# H"older equivalence of the value function for control-affine systems

arXiv: https://arxiv.org/abs/1304.6649 · HTML: https://mlsanity.com/paper/1304.6649/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Apr 2013.

## Built on by (most cited first, 1 of 1)

- [Turnpike in optimal control of PDEs, ResNets, and beyond](https://mlsanity.com/paper/2202.04097/) (2022)

## Abstract (arXiv)

We prove the continuity and the H"older equivalence w.r.t. an Euclidean distance of the value function associated with the L1 cost of the control-affine system dot q = drift(q)+sumj=1m uj fj(q), satisfying the strong H"ormander condition. This is done by proving a result in the same spirit as the Ball-Box theorem for driftless (or sub-Riemannian) systems. The techniques used are based on a reduction of the control-affine system to a linear but time-dependent one, for which we are able to define a generalization of the nilpotent approximation and through which we derive estimates for the shape of the reachable sets. Finally, we also prove the continuity of the value function associated with the L1 cost of time-dependent systems of the form dot q = sumj=1m uj fjt(q).