# Self-Organizing Control of Mega Constellations for Continuous Earth Observation
## 2 Materials and Methods
### Stability Proof of Satellite Control
#### 2.3.1 Stability Proof of RAAN Control and Semi-Major Axis Control  
If there exists one Lyapunov function \(\phi\) that satisfies Equation (44), the control system can be proven to be stable. \[\left\{ \begin{array}{l}\phi(\mathbf{0})=0,\\ \phi(\mathbf{X})>0,\forall\mathbf{X}\
eq\mathbf{0}\\ \phi(\mathbf{0})=0,\\ \phi(\mathbf{X})\leq 0,\phi(\mathbf{X})\
ot\equiv 0,\forall\mathbf{X}\
eq \mathbf{0}\end{array} \right. \tag{44}\]  
In this paper, \(\phi_{i}\) is designated as the Lyapunov function. Referring to Equation (31), it is obvious that \(\phi_{i}\) is positive definite. If the time derivative of \(\phi_{i}\) is negative definite, one can prove that the proposed APF control is stable. For RAAN control and semi-major axis control, \(\mathbf{X}=\mathbf{X}_{i}-\mathbf{X}_{\mathrm{t}}=[\sin i_{i}\sin\Omega_{i}- \sin i_{\mathrm{t}}\sin\Omega_{\mathrm{t}},\sin i_{i}\cos\Omega_{i}-\sin i_{ \mathrm{t}}\cos\Omega_{\mathrm{t}},a_{i}-a_{\mathrm{t}}]^{\mathrm{T}}\). Hence, \(\mathbf{X}\) relates to controlled satellite \(i\) only. According to Equation (31), the time derivative of \(\phi_{i}\) can be written as  
\[\frac{\mathrm{d}\phi_{i}}{\mathrm{d}t}=\frac{\partial\phi_{i}}{\partial \mathbf{X}_{i}^{\mathrm{T}}}\frac{\mathrm{d}\mathbf{X}_{i}^{\mathrm{T}}}{ \mathrm{d}t}+\frac{\partial\phi_{i}}{\partial\mathbf{X}_{\mathrm{t}}^{\mathrm{ T}}}\frac{\mathrm{d}\mathbf{X}_{\mathrm{t}}^{\mathrm{T}}}{\mathrm{d}t} \tag{45}\]  
The time derivative of \(\mathbf{X}_{i}\) is  
\[\frac{\mathrm{d}\mathbf{X}_{i}}{\mathrm{d}t} =\frac{\partial\mathbf{X}_{i}}{\partial\mathbf{R}_{i}^{\mathrm{ T}}}\frac{\mathrm{d}\mathbf{R}_{i}}{\mathrm{d}t}+\frac{\partial\mathbf{X}_{i}}{ \partial\mathbf{V}_{i}^{\mathrm{T}}}\frac{\mathrm{d}\mathbf{V}_{i}}{\mathrm{d}t} \tag{46}\] \[=\frac{\partial\mathbf{X}_{i}}{\partial\mathbf{R}_{i}^{\mathrm{T}} }\frac{\mathrm{d}\mathbf{R}_{i}}{\mathrm{d}t}+\frac{\partial\mathbf{X}_{i}}{ \partial\mathbf{V}_{i}^{\mathrm{T}}}\left(-\mu\frac{\mathbf{R}_{i}}{R_{i}^{3}} +\mathbf{u}\right)\] \[=\left(\frac{\mathrm{d}\mathbf{X}_{i}}{\mathrm{d}t}\right)_{two- body}+\frac{\partial\mathbf{X}_{i}}{\partial\mathbf{V}_{i}^{\mathrm{T}}}\mathbf{u}\]  
where \(\mathbf{R}_{i}\) and \(\mathbf{V}_{i}\) are the position vector and velocity vector in the geocentric inertial frame, respectively. \(\mathbf{u}\) represents the control acceleration vector in the geocentric inertial frame. \(\left(\mathrm{d}\mathbf{X}_{i}/\mathrm{d}t\right)_{two-body}\) represents the time derivative of \(\mathbf{X}_{i}\) under the two-body Keplerian motion condition. Since \(\mathbf{X}_{\mathrm{t}}\) does not change under control, the time derivative of \(\mathbf{X}_{\mathrm{t}}\) is  
\[\frac{\mathrm{d}\mathbf{X}_{\mathrm{t}}}{\mathrm{d}t}=\left(\frac{\mathrm{d} \mathbf{X}_{\mathrm{t}}}{\mathrm{d}t}\right)_{two-body} \tag{47}\]  
According to Equation (32), the control effect generated by \(\phi_{i}\) in geocentric inertial frame is the negative gradient of \(\phi_{i}\) with respect to \(\mathbf{V}_{i}=\left[\hat{X},\hat{Y},\hat{Z}\right]^{\mathrm{T}}\)\[\mathbf{u}=-\frac{\partial\mathbf{X}_{i}^{\mathrm{T}}}{\partial\mathbf{V}_{i}} \frac{\partial\phi_{i}}{\partial\mathbf{X}_{i}}=-\left[ \begin{array}{cccc} \frac{\partial X_{1}}{\partial\bar{X}}&\frac{\partial X_{2}}{\partial\bar{X}}& \cdots&\frac{\partial X_{n}}{\partial\bar{X}}\\ \frac{\partial X_{1}}{\partial\bar{Y}}&\frac{\partial X_{2}}{\partial\bar{Y}}& \cdots&\frac{\partial X_{n}}{\partial\bar{Y}}\\ \frac{\partial X_{1}}{\partial\bar{Z}}&\frac{\partial X_{2}}{\partial\bar{Z}}& \cdots&\frac{\partial X_{n}}{\partial\bar{Z}}\\ \end{array} \right]\left[ \begin{array}{c}\frac{\partial\phi_{i}}{\partial X_{ 1}}\\ \frac{\partial\phi_{i}}{\partial X_{2}}\\ \vdots\\ \frac{\partial\phi_{i}}{\partial X_{n}}\\ \end{array} \right] \tag{48}\]  
Substituting Equations (46)-(48) into Equation (45), the time derivative of \(\phi_{i}\) can be rewritten as  
\[