  In this paper, from the viewpoint of completeness of Marsden-Weinstein
reduction, we illustrate how to give the definitions of a controlled
Hamiltonian (CH) system and a reducible controlled Hamiltonian system with
symmetry; and how to describe the dynamics of a CH system and the controlled
Hamiltonian equivalence; as well as how to give the regular point reduction and
the regular orbit reduction for a CH system with symmetry, by analyzing
carefully the geometrical and topological structures of the phase space and the
reduced phase space of the corresponding Hamiltonian system. We also introduce
briefly some recent developments in the study of reduction theory for the CH
systems with symmetries and applications. These research work reveal the deeply
internal relationships of the geometrical structures of phase spaces, the
dynamical vector fields and controls of the CH systems.
