Some Remarks on Controllability of the Liouville Equation

TL;DR

This paper explores the controllability of the Liouville equation, extending Brockett's results to broader classes of diffeomorphisms and providing quantitative bounds on control complexity for nonlinear systems.

Contribution

It generalizes Brockett's controllability conditions to larger classes of diffeomorphisms and offers a quantitative analysis of control switching complexity for nonlinear systems.

Findings

Controllability extends to a larger class of diffeomorphisms than previously known.

Any diffeomorphism near the identity can be implemented with finitely many switches.

Provides bounds on the number of switchings needed for implementation.

Abstract

We revisit the work of Roger Brockett on controllability of the Liouville equation, with a particular focus on the following problem: Given a smooth controlled dynamical system of the form $\dot{x} = f(x,u)$ and a state-space diffeomorphism $\psi$, design a feedback control $u(t,x)$ to steer an arbitrary initial state $x_0$ to $\psi(x_0)$ in finite time. This formulation of the problem makes contact with the theory of optimal transportation and with nonlinear controllability. For controllable linear systems, Brockett showed that this is possible under a fairly restrictive condition on $\psi$. We prove that controllability suffices for a much larger class of diffeomorphisms. For nonlinear systems defined on smooth manifolds, we review a recent result of Agrachev and Caponigro regarding controllability on the group of diffeomorphisms. A corollary of this result states that, for control-affine systems satisfying a bracket generating condition, any $\psi$ in a neighborhood of the identity can be implemented using a time-varying feedback control law that switches between finitely many time-invariant flows. We prove a quantitative version which allows us to describe the implementation complexity of the Agrachev-Caponigro construction in terms of a lower bound on the number of switchings.