A procedure for calculating the analytical derivatives required for an indirect optimization method of combined, long-duration finite burn and multiple gravity-assist trajectories is presented. The analytical derivatives are calculated using the state transition matrix associated with the complete set of the Euler-Lagrange equations of the optimal control problem on each trajectory segment as well as a state transition matrix that maps perturbations across any discontinuities in the state as a result of a zero-sphere-of-influence patched conic flyby or impulsive maneuver. As applications, the method is used to find Earth-to-Saturn trajectories that use combined low thrust and gravity assists and Earth-to-Saturn trajectories that use impulsive thrust and gravity assists. The state transition matrix derivatives are able to satisfy the trajectory constraints to orders of magnitude greater accuracy than the central difference derivatives. The state transition matrix method also requires less computational time to find an optimal trajectory.
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