## 3 Optimization Problem Statement
\begin{array}{ccc}\cos(\Omega_{0})\cos(u_{0})-&\cos(\Omega_{0}) \sin(u_{0})-\\ \sin(\Omega_{0})\cos(i_{0})\sin(u_{0})&\sin(\Omega_{0})\cos(i_{0})\cos(u_{0}) &\sin(\Omega_{0})\sin(i_{0})\\ \sin(\Omega_{0})\cos(u_{0})+&-\sin(\Omega_{0})\sin(u_{0})+\\ \cos(\Omega_{0})\cos(i_{0})\sin(u_{0})&\cos(\Omega_{0})\cos(u_{0})\cos(u_{0}) &-\cos(\Omega_{0})\sin(i_{0})\\ \sin(i_{0})\sin(u_{0})&\sin(i_{0})\cos(u_{0})&\cos(i_{0})\end{array} \right)\]  
For the purpose of some regularization of the optimization problem, we can consider a third (second in a row) additional velocity impulse (correction)--the so-called apogee velocity impulse \(\mathbf{\Delta V_{2}}\), realized along the transfer trajectory at time \(t_{1}\). Then  
\[\mathbf{r}\big{(}t_{1}{}^{+}\big{)}=\mathbf{r}\big{(}t_{1}{}^{-}\big{)},\ \mathbf{V}\big{(}t_{1}{}^{+}\big{)}=\mathbf{V}\big{(}t_{1}{}^{-}\big{)}+ \mathbf{\Delta V_{2}} \tag{27}\]  
Why we used another impulse here? Because we have thought that the adding of the third (presumably small) impulse will help SC to reach the required vicinity of \(L_{2}\) point (for a problem with a \"full\" or \"complete model of motion\") with some admissible state. We assumed that this impulse should take place somewhere near the apogee of the SC flight trajectory: also, we thought that by using it we would truncate (if needed) possible trajectory declinations right before the SC reaches the close vicinity of the \(L_{2}\) point. The usage of the additional impulse here somehow resemblance with the usage of the parameters (22) in auxiliary problem--it helps us to derive a required solution in more robust way. Despite the fact that the number of unknowns increased, the numerical stability also got better. The value of the vector of the final velocity impulse \(\mathbf{\Delta V_{f}}\) (when the SC reaches a given selenocentric orbit) can be determined as follows:  
\[|\mathbf{\Delta V_{f}}|=\left|\left\|\mathbf{V}\Big{(}t_{f}\Big{)}-\mathbf{V }_{Moon}\Big{(}t_{f}\Big{)}\right\|_{\mathbf{R}^{3}}-\sqrt{\frac{\mu_{Moon}} {R_{Moon}+h_{Moon}}}\right|;\]  
In addition, at the moment \(t_{f}\), the following conditions must be satisfied; they describe the terminal state of the SC (circular selenocentric orbit of a given radius)given altitude):  
\[\left|\mathbf{r}\Big{(}t_{f}\Big{)}-\mathbf{r}_{Moon}\Big{(}t_{f}\Big{)} \right|-\left(R_{Moon}+h_{Moon}\right)=0 \tag{28}\] \[\left.\frac{\sigma_{z}\big{(}\mathbf{r}(t_{f}),\mathbf{v}(t_{f}) \big{)}}{\sqrt{\mu_{Moon}\left(R_{Moon}+h_{Moon}\right)}}-\cos\Big{(}i_{f} \Big{)}=0\]  
In the above expressions (28), the third condition corresponds to the restriction on the inclination of the terminal selenocentric orbit, which should be equal to \(i_{f}\). The optimization problem corresponding to the considered trajectory problem of designing a low-energy trajectory can be formalized as the following non-linear programming problem:\[\left. \begin{array}{l}\underbrace{\min\limits_{\mathbf{z}\in\mathbb{Q}}\Delta V^{ \Sigma}(\mathbf{z})},\\ \Delta V^{\Sigma}(\mathbf{z})=\Delta V_{0}+\left(\mathbf{\Delta V_{2}}^{\mathbf{T }}\cdot\mathbf{\Delta V_{2}}\right)^{\frac{1}{2}}+\left|\mathbf{\Delta V_{f}( \mathbf{z})}\right|\\ \mathbf{z}=\left(\Delta V_{0},\Omega_{0},u_{0},t_{0},t_{1},t_{f},\mathbf{ \Delta V_{2}}\right)^{\mathbf{T}},\,\mathbf{z}\in\mathbb{R}^{9}\end{array} \right\} \tag{29}\]  
The unknown parameters of (29) (components of the vector \(\mathbf{z}\)): the magnitude of the initial velocity impulse \(\Delta V_{0}\); the longitude of the ascending node \(\Omega_{0}\), and the argument of latitude \(u_{0}\) of the SC starting point from the low circular geocentric orbit of a given inclination \(i_{0}\); times of launch \(t_{0}\) and SC insertion into a given selenocentric orbit \(t_{f}\); the moment of application of the apogee velocity impulse \(t_{1}\); components of the apogee velocity impulse vector \(\mathbf{\Delta V_{2}}\).