  In this article, the formation mission design problem for commercial aircraft
is studied. Given the departure times and the departure and arrival locations
of several commercial flights, the relevant weather forecast, and the expected
fuel savings during formation flight, the problem consists in establishing how
to organize them in formation or solo flights and in finding the trajectories
that minimize the overall direct operating cost of the flights. Each aircraft
can fly solo or in different positions inside a formation. Therefore, the
mission is modeled as a switched dynamical system, in which the discrete state
describes the combination of flight modes of the individual aircraft and
logical constraints in disjunctive form establish the switching logic among the
discrete states of the system. The formation mission design problem has been
formulated as an optimal control problem of a switched dynamical system and
solved using an embedding approach, which allows switching decision among
discrete states to be modeled without relying on binary variables. The
resulting problem is a classical optimal control problem which has been solved
using a knotting pseudospectral method. Several numerical experiments have been
conducted to demonstrate the effectiveness of this approach. The obtained
results show that formation flight has great potential to reduce fuel
consumption and emissions.
