On Singularities of Minimum Time Control-Affine Systems

Abstract

Affine control problems arise naturally from controlled mechanical systems. Building on previous results [1,9] we prove that, in the case of time minimization with control on the disk, the extremal flow given by Pontrjagin's maximum principle is smooth along the strata of a well-chosen stratification. We also study this flow in terms of regular-singular transition and prove that the singularity along time-minimizing extremals crossing these strata is at most logarithmic. We then apply these results to mechanical systems, paying special attention to the case of the controlled three body problem.

Optimal control of a solar sail

“…For the sake of simplicity, we assume in the sequel that 𝜓 actually never vanishes. For a detailed discussion on the associated singularities of the dynamics, see 12 .…”

Jacobi fields in optimal control: Morse and Maslov indices

“…However discontinuities can appear even if all of the data is smooth. For example, in [18,5] the authors considered the following optimal control problem: minimize time needed to connect two points on a manifold M via a control system…”

“…and hence w i must form a basis of E. This means that we can reduce (18) to the same equations, but with v, w ∈ E ⊥ . Moreover by definition we have…”