  We consider unmanned aerial vehicle (UAV)-enabled wireless systems where
downlink communications between a multi-antenna UAV and multiple users are
assisted by a hybrid active-passive reconfigurable intelligent surface (RIS).
We aim at a fairness design of two typical UAV-enabled networks, namely the
static-UAV network where the UAV is deployed at a fixed location to serve all
users at the same time, and the mobile-UAV network which employs the time
division multiple access protocol. In both networks, our goal is to maximize
the minimum rate among users through jointly optimizing the UAV's
location/trajectory, transmit beamformer, and RIS coefficients. The resulting
problems are highly nonconvex due to a strong coupling between the involved
variables. We develop efficient algorithms based on block coordinate ascend and
successive convex approximation to effectively solve these problems in an
iterative manner. In particular, in the optimization of the mobile-UAV network,
closed-form solutions to the transmit beamformer and RIS passive coefficients
are derived. Numerical results show that a hybrid RIS equipped with only 4
active elements and a power budget of 0 dBm offers an improvement of 38%-63% in
minimum rate, while that achieved by a passive RIS is only about 15%, with the
same total number of elements.
