Theory of Functional Connections and Nelder-Mead optimization methods in satellite characterization

The growing population of man-made objects with the build up of mega-constellations and the multiplication of space debris not only increases the potential danger to all space vehicles and in-space infrastructures (including space observatories), but also poses a serious threat to astronomy and dark skies. Monitoring of this population requires precise satellite characterization, which is a challenging task that involves analyzing observational data such as position, velocity, and light curves using optimization methods. In this study, the applications of two optimization procedures are proposed and analyzed to determine the parameters associated with the dynamics of a satellite: one based on the Theory of Functional Connections (TFC) and another one based on the Nelder-Mead heuristic optimization algorithm. The TFC performs linear functional interpolation to embed the constraints of the problem into a functional. We propose to use this functional to analytically embed the observational data of a satellite into its equations of dynamics. After that, any solution will always satisfy the observational data. The second proposed procedure takes advantage of the Nelder-Mead algorithm, that does not require the gradient of the objective function, as alternative solution. The accuracy, efficiency, and dependency on the initial guess of each method is investigated, analyzed, and compared for several dynamical models. These methods can be used to obtain the physical parameters of a satellite from available observational data and for space debris characterization contributing to follow-up monitoring activities in space and astronomical observatories.