Homogenization Results for a Deterministic Multi-domains Periodic Control Problem

TL;DR

This paper investigates homogenization in deterministic multi-domain optimal control problems with distinct dynamics and costs, analyzing cases with regular and singular strategies using PDE methods and control techniques.

Contribution

It extends homogenization theory to multi-domain control problems, comparing regular and singular strategy cases with novel control and PDE approaches.

Findings

Homogenization results established for both regular and singular strategies.

Control methods are used for the regular strategy case.

PDE techniques are adapted for the singular strategy case.

Abstract

We consider homogenization problems in the framework of deterministic optimal control when the dynamics and running costs are completely different in two (or more) complementary domains of the space $\R^N$. For such optimal control problems, the three first authors have shown that several value functions can be defined, depending, in particular, of the choice is to use only "regular strategies" or to use also "singular strategies". We study the homogenization problem in these two different cases. It is worth pointing out that, if the second one can be handled by usual partial differential equations method " \'a la Lions-Papanicolaou-Varadhan" with suitable adaptations, the first case has to be treated by control methods (dynamic programming).