## 3 Optimization Problem Statement
**Solving a first-stage problem** (see (21)):  
minimize (19) by varying vector (20), taking into account inequality type constraint in (21); this stage is the simplest one and problem (21) can be solved by using any variant of quasinewtonian algorithm; the set of a parameters (22) should be chosen from some admissible diapason, we can recommend here  
\(c_{L_{2}}=0.95\ldots 0.98,\Delta r_{L_{2}}=2.6\times 10^{-3}\ldots 5.2\times 10^{-2}\)  
**Solving a second-stage problem** (see (23)):  
once again it is necessary to find a constraint minimum of functional (19), but this time by varying vector  
\(\mathbf{y}=\left(\beta_{0},r_{\alpha_{0}},\alpha_{M_{0}},t_{1},\Delta t_{f} \right)^{\mathsf{T}}\)  
and taking into account an equality type constraint which corresponds to a required value of a given radius of final selenacentric orbit; the inequality type constraint from a previous stage also should be used here. To find a numerical solution in that case it is assumed to use a solution of a first stage problem that was derived previously. Adding the equality constraint makes this problem very sensitive to variation of any of the unknown parameters, but as it was shown by our own experience in that problem (usage of a graph-analytical approach), small and slightly variations of (22) provides for us an ability to sufficiently easily satisfy altitude constraint and thus, define the value of \(\Delta t_{f}\). Moreover, for different values of a given altitude of a final orbit it was enough to vary only first coefficient from (22) for a value that not exceeds \(\pm 0.02\). The variation of first coefficient from (22) provides a direct influence to a value of an axial projection of the SC relative velocity vector in RC3 BP Moon-Earth-SC (the direction of an axial projection coincides with the current Moon to Earth direction). Changing of it provides for a SC ability to proceed through the vicinity of \(L_{2}\) libration point inside the Hill sphere and after that approaches to the Moon. **Solving a third-stage problem** (see (25)):  
this time the sum of two impulses  
\(J_{2}(\mathbf{y})=\Delta V_{0}(\mathbf{y})+\Delta V_{f}(\mathbf{y})\)  
should be minimized under the same constraints from the previous second stage problem; the vector of unknowns also is the same. Changing of a criterion does not provide additional complexities in search of a numerical solution (it is interesting by itself). Like previously, as an initial guess was used the solution from the previous (second) stage problem. Next, the solution that we have obtained from the flat auxiliary problem now can be used as an initial approximation for solving the same problem but in a full model (3-dimensional motion, ICRS (J2000.0) reference system, ephemeris model); as well as solving the problems of designing trajectories to the libration points \(L_{1}\) and \(L_{2}\). So, let us briefly describe the formulation of the problem of a designing low-energy Earth-to-Moon flight within the framework of a \"complete\" model of motion. Now, to describe the motion of the SC, as before, the R4 BP is used: Earth--SC--Sun--Moon (system (1), ephemeris model (JPL DE421)). The Earth is the central body and as a main reference ICRS (J2000.0) coordinate system is used. The SC starts its motion at some initial time \(t_{0}\) from a low circular geocentric orbit of a given radius \(r_{0}\) and inclination \(i_{0}\). Then performs a flight to the Moon, in the close vicinity of which a braking impulse is realized, inserting a SC to a circular circumlunar orbit of a given altitude \(h_{Moon}\) and with a given inclination \(i_{Moon}\) (relative to the lunar equator). The initial state of the SC is determined by the following relations:  
\[\mathbf{r}(t_{0})=A_{0}\big{(}r_{0}\quad 0\quad 0\big{)}^{\mathbf{T}},\ \mathbf{V}(t_{0})=A_{0}\Big{(}0\quad\Delta V_{0}+\sqrt{\frac{\mu_{E}}{r_{0}}} \quad 0\big{)}^{\mathbf{T}} \tag{26}\]  
\[A_{0}=\left(