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

Junction conditions for finite horizon optimal control problems on multi-domains with continuous and discontinuous solutions