Optimization Over Families of Quasi-Periodic Orbits

Quasi-periodic orbit families in astrodynamics are usually studied from a global standpoint without much attention to the specific orbits which are computed. Instead, we focus on the computation of particular quasi-periodic orbits and develop tools to do so. These tools leverage the parametric structure of families of quasi-periodic orbits to treat orbits only as a set of orbit frequencies instead of states in phase space. We develop a retraction on the family of quasi-periodic orbits to precisely navigate through frequency space, allowing us to compute orbits with specific frequencies. The retraction allows for movements in arbitrary directions. To combat the effects of resonances which slice through frequency space we develop resonance avoidance methods which detect resonances during continuation procedures and change the step size accordingly. We also develop an augmented Newton's method for root-finding and an augmented gradient descent method for unconstrained optimization over a family of quasi-periodic orbits. Lastly, we implement an augmented Lagrangian method to solve constrained optimization problems. We note that many of the tools developed here are applicable to a wider range of solutions defined implicitly by a system of equations, but focus on quasi-periodic orbits.