Periodic trajectories that satisfy stability-based requirements are useful in many engineering applications. In this work, a time-domain approach using Fourier collocation for solving the Floquet eigenvector boundary value problem, and consequently estimating Floquet exponents and eigenvectors, is introduced. This method is incorporated into a Fourier pseudospectral trajectory optimization framework for generating open-loop stable solution, and for improving the damping of a specific mode identified using the information from Floquet eigenvectors. These applications are illustrated on the dynamic soaring solutions of three-and six-degree-of-freedom aircraft models, and numerical results highlighting the effectiveness of the pseudospectral Floquet computation over the traditional time-march approach are also presented. © 2020 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.