## 1 Introduction  
Small satellite formation missions are widely applied for various important problems and tasks in near-Earth orbits. A number of satellites flying at short relative distances can serve as a distributed instrument, for example, for ionosphere and magnetosphere sampling [1], stereo remote sensing [2], and astrophysical research [3]. Now that Starlink and OneWeb mega constellations are launching satellites in large numbers, satellite flocks have been observed flying over the night sky in different parts of the world, visible to the naked eye [4]. The sunlight reflected by the surface of satellites moving near the terminator line is bright enough to see the flying line of dots. This effect underlies the idea of launching a set of microsatellites equipped with sunlight reflectors to obtain a relative configuration that forms an image visible to an observer on the Earth. Such a formation flying mission can thus function as a space media broadcasting logos or short messages. A recent paper [5] discusses the commercial viability of operating such formations. Surprisingly, the problem of designing a space mission to demonstrate pixel images in the night sky has already been considered in a very practical sense. At least two space advertising projects were discussed in the 20 th century. A string of 100 reflectors to form a ring of light, visible throughout the world, could be sent into orbit in 1989 to mark the centennial of the Eiffel Tower [6]. Then, in the 1990s, the city of Atlanta investigated a Space Billboard concept for the 1996 Olympics [7]. As we have already mentioned, the economic viability of space advertising missions was studied in [5] and was found impossible without reactive propulsion systems for frequent image reconfigurations. However, single-event missions such as the Eiffel Tower anniversary or the Olympics opening ceremony which only require a single demonstration might use the control strategy described in this paper, which allows for the omission of the propulsion system from the mission. Another practical aspect of the problem discussed in this paper is the problem statement itself. We believe that the problem of a formation flying mission to form pixel images can be made a benchmark test for any formation control algorithms, because of very clear visibility requirements, which need to be met and can be met on the up-to-date level of technology. A preliminary feasibility study [5] showed that a formation to meet the visibility requirements under suitable lighting conditions launched at low Earth Sun-synchronous orbits of 400 km can be formed using 12U CubeSats with 2 \(\times\) 2 m\({}^{2}\) sails. The minimum distance between any two reflectors should be greater than 600 m to make the two points distinguishable by the human eye. If the attitude of the image is fixed in the orbital reference frame, it requires continuous control using onboard thrusters and leads to excessive fuel consumption. Hence, the preferred approach is to let the image rotate in the orbital reference frame with the orbital period. Ideally, each satellite can be assigned initial conditions that result in a circle motion when projected onto the plane perpendicular to nadir, so that each \"pixel\" in the image rotates with the same angular velocity. Relative motion control, however, is required to achieve the selected relative trajectories. A hybrid impulsive-continuous control scheme allowing to converge to the required trajectory using continuous thrust was proposed in [8]. This approach relies on a propulsion system and becomes inappropriate if any of the satellites run out of fuel. The recent paper [9] proposes the new transforming equations using topocentric coordinate system allowing to achieve arbitrary advertisement shape, and stable orbits that keep the formation shape without control are found. An approach proposed in this paper is based on using aerodynamic forces, which is acceptable for orbits of 400 km and lower. The sunlight reflectors can function as drag sails.