Calculus of Variations and Geometric Measure Theory

G. Barles - A. Briani - E. Chasseigne

A Bellman approach for two-domains optimal control problems in $R^N$.

created by briani on 22 Mar 2012
modified on 12 Sep 2013


Published Paper

Inserted: 22 mar 2012
Last Updated: 12 sep 2013

Journal: ESAIM: Control Optimisation and Calculus of Variation
Volume: 19
Number: 03
Pages: 710-739
Year: 2013
Doi: 10.1051/cocv/2012030


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 $R^N$. 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 control, Viscosity solutions, Bellman Equation, discontinuous dynamics