  Unmanned aerial vehicle (UAV) is expected to bring transformative improvement
to the integrated sensing and communication (ISAC) system. However, due to
shared spectrum resources, it is challenging to achieve a critical trade-off
between these two integrated functionalities. To address this issue, we propose
in this paper a new integrated \emph{periodic} sensing and communication
mechanism for the UAV-enable ISAC system. Specifically, the user achievable
rate is maximized via jointly optimizing UAV trajectory, transmit precoder, and
sensing start instant, subject to the sensing frequency and beam pattern gain
constraints. Despite that this problem is highly non-convex and involves an
infinite number of variables, we obtain the optimal transmit precoder and
derive the optimal achievable rate in closed-form for any given UAV location to
facilitate the UAV trajectory design. Furthermore, we first prove the
structural symmetry between optimal solutions in different ISAC frames without
location constraints and then propose a high-quality UAV trajectory and sensing
optimization algorithm for the general location-constrained case. Simulation
results corroborate the effectiveness of the proposed design and also unveil a
more flexible trade-off in ISAC systems over benchmark schemes.
