 - 739
Copyright
© EDP Sciences, SMAI, 2013

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