Matched asymptotic solutions for optimum lift controlled atmosphericentry

This paper considers the problem of optimal lift control of a hypersonic lifting body during atmospheric entry for the drag coefficient, a function of the angle of attack, and the atmospheric density, an arbitrary function of altitude. The solution obtained is valid for entering the planetary atmosphere from the Keplerian region, as well as from low altitudes. The method of matched asymptotic expansions was employed, and separate expansions were derived for the Keplerian region and for the aerodynamic region, where the aerodynamic forces are dominant. Lagrange multipliers and state variables obtained in closed form for both expansions were matched in the overlap domain. A method for estimating the order of magnitude of multipliers in various regions was discussed and will be useful in applying singular perturbation methods to a wider class of optimal control problems. For unbounded control, the lift variation can be classified into four different programs depending on the terminal altitude. Results were compared with the numerical solutions obtained by the method of steepest descent. For bounded control, there exist 12 different sequences of arcs which reduce to those obtained in an earlier study as the drag coefficient becomes independent of angle of attack.