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

TL;DR

This paper extends Bellman-based methods for regional optimal control problems in Euclidean spaces, addressing more general domains, finite horizons, and weaker controllability, while characterizing solutions to associated Hamilton-Jacobi-Bellman equations.

Contribution

It generalizes previous results to broader settings, identifies the correct HJB equations at interfaces, and establishes solution maximality, minimality, and stability conditions.

Findings

Identified the correct Hamilton-Jacobi-Bellman equations for regional control problems.

Provided conditions for the uniqueness of solutions.

Established stability results for the HJB equations.

Abstract

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.