# 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
\begin{split}&\frac{\mathrm{d}\phi_{i}}{\mathrm{d}t}=\frac{ \partial\phi_{i}}{\partial\mathbf{X}_{i}^{\mathrm{T}}}\left(\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}\right)+\frac{ \partial\phi_{i}}{\partial\mathbf{X}_{t}^{\mathrm{T}}}\bigg{(}\frac{\mathrm{ d}\mathbf{X}_{t}}{\mathrm{d}t}\bigg{)}_{two-body}\\ &=\frac{\partial\phi_{i}}{\partial\mathbf{X}_{i}^{\mathrm{T}}} \bigg{(}\bigg{(}\frac{\mathrm{d}\mathbf{X}_{i}}{\mathrm{d}t}\bigg{)}_{two- body}-\frac{\partial\mathbf{X}_{i}}{\partial\mathbf{V}_{i}^{\mathrm{T}}} \frac{\partial\mathcal{X}_{i}^{\mathrm{T}}}{\partial\mathbf{V}_{i}}\frac{ \partial\phi_{i}}{\partial\mathbf{X}_{i}}\bigg{)}+\frac{\partial\phi_{i}}{ \partial\mathbf{X}_{t}^{\mathrm{T}}}\bigg{(}\frac{\mathrm{d}\mathbf{X}_{t}}{ \mathrm{d}t}\bigg{)}_{two-body}\\ &=-\Bigg{(}\frac{\partial\mathbf{X}_{i}^{\mathrm{T}}}{\partial\mathbf{V}_{i }}\frac{\partial\phi_{i}}{\partial\mathbf{X}_{i}}\Bigg{)}^{\mathrm{T}}\bigg{(} \frac{\partial\mathbf{X}_{i}^{\mathrm{T}}}{\partial\mathbf{V}_{i}}\frac{ \partial\phi_{i}}{\partial\mathbf{X}_{i}}\bigg{)}+\frac{\partial\phi_{i}}{ \partial\mathbf{X}_{t}^{\mathrm{T}}}\bigg{(}\frac{\mathrm{d}\mathbf{X}_{i}}{ \mathrm{d}t}\bigg{)}_{two-body}+\frac{\partial\phi_{i}}{\partial\mathbf{X}_{t }^{\mathrm{T}}}\bigg{(}\frac{\mathrm{d}\mathbf{X}_{t}}{\mathrm{d}t}\bigg{)}_{ two-body}\\ &=-4\Bigg{[}\frac{\partial\mathbf{X}_{i}^{\mathrm{T}}}{\partial\mathbf{V}_{i }}\mathbf{K}(\mathbf{X}_{i}-\mathbf{X}_{t})\Bigg{]}^{\mathrm{T}}\bigg{[} \frac{\partial\mathbf{X}_{i}^{\mathrm{T}}}{\partial\mathbf{V}_{i}}\mathbf{K}( \mathbf{X}_{i}-\mathbf{X}_{t})\bigg{]}+2(\mathbf{X}_{i}-\mathbf{X}_{t})^{ \mathrm{T}}\mathbf{K}\bigg{(}\frac{\mathrm{d}\mathbf{X}_{i}}{\mathrm{d}t}- \frac{\mathrm{d}\mathbf{X}_{t}}{\mathrm{d}t}\bigg{)}_{two-body}\end{split}  \tag{49}\]  
For RAAN control and semi-major axis control, \(\mathbf{X}_{i}=\left[\sin i_{i}\sin\Omega_{i},\sin i_{i}\cos\Omega_{i},a_{i} \right]^{\mathrm{T}}\) stays constant under the two-body Keplerian motion condition without consideration of perturbations. Hence, \(\big{(}\mathrm{d}\mathbf{X}_{i}/\mathrm{d}t\big{)}_{two-body}\) is zero. Similarly, \(\big{(}\mathrm{d}\mathbf{X}_{t}/\mathrm{d}t\big{)}_{two-body}\) is also zero. The time derivative of \(\phi_{i}\) is  
\[\frac{\mathrm{d}\phi_{i}}{\mathrm{d}t}=-4\Bigg{[}\frac{\partial\mathbf{X}_{i}^{ \mathrm{T}}}{\partial\mathbf{V}_{i}}\mathbf{K}(\mathbf{X}_{i}-\mathbf{X}_{t}) \Bigg{]}^{\mathrm{T}}\Bigg{[}\frac{\partial\mathbf{X}_{i}^{\mathrm{T}}}{ \partial\mathbf{V}_{i}}\mathbf{K}(\mathbf{X}_{i}-\mathbf{X}_{t})\Bigg{]} \tag{50}\]  
The control coefficient matrix \(\mathbf{K}\) is positive definite. Therefore, Equation (50) is a negative definite quadratic function when \(\mathbf{X}_{i}-\mathbf{X}_{t}\
eq 0\). \(\phi_{i}\) is positive definite, and the time derivative of \(\phi_{i}\) is negative definite. According to [PERSON]'s second theory, the RAAN and semi-major axis APFs are stable.