## 5 Conclusions  
In this paper, the performance of an interplanetary spacecraft equipped with a diffractive sail in a phasing maneuver along an elliptical orbit with an assigned semimajor axis and eccentricity has been studied. The mission scenario hypothesized is two-dimensional,and it was assumed that the diffractive sail nominal plane during the flight was kept perpendicular to the Sun-spacecraft line. The sail membrane is covered by a set of electro-optically controlled panels that allow the transverse component of the thrust vector to be reversed simply by applying an adequate electrical voltage. This specific aspect reduces the control terms to a single binary variable and, consequently, facilitates the design of the control law, which, in this work, has been obtained by solving an appropriate optimization problem that allows for minimizing the flight time for a given phasing angle. The optimal control law used is simply related to that of the adjoint variable and is consistent with the results of recent literature regarding this specific type of Sun-facing diffractive sails. Numerical simulations were performed considering a low-performance sail, i.e., assuming a characteristic acceleration one order of magnitude lower than the value used in recent literature results. This value of the characteristic acceleration, which is consistent with the value used in the design of the most recent interplanetary spacecraft equipped with a reflective solar sail, allows us to have an idea of the performance achievable by such an advanced propulsion system in this specific mission scenario considering the technology level hopefully achievable in the near future. In this regard, considering the heliocentric orbit of the Earth, simulation results indicate that a low-performance diffractive sail is able to complete a phasing maneuver with a phasing angle of the order of a few tens of degrees with a flight time of a few hundred days, with a substantial reduction in time in case the phasing is performed behind instead of ahead of the planet. The proposed procedure has also been applied to a Mercury-based mission scenario, where the reference heliocentric orbit has a non-negligible eccentricity, and to the limiting (but important) case of a circular orbit. This last scenario allowed for designing an extremely simple control law, whose performances have been determined by resorting to linearized dynamics of the spacecraft. Although not optimal, such a simplified control law allows for determining potential phasing performances (for circular orbit) using a periodic time variation of the single binary control variable. The concept underlying this simplified control law could be the starting point of a first possible extension of the model discussed in this paper. Such a possible extension could consider linearized dynamics of the spacecraft around an elliptical heliocentric orbit, in which the vehicle's trajectory around the Sun during the maneuver remains close to the reference elliptical orbit. A second, potential continuation of this work, instead, would concern the use of the thrust model recently proposed in the literature capable of modeling the behavior of a diffractive sail with liquid crystal optical arrays. In that case, not only the direction of the transverse thrust component could be changed by varying the electric voltage, but also the thrust vector module itself could be controlled, within certain limits, in order to expand the steering capabilities of a diffractive sail with a Sun-facing attitude. This interesting innovative feature could improve the performance of the diffractive sail in a wide range of mission scenarios. This research received no external funding. Not applicable. The original contributions presented in the study are included in the article. Further inquiries can be directed to the corresponding author. The author declares that he has not used any kind of generative artificial intelligence in the preparation of this manuscript, nor in the creation of images, graphs, tables, or related captions.