  How to free a road from vehicle traffic as efficiently as possible and in a
given time, in order to allow for example the passage of emergency vehicles? We
are interested in this question which we reformulate as an optimal control
problem. We consider a macroscopic road traffic model on networks,
semi-discretized in space and decide to give ourselves the possibility to
control the flow at junctions. Our target is to smooth the traffic along a
given path within a fixed time. A parsimony constraint is imposed on the
controls, in order to ensure that the optimal strategies are feasible in
practice. We perform an analysis of the resulting optimal control problem,
proving the existence of an optimal control and deriving optimality conditions,
which we rewrite as a single functional equation. We then use this formulation
to derive a new mixed algorithm interpreting it as a mix between two methods: a
descent method combined with a fixed point method allowing global
perturbations. We verify with numerical experiments the efficiency of this
method on examples of graphs, first simple, then more complex. We highlight the
efficiency of our approach by comparing it to standard methods. We propose an
open source code implementing this approach in the Julia language.
