  This paper is devoted to the analysis of problems of optimal control of
ensembles governed by the Liouville (or continuity) equation. The formulation
and study of these problems have been put forward in recent years by R.W.
Brockett, with the motivation that ensemble control may provide a more general
and robust control framework.
  Following Brockett's formulation of ensemble control, a Liouville equation
with unbounded drift function, and a class of cost functionals that include
tracking of ensembles and different control costs is considered. For the
theoretical investigation of the resulting optimal control problems, a
well-posedness theory in weighted Sobolev spaces is presented for the Liouville
and transport equations. Then, a class of non-smooth optimal control problems
governed by the Liouville equation is formulated and existence of optimal
controls is proved. Furthermore, optimal controls are characterised as
solutions to optimality systems; such a characterisation is the key to get
(under suitable assumptions) also uniqueness of optimal controls.
