# Low-Thrust Nonlinear Orbit Control for Very Low Lunar Orbits
## 3 Nonlinear Orbit Control
### Feedback Control Law
If we evaluate this expression when \(\mathbf{a}=\mathbf{0}\), we obtain  
\[\dot{b}_{i}=\frac{\partial b_{i}}{\partial x_{6}}\sqrt{\frac{\mu_{m}}{x_{1}^{ 3}}}\eta^{2}+\sum_{j=1}^{5}\frac{\partial b_{i}}{\partial x_{jd}}\dot{x}_{jd}. \tag{51}\]  
For simplicity, let us rename  
\[b_{1}=\Psi_{2}(x_{2},x_{2d},x_{3},x_{3d})\sqrt{\frac{x_{1}}{\mu_{m}}}f_{1}(x _{2},x_{2d},x_{3},x_{3d},x_{6}), \tag{52}\]  
\[b_{2}=\Psi_{1}(x_{1},x_{1d})\sqrt{\frac{x_{1}^{3}}{\mu_{m}}}f_{2}(x_{1},x_{1d },x_{2},x_{3},x_{6})\]  
\[b_{2}=4k_{1}\Psi_{1}\frac{x_{1}-x_{1d}}{\eta}\sqrt{\frac{x_{1}^{3}}{\mu_{m}}}+\]0  
\[b_{2}=4k_{1}\Psi_{1}\frac{x_{1}-x_{1d}}{\eta}\sqrt{\frac{x_{1}^{3}}{\mu_{m}}}+\]1  
\[b_{2}=4k_{1}\Psi_{1}\frac{x_{1}-x_{1d}}{\eta}\sqrt{\frac{x_{1}^{3}}{\mu_{m}}}+\]2  
where  
\[b_{2}=4k_{1}\Psi_{1}\frac{x_{1}-x_{1d}}{\eta}\sqrt{\frac{x_{1}^{3}}{\mu_{m}}}+\]3  
\[b_{2}=4k_{1}\Psi_{1}\frac{x_{1}-x_{1d}}{\eta}\sqrt{\frac{x_{1}^{3}}{\mu_{m}}}+\]4  
\[b_{2}=4k_{1}\Psi_{1}\frac{x_{1}-x_{1d}}{\eta}\sqrt{\frac{x_{1}^{3}}{\mu_{m}}}+\]5  
\[b_{2}=4k_{1}\Psi_{1}\frac{x_{1}-x_{1d}}{\eta}\sqrt{\frac{x_{1}^{3}}{\mu_{m}}}+\]6  
\[b_{2}=4k_{1}\Psi_{1}\frac{x_{1}-x_{1d}}{\eta}\sqrt{\frac{x_{1}^{3}}{\mu_{m}}}+\]7  
Therefore,\[b_{2}=4k_{1}\Psi_{1}\frac{x_{1}-x_{1d}}{\eta}\sqrt{\frac{x_{1}^{3}}{\mu_{m}}}+\]8 \[b_{2}=4k_{1}\Psi_{1}\frac{x_{1}-x_{1d}}{\eta}\sqrt{\frac{x_{1}^{3}}{\mu_{m}}}+\]9 \[2k_{2}\sqrt{\frac{x_{1}}{\mu_{m}}}\frac{\Psi_{2}}{\eta}\Big{(}(x_{2}-x_{2d})[x _{2}+(1+\eta)\cos x_{6}]+(x_{3}-x_{3d})[x_{3}+(1+\eta)\sin x_{6}]\Big{)}, \tag{47}\]0 (61) \[2k_{2}\sqrt{\frac{x_{1}}{\mu_{m}}}\frac{\Psi_{2}}{\eta}\Big{(}(x_{2}-x_{2d})[x _{2}+(1+\eta)\cos x_{6}]+(x_{3}-x_{3d})[x_{3}+(1+\eta)\sin x_{6}]\Big{)}, \tag{47}\]1 \[2k_{2}\sqrt{\frac{x_{1}}{\mu_{m}}}\frac{\Psi_{2}}{\eta}\Big{(}(x_{2}-x_{2d})[x _{2}+(1+\eta)\cos x_{6}]+(x_{3}-x_{3d})[x_{3}+(1+\eta)\sin x_{6}]\Big{)}, \tag{47}\]2 \[2k_{2}\sqrt{\frac{x_{1}}{\mu_{m}}}\frac{\Psi_{2}}{\eta}\Big{(}(x_{2}-x_{2d})[x _{2}+(1+\eta)\cos x_{6}]+(x_{3}-x_{3d})[x_{3}+(1+\eta)\sin x_{6}]\Big{)}, \tag{47}\]3  
It is evident that, as investigated in [23], the subset associated with the rectilinear trajectories can be ruled out as the time derivative \(\dot{b}\) diverges. Instead, it is straightforward to see that, when the target set is reached, the following relations hold:  
\[2k_{2}\sqrt{\frac{x_{1}}{\mu_{m}}}\frac{\Psi_{2}}{\eta}\Big{(}(x_{2}-x_{2d})[x _{2}+(1+\eta)\cos x_{6}]+(x_{3}-x_{3d})[x_{3}+(1+\eta)\sin x_{6}]\Big{)}, \tag{47}\]4  
and therefore, \(\dot{b}=\mathbf{0}\) as well. Since the invariant set does not contain other subsets than the target set, the asymptotic stability is quasi-global. Finally, the quantity \(\Big{[}(\partial\mathbf{\Psi}/\partial x)\ \mathbf{G}\Big{]}^{-1}( \partial\mathbf{\Psi}/\partial t)\) must be investigated to check any singularity issue. Let us denote with \((\overline{d}_{1},\overline{d}_{2},\overline{d}_{3})\) the components of vector \(\boldsymbol{d}-\boldsymbol{a}_{P}\). First of all, component \(\overline{d}_{2}\) is inspected. Using Mathematica [29], one can prove that  
\[2k_{2}\sqrt{\frac{x_{1}}{\mu_{m}}}\frac{\Psi_{2}}{\eta}\Big{(}(x_{2}-x_{2d})[x _{2}+(1+\eta)\cos x_{6}]+(x_{3}-x_{3d})[x_{3}+(1+\eta)\sin x_{6}]\Big{)}, \tag{47}\]5  
which is finite if and only if \(x_{1d}+0\).