Some Remarks on Controllability of the Liouville Equation

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 ẋ = f(x,u) and a state-space diffeomorphism ψ, design a feedback control $u(t,x)$ to steer an arbitrary initial state x₀ to ψ(x₀) 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 ψ. 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 ψ 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.