A procedure for calculating the analytical derivatives required to optimize long duration constant specific impulse finite burns 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. Another transition matrix maps perturbations across any discontinuities in the state due to a zero sphere of influence patched conic flyby or discontinuities in the equations of motion that occur when the engine turns on or off. As applications, the method is used to find optimal Earth to Saturn trajectories. The state transition matrix derivatives are shown to find optimal trajectories from sets of initial conditions where finite difference derivatives fail to converge..
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