# Time-Optimal Trajectories of Generic Control-Affine Systems Have at Worst Iterated Fuller Singularities

## TL;DR

This paper proves that for generic control-affine systems, time-optimal trajectories are mostly smooth except at a finite hierarchy of singularities, which are well-structured and limited in complexity.

## Contribution

It establishes a generic regularity result for time-optimal controls, showing they are smooth except at a finite hierarchy of iterated singularities depending on system dimension.

## Key findings

- Optimal controls are smooth except at isolated points and their iterated accumulations.
- The non-smoothness set is finite and structured, with complexity bounded by system dimension.
- The result applies to generic control-affine systems, providing a detailed regularity characterization.

## Abstract

We consider in this paper the regularity problem for time-optimal trajectories of a single-input control-affine system on a n-dimensional manifold. We prove that, under generic conditions on the drift and the controlled vector field, any control u associated with an optimal trajectory is smooth out of a countable set of times. More precisely, there exists an integer K, only depending on the dimension n, such that the non-smoothness set of u is made of isolated points, accumulations of isolated points, and so on up to K-th order iterated accumulations.