A Bellman approach for regional optimal control problems in $\R^N$

Guy Barles (1, 2) , Ariela Briani (1, 2) , Emmanuel Chasseigne (2, 1)

1 FDP - Fédération de recherche Denis Poisson
2 LMPT - Laboratoire de Mathématiques et Physique Théorique

This article is a continuation of a previous work where we studied infinite horizon control problems for which the dynamic, running cost and control space may be different in two half-spaces of some euclidian space $\R^N$. In this article we extend our results in several directions: $(i)$ to more general domains; $(ii)$ by considering finite horizon control problems; $(iii)$ by weaken the controlability assumptions. We use a Bellman approach and our main results are to identify the right Hamilton-Jacobi-Bellman Equation (and in particular the right conditions to be put on the interfaces separating the regions where the dynamic and running cost are different) and to provide the maximal and minimal solutions, as well as conditions for uniqueness. We also provide stability results for such equations.

Mots clés: viscosity solutions

Mots clés: Optimal control; discontinuous dynamic; Bellman Equation; viscosity solutions.

Domaines: Equations aux dérivées partielles [math.AP]

Guy Barles, Ariela Briani, Emmanuel Chasseigne. A Bellman approach for regional optimal control problems in $\R^N$. 2013. ⟨hal-00825778v1⟩