Restoring Short-Period Oscillations of the Motion of Averaged Optimal Control Systems

Averaging is a valuable technique to gain understanding in the long-term evolution of dynamical systems characterized by slow and fast dynamics. Short period variations of averaged trajectories can be restored a posteriori by means of a near-identity transformation that is a function of both the averaged slow and fast variables. Recent contributions in optimal control theory prove that averaging can be applied to the dynamical system resulting from the necessary conditions for optimality. The present talk extends these results by discussing the evaluation of short-period variations of the adjoint variables. First, the classical approach is shown to be inadequate when applied to the assessment of the adjoints of slow variables because of the peculiar form of their equations of motion. Hence, a consistent transformation is developed, such that variations of the adjoints of fast and slow variables are evaluated in sequence. A simplified transformation is finally obtained when a single fast variable is considered. The methodology is applied to a time-optimal low-thrust orbital transfer.