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

Homogenization Results for a Deterministic Multi-domains Periodic Control Problem