Optimal control of a solar sail

Abstract

SummaryOptimal control of a solar sail orbiting around a fixed center of mass is considered. The sail is modelled as a plane surface with two sides having similar optical properties. The control is assumed to be the attitude of the sail and is represented as an element of the projective plane. Mapping this plane into the three dimensional ambient space to compute the actual force generated by a given attitude results in a generally non‐convex set of admissible control values. A suitable convex relaxation is introduced to study the optimality system associated with maximising the change of the sail orbital parameters in a given direction along one orbit. Existence for both the relaxed and the original problems are deduced. Building on previous works, a refined analysis of the control structure is given, proving rigorously that it contains only switchings, and a global bound of the number of switchings is obtained. In order to compute effective solutions, a multiple shooting approach is retained that is well suited for systems with a sequence of switchings between various control subarcs. Three issues are addressed: initialization of shooting by means a tailored semi‐definite relaxation that takes advantage of good convergence properties of modern convex optimization algorithms; changes in the control structure that are accommodated by coupling shooting with differential continuation; implicit character of the Hamiltonian maximization when using Pontrjagin maximum principle that is taken care of by incorporating the associated equation for the dynamical feedback into the shooting procedure. The method is illustrated on an example from NASA for which the sail inclination is optimally changed over one orbital period.