## 3 Optimization Problem Statement
We formalize this condition in the form of the following inequality-type constraint, which must be satisfied at some time moment \(t_{1}\):  
\[\left. \begin{array}{l}\sqrt{r^{2}+{r_{L_{2}}}^{2}-2\;r\;{r_{L_{2}}}\cos(\beta -\alpha_{M})}\Big{|}_{t_{1}}-\Delta{r_{L_{2}}}\leq 0,\\ {r_{L_{2}}}={r_{M}}+{x_{L_{2}}}{c_{L_{2}}},\;{c_{L_{2}}}\in(0,1)\end{array} \right\} \tag{18}\]In the above expression, \(r_{L_{2}}\) (undim., as a unit of length here and everywhere later 384,400 km is used)--the modulus of the radius vector plotted along the current direction of the Earth-Moon axis and characterizing the position of the center of some neighborhood of the libration point \(L_{2}\). As can be seen from the second relation in (18), the value of the latter is determined by setting the parameter \(c_{L_{2}}\), and when its value is equal to 1, it exactly coincides with the position of the libration point. The value \(\Delta r_{L_{2}}\geq 0\) specifies the radius of the neighborhood of the point \(L_{2}\) in which the SC should be at the time \(t_{1}\). In expression (18), the position of the libration point is determined by the value of the corresponding coordinate (undim. ), counted from the center of mass of the Moon. The latter is taken equal to \(x_{L_{2}}=0.21197817909\). Now, in full accordance with the above reasoning, let us finally formulate an auxiliary problem of trajectory optimization of the first stage: we will solve the problem of designing the spacecraft's transfer trajectory from the Earth's low circular reference orbit up to a given neighborhood of the libration point \(L_{2}\). In this case, the radius of the perigee of the transfer orbit when SC reaches the designated neighborhood is considered as criterion. The latter obviously needs to be maximized. Let us formalize the optimization problem of the first stage. The target functional is defined as:  
\[J_{1}(\mathbf{y})=\left.-\frac{\sigma^{2}}{1+\sqrt{\Lambda_{\mathrm{g}}+ \Lambda_{\eta}}}\right|_{t_{1}}, \tag{19}\]  
and as the only constraint of the inequality type here we use relation (18). The vector of unknown parameters \(\mathbf{y}\) of the problem:  
\[\mathbf{y}=\left(\beta_{0},r_{\alpha_{0}},\alpha_{M_{0}},t_{1}\right)^{ \mathrm{T}},\,\mathbf{y}\in\mathbf{R}^{4} \tag{20}\]  
In the above expression (20): \(\beta_{0}\) is the polar angle characterizing the position of the SC at the moment of its launch from a low Earth reference circular orbit (altitude 200 km); it is started to counting from the current direction to the Sun (at the time of launch). It is important to note that the departure impulse is applied at the point of the circular orbit corresponding to the perigee of the transfer orbit at the time when mission starts, therefore, the initial position of the apsidal line of the transfer orbit is determined by the same angle; \(r_{\alpha_{0}}\)--initial value of the apogee radius (undim.) of the transfer orbit; \(\alpha_{M_{0}}\)--polar angle characterizing the position of the Moon at start; counted from the current direction to the Sun; \(t_{1}\) is the moment of time (undim.) of reaching a given neighborhood of the point \(L_{2}\). At the current stage of solving the problem of designing a low-energy flight trajectory (17), the motion of the center of mass of the SC is considered on a closed time interval. Finally, the auxiliary trajectory problem of the first stage of the form (18)-(20) can be represented in the following form:  
\[\left.\left. \begin{array}{l}\underset{\mathbf{y}\in\mathbf{Q}}{\min}J_{1}( \mathbf{y})\\ Q=\left\{\mathbf{y}\in\mathbf{R}^{4}:\sqrt{r^{2}+{r_{L_{2}}}^{2}-2\,r\,r_{L_{2} }\cos(\beta-\alpha_{M})}\right|_{t_{1}}-\Delta r_{L_{2}}\leq 0,\right\}\\ \mathbf{y}=\left(\beta_{0},r_{\alpha_{0}},\alpha_{M_{0}},t_{1}\right)^{ \mathrm{T}},\,\mathbf{y}\in\mathbf{R}^{4}\end{array}