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In this paper we provide a thorough, rigorous theoretical framework to assessoptimality guarantees of sampling-based algorithms for drift control systems:systems that, loosely speaking, can not stop instantaneously due to momentum.We exploit this framework to design and analyze a sampling-based algorithm (theDifferential Fast Marching Tree algorithm) that is asymptotically optimal, thatis, it is guaranteed to converge, as the number of samples increases, to anoptimal solution. In addition, our approach allows us to provide concretebounds on the rate of this convergence. The focus of this paper is on mixedtime/control energy cost functions and on linear affine dynamical systems,which encompass a range of models of interest to applications (e.g.,double-integrators) and represent a necessary step to design, via successivelinearization, sampling-based and provably-correct algorithms for non-lineardrift control systems. Our analysis relies on an original perturbation analysisfor two-point boundary value problems, which could be of independent interest.

Optimal Sampling-Based Motion Planning under Differential Constraints: the Drift Case with Linear Affine Dynamics