  We study an optimal control problem for traffic regulation via variable speed
limit. The traffic flow dynamics is described with the
Lighthill-Whitham-Richards (LWR) model with Newell-Daganzo flux function. We
aim at minimizing the $L^2$ quadratic error to a desired outflow, given an
inflow on a single road. We first provide existence of a minimizer and compute
analytically the cost functional variations due to needle-like variation in the
control policy. Then, we compare three strategies: instantaneous policy; random
exploration of control space; steepest descent using numerical expression of
gradient. We show that the gradient technique is able to achieve a cost within
10% of random exploration minimum with better computational performances.
