  We consider the problem of planning trajectories for a group of $N$ vehicles,
each aiming to reach its own target set while avoiding danger zones of other
vehicles. The analysis of problems like this is extremely important
practically, especially given the growing interest in utilizing unmanned
aircraft systems for civil purposes. The direct solution of this problem by
solving a single-obstacle Hamilton-Jacobi-Isaacs (HJI) variational inequality
(VI) is numerically intractable due to the exponential scaling of computation
complexity with problem dimensionality. Furthermore, the single-obstacle HJI VI
cannot directly handle situations in which vehicles do not have a common
scheduled arrival time. Instead, we perform sequential path planning by
considering vehicles in order of priority, modeling higher-priority vehicles as
time-varying obstacles for lower-priority vehicles. To do this, we solve a
double-obstacle HJI VI which allows us to obtain the reach-avoid set, defined
as the set of states from which a vehicle can reach its target while staying
within a time-varying state constraint set. From the solution of the
double-obstacle HJI VI, we can also extract the latest start time and the
optimal control for each vehicle. This is a first application of the
double-obstacle HJI VI which can handle systems with time-varying dynamics,
target sets, and state constraint sets, and results in computation complexity
that scales linearly, as opposed to exponentially, with the number of vehicles
in consideration.
