ICNPAA 2010: Mathematical Problems in
Engineering, Aerospace and Sciences

In this work, an analytical theory of optimal low-thrust limited-power trajectories is presented. This theory is based on properties of generalized canonical systems, Lie-Hori canonical perturbation method and Hamilton-Jacobi canonical transformation theory. The optimization problem associated to the general space transfer problem is formulated as a Mayer problem of optimal control theory with Cartesian elements – position and velocity vectors – as state variables. After applying the Pontryagin Maximum Principle and determining the maximum Hamiltonian, classical orbital elements are introduced through a canonical transformation derived from the properties of generalized canonical systems. Short periodic terms are then eliminated from the maximum Hamiltonian function through an infinitesimal canonical transformation built by applying Lie-Hori canonical perturbation method. The new Hamiltonian function, resulting from the infinitesimal canonical transformation, describes the extremal trajectories associated with the long duration maneuvers for simple transfers (no rendez-vous). This new Hamiltonian function can be simplified for four special classes of maneuvers: transfers between elliptical coplanar orbits, transfers between elliptical non-coplanar coaxial orbits, transfers between elliptical non-coplanar co-parameters orbits and transfers between orbits with very small eccentricities, and, closed-form analytical solutions can be obtained through Hamilton-Jacobi theory. Using Lie-Hori algorithm, first order approximated analytical solutions, which include short periodic terms, are obtained for these particular problems.