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 of 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.
"Controlled equilibrium selection in stochastically perturbed dynamics." Ann. Probab. 46 (5) 2749 - 2799, September 2018. https://doi.org/10.1214/17-AOP1238