  We consider the problem of incentivization and optimal control of autonomous
vehicles for improving traffic congestion. In our scenario, autonomous vehicles
must be incentivized in order to participate in traffic improvement. Using the
theory and methods of optimal transport, we propose a constrained optimization
framework over dynamics governed by partial differential equations, so that we
can optimally select a portion of vehicles to be incentivized and controlled.
  The goal of the optimization is to obtain a uniform distribution of vehicles
over the spatial domain. To achieve this, we consider two types of penalties on
vehicle density, one is the $L^2$ cost and the other is a multiscale-norm cost,
commonly used in fluid-mixing problems. To solve this non-convex optimization
problem, we introduce a novel algorithm, which iterates between solving a
convex optimization problem and propagating the flow of uncontrolled vehicles
according to the Lighthill-Whitham-Richards model. We perform numerical
simulations, which suggest that the optimization of the $L^2$ cost is
ineffective while optimization of the multiscale norm is effective. The results
also suggest the use of a dedicated lane for this type of control in practice.
