  We consider the problem of optimizing the trajectory of an Unmanned Aerial
Vehicle (UAV). Assuming a traffic intensity map of users to be served, the UAV
must travel from a given initial location to a final position within a given
duration and serves the traffic on its way. The problem consists in finding the
optimal trajectory that minimizes a certain cost depending on the velocity and
on the amount of served traffic. We formulate the problem using the framework
of Lagrangian mechanics. We derive closed-form formulas for the optimal
trajectory when the traffic intensity is quadratic (single-phase) using
Hamilton-Jacobi equations. When the traffic intensity is bi-phase, i.e. made of
two quadratics, we provide necessary conditions of optimality that allow us to
propose a gradient-based algorithm and a new algorithm based on the linear
control properties of the quadratic model. These two solutions are of very low
complexity because they rely on fast convergence numerical schemes and closed
form formulas. These two approaches return a trajectory satisfying the
necessary conditions of optimality. At last, we propose a data processing
procedure based on a modified K-means algorithm to derive a bi-phase model and
an optimal trajectory simulation from real traffic data.
