  We consider an optimal control on networks in the spirit of the works of
Achdou et al. (2013) and Imbert et al. (2013). The main new feature is that
there are entry (or exit) costs at the edges of the network leading to a
possible discontinuous value function. We characterize the value function as
the unique viscosity solution of a new Hamilton-Jacobi system. The uniqueness
is a consequence of a comparison principle for which we give two different
proofs, one with arguments from the theory of optimal control inspired by
Achdou et al. (2014) and one based on partial differential equations techniques
inspired by a recent work of Lions and Souganidis (2016).
