  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.
