A Physics-Informed Indirect Method for Trajectory Optimization

The indirect method has long been favored for solving trajectory optimization problems due to its ability to reveal the structure of the corresponding optimal control. However, providing an appropriate initial guess of the costate vector remains challenging due to its lack of physical significance. To address this issue, this article introduces a physics-informed indirect method (PIIM), for which all shooting variables are constrained into a narrow space by utilizing their physics-informed information. Starting from the time-optimal soft landing problem (TOSLP), an analytical estimation method for the minimum flight time is provided; we show that the costate vector at the final time can be constrained on a unit 3-D hypersphere by eliminating the mass costate and the numerical factor used in the initial costate vector normalization; and the physical significance of the optimal control at the final time is exploited, allowing one to further narrow down the solution space for the costate vector to a unit 3-D octant sphere. Then, by incorporating the above physics-informed information into a shooting function that involves of propagating the dynamics of both states and costates backward, the PIIM achieves faster and more robust convergence for the TOSLP compared with existing indirect methods. In addition, the combination of the PIIM with a homotopy approach is proposed, allowing one to solve the fuel-optimal soft landing problem robustly. Finally, numerical simulations are presented to demonstrate and validate the developments of this article.