# Low-Thrust Transfer to Quasi-Synchronous Martian Elliptic Orbit via Nonlinear Feedback Control
## 3 Nonlinear Orbit Control
### Feedback Law and Related Stability Analysis  
This section uses the Lyapunov direct method to identify a feedback control law aimed at driving the spacecraft toward the desired orbit injection conditions. Orbital motion is governed by Equations (2), (3), and (5). In particular, the non-Keplerian acceleration is written as the sum of thrust acceleration and perturbing acceleration, and Equation (2) becomes  
\[\dot{\mathbf{z}}=\mathrm{G}(\mathbf{z},\mathbf{x}_{6})\left(\frac{\mathbf{u}_{T}}{\mathbf{x}_{7}}+ \mathbf{a}_{P}\right) \tag{28}\]  
For systems governed by Equation (28) with \(\mathbf{a}_{P}=\mathbf{0}\), the Jurdjevic-Quinn theorem [15; 21] provides a feedback control law that drives the dynamical system to an arbitrary target state, making the controlled system Lyapunov-stable. On the other hand, if orbit perturbations are included in the model, a different approach can be adopted. A candidate Lyapunov function \(V\) is introduced as  
\[V=\frac{1}{2}\mathbf{\psi}^{T}\mathrm{K}\mathbf{\psi} \tag{29}\]  
where \(\mathrm{K}\) is a symmetric positive definite constant matrix. It is convenient to choose it as diagonal with positive elements that play the role of weights, to properly select in order to achieve the desired performance. It is straightforward to recognize that \(V>0\) unless \(\mathbf{\psi}=\mathbf{0}\). However, to be a Lyapunov function, \(V\) must have a nonpositive time derivative, and this can be ensured through proper selection of the control action \(\mathbf{u}_{T}\). The \((3\times 1)\)-vectors \(\mathbf{b}\) and \(\mathbf{d}\) are introduced as  
\[\mathbf{d}:=\mathbf{a}_{P}+\left[\frac{\partial\mathbf{\psi}}{\partial\mathbf{z}}\mathrm{G} \right]^{-1}\frac{\partial\mathbf{\psi}}{\partial t} \tag{30}\]  
\[\mathbf{b}:=\mathrm{G}^{T}\left(\frac{\partial\mathbf{\psi}}{\partial\mathbf{z}}\right)^{T }\mathrm{K}\mathbf{\psi} \tag{31}\]  
Two propositions, proven in ref. [14], establish the conditions for \(V\) to be a Lyapunov function. Proposition 1: _If \(\mathbf{\psi}\) and \((\partial\mathbf{\psi}/\partial\mathbf{z})\) are continuous, \(\left[(\partial\mathbf{\psi}/\partial\mathbf{z})\mathrm{G}\right]^{-1}(\partial\mathbf{ \psi}/\partial t)\) is finite, \(|\mathbf{b}|>0\) unless \(\mathbf{\psi}=\mathbf{0}\), and \(u_{T}^{(max)}\geq\mathbf{x}_{7}|\mathbf{b}+\mathbf{d}|\), then the feedback control law_  
\[\mathbf{u}_{T}=-\mathbf{x}_{7}(\mathbf{b}+\mathbf{d}) \tag{32}\]  
_leads the dynamical system governed by Equations (2), (3), and (5) to converge asymptotically to the target set \(\mathbf{\psi}=\mathbf{0}\)._  
The previous proposition includes the hypothesis \(u_{T}^{(max)}\geq\mathbf{x}_{7}|\mathbf{b}+\mathbf{d}|\). If this condition is violated, the feedback control law is not feasible because \(|\mathbf{u}_{T}|=\mathbf{x}_{7}|\mathbf{b}+\mathbf{d}|\) would exceed the maximal value \(u_{T}^{(max)}\), i.e., the propulsive capability of the system. For this reason, when this occurs, an alternative saturated feedback law can be used. Proposition 2: _If \(\mathbf{\psi}\) and \((\partial\mathbf{\psi}/\partial\mathbf{z})\) are continuous, \(\left[(\partial\mathbf{\psi}/\partial\mathbf{z})\mathrm{G}\right]^{-1}(\partial\mathbf{ \psi}/\partial t)\) is finite, \(|\mathbf{b}|>0\) unless \(\mathbf{\psi}=\mathbf{0}\), and \(\mathbf{x}_{7}|\mathbf{d}|<u_{T}^{(max)}<\mathbf{x}_{7}|\mathbf{b}+\mathbf{d}|\), then the feedback control law_  
\[\mathbf{u}_{T}=-u_{T}\frac{\mathbf{b}+\mathbf{d}}{|\mathbf{b}+\mathbf{d}|},\;\text{with}\;\left\{ \begin{array}{ll}(\mathrm{a})\;u_{T}=0,&\text{if}\;\mathbf{b}^{T}\mathbf{b}+\mathbf{b}^ {T}\mathbf{d}<0\\ (\mathrm{b})\;x_{7}|\mathbf{d}|<u_{T}\leq u_{T}^{(max)},&\text{if}\;\mathbf{b}^{T} \mathbf{b}+\mathbf{b}^{T}\mathbf{d}\geq 0\end{array} \right. \tag{33}\]  
_leads a dynamical system governed by Equations (2), (3), and (5) to converge asymptotically to the target set \(\mathbf{\psi}=\mathbf{0}\)._In Equation (33) it is convenient to choose \(u_{T}=u_{T}^{(max)}\) because this corresponds to the least value of \(\dot{V}\).