We consider a hyperbolic conservation law with discontinuous flux. Such a partial differential equation arises in different applications, in particu- lar we are motivated by a model of traffic flow. We provide a new formulation in terms of Riemann Solvers. Moreover, we determine the class of Riemann Solvers which provide existence and uniqueness of the corresponding weak en- tropic solutions.

We construct a model of traffic flow with sources and destinations on a road s network. The model is based on a conservation law for the density of traffic and on semilinearequations for traffic-type functions, i. e. functions describing paths for cars.

We consider a hybrid control system and general optimal control problems for this system. We suppose that the switching strategy imposes restrictions on control sets and we provide necessary conditions for an optimal hybrid trajectory, stating a Hybrid Necessary Principle (HNP). Our result generalizes various necessary principles available in the literature.