## 3 Lossless Convex Optimization Algorithm
### Discussion about Optimal Terminal Time  
Section 3.1 uses the LCvx to convert the problem into an SOCP and efficiently solve it. However, the algorithm requires a given terminal time in advance, which is also the main shortcoming of the algorithm in Section 3.1. Nevertheless, few scholars have studied the impact of terminal time on the solution in detail. This section will analyze the relationship between the terminal mass and the terminal time through the simulation.  
First, set the rocket parameters and environment parameters, as shown in Table 1.  
Figure 5: Description of the discrete convex problem.  
The relationship can be divided into three intervals:  
1. LCvx failure interval (\(t_{\text{f}}\in[0,t_{\text{cr}}]\)): Although it is possible for the numerical solver to obtain the optimal solution, the convex relaxation is no longer valid, which means the solutions do not satisfy \(\|u(t)\|=\sigma(t)\). Therefore, the solution is infeasible;
2. Increasing interval (\(t_{\text{f}}\in(t_{\text{cr}},t_{\text{opt}}]\)): The algorithm can successfully obtain the optimal solution, and the terminal mass increases monotonously with the terminal time. Numerical simulations show that this interval is usually narrow. In Figure 6, the interval length is only 0.64 s. When the initial state variables are set to other values, the interval length is generally less than 2 s;
3. Decreasing interval (\(t_{\text{f}}\) is slightly larger than \(t_{\text{opt}}\)): The algorithm can also successfully obtain the optimal solution, and the terminal mass decreases monotonously with the terminal time. The rocket will increase the landing time by consuming more fuel in this interval. If \(t_{\text{f}}\) is very large, the altitude might increase during the powered landing.  
Combining the above three situations, the following conclusions can be drawn. The feasibility and optimality of the algorithm in Section 3.1 are closely related with \(t_{\text{f}}\). If \(t_{\text{f}}\) is too small, a feasible trajectory cannot be obtained. If the value is too large, it will consume more fuel, and even lead to increasing the altitude. In addition, it can be found  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Amount** & **Parameter** & **Amount** \\ \hline \(T_{\text{min}}\) & 845.2 kN & \(I_{\text{sp}}\) & 282 s \\ \(T_{\text{max}}\) & 169.0 kN & \(\gamma_{\text{max}}\) & 80\({}^{\circ}\) \\ \(g_{0}\) & 9.80665 m/s\({}^{2}\) & \(\theta_{\text{max}}\) & 30\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Rocket and environmental parameters.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Amount** & **Parameter** & **Amount** \\ \hline \(r_{x0}\) & \(5\times 10^{3}\) m & \(v_{x0}\) & \(-150\) m/s \\ \(r_{y0}\) & \(5\times 10^{2}\) m & \(v_{y0}\) & \(-30\) m/s \\ \(r_{z0}\) & \(5\times 10^{2}\) m & \(v_{z0}\) & 30 m/s \\ \(m_{0}\) & \(3.8\times 10^{4}\) kg & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Setting of initial state variables.  
Figure 6: The relationship between terminal mass and terminal time.  
that the \(t_{\text{opt}}\) and \(t_{\text{cr}}\) are very close, so choosing an inappropriate \(t_{\text{f}}\) can easily lead to an infeasible solution.  
Many scholars used the one-dimensional search and SCvx method to solve the \(t_{\text{opt}}\), which have achieved good results. However the two methods both need to solve the SOCP many times, and the number of iterations is related to the initial selection of \(t_{\text{f}}\). Since the number of iterations cannot be predicted in advance when running onboard, and the iteration significantly increases the computation cost, the above iteration-based method limits the algorithm's efficiency in Section 3.1.