## 3 Optimization Problem Statement  
Let us start with the formulation of the main problem. It is required to design a low-energy trajectory of the SC flight from the Earth to the Moon with insertion to a selenacentric low circular orbit of a given height; and at the same time, the total energy costs (total characteristic velocity of a SC) for the flight should minimized. We assume that the impulse approximation is used to describe the active thrust arcs of the motion of the center of mass of the SC. As a motion model, we consider a RC4 BP, described by a system (11)-(14). From the mechanical \"essence\" of the problem, it is obvious that the Sun and the Moon have a significant influence on the trajectory motion of the SC. Especially this concerns to the influence of Sun. Therefore, the design of a low-energy trajectory directly relies on the active usage of lunisolar perturbations. Their influence should help to reduce the total characteristic velocity of the trajectory. The ideal (and well-known) variant of the solution consist of only two impulses to be applied--the initial one from the low Earth reference orbit, and the final one, at the moment when SC inserts the required selenocentric orbit with given parameters. Everything else depends on using the influence of Sun and Moon. Therefore, as a functional (criterion) of the main problem, we consider the total characteristic velocity of SC which is needed for a flight; consider it as a sum of two corresponding impulses. Then the objective functional of the main problem we can write in the following (rather general) form:  
\[J_{0}(\mathbf{y})=\sum_{i=1}^{2}\Delta V_{i}(\mathbf{y}),\ \mathbf{y}\in \mathbf{R}^{n} \tag{15}\]  
Let the vector \(\mathbf{y}\) consist of unknown parameters of the problem, until we set them explicitly. Its composition will be determined in the future, in the course of the formation and formalization of a sequence of auxiliary problems. Naturally, it is assumed that the functional in expression (15)  
\[J_{0}(\cdot),J_{0_{\mathbf{y}}}(\cdot)\in C(Q)\]  
where \(Q\) is the set of values of parameters \(\mathbf{y}\) which are admissible under constraints. The latter is formed by the corresponding system, consisting of equalities and inequalities:  
\[\mathbf{g}(\mathbf{y})=\mathbf{0},\ \mathbf{h}(\mathbf{y})\leq\mathbf{0}\]  
Here the functions \(\mathbf{g}\) and \(\mathbf{h}\) are also assumed to be continuous and continuously differentiated. As an example, one can explicitly write down a constraint that describes the condition for entering a given selenocentric circular orbit at the end of the flight (this moment of time obviously coincides with the moment of application of the second (last impulse) impulse, inserting the SC to a circular orbit). Within the framework of the planar problem, only its altitude determines the terminal orbit. Therefore, next equality type constraint must be satisfied:  
\[g(\mathbf{y})=\sqrt{1+r\Big{(}t_{f}\Big{)}^{2}-2\ r\Big{(}t_{f}\Big{)}\cos \Big{(}\beta\Big{(}t_{f}\Big{)}-\alpha_{M}\Big{(}t_{f}\Big{)}\Big{)}}-(R_{ Moon}+h_{Moon})=0 \tag{16}\]  
In the above expression (16) \(R_{Moon}\) is the radius of the Moon, it is taken equal to 1738 km; \(h_{Moon}\), the altitude of the final circular orbit assumed to be 100 km. Thus, the optimization problem corresponding to the main problem can be formulated (in a \"minimal form\") as the following non-linear programming (NLP) problem:  
\[\min_{\mathbf{g}(\mathbf{y})=0}J_{0}(\mathbf{y}), \tag{17}\]Those, at least, it is required to minimize the functional (15) taking into account the constraint (16). Such a \"minimalistic\" formulation of the optimization problem turns out to be too general and multivariant, the corresponding problem has a huge number of local minima. Moreover, it is not very obvious, for example, how to set additional restrictions in the framework of statement (17) that would make it possible to relate the positive effects of the influence of lunisolar perturbations, which are determined primarily by the configurations of celestial bodies.