  We develop aspects of geometric control theory on Lie groups G which may be
infinite dimensional, and on smooth G-manifolds M modelled on locally convex
spaces. As a tool, we discuss existence and uniqueness questions for
differential equations on M given by time-dependent fundamental vector fields
which are L^1 in time. We then discuss the closures of reachable sets in M for
controls in the Lie algebra of G, or within a compact convex subset of the Lie
algebra. Regularity properties of the Lie group G play an important role.
