Smooth Approximations of Single-Input Optimal Orbital Transfer using Continuation and Averaging Techniques
ABSTRACT. This article deals with the optimal transfer of a satellite between Keplerian orbits when the control is oriented along the tangential direction. We consider the time-optimal control problem and the energy minimization problem. The optimal controls laws have discontinuities and continuations and averaging techniques are applied to smooth the discontinuities. The smooth approximations of the solutions are computed using a shooting method taking into account second-order optimality conditions.
KEYWORDS: Optimal control, shooting method, second-order optimality condition, single-input orbit transfer, continuation method, averaging
The orbital transfer is described by the controlled KEPLER equation
and the mass variation equation
where the gravitation constant μ is normalized to 1, q ∈ R3 is the position of the satellite in a fixed frame whose origin is the Earth center, m is the mass, β > 0 and u ∈ R3 represents the thrust, satisfying |u| ≤ε , and the maximal thrust is small for ...