An intrusive polynomial algebra multiple shooting approach to the solution of optimal control problems

This paper proposes an approach to the solution of optimal control problems under uncertainty, that extends the classical direct multiple shooting transcription to account for random variables defined on extended sets. The proposed approach employs a Generalised Intrusive Polynomial Expansion to model and propagate uncertainty. The development of a generalised framework for a direct multiple shooting transcription of the optimal control problem starts with the discretisation of the time domain in sub-segments. At the beginning of each segment, the state spatial distribution is modelled with a multivariate polynomial and then propagated to the sub-interval final time. Continuity conditions are implicitly imposed at the boundary of two adjacent segments, a critical operation because it requires the continuity of two extended sets. The Intrusive Polynomial Algebra aNd Multiple shooting Approach (IPANeMA) developed in this paper can handle optimal control problems under a wide range of uncertainty models, e.g. nonparametric, expensive to sample, and imprecise probability distributions. In this paper, the approach is applied to the design of a low-thrust trajectory to a Near-Earth Object with uncertain initial conditions.