A Hamiltonian approach to small time local attainability of manifolds for nonlinear control systems

TL;DR

This paper introduces a Hamiltonian-based method to analyze small time local attainability of manifolds in nonlinear control systems, providing explicit conditions and proving Hölder continuity of the minimum time function.

Contribution

It develops a novel Hamiltonian approach for small time local attainability of manifolds, extending classical results and providing explicit higher order conditions for complex targets.

Findings

Explicit pointwise conditions using higher order Hamiltonians.

Extension of controllability results to higher-dimensional targets.

Proven Hölder continuity of the minimum time function.

Abstract

This paper develops a new approach to small time local attainability of smooth manifolds of any dimension, possibly with boundary and to prove H\"older continuity of the minimum time function. We give explicit pointwise conditions of any order by using higher order hamiltonians which combine derivatives of the controlled vector field and the functions that locally define the target. For the controllability of a point our sufficient conditions extend some classically known results for symmetric or control affine systems, using the Lie algebra instead, but for targets of higher dimension our approach and results are new. We find our sufficient higher order conditions explicit and easy to use for targets with curvature and general control systems.