Minimum time loop maneuver with no speed loss

A problem of how to move the center of gravity of the maneuverable aircraft from one given point in the three-dimensional space to another point as quickly as possible with the fixed vectors of the respective velocities is considered. Numerical solutions for coinciding initial and final conditions for trajectories entirely lying in the vertical plane are found. The solution is shown to be non-unique in the general case, with the change of the sign of curvature of the best trajectory being possible. Locally optimal solutions are found. The problem that takes into account the restriction on the sign of curvature of the trajectory is considered, with the respective optimal controls obtained both for fixed and free end point. Examples are given for cases when the velocity vector equals the initial one at the end of motion.