# Low-Thrust Transfer to Quasi-Synchronous Martian Elliptic Orbit via Nonlinear Feedback Control
## 3 Nonlinear Orbit Control
### Feedback Law and Related Stability Analysis
In conclusion, the attracting set contains two subsets:  
1. rectilinear trajectories \(x_{1}=0\)
2. target set, i.e., \(\mathbf{\psi}=\mathbf{0}\)  
Because the attracting set contains another subset other than the target set, asymptotic convergence toward the latter is only local, based on Lyapunov stability theorem. However,the equality \(\mathbf{b}=\mathbf{0}\) can be considered again, to rule out, if possible, subset 1. The condition \(\mathbf{b}=\mathbf{0}\) implies \(\dot{\mathbf{b}}=\mathbf{0}\), i.e., \(\dot{b}_{1}=\dot{b}_{2}=\dot{b}_{3}=0\) while \(\mathbf{a}\equiv\mathbf{0}\), where  
\[\dot{b}_{j}=\frac{\partial b_{j}}{\partial x_{6}}\sqrt{\frac{\mu_{M}}{x_{1}^{3 }}}\eta^{2}+\frac{\partial b_{j}}{\partial\Omega_{d}}\dot{\Omega}(a_{d},e_{d},i_{d})\ \ \ (j=1,2,3) \tag{40}\]  
Equation (40) yields three relations. Inspection of their closed-form expressions (obtained with MATLAB symbolic toolbox [22] and not reported for the sake of conciseness) leads to ruling out subset 1, associated with rectilinear trajectories. Therefore, only subset 2 (i.e., the target set) corresponds to an equilibrium condition. Therefore, global asymptotic convergence toward the target set is demonstrated. As a final step, the vector \([(\partial\mathbf{\psi}/\partial\mathbf{z})\mathrm{G}]^{-1}(\partial\mathbf{\psi}/ \partial t)\), which appears in the definition of \(\mathbf{d}\), is to be analyzed, in order to verify the existence of possible singularity issues. Using the definitions of \(\mathbf{\psi}\) and \(\mathrm{G}\), the limit of \([(\partial\mathbf{\psi}/\partial\mathbf{z})\mathrm{G}]^{-1}(\partial\mathbf{\psi}/ \partial t)\) as \(\{i\to i_{d},\Omega\to\Omega_{d},e\to e_{d},p\to p_{d}\}\) yields three closed-form analytical expressions for the three components. The second and the third component of the vector \([(\partial\mathbf{\psi}/\partial\mathbf{z})\mathrm{G}]^{-1}(\partial\mathbf{\psi}/ \partial t)\) turn out to tend to zero, while the first component tends to  
\[\frac{e_{d}\dot{\Omega}_{d}\sin(\omega-\omega_{d})}{\sqrt{\frac{x_{1}}{\mu}}( \sin(\omega-\omega_{d}+f)-\sin(f))} \tag{41}\]  
when also the argument of periapse \(\omega\) tends to its desired value one obtains  
\[\lim_{\omega\to\omega_{d}}\frac{e_{d}\dot{\Omega}_{d}\sin(\omega-\omega_{d})} {\sqrt{\frac{x_{1}}{\mu}}(\sin(\omega-\omega_{d}+f)-\sin(f))}=\frac{e_{d} \dot{\Omega}_{d}}{\sqrt{\frac{x_{1}}{\mu}}\cos(f)} \tag{42}\]  
which is finite almost everywhere. Because \([(\partial\mathbf{\psi}/\partial\mathbf{z})\mathrm{G}]^{-1}(\partial\mathbf{\psi}/ \partial t)\) is finite almost everywhere, the feedback law (34) is feasible. This section establishes some analytical conditions for convergence toward the operational orbit while neglecting the spacecraft shadowing. The unavailability of solar illumination may limit the onboard electrical power needed to operate the low-thrust system. If this occurs, electrical power must be switched off during eclipse intervals. The feedback law can easily take this into account. This is shown in ref. [13] for low Earth orbits. The major effect is an occasional and temporary violation of the boundary conditions along eclipse arcs. However, in the present research, the shadowing effect on the available electrical power is neglected, also because the initial orbit has a very high apoapse radius (while the final orbit is also highly eccentric), and the eclipse intervals seldom occur as a result.