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

TL;DR

This paper establishes the continuity and H"older equivalence of the value function for control-affine systems satisfying the strong H"ormander condition, extending geometric control theory techniques to these systems.

Contribution

It introduces a novel approach to analyze control-affine systems by reducing them to linear, time-dependent systems and generalizing nilpotent approximations, leading to new estimates of reachable sets.

Findings

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

Develops a reduction technique to linear, time-dependent systems.

Establishes continuity of the value function for time-dependent control systems.

Abstract

We prove the continuity and the H\"older equivalence w.r.t.\ an Euclidean distance of the value function associated with the $L^1$ cost of the control-affine system $\dot q = \drift(q)+\sum_{j=1}^m u_j f_j(q)$, 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 $L^1$ cost of time-dependent systems of the form $\dot q = \sum_{j=1}^m u_j f_j^t(q)$.