  This paper considers an unmanned aerial vehicle (UAV)-enabled wireless power
transfer (WPT) system, in which a UAV equipped with a directional antenna is
dispatched to deliver wireless energy to charge two energy receivers (ERs) on
the ground. Under this setup, we maximize the common (or minimum) energy
received by the two ERs over a particular finite charging period, by jointly
optimizing the altitude, trajectory, and transmit beamwidth of the UAV, subject
to the UAV's maximum speed constraints, as well as the maximum/minimum altitude
and beamwidth constraints. However, the common energy maximization is a
non-convex optimization problem that is generally difficult to be solved
optimally. To tackle this problem, we first ignore the maximum UAV speed
constraints and solve the relaxed problem optimally. The optimal solution to
the relaxed problem reveals that the UAV should hover above two symmetric
locations during the whole charging period, with the corresponding altitude and
beamwidth optimized. Next, we study the original problem with the maximum UAV
speed constraints considered, for which a heuristic hover-fly-hover trajectory
design is proposed based on the optimal symmetric-location-hovering solution to
the relaxed problem. Numerical results validate that thanks to the employment
of directional antenna with adaptive beamwidth and altitude control, our
proposed design significantly improves the common energy received by the two
ERs, as compared to other benchmark schemes.
