Numerical computation of optimal trajectories for coplanar, aeroassisted orbital transfer.

*(English)* Zbl 1168.49310
Summary: This paper is concerned with the problem of the optimal coplanar aeroassisted orbital transfer of a spacecraft from a high Earth orbit to a low Earth orbit. It is assumed that the initial and final orbits are circular and that the gravitational field is central and is governed by the inverse square law. The whole trajectory is assumed to consist of two impulsive velocity changes at the begin and end of one interior atmospheric subarc, where the vehicle is controlled via the lift coefficient. The problem is reduced to the atmospheric part of the trajectory, thus arriving at an optimal control problem with free final time and lift coefficient as the only (bounded) control variable. For this problem, the necessary conditions of optimal control theory are derived. Applying multiple shooting techniques, two trajectories with different control structures are computed. The first trajectory is characterized by a lift coefficient at its minimum value during the whole atmospheric pass. For the second trajectory, an optimal control history with a boundary subarc followed by a free subarc is chosen. It turns out, that this second trajectory satisfies the minimum principle, whereas the first one fails to satisfy this necessary condition; nevertheless, the characteristic velocities of the two trajectories differ only in the sixth significant digit. In the second part of the paper, the assumption of impulsive velocity changes is dropped. Instead, a more realistic modeling with twofinite-thrust subarcs in the nonatmospheric part of the trajectory is considered. The resulting optimal control problem now describes the whole maneuver including the nonatmospheric parts. It contains as control variables the thrust, thrust angle, and lift coefficient. Further, the mass of the vehicle is treated as an additional state variable. For this optimal control problem, numerical solutions are presented. They are compared with the solutions of the impulsive model.

##### MSC:

49N90 | Applications of optimal control and differential games |

49K15 | Optimal control problems with ODE (optimality conditions) |

93C95 | Applications of control theory |