Unified Formulation for Element-Based Indirect Trajectory Optimization

A general mathematical framework is presented to treat low thrust trajectory optimization problems using the indirect method and employing a generic set of orbital elements (e.g. classical elements, equinoctial, etc.). An algebraic manipulation of the optimality conditions stemming from Pontryagin Maximum Principle reveals the existence of a new quadratic form of the costate, which governs the costate contribution in all the equations of the first order necessary optimality conditions. The quadratic form provides a simple tool for the mathematical development of the optimality conditions for any chosen set of orbital elements and greatly simplifies the computation of a state transition matrix needed in order to improve the convergence of the associated two-point boundary value problem. Objective functions corresponding to minimum-time, minimum-energy and minimum-fuel problems are considered.