  We consider a dynamical system with finitely many equilibria and perturbed by
small noise, in addition to being controlled by an `expensive' control. The
controlled process is optimal for an ergodic criterion with a running cost that
consists the sum of the control effort and a penalty function on the state
space. We study the optimal stationary distribution of the controlled process
as the variance of the noise becomes vanishingly small. It is shown that
depending on the relative magnitudes of the noise variance and the `running
cost' for control, one can identify three regimes, in each of which the optimal
control forces the invariant distribution of the process to concentrate near
equilibria that can be characterized according to the regime. We also obtain
moment bounds for the optimal stationary distribution. Moreover, we show that
in the vicinity of the points of concentration the density of optimal
stationary distribution approximates the density of a Gaussian, and we
explicitly solve for its covariance matrix.
