## Inclusion of Known Integrals in the Optimal Trajectory Problem |

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achieved adjoint equations Aeronautics and Space Alabama algebraic angular momentum Assuming calculate defines denominator of equation differential equation direction and magnitude dot product Dotting equation 56 eliminate equation 18 equation 21 equation 33 equation 50 equations represented f r x important integrals INCLUSION OF KNOWN initial value integration constant iteration known integrals KNOWN RESULTS Marshall Space Flight minimizes mass expenditure NAME AND ADDRESS National Aeronautics nonplanar nonzero numerical optimal trajectory problem orthogonal PERFORMING ORGANIZATION planar r x r r x v r x v)2 reduction regularization represented by equations rocket Rowland scalar functions SECURITY CLASSIF seventh-order adjoint system single attracting center single gravitating singularity Space Administration Space Flight Center SPONSORING AGENCY steering third-order adjoint system three-dimensional trajectories thrust Tmax trajectory which minimizes transformation two-point boundary value Unclassified v x r variables vector velocity written yields