Implicit Central Difference Approximations of Averaged Dynamics of Highly Oscillatory Systems With Applications to Direct Optimal Control

ABSTRACT Trajectory optimization of highly oscillatory systems can require huge computational efforts if the horizon of the problem is much larger than the duration of a single cycle. To alleviate this effort, stroboscopic averaging methods can be used, which utilize local single‐cycle simulations of the oscillatory dynamics to then approximate the average dynamics of the system and integrate them with integration steps much larger than a single cycle. Targeting especially the field of direct optimal control, where, after discretization, the simulation of the dynamics is often included implicitly in the equality constraints of the nonlinear programming problem, we introduce an implicit central difference scheme for the approximation of the average dynamics. In the introduced implicit central difference methods, we solve a nonlinear system of equations that connects single‐cycle simulations of the oscillatory dynamics in order to approximate the average dynamics from a linear combination of points in state‐space. We derive the accuracy order for this generalized ‐point central difference approximation of the average dynamics and confirm the claims in numerical experiments. Compared to existing explicit approaches, the introduced methods are up to twice as efficient while maintaining the same level of accuracy.