Optimal reentry trajectories by asymptotic matching

In this paper, we use the matched asymptotic expansions method coupled with the full use of the known exact integrals of motion to derive the explicit optimal control laws for lift and bank modulations during entry. The outer solutions are the limiting solutions in the Keplerian region where the gravitational force is predominant. The inner solutions are the solutions near the planetary surface where the aerodynamic force is predominant. Since the adjoint equations can be integrated in closed forms near these extreme boundaries by retaining the dominant forces, the optimal aerodynamic controls can be expressed explicitly in terms of the state variables and two sets of constants. The results from this control strategy by asymptotic matching have been compared with the exact numerical solutions for a variety of skip and glide entry trajectories. Its performance is excellent in the sense that the control exhibits the same characteristic behavior as the exact solution and as such it leads to nearly the same cost function.