## 4 Results
As a result of solving problem (25), the total characteristic velocity of the flight turned out to be 3.829224 km/s, the value of the departure impulse from the Earth \(\Delta V_{0}=\) 3.194456 km/s; the magnitude of the braking impulse near the Moon is \(\Delta V_{f}=\) 0.634768 km/s. The total duration of the flight was \(t_{1}+\Delta t_{f}=\) 23.852629 (undim. ), which corresponds to 103.579628 days. The initial value of the polar angle \(\beta_{0}\) was: \(-\)10.636105 degrees, the optimal initial value of the apogee radius of the transfer orbit: 1.138830 million km; the angle characterizing the initial position of the Moon relative to the current direction to the Sun (at the time of launch) was 243.886838 degrees. Further, Figures 27-31 (all undim.) show the optimal trajectory of a low-energy spacecraft flight obtained by solving problem (25) as well as a number of dependencies characterizing it. The optimal two-impulse low-energy spacecraft flight trajectory obtained by solving problem (25) is shown in Figure 3:6 It is clearly seen that the resulting trajectory of problem (25) with a functional expressing the total characteristic velocity of a two-impulse flight almost exactly coincides with the trajectories obtained by solving problems (21) and (23) with a functional (19). This is also confirmed by a direct comparison of the corresponding values of the vector of unknown parameters \(\mathbf{y}\) for the three considered problems. In general, this indicates the correctness of the approach, in which the perigee radius of the transfer orbit was chosen as the criterion for the optimality of the auxiliary problem; it also once  
Figure 3:7 Selenocentric \(r_{M}\) [undim.] distance along the optimal flight trajectory. Figure 3:8 Relative selenocentric distance \(r_{M}\) [undim.] of the SC during its trajectory motion in the “near-moon” section up to a circular orbit of a given radius. again emphasizes the role of the Sun, because the considered formulations of the trajectory problem are aimed at maximizing the effect of perturbations (mainly from the Sun). Finally, let us consider the solution of problem (29). The solution of (29) for a given value of the initial inclination \(i_{0}=51.8^{0}\) (to the geoequator) was found: from the solution of the flat auxiliary problem, approximations to the parameters (\(\Delta V_{0}\), \(t_{0}\), \(t_{i}\)) and optimal configuration of celestial bodies (Earth--Sun--Moon) were obtained (corresponds to \(\beta_{0}\), \(\alpha_{M0}\)). Next, a numerical solution of the optimization problem was determined--for this, the nonlocal CMA-ES algorithm [12] was used, which is a variant of the evolutionary strategy with adaptation of the covariance matrix. The numerical integration of the equations of motion was carried out using the Dorman and Prince DOPRI853 algorithm [13]. Below the results of the obtained solution (Figure 31 and Table 1) are presented (briefly). Figure 3:9 Low-energy trajectory of the center of mass of the SC in the framework of the full model [undim.] (ICRS J2000.0). Figure 2:0 Section of the spacecraft’s selenocentric motion ending with the moment of entering the terminal orbit. The blue cross shows the point of application of the braking impulse. A fragment of the trajectory is shown in the selenocentric coordinate system with fixed axes oriented along the axes of the system \(\xi\)\(\eta\)(\(\xi_{M}\)C\(\eta_{M}\)(\(\xi\))). It is easy to see that the derived results are quite close to each other but the solution of the flat auxiliary problem is better as it was expected.