Optimal trajectories for spacecraft with low electric-jet thrust in mission to asteroid Apophis

There is considered the problem of space flights to the asteroid Apophis. We analyze the flight scheme, which includes the geocentric stage, when a spacecraft with a high-thrust engine is accelerated; the heliocentric stage, in which the spacecraft moves using a low-thrust engine; and, finally, the deceleration stage, when the spacecraft becomes an artificial satellite orbiting the asteroid. We solve the problem of optimal control for the ideal and piecewise-constant low thrust, as well as determine the optimal value and direction of the hyperbolic velocity at "infinity" achieved by the spacecraft when it leaves the Earth sphere of influence. There is defined the set of space trajectories for a wide range of start dates and various flight durations using a complex method of optimization. We estimate the final mass of the spacecraft and the mass of the payload that can be delivered to the asteroid using the Soyuz-Fregat launcher.