  In this paper, we first study the Poisson reductions of controlled
Hamiltonian (CH) system and symmetric CH system by controllability
distributions. These reductions are the extension of Poisson reductions by
distribution for Poisson manifolds to that for phase spaces of CH systems with
external force and control. We give Poisson reducible conditions of CH system
by controllability distribution, and prove that the Poisson reducible property
for CH systems leaves invariant under the CH-equivalence. Moreover, we study
the Poisson reduction of symmetric CH system by G-invariant controllability
distribution. Next, we consider the singular Poisson reduction and
SPR-CH-equivalence for CH system with symmetry, and prove the singular Poisson
reduction theorem of CH system. We also study the relationship between Poisson
reduction for singular Poisson reducible CH systems by G-invariant
controllability distribution and that for associated reduced CH system by
reduced controllability distribution. At last, some examples are given to state
the theoretical results.
