  We consider long term average or `ergodic' optimal control poblems with a
special structure: Control is exerted in all directions and the control costs
are proportional to the square of the norm of the control field with respect to
the metric induced by the noise. The long term stochastic dynamics on the
manifold will be completely characterized by the long term density $\rho$ and
the long term current density $J$. As such, control problems may be
reformulated as variational problems over $\rho$ and $J$. We discuss several
optimization problems: the problem in which both $\rho$ and $J$ are varied
freely, the problem in which $\rho$ is fixed and the one in which $J$ is fixed.
These problems lead to different kinds of operator problems: linear PDEs in the
first two cases and a nonlinear PDE in the latter case. These results are
obtained through through variational principle using infinite dimensional
Lagrange multipliers. In the case where the initial dynamics are reversible we
obtain the result that the optimally controlled diffusion is also
symmetrizable. The particular case of constraining the dynamics to be
reversible of the optimally controlled process leads to a linear eigenvalue
problem for the square root of the density process.
