Book excerpt: This thesis is concerned with the study of pseudospectral discretizations of optimal control problems governed by ordinary differential equations and with their application to the solution of the International Space Station (ISS) momentum dumping problem. Pseudospectral methods are used to transcribe a given optimal control problem into a nonlinear programming problem. Adjoint estimates are presented and analyzed that provide approximations of the original adjoint variables using Lagrange multipliers corresponding to the discretized optimal control problem. These adjoint estimations are derived for a broad class of pseudospectral discretizations and generalize the previously known adjoint estimation procedure for the Legendre pseudospectral discretization. The error between the desired solution to the infinite dimensional optimal control problem and the solution computed using pseudospectral collocation and nonlinear programming is estimated for linear-quadratic optimal control problems. Numerical results are given for both linear-quadratic and nonlinear optimal control problems. The Legendre pseudospectral method is applied to formulations of the ISS momentum dumping problem. Computed solutions are verified through simulations using adaptive higher order integration of the system dynamics.