A control version of the Freidlin-Wentzell Theorem

The Freidlin-Wentzell theorem gives the probability of large deviations under small noise. Very roughly, it gives the probability that a particle undergoing diffusion inside a box, under very small but positive noise, will eventually hit the boundary of the box in a given time. Exit always happens at a particular point irrespective of the starting position, and this point of exit and time to exit can be calculated using the drift and volatility terms. We present an analogous control version of the theorem, where the goal of the controller is to keep the particle inside the box, as long as possible.