  Reconfigurable Intelligent Surface (RIS) technology has gained significant
attention due to its ability to enhance the performance of wireless
communication systems. The main advantage of RIS is that it can be
strategically placed in the environment to control wireless signals, enabling
improvements in coverage, capacity, and energy efficiency. In this paper, we
investigate a scenario in which a drone, equipped with a RIS, travels from an
initial point to a target destination. In this scenario, the aerial RIS (ARIS)
is deployed to establish a direct link between the base station and obstructed
users. Our objective is to maximize the energy efficiency of the ARIS while
taking into account its dynamic model including its velocity and acceleration
along with the phase shift of the RIS. To this end, we formulate the energy
efficiency problem under the constraints of the dynamic model of the drone. The
studied problem is challenging to solve. To address this, we proceed as
follows. First, we introduce an efficient solution that involves decoupling the
phase shift optimization and the trajectory design. Specifically, the
closed-form expression of the phase-shift is obtained using a convex
approximation, which is subsequently integrated into the trajectory design
problem. We then employ tools inspired by economic model predictive control
(EMPC) to solve the resulting trajectory optimization.
  Our simulation results show a significant improvement in energy efficiency
against the scenario where the dynamic model of the UAV is ignored.
