Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems

In this paper we are interested in the numerical solution of optimal control problems for non-linear hyperbolic conservation laws. To this aim, we consider relaxation approximations to the conservation laws coupled with the optimal control problem. Following a semi--Lagrangian interpretation of the hyperbolic relaxation system, and its adjoint counterpart, we solve efficiently the time discretization introducing a multi--step scheme in the class of BDF methods. Computational results illustrating the theoretical findings with applications to traffic flow models are presented.