  We consider an infinite horizon control problem for dynamics constrained to
remain on a multidimensional junction with entry costs. We derive the
associated system of Hamilton-Jacobi equations (HJ), prove the comparison
principle and that the value function of the optimal control problem is the
unique viscosity solution of the HJ system. This is done under the usual strong
controllability assumption and also under a weaker condition, coined 'moderate
controllability assumption'.
