Solving optimal continuous thrust rendezvous problems with generating functions

The optimal control of a spacecraft as it transitions between specified states using continuous thrust in a fixed amount of time is studied using a recently developed technique based on Hamilton-Jacobi theory. Start-ing from the 1st order necessary conditions for optimality, we derive a Hamiltonian system for the state and adjoints with split boundary conditions. Then, recognizing the two point boundary value problem as a canoni-cal transformation, we employ generating functions to find the optimal feedback control as well as the optimal trajectory. Though we formulate the optimal control problem in the context of the necessary conditions for optimality, our closed-loop solution also formally satisfies the sufficient conditions for optimality via the fun-damental connection between the optimal cost function and generating functions. A solution procedure for these generating functions is posed and numerically tested on a non-linear optimal rendezvous problem in the vicinity of a circular orbit. Generating functions are developed as series expansions, and the optimal trajecto-ries obtained from them are compared favorably with those of a numerical solution to the two point boundary value problem using a forward shooting method. Nomenclature A linearized system dynamics about the circular reference trajectory argmin(·) argument minimum with respect to the variable (·