Analytical Treatise on Endo-Atmospheric Fuel-Optimal Rocket Landings

While there is a strong current interest in and an increasing body of the latest numerical work on fuel-optimal powered descent inside an atmosphere, little recent analytical understanding of the problem has been reported. This paper analyzes the endo-atmospheric fuel-optimal rocket landing problem in three-dimensional motion. The necessary conditions for the problem that account for both propulsive and aerodynamic forces are derived. They include the full set of costate equations and the optimality conditions of the body attitude in terms of the angle of attack and sideslip angle. In contrast to a long-standing assumption of coordinated flight in the literature, it is shown that in general coordinated flight is not optimal if sideslip modulation is allowed. Special cases, such as small-angle and in-plane motion, are analyzed and harmonized with the previous results that are well known in the literature. The analytical conditions derived in this paper can also serve as independent verification for numerical solutions obtained by direct methods in optimal control or can be exploited for trajectory-design purposes.