A Continuous Implementation of a Second-Variation Optimal Control Method for Space Trajectory Problems

The paper describes a continuous second-variation method to solve optimal control problems with terminal constraints where the control is defined on a closed set. The integration of matrix differential equations based on a second-order expansion of a Lagrangian provides linear updates of the control and a locally optimal feedback controller. The process involves a backward and a forward integration stage, which require storing trajectories. A method has been devised to store continuous solutions of ordinary differential equations and compute accurately the continuous expansion of the Lagrangian around a nominal trajectory. Thanks to the continuous approach, the method adapts implicitly the numerical time mesh and provides precise gradient iterates to find an optimal control. The method represents an evolution to the continuous case of discrete second-order techniques of optimal control. The novel method is demonstrated on bang–bang optimal control problems, showing its suitability to identify automatically optimal switching points in the control without insight into the switching structure or a choice of the time mesh. A complex space trajectory problem is tackled to demonstrate the numerical robustness of the method to problems with different time scales.