  Kullback-Leibler (KL) control enables efficient numerical methods for
nonlinear optimal control problems. The crucial assumption of KL control is the
full controllability of the transition distribution. However, this assumption
is often violated when the dynamics evolves in a continuous space.
Consequently, applying KL control to problems with continuous spaces requires
some approximation, which leads to the lost of the optimality. To avoid such
approximation, in this paper, we reformulate the KL control problem for
continuous spaces so that it does not require unrealistic assumptions. The key
difference between the original and reformulated KL control is that the former
measures the control effort by KL divergence between controlled and
uncontrolled transition distributions while the latter replaces the
uncontrolled transition by a noise-driven transition. We show that the
reformulated KL control admits efficient numerical algorithms like the original
one without unreasonable assumptions. Specifically, the associated value
function can be computed by using a Monte Carlo method based on its path
integral representation.
