  This is a survey article, from the viewpoint of the completeness of the
Marsden- Weinstein reduction, to introduce briefly some recent developments of
the symmetric reductions and Hamilton-Jacobi theory of the regular controlled
Hamiltonian systems, the nonholonomic controlled Hamiltonian systems and the
controlled magnetic Hamiltonian systems. These research reveal the deeply
internal relationships of the geometrical structures of phase spaces, the
nonholonomic constraint, the dynamical vector fields and the controls of these
systems.
