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