Analytical Solutions of a Class of Optimum Orbit Modifications 1

This paper presents the complete analytical solution of several fundamental problems in orbital correction. The initial orbit is represented by a given point in the phase space, while the final orbit is constrained to stay in a given curve which can be bounded, unbounded, or composed of a finite number of segments of different curves. The inclusion of atmospheric maneuver as part of the optimum process is discussed; its analytical treatment can be carried out by modifying the final state to include the set of orbits having their perigee at the boundary of the atmosphere. The selection of the apogee and perigee distances as state variables gives a symmetric form to the problem and results in a linear differential equation of the first order for the ratio of the adjoint variables. The introduction of a curve of comparison, called the separatrix, facilitates the discussion of the existence of a corner on an optimal trajectory.