Averaging is a valuable technique to gain understanding in the long-term evolution of dynamical systems characterized by slow and fast dynamics. Recent contributions proved that averaging can be applied to the extremal flow of optimal control problems. The present work extends these results by tackling averaging of time optimal systems with two fast variables.
The first part of the talk is devoted to the justification of the application of the averaging principle to this problem. The key role of the adjoints of fast variables is then disclosed, which yields the assessment of a compatibility condition between their boundary values in the original and averaged systems.
The second part is devoted to the a posteriori reconstruction of short-period variations. The classical near-identity transformation exploited in dynamical system theory is shown to be inadequate to restore the adjoints of slow variables because of the peculiar form of their equations of motion. Hence, a consistent transformation is developed. Effects due to the crossing of resonances are finally discussed. Developments are applied to a time-optimal low-thrust orbital transfer in the Earth-Moon system.
