Locating Extremal Periodic Orbits for the Planar Circular Restricted Three Body Problem using Polynomial Sum-of-Squares Optimization

Abstract

With an increasing interest in the design of long and complex space missions, the search for orbits that require the least amount of fuel is of fundamental interest. This paper develops existing computational models for locating Unstable Periodic Orbits (UPOs) in polynomial dynamical systems using Sum-of-Squares (SOS) optimization technique and proposes a numerical framework to converge UPOs for the Planar Circular Restricted Three-Body Problem (PCR3BP) in astrodynamics. This is done by developing the polynomial SOS optimization technique with extension to systems with non-polynomial and Hamiltonian dynamics. First, we demonstrate and exploit the dependency of convergence of tight bounds on an observable of interest with varying scaling factors for large polynomial degrees. SOS optimization is then used to compute nonnegative polynomials, the minimization sublevel sets of which, approximately localise parts of the corresponding UPO. Improvements in current non-linear optimization techniques are suggested to compute a large number of points inside the relevant sublevel sets. Such points provide good initial conditions for UPO computations with existing algorithms. The distinguishing feature of such UPOs is that they optimize the long-time average of an input observable of interest which is a function of state variables. For the PCR3BP this means that such orbits in space can be traversed indefinitely in time without continuous fuel expenditure. As practical applications to space mission designs, we converge UPOs that minimise transmitted power required by satellites for the Earth-Moon system in a communication relay problem by minimizing the infinite-time average of sum of squares of distances of a satellite from Earth and the Moon.