Constructing Pontryagin extremals for the optimal control problem of asteroid fly-by

Fuel optimal low thrust rendezvous with outer planets via gravity assist

“…One has to admit, however, that the above admissible domain is larger than the actual one by noting that various main is larger than the actual one by noting that various constraints on velocity before and after GA, i.e., eqs. (18)- (20) are relaxed at this moment. It is one of the advantages of indirect method in trajectory optimization, i.e., the formulation of shooting equations, i.e.,   ,  0 q  guarantees the optimality of the original problem and the satisfaction of various constraints at the same time, that one can assume here that the relaxation of some constraints when setting admissible domain will not affect the optimality, but bring much convenience.…”

“…(20), that is the decrease of v  , periapsis r is always accompanied by the increase of turn angle . However, with the hyperbolic excess velocity v  specified, the maximum turn angle is always limited by the minimum periapsis radius min r of the GA hyperbolic orbit, if the surface or the extent of the atmosphere of the planet are considered.…”

“…Actually some missions like Mariner 10 and Voyager 1 have utilized intermediate GA and various problems associated with GA are treated in refs. [19][20][21][22]. However, most of the researchers formulate the problem as a parameter optimization problem or consider only the impulsive case.…”

Choosing promising sequences of asteroids

“…Δt k and Δr k are parameters of the continuation method, and they may be different for different k. In general, we used the following values in adding the levels: Δr k = 0.1 a.u., Δt k = 100 days. 3 To find target asteroids and flyby moments we used the following algorithm. On the passive trajectory of SV flight, with step 1-2 days we computed the distances between the SV and each admissible asteroid.…”

“…For instance, in choosing promising trajectories we could use the "easy implementability" condition that states that active segments corresponding to impulse impacts during asteroid flybys should be disjoint; these conditions were used in asteroid selection in previous similar competitions [2,3]. However, such conditions would increase the average flight duration between asteroids.…”