## 2 Problem Statement and Equations of Motion  
A number of small satellites after a cluster launch in LEO are considered. All satellites are equipped with flat sunlight reflectors fixed to the satellite body. An active attitude control system is used to achieve the required angular position relative to the Sun's direction or to the incoming airflow. Sunlight is reflected to the observer's position on the Earth's surface under the required attitude of the reflector relative to the Sun's direction, and it can be visible to the naked eye in the night sky (see Figure 1). Meanwhile, the sunlight reflector is used as an aerodynamic wing for relative motion control in low-Earth orbits. A number of sunlight reflectors form a specified image, functioning like pixels. It is assumed that all the satellites are provided with information about the relative motion of all the other satellites, i.e., the relative motion is assumed to be known. The main goal of the aerodynamic-force-based relative motion control is to achieve the required configuration to compose the chosen image. The linear model is used for the reference relative trajectories calculation. It can be described by Hill-Clohessy-Wiltshire equations [26], written in the local-vertical local-horizontal (LVLH) reference frame. The origin \(O\) of the LVLH reference frame moves along a circular orbit, the \(Oz\) axis is aligned along the vector from the Earth center \(O_{I}\) to \(O\), the \(Oy\) axis is directed along the orbital angular momentum vector, and the \(Ox\) axis completes the right-handed triad. The HCW equations are as follows:  
\[\left\{ \begin{array}{l}\ddot{x}_{ij}\,+\,2\omega\ddot{z}_{ij}\,=\,u_{x}^{ij}, \\ \ddot{y}_{ij}\,+\,\omega^{2}y_{ij}\,=\,u_{y}^{ij},\\ \ddot{z}_{ij}\,-\,2\omega\dot{x}_{ij}\,-\,3\omega^{2}z_{ij}\,=\,u_{z}^{ij}. \end{array} \right. \tag{1}\]  
Here \([x_{ij},y_{ij},z_{ij}]^{\mathrm{T}}=\mathbf{r}_{ij}=\mathbf{r}_{j}-\mathbf{r} _{i}\) is the difference between the positions of the \(j\)-th and \(i\)-th satellites in the LVLH reference frame, \(\omega\) is the orbital angular velocity, and \([u_{x}^{ij},u_{y}^{ij},u_{z}^{ij}]^{\mathrm{T}}=\mathbf{u}^{ij}=\mathbf{u}^{i}\) is a vector of control acceleration difference. If the control input in Equation (1) is zero, the system becomes homogeneous and its solution is:  
\[\left\{ \begin{array}{l}x_{ij}(t)=-3C_{1}^{ij}\omega t+2C_{2}^{ij}\cos(\omega t )-2C_{3}^{ij}\sin(\omega t)+C_{4}^{ij},\\ y_{ij}(t)=C_{5}^{ij}\sin(\omega t)+C_{6}^{ij}\cos(\omega t),\\ z_{ij}(t)=2C_{1}^{ij}+C_{2}^{ij}\sin(\omega t)+C_{3}^{ij}\cos(\omega t), \end{array} \right. \tag{2}\]  
Figure 1: Scheme of the mission to demonstrate images with reflected sunlight. where constants \(C_{1}^{ij}-C_{6}^{ij}\) depend on the initial conditions at time \(t_{0}=0\) as follows:  
\[ \begin{array}{l}C_{1}^{ij}=\frac{\dot{x}_{ij}(0)}{\omega}+2z_{ij}(0),\;C_{2}^{ ij}=\frac{\dot{z}_{ij}(0)}{\omega},\;C_{3}^{ij}=-2\frac{\dot{x}_{ij}(0)}{\omega}-3z_{ ij}(0),\\ C_{4}^{ij}=-2\frac{\dot{x}_{ij}(0)}{\omega}+x_{ij}(0),\;C_{5}^{ij}=\frac{y_{ij }(0)}{\omega},\;C_{6}^{ij}=y_{ij}(0).\end{array}  \tag{3}\]  
If all the constants \(C_{1}^{ij}\) are equal to zero, then all the relative trajectories are closed ellipses. For certain sets of the constant values, the relative orbits become projected circular orbits, which means that their projections onto the local horizontal plane _Oxy_ are circular. In this paper, all the reference trajectories for each satellite are considered to be projected circular orbits (PCOs). The constants \(C_{1}^{ij}-C_{6}^{ij}\) determine the size and the phase parameters of each \"pixel\" motion relative to the image center. It is known that the Hill-Clohessy-Wiltshire relative motion Equation (1) expressed in Cartesian coordinates is valid for short relative distances between the satellites and for short-term motion propagation. The comparison of the relative motion model for formation flying was carried out in [27].