# 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, Propositions 1 and 2 lead to defining a feedback control law that identifies direction and magnitude of the thrust acceleration at each instant. This law can be written in compact form as  
\[\mathbf{u}_{T}=-u_{T}\frac{\mathbf{b}+\mathbf{d}}{|\mathbf{b}+\mathbf{d}|},\ \text{with}\ u_{T}= \left\{ \begin{array}{ll}x_{\gamma}|\mathbf{b}+\mathbf{d}|,&\text{if}\ x_{\gamma}|\mathbf{b}+\mathbf{d}| \leq u_{T}^{(max)}\\ 0,&\text{if}\ \ x_{\gamma}|\mathbf{b}+\mathbf{d}|>u_{T}^{(max)}\quad\text{and}\ \ \mathbf{b}^{T}\mathbf{b}+\mathbf{b}^{T}\mathbf{d}<0\\ u_{T}^{(max)},&\text{if}\ \ x_{\gamma}|\mathbf{b}+\mathbf{d}|>u_{T}^{(max)}\quad\text{and}\ \ \mathbf{b}^{T}\mathbf{b}+\mathbf{b}^{T}\mathbf{d}\geq 0 \end{array} \right. \tag{34}\]  
when the condition \(x_{\gamma}|\mathbf{d}|<u_{T}^{(max)}\) is violated, the thrust is turned off, i.e., \(\mathbf{u}_{T}=\mathbf{0}\). It is worth remarking that Propositions 1 and 2 provide some sufficient conditions for stabilizing the dynamical system of interest. This circumstance implies that the assumptions of Propositions 1 and 2 can be violated (in some time intervals) without necessarily compromising asymptotic convergence to the desired final condition. To complete the stability analysis, the expressions for the components of \(\mathbf{b}\) must be found. After several analytical steps, omitted for the sake of brevity, one obtains  
\[b_{1}=2k_{2}\sqrt{\frac{x_{1}}{\mu}}\{[x_{2}-e_{d}\cos(\omega_{d}+\Omega_{d})] \sin x_{6}-[x_{3}-e_{d}\sin(\omega_{d}+\Omega_{d})]\cos x_{6}\}\psi_{2} \tag{35}\]  
\[ \begin{array}{l}b_{2}=2\frac{k_{1}}{\eta}x_{1}\sqrt{\frac{x_{1}}{\mu}}\psi_{ 1}+2\frac{k_{2}}{\eta}\sqrt{\frac{x_{1}}{\mu}}\{[x_{2}-e_{d}\cos(\omega_{d}+ \Omega_{d})][x_{2}+(1+\eta)\cos x_{6}]\\ \quad\quad\quad+[x_{3}-e_{d}\sin(\omega_{d}+\Omega_{d})][x_{3}+(1+\eta)\sin x _{6}]\}\psi_{2}\end{array}  \tag{36}\]  
\[ \begin{array}{l}b_{3}=-2e_{d}\frac{k_{2}}{\eta}\sqrt{\frac{x_{1}}{\mu}}[x_{ 3}\cos(\omega_{d}+\Omega_{d})-x_{2}\sin(\omega_{d}+\Omega_{d})](x_{5}\cos x_{ 6}-x_{4}\sin x_{6})\psi_{2}\\ \quad\quad\quad-\frac{k_{3}}{\eta}\sqrt{\frac{x_{1}}{\mu}}(x_{4}\cos x_{6}+x_ {5}\sin x_{6})(1+\cos i_{d})\big{(}x_{4}^{2}+x_{5}^{2}+1\big{)}\psi_{3}\\ \quad\quad\quad+\frac{k_{3}}{\eta}\sqrt{\frac{x_{1}}{\mu}}\sin i_{d}(\sin \Omega_{d}\sin x_{6}+\cos\Omega_{d}\cos x_{6})\big{(}x_{4}^{2}+x_{5}^{2}+1 \big{)}\psi_{3}\end{array}  \tag{37}\]  
The attracting set collects all the dynamical states where \(\dot{V}=0\). This condition is met if \(\mathbf{b}=\mathbf{0}\), i.e., if all the three components \(\{b_{1},\ b_{2},\ b_{3}\}\) vanish, for any choice of the positive coefficients \(\{k_{1},\ k_{2},\ k_{3}\}\). While looking for conditions for states related to the attracting set, one must rule out those depending on \(x_{6}\), which is time-varying (also along the desired orbit). Clearly, if \(x_{1}=0\), then \(b_{1}=b_{2}=b_{3}=0\). Therefore, the attracting set certainly contains rectilinear trajectories (\(p=0\)). It is straightforward to recognize that \(\mathbf{b}=\mathbf{0}\) also in the target set \(\mathbf{\psi}=\mathbf{0}\). Moreover, \(b_{1}=0\) regardless of \(x_{6}\) if \((x_{2}-e_{d}\cos(\omega_{d}+\Omega_{d})=0)\) and \((x_{3}-e_{d}\sin(\omega_{d}+\Omega_{d})=0)\), whose fulfillment is equivalent to \(\psi_{2}=0\). Moreover, the preceding two equalities also imply \((x_{3}\cos(\omega_{d}+\Omega_{d})-x_{2}\sin(\omega_{d}+\Omega_{d})=0\) in Equation (37). In the latter equation, the remaining term  
\[(x_{4}\cos x_{6}+x_{5}\sin x_{6})(1+\cos i_{d})-\sin i_{d}(\sin\Omega_{d}\sin x _{6}+\cos\Omega_{d}\cos x_{6}) \tag{38}\]  
can be rewritten as  
\[\tan\frac{i}{2}\cos(\Omega-x_{6})-\tan\frac{i_{d}}{2}\cos(\Omega_{d}-x_{6}) \tag{39}\]  
and is equal to zero, regardless of \(x_{6}\), only if \(i=i_{d}\) and \(\Omega=\Omega_{d}\), i.e., \(\psi_{3}=0\).