  This work presents a Physics-Informed Indirect Method (PIIM) that propagates
the dynamics of both states and co-states backward in time for trajectory
optimization problems. In the case of a Time-Optimal Soft Landing Problem
(TOSLP), based on the initial co-state vector normalization technique, we show
that the initial guess of the mass co-state and the numerical factor can be
eliminated from the shooting procedure. As a result, the initial guess of the
unknown co-states can be constrained to lie on a unit 3-D hypersphere. Then,
using the PIIM allows one to exploit the physical significance of the optimal
control law, which further narrows down the solution space to a unit 3-D octant
sphere. Meanwhile, the analytical estimations of the fuel consumption and final
time are provided. Additionally, a usually overlooked issue that results in an
infeasible solution with a negative final time, is fixed by a simple remedy
strategy. Consequently, the reduced solution space becomes sufficiently small
to ensure fast, robust, and guaranteed convergence for the TOSLP. Then, we
extend the PIIM to solve the Fuel-Optimal Soft Landing Problem (FOSLP) with a
homotopy approach. The numerical simulations show that compared with the
conventional indirect method with a success rate of 89.35%, it takes a shorter
time for the proposed method to find the feasible solution to the FOSLP with a
success rate of 100%.
