## 3 Optimization Problem Statement
Many papers present various approaches and techniques for assessing the influence of the latter [1; 2; 3; 4], but there may be no rigorous constructions or formalization in the form of a finite set of additional restrictions for problem (17), with very rare exceptions. Therefore, in this paper, we propose a step-by-step approach to solving the main problem, in which the emphasis is on maximizing some positive effect of the influence of lunisolar perturbations (mainly solar) on the flight trajectory. Its essence is as follows. Based on the study of works on this problem [1; 2; 3; 4; 5; 6; 7; 8; 9; 10], it is not difficult to draw the following conclusion: a low-energy trajectory is characterized by monotonical increasing of the perigee of the osculating orbit of the SC to a value that exceeds the radius of the Moon's orbit. The main role here is played by the Sun; the SC trajectory belongs to the outer region (relative to the Moon's orbit). Let us try to figure this out. Based on the analysis of works on the subject of low-energy trajectories [1; 2; 5; 6] and the assessment of the influence of the Sun, we can draw the following conclusion. The departure impulse transfers the SC to a highly elliptical orbit, the apogee of which is far from the Earth--while providing the required and long-term effect of solar perturbations (with the proper location of the Sun, of course), in which the perigee radius of the orbit increases. Otherwise, reaching the vicinity of the Moon as a whole would be very problematic, or would be realized with too large a difference in velocities. In general, the very fact of reaching the Moon in the vicinity of the perigee of the transfer orbit is understandable, since in this case, the general scheme of the flight resembles a three-impulse one, with the only difference that the second impulse is \"realized\" by the influence of the Sun. In order to ensure passive gravitational capture of the SC by the Moon at the moment when SC passes the vicinity of the perigee of the transfer orbit, it is clear that it is necessary to minimize the modulus of their relative velocity, so the value of the latter must be greater than the radius of the Moon's orbit. Therefore, when designing a low-energy trajectory by considering an auxiliary problem, it seems appropriate to consider as an optimality criterion not the sum of impulses (15), but the value of the perigee radius of the transfer orbit when SC approaches the Moon. Obviously, the latter must be maximized. Thus, the question of introducing additional conditions into the problem (17), formalizing the effective usage of the influence of lunisolar perturbations, can be removed. Since, based on simple logic, along the flight path of the SC the influence of the Sun will be the most effective. Another question arises with the \"mechanism\" of reaching the SC the vicinity of the Moon and its further insertion to the terminal selenocentric orbit. As is known, the dynamical gravity sphere of the Moon itself is relatively small, and on the general scale of the flight trajectory it is a very small area. The size of the latter is known; the entrance to it from the side of the outer region with respect to the Moon's orbit can be carried out only through the neck in the vicinity of the \(L_{2}\) point of the Earth-Moon system. In this case, it is assumed that, generally speaking, the size of the \"inlet neck\" is relatively small; otherwise, the selenocentric energy of the SC would be too high, and it would hardly be possible to carry out a passive gravitational capture of the SC by the Moon into a selenocentric elliptical orbit. These reasonings are confirmed by the analysis of Hill surfaces constructed for the derived solutions. The latter will be given in the course of further presentation. Thus, at the current stage of consideration of the auxiliary problem, it seems appropriate to use the condition that ensures that the SC enters some neighborhood of the libration point \(L_{2}\).