Constructive controllability for incompressible vector fields

TL;DR

This paper provides a constructive proof of global controllability for incompressible vector fields guiding autonomous ODEs, under specific conditions, and extends controllability results on Riemannian manifolds with minimal observation requirements.

Contribution

It introduces a constructive approach to controllability for divergence-free vector fields and strengthens existing theorems by allowing control with minimal state observations and small vector field modifications.

Findings

Constructive proof of global controllability for incompressible vector fields.

Control can be achieved by slight modifications of the vector field.

Results extend to Riemannian manifolds with minimal observation requirements.

Abstract

We give a constructive proof of a global controllability result for an autonomous system of ODEs guided by bounded locally Lipschitz and divergence free (i.e.\ incompressible) vector field, when the phase space is the whole Euclidean space and the vector field satisfies so-called vanishing mean drift condition. For the case when the ODE is defined over some smooth compact connected Riemannian manifold, we significantly strengthen the assertion of the known controllability theorem in absence of nonholonomic constraints by proving that one can find a control steering the state vector from one given point to another by using the observations of only the state vector, i.e., in other words, by changing slightly the vector field, and such a change can be made small not only in uniform, but also in Lipschitz (i.e. $C^1$) topology.