This paper deals with junction conditions for Hamilton-Jacobi-Bellman (HJB)
equations for finite horizon control problems on multi-domains. We consider two
different cases where the final cost is continuous or lower semi-continuous. In
the continuous case we extend the results of "Hamilton-Jacobi-Bellman equations
on multi-domains" by the second and third authors in a more general framework
with switching running costs and weaker controllability assumptions. The
comparison principle has been established to guarantee the uniqueness and the
stability results for the HJB system on such multi-domains. In the lower
semi-continuous case, we characterize the value function as the unique lower
semi-continuous viscosity solution of the HJB system, under a local
controllability assumption
