Approximate solutions to minimax optimal control problems for aeroassisted orbital transfer

Abstract

This paper considers minimax problems of optimal control arising in the study of aeroassisted orbital transfer. The maneuver considered involves the coplanar transfer from a high planetary orbit to a low planetary orbit. An example is the HEO-to-LEO transfer of a spacecraft, where HEO denotes high Earth orbit and LEO denotes low Earth orbit. In particular, HEO can be GEO, a geosynchronous Earth orbit. The basic idea is to employ the hybrid combination of propulsive maneuvers in space and aerodynamic maneuvers in the sensible atmosphere. Hence, this type of flight is also called synergetic space flight. With reference to the atmospheric part of the maneuver, trajectory control is achieved by means of lift modulation. The presence of upper and lower bounds on the lift coefficient is considered. The following minimax problems of optimal control are investigated: (i) minimize the peak heating rate, problem P1; and (ii) minimize the peak dynamic pressure, problem P2. It is shown that problems P1 and P2 are approximately equivalent to the following minimax problem of optimal control: (iii) minimize the peak altitude drop occurring in the atmospheric portion of the trajectory, problem P3. Problems P1–P3 are Chebyshev problems of optimal control, which can be converted into Bolza problems by suitable transformations. However, the need for these transformations can be bypassed if one reformulates problem P3 as a two-subarc problem of optimal control, in which the first subarc connects the initial point and the point where the path inclination is zero, and the second subarc connects the point where the path inclination is zero and the final point: (iv) minimize the altitude drop achieved at the point of junction between the first subarc and the second subarc, problem P4. Note that problem P4 is a Bolza problem of optimal control. Numerical solutions for problems P1–P4 are obtained by means of the sequential gradient-restoration algorithm for optimal control problems. Numerical examples are presented, and their engineering implications are discussed. In particular, it is shown that, from an engineering point of view, it is desirable to solve problem P3 or P4, rather than problems P1 and P2.